Je Souhaite Remercier

Didina Serban

Jörg Teschner

Jean Michel Maillet

Giuliano Niccoli
email: giuliano.niccoli@ens-lyon.fr

Baptiste Pezelier
email: baptiste.pezelier@ens-lyon.fr

Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I

pour la tâche de président du jury.

Introduction

As physicists we are interested in discovering and understanding the laws of nature. This is a rather complex process that goes along observing the multitude of phenomena around us, performing related experiments to test various contexts and parameters and finally building new models or theories that can explain and predict the corresponding phenomena. Going back and forth between theory and experiments, we try to bring out the elementary principles governing the laws of nature, and to gather them in different classes. All along such a path, the interactions between Physics and Mathematics play a fundamental role. While Mathematics is providing a natural framework for the formulation of new theories, Physics is bringing to light new questions that could have great influence in the creation of new concepts and domains in Mathematics. This fruitful interaction goes back certainly to the early history of sciences.

One of the paradigmatic examples of such a tight interaction between Physics and Mathematics is given by the modelling of the celestial bodies motion. This problem, which goes back to the Antiquity, has first been "solved" in modern phenomenological terms by Kepler [1,2], formulating his famous three laws building on Copernic heliocentric system [3] and on the careful interpretation of the first observations of Galilee and of the impressive amount of datas collected by and together with Tycho-Brahé [4]. But if these laws could effectively describe the motion of planets of the solar system, they do not explain the deep universal physical principle behind such behaviour. The true conceptual breakthrough came with Newton in his famous Principia [5], with the formulation of the fundamental laws of mechanics and gravity and their universal application from the falling bodies on earth to the celestial movements of planets around the sun. These progresses were made possible on the one hand by the formulation of the laws of mechanics with Descartes, Galilee and then Newton, and by the identification of the inverse square law for the gravity forces 1 , and on the other hand by the developments of differential calculus with Newton and Leibniz [5,6]. These advances enabled Newton to give a description of the motion of celestial bodies in terms of dynamical equations that eventually can be solved explicitly. In particular he was able to give a proof of the three Kepler phenomenological laws from the few universal principles he stated for mechanics and gravity. Hence the great success of Newton was not only to build up a new theory from a few general principles but also to develop the mathematical tools to make it predictive, namely by providing solutions of the dynamical equations determining the motion. Although the solutions of the equations of motion were formulated in more conventional geometrical terms at the time of first publication of the Principia [5], it became clear soon after that the differential calculus of Newton and Leibniz was a powerful tool in effectively getting their analytical solutions (see e.g. the translation of Principia to french by Emilie du Châtelet [7], and especially her own appendix using differential calculus to derive Newton's results).

This feature of solvability in Newton's theory, enabling its full comparison to observations, was fundamental in convincing other scientists that this theory was indeed describing the fundamental laws of nature governing gravity. It certainly had a great influence on many further developments and in particular in the reformulations of Newton's theory by Lagrange and by Hamilton [8,9]. Rapidly however, it turned out that most of the dynamical equations of mechanics cannot be solved exactly. For instance, the two bodies interaction with an 1/r potential admits an exact solution, but this is generically not the case for more than two bodies. Even worse, in most of the situations of classical mechanics there is no hope INTRODUCTION to get an exact solution, showing that the explicitly solvable models, or so-called integrable models, are exceptions [10].

Nevertheless, the knowledge of an exact solution, even if restricted to a very particular situation, is of prime interest. Indeed, although the motion of planets in the solar system cannot be reduced to two body motion involving the sun and each individual planet, this first approximation is an interesting starting point to perform perturbations around such a simplified picture. And if one is interested in time interval not too large2 , the scheme is good enough to get rather high accuracy in the determination of the trajectories of planets in the solar systems. Hence, in many situations, very specific integrable models can be used as a starting point for the description of many actual realistic models, mainly by a perturbation process. There are numerous examples of the impact of exact solutions on the understanding of more general (non integrable) systems. The resolution of Schrödinger equation for the hydrogen atom is one of them, revealing the layered and discrete structure in its energy levels. It led to a theory of atoms with several electrons which was able, among others, to explain the Mendeleev classification. Moreover, there is many situations where integrable models, and well defined perturbation theory around them [13] (and references therein), give access to a whole class of non perturbative phenomena, allowing to get an understanding of complex and non-linear behaviours (strong couplings, collective effects,...).

Historically, the field of integrable systems appeared just as the search for exact solutions of the dynamical equations of classical mechanics. Because not all systems are integrable, the first question was to determine what were the necessary and sufficient conditions for integrability. In this quest for constructing and solving integrable systems, important developments emerged with the works of Lagrange, Hamilton, Jacobi, and Liouville [8,9,14,15]. There, the notion of conserved quantities became central, leading in particular to the first rigorous definition of integrability by Liouville [15] (see for its modern formulation Arnold in [16]). The systematic link to symmetries was elucidated later on in the works of Emmy Noether [17]. A system with n degrees of freedom is said to be Liouville integrable if it possesses n commuting (under Poisson brackets) independent (in the sense of differentials) integrals of motion; then the equations of motion can be solved (at least locally) by quadratures and, with compactness and connectivity hypothesis of the level manifold, action-angle variables can be determined through multivariables curvilinear integrals (see e.g. [16,18]). It should be noted however that, from a practical point of view, their effective construction in a given model has usually to rely on additional (e.g., algebraic) constructions. An important step towards this goal is played by the notion of separation of variables in Hamilton-Jacobi theory. In short, the key idea is that to solve a system with n degrees of freedom effectively we use the n independent integrals of the motion in such a way that in the adequate (separate) variables, the system of n coupled differential equations giving the dynamics ultimately separates into n independent differential equations each involving only one variable together with the fixed values of the conserved quantities. As we shall see later, this notion of separation of variables admits an interesting quantum counterpart which will be a central tool in this thesis.

From these early developments in classical mechanics, the notion of solvability has been explored and generalised within several domains of physics: continuous models (e.g. in hydrodynamics), statistical mechanics, quantum mechanics and classical and quantum field theories.

One of the most famous example of continuous model suspected to be solvable in the end of the nineteenth century was certainly the Korteweg-de Vries (KdV) equation and the discovery of its soliton solutions (after investigations of Boussinesq [19] and Rayleigh [20] in the 1870's, which lead to the formulation of Korteweg and de Vries [21] in 1895) that were first observed by Scott-Russell in 1834 [22]. Although the existence of such solutions was pointing towards the existence of (an infinite number of) conserved quantities explaining the shape conservation of the solitons, the effective construction of those has been long awaited. The main difficulty there is that contrary to the two-body gravity problem, there is no obvious geometrical symmetry leading through Noether theorem to the corresponding conserved quantities. Rather, the conservation laws are direct consequences of the special dynamics of the system, i.e. of INTRODUCTION the specific interacting potential. These symmetries and conserved quantities are of dynamical type, and are very similar to the Runge-Lenz vector, associated to the Kepler problem [23,24]. The possibility to construct and to be able to solve such systems, owning a dynamical symmetry, originated from the work of Gardner, Green, Kruskal, and Miura [25] in 1967 and then Lax [26] for the KdV equation, then generalised by Zakharov and Shabat [27,28] in the form of zero curvature equations. The main idea is to equivalently rewrite the equation of motion as a particular iso-spectral evolution equation for a (Lax) matrix, allowing to write the conserved quantities from its spectral invariants. Combining these settings with the inverse scattering transform of Gel'fand, Levitan and Marchenko [29,30], several solvable continuum models like the KdV equation can be constructed and their (multi) soliton solutions derived [25,[31][32][33]. The Hamiltonian interpretation of the scheme was first given by Zakharov and Faddeev [34]. The solitons, their semi-classical quantisation as particles and their scattering were intensively studied in two-dimensional relativistic quantum field theories, with the hope to describe their bound-states in strongly interacting regimes [35]. Further progress was to come with the merging of this line of research with another one stemming from solvable models in classical and quantum statistical mechanics originating from the search for a description of magnetic properties of solids.

Motivated notably by the study of phase transitions in magnetic materials, Lenz proposed a lattice model [36], solved by Ising in the one-dimensional case [37] and that became famous as the Ising model. The two-dimensional case was much more tedious to solve, and was successfully considered by Onsager and then Onsager and Kaufman twenty years later [38,39]. They succeeded to compute the partition function, the critical exponents and the magnetisation [40,41]. Long after, the spontaneous magnetisation (the model original raison d'être) could be achieved by Yang [42]. Their method relies on the so-called Onsager and Clifford algebras, which marked the early history of exactly solvable statistical systems. This example of the Ising model is indeed an archetype of the current philosophy behind integrability, aiming to embed the problem in algebraic structures giving the tools towards their resolution. The one-dimensional Ising solution not showing a phase transition, and Ising (wrongly) conjecturing that it should hold for higher dimensions, physicists started to study quantum models. This gave birth to the XXX spin-1/2 chain, due to Heisenberg [43], a lattice quantum model where the interactions between nearest neighbours are isotropic. In 1931 Bethe could express the spectrum of this chain thanks to his famous coordinate ansatz [44], now called after his name. Then the works of Orbach [45] and Walker [46] allowed to apply this ansatz to different models, like the XXZ Heisenberg spin chain, which introduces an anisotropy in the z direction compared to the XXX chain. For this anisotropic chain, Yang and Yang could determine the energy of the ground state for finite [47] and infinite chains [48]. Since then, XXZ spin chain has been at the heart of many studies, as it represents a simple model which allowed to lay the foundations of several fruitful techniques. Let us mention the considerable work of Baxter on the lattice systems, in particular his studies of the 6-vertex and 8-vertex models. His approach, using the notion of Q-operator and the so-called Baxter equation [49], has shown to be applicable in a large range of models and to allow a more universal study of lattice models [50][51][52][53][54][55][56][57][58]. It is in particular worth noticing that the 8-vertex model contains, as particular cases, the 6-vertex and Ising models, see e.g. [59][60][61]. Another important model was shown to belong to this class, the 1D Bose gas and associated non-linear Schrödinger equation [62,63], which was solved by Lieb and Liniger using a technique [64,65] similar to the Bethe ansatz. On the other hand, the classical version of this model was solvable by mean of the Inverse Scattering Method, immediately rising the question of the connection between these two approaches and of the existence of a quantised version of the Classical Inverse Problem Method.

The conceptual efforts to understand this problem led to the discovery of the Quantum Inverse Scattering Method, or Algebraic Bethe Ansatz [66][67][68][69][70], realising an impressive breakthrough for the field of integrable systems. This is both a quantum version of the Classical Inverse Scattering Method and an algebraic version of the coordinate Bethe ansatz. It was very soon applied on discretised (lattice) version of the sine-Gordon relativistic field theory [67], and then became a quite universal tool applicable to many integrable lattice models [71,72]. In the first chapter of this thesis we will review in more details the working of this method, but let us emphasise on the structure. The main point is the construction of the so-called monodromy matrix, a matrix defined on an auxiliary space which gathers operators on the quantum space acting on the Hilbert space of states of the chain. This monodromy matrix satisfies INTRODUCTION commutation relations whose structure constants are given by an R-matrix, solution of the Yang-Baxter equation (it is the R-matrix of Baxter appearing as Boltzmann weights of the 2D-lattice models, e.g. the 6-vertex model if we consider the XXZ Heisenberg spin chain). From this so-called Yang-Baxter algebra, it turns out that the transfer matrix, defined as the trace of the monodromy matrix on the auxiliary space, defines a commuting family of operators. This is the mark of the integrability, the Hamiltonian being then expressed as a function (trace identity) of the transfer matrix. Thereafter, the Algebraic Bethe Ansatz allows one to find eigenvectors of the transfer matrix, by the repeated action of creation operators (some entries of the monodromy matrix) on a so-called reference state. This latter is the cornerstone of the method, it is in fact a first eigenvector of the transfer matrix, from which one can construct new eigenvectors. This framework finally proved to be able to tackle a much more difficult problem for Heisenberg chains: the computation of correlation functions. To achieve this goal, it was necessary to actually solve the Quantum Inverse Scattering Problem, namely to reconstruct the local quantum operators (like spins at a given lattice site) from the scattering data, i.e. from the entries of the quantum monodromy matrix [73,74]. In the classical case this procedure leads to the computation of soliton solutions of the non-linear integrable differential equations. Here it allows to compute the action of local fields on eigenstates of the transfer matrix and then to access to their form factors and correlation functions via scalar products determinant formula [75][76][77][78][79][80] as we shall describe later on.

One very interesting feature of this breakthrough for the quantum case is its impact on the study of the classical integrable systems. Indeed, it was almost immediately realised by Sklyanin [81] that the concept of R-matrix, Yang-Baxter equations and algebras have classical counterparts, namely the classical r-matrix and the classical Yang-Baxter equations and algebras, shading completely new light on the algebraic structure of classical integrable models. This emergence of Yang-Baxter algebraic structures in classical integrable systems paved the way for their classification using Lie algebra representations, thanks to the works of Belavin and Drinfel'd [82][83][84]. What is more, the interpretation of this classification states the importance of the study of Lie algebras, which reached an height with the use of Lie-Poisson groups and Lie bi-algebras [85][86][87][88], as it allows to construct and solve classical integrable models. We can cite in this context the fundamental works of Adler [89,90], Kostant [91], Symes [92] and of Reymann and Semenov-Tian-Shansky [93][94][95].

In parallel and somehow motivated by these progresses in the classical context, a considerable endeavour has been pursued to find general solutions of the Yang-Baxter equation and representations of Yang-Baxter algebras, in order to construct and to classify quantum integrable systems. The key idea was to try to promote to the quantum case the relation between Lie algebras and classical integrable models. Thanks to the pioneering works of Kulish and Reshetikhin [96,97], Jimbo [98] and Drinfel'd [99] discovered the notion of quantum groups, and showed how it was replacing the role played by Lie algebras in the classical case. Quantum group structures written in terms of Hopf algebras [100] realise a quantification of Lie-Poisson groups. Previously known R-matrices for specific models appeared then as special representations of a universal R-matrix, an object which is at the heart of the intertwining properties of coproduct (and hence of tensor products of representations) in quantum groups. It opened a completely new field of Algebra with numerous applications ranging from Combinatorics, Algebra and Topology, see e.g. [101].

In this thesis, we especially focus on one dimensional quantum systems. Even if they are simplified models, they very often contain the essence of the physics of fully 3D systems, and can describe their relevant properties. Baxter, in the preface of his book [59], even writes: "Basically, I suppose the justification for studying these lattice models is very simple: they are relevant and they can be solved, so why not do so and see what they tell us?". In this context of lattice quantum integrable systems, spin chains are among the most studied examples with applications which range from condensed matter to high energy physics. However, to successfully apply these models to physical situations of interest, we have to be able to compare their theoretical predictions to experiments, for example performed on magnetic materials. Linear response theory give a precise link between the response of a system submitted to an external perturbation and its correlation functions. For instance, dynamical structure factors, which are given by the Fourier transform of dynamical correlation functions, can describe the response of a magnetic system scattered with neutrons [102][103][104]. Moreover the critical properties of such systems are also encoded in INTRODUCTION the asymptotic behaviour of their correlation functions. As they ultimately contain all the information on their dynamics, there have been strong motivations to compute these objects. Indeed, they are also at the heart of statistical mechanics, as in principle any physical observable can be expressed through these functions. It turned out however that the computation of the correlation functions is a very involved problem. The first results were obtained for the Ising model, heavily using its free fermionic structure. Even there, although the free fermion algebra is quite simple (the interaction vanishes), a considerable amount of work was necessary to obtain satisfactory results since the pioneering work of Lieb, Shultz and Mattis [105]. Several outstanding groups, including Barouch, McCoy, Tracy and Wu [106], Jimbo, Miwa, Môri and Sato [107], and also Creamer, Thacker and Wilkinson [108] leading to the Painlevé description of correlation functions in the thermodynamic limit, worked in this direction. The similar question for non-free fermionic models, like the Heisenberg spin chains, was for a long time impossible to tackle in reasonable terms [109][110][111]. This was mainly due to the intricate structure of Bethe eigenstates with respect to the action of local (spin) operators. In a nutshell, Bethe states are highly non-local states and the action of local operators is very complicated to characterise in handleable terms. As already anticipated above, the full solution to this problem was obtained by solving the Quantum Inverse Scattering Problem, namely by obtaining the construction of local spin operators in an arbitrary lattice site in terms of the entries of the quantum monodromy matrix. The fact that Bethe eigenstates are obtained within the Algebraic Bethe Ansatz as repetitive actions of one of the entries of the monodromy matrix on a reference state, together with the knowledge of the Yang-Baxter algebra, enabled then to compute the action of local operators on Bethe states. This action being again written in terms of (off shell) Bethe states, the computation of form factors (matrix elements of local operators in the eigenstates basis of the transfer matrix) and correlation functions reduces to the computation of scalar products of a Bethe eigenstate with an arbitrary (off shell) Bethe state. Fortunately such formulas exists for XXX and XXZ model, leading in [73,112] to the famous determinant formula for the form factors and to the multiple integral formula for the elementary blocks for the correlation functions of the XXZ spin-1/2 chain. Thereby it gave a full derivation of the expressions found earlier by Jimbo, Miwa and collaborators using q-vertex algebra approach to the infinite volume chains [113][114][115][116]. Moreover, because the solution to this problem within Algebraic Bethe Ansatz was obtained for finite volume, numerical techniques can be applied to compute dynamical structure factors essentially by summing up the form factor series. Those being written in terms of explicit determinants involving Bethe roots, it has been possible to use efficient (and fully controlled) numerical algorithms to obtain magnetic dynamical structure factors of the XXZ Heisenberg spin chain and to successfully compare the results to actual neutron scattering experiments [117,118]. Moreover, subsequent works led to the full determination of the asymptotic behaviour of correlation functions and explicit contact with conformal field theory [75][76][77][78][79][80]. One should also mention the works towards temperature dependent case, with a great contribution of Goehmann et al. since 2004 [119][120][121] (see [122] for a pedagogical review and [123,124] for more recent works) and other groups [125,126], and the approach using hidden fermionic structures by Jimbo, Miwa, Smirnov et al. [START_REF] Boos | Hidden Grassmann Structure in the XXZ Model[END_REF][128][129][130][131][132].

Although the situation for Heisenberg spin chains with periodic boundary conditions is rather satisfactory, the questions concerning correlation functions for more general models remains largely open. It concerns in particular integrable systems associated to higher rank algebras (the Hubbard model is one of the prominent representative), systems with non periodic boundary conditions and most of the discretised quantum field theories. While in general the solution of the Quantum Inverse Scattering Problem can be obtained in rather general terms for finite dimensional cases [74], the other ingredients are not always available : lack of a reference state preventing the use of Algebraic Bethe Ansatz, lack of determinant expressions for the scalar product of states or even worse, lack of a proper algebraic structure to handle the transfer matrix spectral problem. Hence, in some of these generalised situations, even the first step of characterising the transfer matrix spectrum along some algebraic scheme is not available yet. This is a strong motivation for developing a more general method than Algebraic Bethe Ansatz that would be applicable to these general cases. Such a method has been first proposed in the framework of the Quantum Inverse Scattering Method by Sklyanin [133,134] (see [135] for a review). It is a quantum version of the separation of variables method of Hamilton-Jacobi in classical mechanics. The key idea of the separation of variables method for quantum integrable models having N degrees of freedom is to map the (in general) INTRODUCTION highly coupled N -variables spectral problem to a set of N one-variable (hence decoupled) soluble spectral problems. In that case the map takes the form of a change of basis to the so-called separate basis. This procedure generalises to the quantum case the well-known Hamilton-Jacobi method of classical mechanics [18]. In Sklyanin approach [133] to quantum integrable models, the separate basis is identified with the eigenstate basis of some particular operator of the Yang-Baxter algebra having simple spectrum; for lattice integrable models this amounts in general to introduce inhomogeneity parameters on each site of the lattice. In the separate basis, the transfer matrix wave functions take a simple factorised form over the spectrum of the separate variables. This simplification in the spectral problem represents thus a natural motivation to develop quantum separation of variables in the analysis of integrable quantum models. This is the point of view taken in this thesis. Further motivations to work in this framework come naturally as this separation of variables approach has proven to be an efficient tool to solve the spectral problem of a large class of integrable models [136][137][138][START_REF] Amico | Separation of variables for integrable spin-boson models[END_REF][START_REF] Amico | Integrable spin-boson models descending from rational six-vertex models[END_REF], even not analysable by other methods, leading to the characterisation of both the eigenvalues and the eigenstates [START_REF] Niccoli | Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators[END_REF][START_REF] Faldella | SOV approach for integrable quantum models associated with general representations on spin-1/2 chains of the 8-vertex reflection algebra[END_REF][START_REF] Faldella | The complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms[END_REF][START_REF] Kitanine | Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables[END_REF]. Contrary to other methods, the proof that the spectrum characterisation is complete is rather straightforward and, in this approach, no reference state is needed unlike the Algebraic Bethe Ansatz. Moreover, this approach proved already to be very efficient for the study of the dynamics of quantum integrable models, as universal determinant formula has emerged (first shown in [START_REF] Grosjean | On form factors of local operators in the lattice sine-Gordon model[END_REF]) [START_REF] Niccoli | Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators[END_REF][START_REF] Kitanine | The open XXX spin chain in the SoV framework: scalar product of separate states[END_REF][START_REF] Kitanine | On determinant representations of scalar products and form factors in the SoV approach: the XXX case[END_REF][START_REF] Niccoli | Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Form factors and complete spectrum[END_REF][START_REF] Niccoli | Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables[END_REF][START_REF] Niccoli | An antiperiodic dynamical six-vertex model : I. Complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic eight-vertex model[END_REF][START_REF] Grosjean | On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models[END_REF][START_REF] Levy-Bencheton | Antiperiodic dynamical 6-vertex model by separation of variables II: Functional equations and form factors[END_REF] for the scalar products of the so-called separate states and for the matrix elements of local operators on transfer matrix eigenstates. It has also been shown recently that the question of homogeneous limit (the limit in which all inhomogeneity parameters, necessary for the separation of variables to apply, are set to the same value) can be tackled by rewriting the determinants found in this approach in terms of Izergin type and then Slavnov type determinants, where homogeneous limit can be taken [START_REF] Kitanine | On determinant representations of scalar products and form factors in the SoV approach: the XXX case[END_REF].

This thesis is willing to belong to the research effort taken in the development of the separation of variables method for integrable systems having a potential impact for physical applications in condensed matter, statistical physics or quantum field theory. We will be considering quantum one-dimensional systems associated to the 6-vertex R-matrix and to general cyclic representations of the Yang-Baxter algebra (and of the reflection algebra as we aim to consider general integrable boundary conditions). Namely, they are associated to representations of the quantum group U q p ŝl 2 q for q root of unity. Their study is mathematically interesting but also physically attractive since they contain, as a particular case, the chiral Potts model 3 . This model was first considered by Potts as a N-states extension of the Ising model, has then been generalised by Wu and Wang in 1976 [START_REF] Wu | Duality transformation in a many-component spin model[END_REF] with a dependence of the interaction energy depending on the direction, and then by Ästlund and Huse [START_REF] Ostlund | Incommensurate and commensurate phases in asymmetric clock models[END_REF][START_REF] Huse | Simple three-state model with infinitely many phases[END_REF], who considered a chiral version of the two-dimensional Potts model, in which the interaction differs depending on the x or y axis of the lattice. The study of this system has been carried for the superintegrable Z N -symmetrical chiral Potts quantum chain [START_REF] Howes | Quantum model for commensurate-incommensurate transitions[END_REF][START_REF] Von Gehlen | Z n -symmetric quantum chains with an infinite set of conserved charges and Z n zero modes[END_REF], which was shown to be integrable [START_REF] Von Gehlen | Z n -symmetric quantum chains with an infinite set of conserved charges and Z n zero modes[END_REF] because it forms representations of the Onsager algebra. Moreover, few years later, Bazhanov and Stroganov have show the integrability of the chiral Potts model [START_REF] Bazhanov | Chiral Potts model as a descendant of the six vertex model[END_REF] within the framework of the Yang-Baxter algebra, embedding it in a 6-vertex model (see also the works of Tarasov [START_REF] Tarasov | Transfer matrix of the superintegrable chiral Potts model. Bethe ansatz spectrum[END_REF][START_REF] Tarasov | Cyclic monodromy matrices for the R-matrix of the six-vertex model and the chiral Potts model with fixed spin boundary conditions[END_REF][START_REF] Tarasov | Cyclic monodromy matrices for slpnq trigonometric R-matrices[END_REF]). This is the setting that we will be using in this thesis. The important feature with the chiral Potts model is that it is associated to high genus curves, making much more involved its resolution since the dependence on spectral parameter becomes more involved (it does not have the difference property). Despite this difficulty, Baxter tackled this problem and could express several years later, inspired by a technique from Jimbo, Miwa, and Nakayashiki [START_REF] Jimbo | Difference equations for the correlation functions of the eight-vertex model[END_REF], the order parameter of the model [START_REF] Baxter | Derivation of the order parameter of the chiral Potts model[END_REF]. The question of correlation functions however is still open for this model and motivated the application of separation of variables to compute the form factors [START_REF] Grosjean | On form factors of local operators in the lattice sine-Gordon model[END_REF][START_REF] Grosjean | On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models[END_REF]. Eventually, let us mention that the study of these cyclic models in their generality is also motivated by the fact that the lattice sine Gordon model, in particular at its q-root of unity reduction points, can be obtained, with a special set of parameter, from the Bazhanov-Stroganov cyclic solution of Yang-Baxter algebras. One related interesting question here concerns the asymptotic behaviour of the correlation functions and the link with minimal models of Conformal Field Theory [START_REF] Lukyanov | Exact expectation values of local fields in the quantum sine-Gordon model[END_REF].

In this thesis we will be considering an additional feature for such models, namely their resolution INTRODUCTION in the presence of general boundary conditions preserving integrability for the finite chain and the construction of the corresponding integrable local Hamiltonians. The description of integrable boundaries has attracted a large research enthusiasm as they can describe both equilibrium and out of equilibrium physics. Some interesting applications concern the description of classical stochastic relaxation processes, like the asymmetric exclusion process (ASEP) [START_REF] Derrida | An exactly soluble non-equilibrium system: the asymmetric simple exclusion process[END_REF][START_REF] Schutz | Phase transitions and critical phenomena[END_REF][START_REF] Shiroishi | Bethe Ansatz equation for the Hubbard model with boundary fields[END_REF][START_REF] Shiroishi | Integrable boundary conditions for the one-dimensional Hubbard model[END_REF][START_REF] Alcaraz | Reaction-diffusion processes, critical dynamics, and quantum chains[END_REF][170][START_REF] De Gier | Bethe Ansatz solution of the asymmetric exclusion process with open boundaries[END_REF][START_REF] De Gier | Exact spectral gaps of the asymmetric exclusion process with open boundaries[END_REF] and quantum transport properties in spin systems [START_REF] Sirker | Diffusion and ballistic transport in one-dimensional quantum systems[END_REF][START_REF] Prosen | Open XXZ spin chain: nonequilibrium steady state and a strict bound on ballistic transport[END_REF].

Since the paper of Gaudin [START_REF] Gaudin | Boundary energy of a Bose gas in one dimension[END_REF], the coordinate Bethe ansatz [START_REF] Alcaraz | Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models[END_REF][START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF][START_REF] Cherednik | Factorizing particles on a half-line and root systems[END_REF][START_REF] Kulish | The general Uq(sl(2)) invariant XXZ integrable quantum spin chain[END_REF] has been applied to tackle the question of boundaries, as well as a development concerning the algebraic point of view [START_REF] Pasquier | Common structures between finite systems and conformal field theories through quantum groups[END_REF][START_REF] Batchelor | q-deformations of the O(3) symmetric spin-1 Heisenberg chain[END_REF][START_REF] Kulish | Algebraic structures related to the reflection equations[END_REF][START_REF] Vega | The highest weight property for the SUq(n) invariant spin chains[END_REF][START_REF] Ghoshal | Boundary s matrix and boundary state in two-dimensional integrable quantum field theory[END_REF][START_REF] Ghoshal | Boundary s matrix and boundary state in two-dimensional integrable quantum field theory[END_REF]. In the framework of the Quantum Inverse Scattering Method, Sklyanin has shown how to construct classes of quantum integrable models with non-trivial boundaries [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF]. His method relies on the so-called reflection algebras, generated by the elements of a monodromy matrix satisfying the so-called reflection equations, introduced by Cherednik [START_REF] Cherednik | Factorizing particles on a half-line and root systems[END_REF]. They have been derived in the context of the study of the scattering matrix factorisation in the presence of reflecting walls, i.e. on the interval instead of on the infinite line. This approach belongs to the "bootstrap method" that was developed to study relativistic quantum field theories, where the scattering of n particles factorises in terms of products of 2 particles scattering [START_REF] Berg | Factorized U pN q symmetric S matrices in two dimensions[END_REF][START_REF] Karowski | On the bound state problem in (1+1)-dimensional field theories[END_REF][START_REF] Karowski | Complete S matrix of the Op2N q Gross-Neveu model[END_REF][START_REF] Zamolodchikov | Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models[END_REF], related to the Yang-Baxter equation.

The pioneering work of Sklyanin on integrable models with boundaries states that when one knows a representation of the reflection algebra in the quantum space and a scalar representation of its dual, one can compute a commuting family of operators, the so-called boundary transfer matrices, and extract from them local integrable Hamiltonians with general integrable boundaries (for fundamental models). Moreover, he gives a way to use solutions of the standard Yang-Baxter equation to generate solutions of the reflection equation, once a scalar solution to the latter is known. The main point for the integrability, as well as for the computation of the Hamiltonians, is the introduction of the boundary transfer matrix, which generalises the standard transfer matrix introduced for Yang-Baxter algebras. Similarly to the Yang-Baxter case, the Hamiltonians are obtained via some derivatives of the boundary transfer matrix; this technique being based on the reduction of the quantum Lax operator to the permutation operator (in a specific point), it holds for fundamental models only, i.e. for models where the auxiliary and quantum spaces are isomorphic. Let us comment that the boundaries of these integrable Hamiltonians are encoded into two boundary matrices, scalar solutions of the reflection equation (and its dual). As it is the case for the Yang-Baxter algebra, the most representative model to begin to understand this class of systems with boundaries is probably the XXZ spin-1/2 quantum chain with general integrable boundary conditions. To fully characterise the spectrum of this Hamiltonian, the Algebraic Bethe Ansatz [66,67] has been extensively used, leading for diagonal boundary matrices, to a characterisation from the spectrum [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF] up to the correlation functions [START_REF] Kitanine | Correlation functions of the open XXZ chain: I[END_REF][START_REF] Kitanine | Correlation functions of the open XXZ chain: II[END_REF]. This method faces however some technical difficulties when nondiagonal boundaries are considered, like the non existence of a reference state or the completeness of the spectrum description, resulting to its characterisation only under special constraints. More specifically, the method can be applied to non-diagonal boundary matrices which satisfy certain constraints which allow the definition of reference states by a gauge transformation [START_REF] Fan | Algebraic Bethe ansatz for eight vertex model with general open-boundary conditions[END_REF][START_REF] Cao | Exact solution of XXZ spin chain with unparallel boundary fields[END_REF][START_REF] Yang | On the second reference state and complete eigenstates of the open XXZ chain[END_REF]. The completeness is then a complicated task to prove, involving notably two sets of Bethe ansatz equations to evidence some numerical results [START_REF] Nepomechie | Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms[END_REF]. In fact, whereas the first steps of the Quantum Inverse Scattering Method (leading to the boundary transfer matrix) are independent on the nature of the boundaries, the effective construction of the eigenstates is more difficult, if not impossible. Numerous models were however solved in this context, for example the spin s chains [START_REF] Doikou | Fused integrable lattice models with quantum impurities and open boundaries[END_REF][START_REF] Frappat | A complete Bethe ansatz solution for the open spin-s XXZ chain with general integrable boundary terms[END_REF] or also higher rank models with integrable boundaries [START_REF] Arnaudon | Analytical Bethe Ansatz for closed and open gl(n)-spin chains in any representation[END_REF][START_REF] Arnaudon | Spectrum and Bethe ansatz equations for the U q pglpN qq closed and open spin chains in any representation[END_REF][START_REF] Ragoucy | Analytical Bethe ansatz for closed and open gl(M|N) super-spin chains in arbitrary representations and for any Dynkin diagram[END_REF][START_REF] Belliard | The nested Bethe ansatz for 'all' open spin chains with diagonal boundary conditions[END_REF], in particular concerning the Hubbard model [START_REF] Shiroishi | Bethe Ansatz equation for the Hubbard model with boundary fields[END_REF][START_REF] Shulz | Hubbard chain with reflecting ends[END_REF][START_REF] Zhou | Quantum integrability for the one-dimensional hubbard open-chain[END_REF][START_REF] Zhou | Graded reflection equations and the one-dimensional Hubbard open chain[END_REF][START_REF] Asakawa | Finite-size corrections in the XXZ model and the Hubbard model with boundary fields[END_REF][START_REF] Guan | Lax pair and boundary K-matrices for the onedimensional Hubbard model[END_REF][START_REF] Guan | Algebraic Bethe ansatz for the one-dimensional Hubbard model with open boundaries[END_REF].

Several methods were introduced to overcome the problems encountered in the description of the spectrum of quantum models with general integrable boundary conditions, examples are generalised Bethe ansatz [START_REF] Cao | Exact solution of XXZ spin chain with unparallel boundary fields[END_REF][START_REF] Yang | On the second reference state and complete eigenstates of the open XXZ chain[END_REF][START_REF] Crampe | Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions[END_REF][START_REF] Crampe | Generalized coordinate Bethe ansatz for non diagonal boundaries[END_REF], the fusion procedure [START_REF] Nepomechie | Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms[END_REF][START_REF] Mezincescu | Bethe ansatz solution of the Fateev-Zamolodchikov quantum spin chain with boundary terms[END_REF][START_REF] Mezincescu | Integrability of open spin chains with quantum algebra symmetry[END_REF][START_REF] Nepomechie | Solving the open XXZ spin chain with nondiagonal boundary terms at roots of unity[END_REF][START_REF] Nepomechie | Functional relations and Bethe Ansatz for the XXZ chain[END_REF][START_REF] Murgan | Exact solution of the open XXZ chain with general integrable boundary terms at roots of unity[END_REF][START_REF] Galleas | Functional relations from the Yang-Baxter algebra: eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions[END_REF][START_REF] Yang | Q-operator and T-Q relation from the fusion hierarchy[END_REF][START_REF] Cao | Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fields[END_REF] (for spin chains) or the q-Onsager algebra [START_REF] Baseilhac | A deformed analogue of Onsager's symmetry in the XXZ open spin chain[END_REF][START_REF] Baseilhac | Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory[END_REF], which led to the same constraint on the boundaries to be able to express the eigenvalues. The separation of variables method has been shown in the recent years to be able to overcome these difficulties and to characterise completely the spectrum of such models in the cases of XXX, XXZ and XYZ Heisenberg chains with general integrable boundaries [START_REF] Niccoli | Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators[END_REF][START_REF] Faldella | The complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms[END_REF][START_REF] Kitanine | Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables[END_REF]. Hence, for lattice integrable models associated to cyclic representations, which are at the heart of this thesis, the lack of a reference INTRODUCTION state and the willing to address the general integrable boundary conditions also require the use of the quantum separation of variables.

We will first solve the spectral problem for models associated to Bazhanov-Stroganov Lax operators with the most general integrable boundary conditions. These models are associated to cyclic representations of the 6-vertex Yang-Baxter algebra, hence there is no obvious reference state, even for the periodic boundary condition. To solve this problem we develop the separation of variables method adapted to these representations; both eigenstates and eigenvalues are characterised. Furthermore, writing the reflection equation in mixed representations, involving both spin 1/2 and cyclic ones, we are able to determine new integrable local Hamiltonians with boundaries, acting in tensor products of any cyclic representations. The current manuscript is divided into five chapters.

The first chapter briefly reminds the key classical integrable structures and presents then the Quantum Inverse Scattering Method, introducing the Yang-Baxter algebra and the transfer matrices. It explicitly focuses on the 6-vertex Yang-Baxter algebra, and presents the Algebraic Bethe Ansatz method on the example of the XXZ spin 1/2 chain. The chapter ends introducing cyclic representations of the 6-vertex Yang-Baxter algebra and thus the Lax operator at the heart of our study, the so-called Bazahnov-Stroganov Lax operator.

In the second chapter, the implementation of general integrable boundaries is addressed. On the example of the XXZ spin 1/2 chain, a twist matrix is introduced, which allows to deal with quasi-periodic boundary conditions. Then, the true generalisation due to Sklyanin [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF] is considered, namely the reflection algebra and in particular the 6-vertex reflection algebra. V ia two boundary matrices, one can compute integrable local Hamiltonians with general boundaries for fundamental models, i.e. for which the auxiliary and quantum spaces are isomorphic. Both the general construction and the explicit example of the XXZ spin 1/2 chain are given. At the end, the fusion procedure is briefly introduced in order to present a method allowing the computation of integrable local Hamiltonians (with general integrable boundaries) for some non fundamental models.

The third chapter is dedicated to the quantum separation of variables. In a first part, the classical notion of separation of variables is briefly recalled. In particular, it is shown how to construct separate variables when the integrability is characterised thanks to a two dimensional Lax matrix and a Yang-Baxter classical r-matrix. Then the quantum separation of variables method is defined. An analogy with the classical method is emphasised, particularly regarding the use of the elements of the monodromy matrix. Following [START_REF] Niccoli | Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Form factors and complete spectrum[END_REF], an explicit construction of the separate basis for the XXZ spin 1/2 antiperiodic chain is done, allowing us to bring to light the promising features of this method.

The new results brought by this thesis are exposed in chapters four and five. Chapter four focuses on the characterisation of the transfer matrix spectrum associated to cyclic representations of the 6-vertex reflection algebra, with general integrable boundary conditions. It is mainly decomposed in two parts. In the first one, we consider the system with a constraint on the right boundary, explicitly one of the associated boundary matrix is taken triangular. It allows us to present the separation of variables for this reflection algebra, and to state a characterisation of the spectrum. The other part is dedicated to the general integrable boundary conditions. To this aim, we introduce a generalisation of Baxter's gauge transformations and construct a separate basis, choosing an appropriate value for the gauge parameter. We can then state a similar characterisation of the spectrum, valid for any integrable boundaries. Moreover, an equivalent functional characterisation via a functional Baxter T -Q type equation is given, establishing a link with the Algebraic Bethe Ansatz technique. The chapter ends with the computation of the scalar products of so-called separate states, which is a first step toward the description of the dynamics.

The aim of the fifth and last chapter is the computation of integrable local Hamiltonians associated to the non fundamental models described by the Bazhanov-Stroganov Lax operator. Starting from the known fundamental R-matrix of the chiral Potts model [START_REF] Bazhanov | Chiral Potts model as a descendant of the six vertex model[END_REF], model which is obtained in our study as a particular case, we extend the knowledge of such a matrix beyond chiral Potts. Then, using (ABCD)type quantum algebras [START_REF] Freidel | Quadratic algebras and integrable systems[END_REF] we define reflection equations and their duals. Considering mixed equations, namely involving representation spaces of different dimensions, spin 1/2 and arbitrary cyclic ones, we are able to define (quantum) boundary matrices and thus a fundamental transfer matrix. We also find explicit INTRODUCTION solutions for these boundaries (the one diagonalising the v-basis, where v is one of the elements of the dynamical Weyl pair pû, vq). At the end of this chapter, using the commutation of these multi spectral parameter fundamental transfer matrices between themselves and with the 6-vertex transfer matrix, we can define multi parameter families of integrable local Hamiltonians, with integrable boundaries. The chapter ends with some explicit models of interest, as the XXZ spin 1 chain at root of unity, the sine Gordon model or the (superintegrable) chiral Potts model, stating a symmetry on the boundaries which has to be investigated.

We conclude this thesis by discussing several open questions that could be investigated now building on our results.

Chapter 1

Quantum integrable models

As we briefly described in the Introduction, for finite number of degrees of freedom, there is a rigorous definition of the notion of integrability in classical mechanics due to Liouville. However, already for classical continuum models, there is no universal analogue of the Liouville theorem as the notion of "an infinite number of independent conserved charges" has to be examined case by case carefully. The situation for quantum models is not better, even for a finite number of degrees of freedom, as there is no analogue of Liouville theorem either. Nevertheless, since the pioneering work of Bethe [44], a lot has been achieved to design powerful methods to solve the spectrum and even the dynamics of many quantum models of interest. The most successful framework in this respect is provided by the so-called Quantum Inverse Scattering Method (QISM). It can be considered as a quantised version of the Classical Inverse Scattering Method, that helps solving classical continuum theories with soliton solutions, while providing also an algebraic version of the Bethe ansatz. It also inherits the strategy developed in 2D solvable lattice models using the notion of commuting transfer matrices. In fact, as we will describe in Chapter 3, it also contains the necessary ingredients for defining a quantum analogue of the Separation of Variables (SoV) method, that, in some sense, could be considered as a practical definition of quantum integrability.

As in the classical case, the first question to be able to diagonalise a quantum Hamiltonian is to find a complete set of commuting conserved charges, having common simple spectrum, hence leading to a complete description of eigenstates using their corresponding quantum numbers. Of course this is a priori not enough as one has still to construct the corresponding eigenvectors and compute their energy levels. In fact one would also be interested in computing dynamical correlation functions. The QISM provides the algebraic framework to tackle such problems for a large class of systems, mainly defined on 1D lattices. We will present this method in the particular example of the 6-vertex Yang-Baxter algebra, which can describe a large class of models of physical interest, such as Heisenberg spin chains, the lattice sine-Gordon model or the chiral Potts model. Then we will focus on the Algebraic Bethe Ansatz on the example of the XXZ spin-1/2 chain to give a concrete example. A short section of the chapter is then dedicated to the computation of the form factors of local operators and correlation functions of that model. Then we will define the class of models under study in this thesis, namely, associated to the cyclic representations of the 6-vertex Yang-Baxter algebra. Before doing that, let us very briefly describe the corresponding classical settings in a way that will make clear the classical-quantum correspondence.

Classical integrable structures

The modern starting point in classical integrable models is the existence of an auxiliary linear system (Lax pair) the compatibility condition of which (Lax equation or zero-curvature equation) is equivalent to the initial equations of motion. The original Lax equation takes the form [26],

dL dt " rM, Ls (1.1)
where L and M are size n matrices, their entries being functions of the dynamical variables of the system at hand. The very nice feature of this equation is that one can immediately extract conserved quantities under time evolution from the invariants of the L-matrix. Furthermore, the fact that these invariants, as functions on the phase space of the system, are in involution under some Poisson brackets structure is equivalent to the existence of an r-matrix [START_REF] Faddeev | The quantum method of the inverse problem and the Heisenberg xyz model[END_REF][START_REF] Faddeev | Les Houches lectures of 1982[END_REF] such that [START_REF] Babelon | Hamiltonian structures and Lax equations[END_REF]:

tL 1 , L 2 u " rr 12 , L 1 s ´rr 21 , L 2 s (1.2)
where we have introduced standard tensor product notations, with an n-dimensional vector space V such that L 1 " Lpλq b 1, L 2 " 1 b Lpµq and r 12 " r 12 pλ, µq are endomorphisms of V b V and λ and µ stands for the spectral parameters (they are in many cases just free complex numbers but could eventually be more sophisticated objects). In the above equation r 21 stands for the same object as r 12 in which the two vector spaces copies in the tensor product V b V have been exchanged together with the corresponding spectral parameters. While in general the matrix r 12 is a function on the phase space, it so happens that in many classical integrable systems it is just a constant matrix function only of the spectral parameters, moreover having the following antisymmetry property r 21 " ´r12 . The Jacobi identity of the Poisson brackets then turns into the following (sufficient)1 quadratic relations for the r-matrix known as the classical Yang-Baxter equation [START_REF] Faddeev | The quantum method of the inverse problem and the Heisenberg xyz model[END_REF][START_REF] Sklyanin | Inverse scattering method and quantum nonlinear Schrödinger equation[END_REF]: Originally designed by Gardner,Greene, Kruskal and Miura [25] to solve the Korteweg-de-Vries equation [21], the Classical Inverse Scattering Method consists in constructing the so-called scattering data, which turns out to contain the analogue of the action-angle variables for the continuum model. Those are given by the monodromy matrix T px, y; tq of this linear system. It is defined at fixed time t to transport the solution Ψ from the point y to the point x as Ψpx, tq " T px, y; tqΨpy, tq and reduces to the identity if x " y. It contains, e.g. for periodic boundary conditions on the interval ra, bs, i.e. such that M pa, tq " M pb, tq for any t, conserved quantities under time evolution given by its trace trrT pb, a; tqs. If one goes to the real line instead of the interval, the off-diagonal elements of T lead to the determination of angle type variables first achieved in these settings by Faddeev and Zakharov [34] for KdV. It needs to compute the Poisson brackets structure of the matrix elements of T . They are given in terms of the linear Poisson bracket for the L's which for a large class of models reads [START_REF] Sklyanin | Quantum version of the method of inverse scattering problem[END_REF]: As we will see now, the structure for quantum integrable lattice models is very closely related to the classical one.

rr 12 ,
tL 1 px

Quantum Inverse Scattering Method

The QISM discovery originates from the study of the non-linear Schrödinger model that on the one hand was solvable through the Classical Inverse Scattering Method and on the other hand admits a quantisation (the 1D quantum Bose gas) solvable by coordinate Bethe ansatz. Adapting transfer matrix techniques from Baxter's corner transfer matrix [56,57], Sklyanin was able to understand how to quantise the classical inverse scattering scheme. The main feature of QISM is to embed the quantum Hamiltonian at hand into an algebra of operators (the entries of the quantised version of the monodromy matrix) that on the one hand provides a large abelian sub-algebra of conserved charges (containing the Hamiltonian) generated by the transfer matrix (the trace of the monodromy matrix), while on the other hand it gives access through the off diagonal elements of the same monodromy matrix to creation and annihilation operators of the common eigenstates of the transfer matrix (and hence of the Hamiltonian). The whole scheme is made possible thanks to the knowledge of the quadratic Yang-Baxter algebra of the monodromy matrices entries governed by an R-matrix solving the Yang-Baxter cubic equation that appeared previously in 2D solvable models [50,[52][53][54][START_REF] Yang | Some exact results for the many-body problem in one dimension with repulsive deltafunction interaction[END_REF].

Yang-Baxter algebra and transfer matrix

Let us consider a quantum system on a one dimensional lattice having N sites. To each site n of the chain, we associate a local Hilbert space of states H n and an algebra of local operators acting on H n . The total Hilbert space H of the chain is the tensor product of the local spaces:

H " N b n"1 H n (1.11)
The quantised version of the Lax matrix is the so-called quantum Lax operator. It is a m ˆm matrix, its entries being operators on the local Hilbert space H n . We can introduce the auxiliary space V a of dimension m, such that the quantum Lax operator is an element of EndpV a b H n q denoted as L an pλ, ξ n q, CHAPTER 1. QUANTUM INTEGRABLE MODELS with λ standing generically for the spectral parameter and ξ n for the inhomogeneity parameter2 associated to the site n. In that way, the entries of two Lax operators defined in two different sites commute. At the same site they obey quadratic commutation relations with structure constants given by an R-matrix: R ab pλ, µqL an pλ, ξ n qL bn pµ, ξ n q " L bn pµ, ξ n qL an pλ, ξ n qR ab pλ, µq (1.12)

while the R-matrix satisfies the Yang-Baxter cubic equation:

R ab pλ, µqR ac pλ, νqR bc pµ, νq " R bc pµ, νqR ac pλ, νqR ab pλ, µq (1.13)

These equations hold respectively in the spaces

V a b V b b H n and V a b V b b V c
and we use the standard tensor product notation with lower indices to specify in which spaces the various objects act. For example, the notation R ab means that the matrix R acts non trivially only on the space V a b V b while it acts as the identity in all other spaces. It means in particular that it is a pure number matrix (it acts on H n as the identity).

We now define the quantum monodromy matrix of the chain, denoted M aQ pλq, as the ordered product of the Lax operators all along the chain. It is an m ˆm matrix with entries that are quantum operators acting on the space H symbolised by the subindex Q, i.e. it is an element of EndpV a b Hq:

M aQ pλ; ξ 1 , . . . ξ N q " L aN pλ, ξ N q...L a1 pλ, ξ 1 q (1.14)
For simplicity, in the following we will omit writing explicitly the dependency of M aQ on the inhomogeneity parameters ξ j , namely M aQ pλ; ξ 1 , . . . ξ N q " M aQ pλq. One can show that the elements of the monodromy matrix satisfy quadratic commutation relations straightforwardly coming from (1.12) and named Yang-Baxter algebra: R ab pλ, µqM aQ pλqM bQ pµq " M bQ pµqM aQ pλqR ab pλ, µq (

It is the associative algebra generated by the elements of the monodromy matrix thanks to the Yang-Baxter equation for the R-matrix (1.13). The above relations degenerate in the classical limit in which R " 1 `i r `op q to the above classical Poisson bracket relations.

A very important object at the heart of the Inverse Scattering Method is the transfer matrix T pλq, which is defined as the trace over the auxiliary space of the monodromy matrix:

T pλq " tr a tM aQ pλqu (1.16)

Thus T pλq is an operator acting on the total Hilbert space H of the chain. As we shall see in particular examples, the Hamiltonian can be computed through trace identities in terms of the transfer matrix at the homogeneous limit. For an invertible R-matrix, the equation (1.15) gives, taking the trace over the two auxiliary spaces V a and V b :

@pλ, µq P C 2 , rT pλq, T pµqs " 0 (1.17)

Hence the transfer matrix generates via an expansion in the spectral parameter λ a full series of commuting operators that also commute with the Hamiltonian. Therefore it leads to a full series of commuting conserved operators generating the symmetries responsible for the integrability properties of the system. The main question is then to find a common basis of eigenvectors for the transfer matrix T pλq, namely a basis independent of the spectral parameter λ, and to compute the corresponding eigenvalues of T pλq, leading to the spectrum of the Hamiltonian. To effectively construct the eigenvectors and eigenvalues, one has to go one step further. The two main approaches we will consider in this thesis are the Algebraic Bethe Ansatz (ABA), developed in paragraph 1.4 and the quantum separation of variables, developed in Chapter 3. These two methods make use the above Yang-Baxter algebra. The main difference between these two methods is, as we will see later on, that the quantum separation of variables, for all the studied models for which this technique can be applied, has so far led to the full characterisation (and proof of the sim-1.3. THE 6-VERTEX YANG-BAXTER ALGEBRA plicity) of spectrum of the transfer matrix, while such a completeness is in general out of the reach of ABA.

A quantum integrable chain is thus completely characterised by the data of an R-matrix solution of the Yang-Baxter equation and a Lax operator giving a representation in H n of the Yang-Baxter algebra associated to R. In general, starting from a given system, there is no standard procedure to show its integrability finding such an R-matrix and a corresponding Lax operator. However, there are quite general techniques to construct R-matrices and Lax operators satisfying (1.12) from representation theory of quantum groups, and a considerable literature has been established classifying the solutions and the physical models which can be described [101,[START_REF] Jimbo | Topics from representations of U q pgq -An introductory guide to physicists. Quantum groups and quantum integrable systems[END_REF]. The equation (1.12) appears this way as a pivotal relation: it is the starting point to find Lax operators, i.e. integrable models, and it also provides the key to their solving.

The 6-vertex Yang-Baxter algebra

In this section, we focus on the 6-vertex R-matrix, solution of the Yang-Baxter equation (1.13), and on its associated algebra, the 6-vertex Yang-Baxter algebra. Very important models are described thanks to this matrix: the XXX and XXZ spin 1/2 chains, the discretised sine-Gordon model or the chiral Potts model. The main model studied in this thesis, namely the τ 2 model (with general integrable boundary conditions), is also based on the 6-vertex R-matrix.

This matrix is a solution of the Yang-Baxter equation (1.13) with an auxiliary space of dimension 2, it is explicitly written:

Rpλ, µq " ¨apλ, µq 0 0 0 0 bpλ, µq c 0 0 c bpλ, µq 0 0 0 0 apλ, µq ‹ ‹ ' (1.18)
with apλ, µq " λq µ ´µ λq , bpλ, µq " λ µ ´µ λ and c " q ´1{q (1.19)

The parameters λ and µ are the spectral parameters, while q is a free parameter. One of the properties of this matrix is its dependence on the ratio of the spectral parameters. In the next, the 6-vertex R-matrix will be often denoted Rpλq, keeping the same notation R:

Rpλ, µq " Rpλ{µq (1.20)
It is standard to denote such a monodromy matrix, with an auxiliary space V 0 of dimension 2, by:

M 0Q pλq " ˆApλq Bpλq Cpλq Dpλq ˙(1.21)
Note that according to (1. where we introduced the structure constants: bpλ, µq " bpλ, µq{apλ, µq and cpλ, µq " c{apλ, µq (1.36) In particular, let us comment that the four families Apλq, Bpλq, Cpλq and Dpλq are each commuting families of operators. Let us also notice here that the inhomogeneity parameters ξ j being attached to the lattice sites, they are the same in Apλq and in Apµq.

Quantum determinant Let us complete the description of the 6-vertex Yang-Baxter algebra by introducing the quantum determinant q-det M aQ pλq. It is a central element of the algebra:

@pλ, µq P C 2 , rq-det M aQ pλq, M bQ pµqs " 0 (1.37)
which can be defined for any representation (not necessarily associated to the monodromy matrix, but also for the Lax matrices, representations of the Yang-Baxter algebra in the local Hilbert space H n ). Due to the particular form of the 6-vertex R-matrix, it can be written in terms of the generators as:

q-det M 0Q pλq " ApλqDpλ{qq ´BpλqCpλ{qq (1.38)
Let us mention that in order to find quantum determinants, Kulish and Sklyanin [START_REF] Kulish | Quantum spectral transform method : recent developments[END_REF] make use of the degeneracy of the R-matrix (1.20) at the points λ " 1{q. This one reduces indeed to the projector P

´:

Rp1{qq " ´2pq ´1{qq P ´with P

´" ¨0

0 0 0 0 1{2 ´1{2 0 0 ´1{2 1{2 0 0 0 0 0 ‹ ‹ ' (1.39)
Then the 6-vertex R-matrix satisfies moreover the identity (P ´is rank 1):

R 12 p1{qqR 13 pλ{qqR 23 pλq " cpλqR 12 p1{qq (1.40)
with cpλq a scalar function, leading to the centrality of the element q-det M 0Q pλq " tr 00

1 P 00 1 M 0Q pλ{qqM 0 1 Q pλq ( (1.41)
The scalar cpλq can thus be understood as the quantum determinant of the R-matrix itself. What is more, using the Yang-Baxter equation and the commutation relations (1.33) and (1.32), one can easily check the different expressions:

q-det M 0Q pλq " DpλqApλ{qq ´CpλqBpλ{qq (1.42) q-det M 0Q pλq " Dpλ{qqApλq ´Bpλ{qqCpλq (1.43) q-det M 0Q pλq " Apλ{qqDpλq ´Cpλ{qqBpλq (1.44)
Lastly, let us mention that the quantum determinant of the monodromy matrix factorises in the q-det M 0Q pλq " q-det L 0N pλq ... q-det L 01 pλq (1.45)

Inversion formula Among other properties that will be used later, the quantum determinant is useful to express the inverse of the monodromy matrix. Indeed, it holds the following inversion formula:

σ y 0 M t0 0Q pλ{qqσ y 0 M 0Q pλq " M 0Q pλqσ y 0 M t0 0Q pλ{qqσ y 0 " q-det M 0Q pλq (1.46)
where t 0 denotes the transposition in the auxiliary space V 0 and σ y is the Pauli matrix.

The use of different representations of the 6-vertex Yang-Baxter algebra leads to different quantum systems. Let us just cite for example the XXZ spin-s chains, the discretised sine-Gordon model or the chiral Potts model for cyclic representations (the basic operators on H n are the pairs pû n , vn q; they satisfy a Weyl algebra ûn vn " qv n ûn with q is a root of unity. ûn and vn are exponentials of conjugated variables, i.e. a field φ and its conjugate momentum π). We will focus in this thesis on the τ 2 model and its generalisations, associated to cyclic representations.

Spectrum characterisation by the Algebraic Bethe Ansatz: example of the XXZ spin 1/2 chain

As mentioned previously, the integrability can be characterised in the framework of Quantum Inverse Scattering Method. The next step is thus to solve the system, i.e. to characterise the spectrum. Several techniques have been introduced, as Baxter Q-operators, Algebraic Bethe Ansatz or quantum separation of variables to achieve this goal. We present here the Algebraic Bethe Ansatz, whereas the chapter 3 is dedicated to the quantum separation of variables. The Algebraic Bethe Ansatz has been developed in the late 70's. The idea is to use the Yang-Baxter algebra commutation relations and the existence of a so-called reference state |0y, which is an eigenstate of the operators Apλq and Dpλq and such that the action of Cpλq gives 0. Then the action of the operator Bpλq on |0y allows to construct eigenvectors of the transfer matrix. This method led to the spectrum of a large class of models, here we present it on the example of the XXZ spin 1/2 chain.

The XXZ 1/2 Heisenberg spin chain

The XXZ 1/2 Heisenberg spin chain is composed of N sites, to each one is associated a local Hilbert space of dimension 2. This model is characterised by the 6-vertex R-matrix, and the following Lax operators:

L 0n pλq " ˆx`p λq `x´p λqσ z n pq ´1{qqσ ń pq ´1{qqσ ǹ x ´pλq `x`p λqσ z n ˙(1.47)
with the scalar function x ˘pλq " 1 2 pλ q ´1{pλ qq ˘pλ ´1{λqq (1.48)

Let us comment that this Lax operator satisfies (1.12) as it is just proportional to the R-matrix itself in V 0 b H n , and then this relation is simply the Yang-Baxter equation (1.13). We introduce thereafter the monodromy matrix, with the inhomogeneities pξ n q 1ďnďN :

M 0Q pλq " L 0N pλ{ξ N q...L 01 pλ{ξ 1 q " ˆApλq Bpλq Cpλq Dpλq ˙(1.49)
According to the standard procedure described in the first section, the transfer matrix is given by: T pλq " tr 0 tM 0Q pλqu " Apλq `Dpλq (1.50) CHAPTER 1. QUANTUM INTEGRABLE MODELS Eventually, at the homogeneous limit (for which it holds ξ i " 1 for all the inhomogeneities), one can show that:

H " pq ´q´1 q d ln pT pλqq dλ

ˇˇˇλ "1 ´N q `q´1 2 (1.51)
is indeed the Hamiltonian of the model under consideration, namely:

H " N ÿ n"1 ˆσx n σ x n`1 `σy n σ y n`1 `q `q´1 2 σ z n σ z n`1 ˙(1.52)
with periodic boundary conditions:

σ s N `1 " σ s 1 for s " x, y or z (1.53)

The Algebraic Bethe Ansatz

To construct eigenvectors of the transfer matrix T pλq, the Algebraic Then, starting from the reference state, eigenvectors of the transfer matrix can be computed using the operators Bpλq as creator operators. Explicitly, let us define the following "n-states" vectors:

|λ 1 , ..., λ n y " Bpλ 1 q...Bpλ n q |0y (1.58)
where the pλ i q 1ďiďn is a family of complex number.

The method consists in searching the eigenvectors of T pλq among these n-states vectors. Let us remark that from the 6-vertex Yang-Baxter algebra, Bpλq is a commuting family of operators, so the order of the pλ i q 1ďiďn in the n-states vectors is not important.

Then, thanks to the 6-vertex Yang-Baxter commutation relations, one can show the following:

Apλq |λ 1 , ..., λ n y " Λ |λ 1 , ..., λ n y `n ÿ k"1 Λ k Bpµq λ 1 , ..., x λ k , ..., λ n E (1.
ź j"1 b ´1pλ j , µq , Λ k " apλ k q c bpµ, λ k q n ź j"1 j‰k b ´1pλ j , λ k q (1.61) Λ " dpµq n ź j"1 b ´1pµ, λ j q , Λk " dpλ k q c bpλ k , µq n ź j"1 j‰k b ´1pλ k , λ j q (1.62)
This way, a sufficient condition for |λ 1 , ..., λ n y to be an eigenstate of the transfer matrix (1.50) is:

@k P 1, n , Λ k `Λ k " 0 (1.63)
Due to the skew symmetry property bpλ k , µq " ´bpµ, λ k q, this condition is independent of λ, and can be put under the form:

@k P 1, n , apλ k q dpλ k q " n ź j"1 j‰k bpλ j , λ k q bpλ k , λ j q (1.64)
It means that if such a condition is satisfied, the corresponding n-state vector |λ 1 , ..., λ n y is a common eigenvector of T pλq, @λ P C (provided of course it is not zero). These are the famous Bethe equations, first obtained by Bethe [44] for the XXX-1/2 spin chain (using coordinate ansatz), obtained from XXZ in the limit q Ñ 1, while rescaling all objects. Similar forms for this system of equations occur for other integrable models. One can see that, up to the existence of the reference state |0y, the system (1.64) is valid for all the models described thanks to the 6-vertex R-matrix. This method has been generalised to other R-matrix.

Eventually, let us mention that to construct eigenvectors on the left, one can use the same procedure as the one describe above but using the following "n-states", constructed thanks to the C operators:

xλ 1 , ..., λ n | " x0| Cpλ 1 q...Cpλ n q (1.65)
The reference state x0| satisfies:

x0| Apλq " apλq |0y , x0| Dpλq " dpλq (1.66)

x0| Cpλq ‰ 0 , x0| Bpλq " 0 (1.67)

Limitations and advantages

The ABA method is quite a powerful tool. However it suffers from two main limitations, the first one being notably the need of a reference state, that can be problematical. Indeed, for some models like the antiperiodic XXZ spin-1/2 chain, there is no obvious reference state. And then, even if a reference state has been identified, it doesn't ensure the completeness of the description of the eigenvectors. Moreover, not all solutions of Bethe equations provide a non zero state |λ 1 , ..., λ n y.

Let us simply mention here that the Bethe equations have been extensively studied. For example, there is the possibility to describe the ground state and the excited states in the thermodynamic limit (for which N Ñ `8) in terms of linear integral equations. [47,48]

Form factors and correlation functions

To solve completely a quantum integrable model it is not sufficient to get its spectrum. We also need to compute its dynamical correlation functions. Moreover, if we would like to consider perturbations around such an integrable model, we would need the matrix elements of the perturbation potential in the basis of eigenvectors of the transfer matrix. These quantities are at the heart of statistical physics, as in principle any physical and experimentally observable can be expressed through these functions.

CHAPTER 1. QUANTUM INTEGRABLE MODELS

An n-point correlation function is given by the following normalised trace:

tr φ 1 ...φ n e ´βH ( tr te ´βH u (1.68)
where H is the Hamiltonian of the system, β " pk B T q ´1 with T the absolute temperature and k B the Boltzmann constant, and the φ i are a set of local operators. In the zero temperature limit, if the ground state is non degenerated and denoted |Ψ g y, the trace reduces to the mean value of the local operators on this ground state :

xΨ g | φ 1 ...φ n |Ψ g y xΨ g |Ψ g y (1.69)
One way to compute such a quantity is to introduce the decomposition of the identity in terms of the full set of eigenvectors xt| and |ty of the Hamiltonian:

1 " ÿ tPSppHq |ty xt| xt|ty (1.70)
leading to the expression:

xΨ g | φ 1 ...φ n |Ψ g y xΨ g |Ψ g y " ÿ t1PSppHq ÿ t2PSppHq ... ÿ tn´1PSppHq xΨ g | φ 1 |t 1 y xt 1 | φ 2 |t 2 y ... xt n´1 | φ n |Ψ g y xΨ g |Ψ g y xt 1 |t 1 y ... xt n´1 |t n´1 y (1.71)
To compute such a quantity we need to solve the following problems :

• Determine the eigenstates |ty of the Hamiltonian using Algebraic Bethe Ansatz and solving the Bethe equations.

• Then determine the action, say to the right, of the local operators φ j on the right Bethe states. This can be achieved by first solving the so-called Quantum Inverse Scattering Problem [74], namely by reconstructing any local operator φ j in terms of the elements of the monodromy matrix ; then using the Yang-Baxter algebra, one can compute the action of any element of the monodromy matrix on any Bethe state. It gives again a state that is a combination of states of the form (1.58), namely the action of a product of B operators on the reference state, but which are no longer eigenstates of the transfer matrix. Those are called off shell Bethe states.

• Then we need to compute the resulting scalar products between an arbitrary left Bethe eigenstate with an off-shell right state. In the framework of the Algebraic Bethe Ansatz, determinant formulas have been obtained for such scalar products [73,[START_REF] Slavnov | Calculation of scalar products of wave functions and form factors in the framework of the algebraic Bethe ansatz[END_REF]. For the square norm xt|ty of an eigenstate, this is given by the Gaudin determinant while for the scalar products between an eigenstate and a general state, it is the ratio of two determinants; we need both in (1.71).

• Then one would have to compute the resulting sums at least in the thermodynamic limit and in the asymptotic regime where the local operators φ j are located in points n j of the chain such that |n i ´nj | ąą 1 for any different i and j.

These different steps have been accomplished for the XXZ spin-1/2 chain in the massless regime in a series of works [73,[76][77][78][79][80][START_REF] Kitanine | A form factor approach to the asymptotic behavior of correlation functions in critical models[END_REF][START_REF] Kitanine | Form factor approach to dynamical correlation functions in critical models[END_REF][START_REF] Kitanine | Large-distance asymptotic behaviour of multi-point correlation functions in massless quantum models[END_REF][START_REF] Kozlowski | Microscopic approach to a class of 1d quantum critical models[END_REF], where one can also find the expression of the form factors associated to the spin operators σ ń , σ ǹ , σ z n . These results constitute a real breakthrough as before, the only known exact results obtained from first principles were the one for the free fermion point, namely for q " i. Combined with powerful numerical summation techniques, it led to experimentally verifiable forecasts for scattering neutrons on magnetic materials, well described by a XXZ spin chain [117,118].

Let us mention another way to compute quantities of the form (1.69), which consists in first computing ΨD " φ 1 ...φ n |Ψ g y and then in computing the scalar product @ Ψ g ˇˇΨ D . This is the way taken in [75,112].

1.6. CYCLIC REPRESENTATIONS OF THE 6-VERTEX YANG-BAXTER ALGEBRA

Cyclic representations of the 6-vertex Yang-Baxter algebra

In this thesis, we study the τ 2 model with general integrable boundary conditions. In this paragraph, we aim to describe this model for periodic boundary conditions, as the notation of this section will be used all along the manuscript. Historically, this model is obtained from the Lax operator which is a general cyclic solution of the equation (1.12) with the 6-vertex R-matrix [START_REF] Bazhanov | Chiral Potts model as a descendant of the six vertex model[END_REF], for q a root of unity. It is defined by the following Bazhanov-Stroganov Lax operator:

L 0n pλ, p n q " ¨λα n vn ´λ´1 β n v´1 n ûn pq ´1{2 a n vn `q1{2 b n v´1 n q û´1 n pq 1{2 c n vn `q´1{2 d n v´1 n q λ ´1γ n vn ´λδ n v´1 n 'r0s (1.72)
where p n " pα n , β n , γ n , δ n , a n , b n , c n , d n q denotes the 8 parameters associated to the site n. In order for this matrix to satisfy the Yang-Baxter equations (1.12), the parameters on a given site are not independent and have to satisfy :

@ n P 1, N , α n γ n " a n c n and β n δ n " b n d n (1.73)
We can moreover emphasise a global normalisation, with the parameter γ n for instance. What is more, let us highlight that the inhomogeneities ξ n can be absorbed into the parameters p n . Moreover, the hermitian operators ûn and vn are the generators of the following local Weyl algebra on H n : @pn, mq P 1, N , ûn vm " q δmn vm ûn (1.74) Through all this thesis, we will study the case where q is the following particular p th -root of unity:

q " e 2ikπ{p , k P N ˚, p " 2l `1 , l P N ˚, with k and p coprime integers, (

and we will assume

@n P 1, N , ûp n " vp n " 1 (1.76)
We consider this choice to handle irreducible representations of the local Weyl algebras, as reducible representations appear under different possible choices of q. As a consequence, we can use finite-dimensional representations of dimension p for each Weyl algebra, referred as cyclic representations. We will use mainly two different basis:

• The so-called v-basis, where we introduce the left states txk, n| vu kP 0,p´1 , with the cyclicity xk `p, n| v " xk, n| v and the right states t|k, ny vu kP 0,p´1 with the cyclicity |k `p, ny v " |k, ny v

The action of the operators read:

xk, n| v ûn " xk `1, n| v xk, n| v vn " q k xk, n| v ; ûn |k, ny v " |k ´1, ny v vn |k, ny v " q k |k, ny v (1.77)
• The so-called û-basis, where we introduce the left states txk, n| ûu kP 0,p´1 , with the cyclicity xk `p, n| û " xk, n| û and the right states t|k, ny ûu kP 0,p´1 with the cyclicity |k `p, ny û " |k, ny û

The action of the operators read:

xk, n| û ûn " q ´2k xk, n| û xk, n| û vn " xk ´l, n| û ; ûn |k, ny û " q ´2k |k, ny û vn |k, ny û " |k `l, ny û (1.78)

In a very standard way, we define the bulk monodromy matrix:

M 0Q pλ, p Q q " L 0N pλ, p N q...L 01 pλ, p 1 q (1.79)
where p Q denotes the parameters in the whole quantum space p Q " pp 1 , ..., p N q. The Yang-Baxter CHAPTER 1. QUANTUM INTEGRABLE MODELS equation implies that the transfer matrix:

τ 2 pλq " tr 0 tM 0Q pλ, p Q q u " Apλq `Dpλq (1.80)
defines a one parameter family of commuting operators. Moreover, for this model it is possible to define an operator:

Θ " N ź n"1 vn (1.81)
which commutes with the transfer matrix:

rτ 2 pλq, Θs " 0 (1.82)
The eigenvalues of the operator Θ being p th roots of unity, it is possible to associate to each eigenvector of the transfer matrix its eigenvalue by this operator. It notably allows a foliation of the transfer matrix spectrum Σ τ2 in terms of Σ k τ2 (with k P 0, p ´1 ), the spectrum of the eigenvectors whose Θ-eigenvalue is

q k : Σ τ2 " p´1 ď k"0 Σ k τ2 (1.83)
Let us complete the description of cyclic representations of the 6-vertex Yang-Baxter algebra by making explicit the quantum determinant. Using the factorisation property (1.45), one can write:

q-det M 0Q pλ, p Q q " N ź n"1 k n p λ µ n,`´µ n,λ qp λ µ n,´´µ n,λ q " apλqdpλ{qq (1.84)
where the constants k n and the zeros µ n,˘o f the quantum determinant are given by:

k n " pa n b n c n d n q 1{2 and µ n,h " # iq 1{2 pa n β n {α n b n q 1{2 h " ìq 1{2 pc n β n {α n d n q 1{2 h "
´(1.85)

As for the functions apλq and dpλq, they are defined by:

apλq " a 0 N ź n"1 ˆβn λ `q´1 b n α n a n λ ˙and dpλq " p´1q N a 0 N ź n"1 a n c n α n ˆ1 λ `q d n α n c n β n λ ˙(1.86)
Let us mention the close link existing between this τ 2 model and the chiral Potts model [START_REF] Bazhanov | Chiral Potts model as a descendant of the six vertex model[END_REF]. Indeed, the transfer matrix of the chiral Potts model is a Baxter Q-operator for the τ 2 model, a property that allows to obtain results concerning the chiral Potts model. For example, the spectrum Σ τ2 is completely characterised (via the construction of a Baxter Q-operator by separation of variables [START_REF] Grosjean | The τ 2 -model and the chiral Potts model revisited : Completeness of Bethe equations originated from Sklyanin SOV[END_REF]).

Another interesting feature with the Lax operator (1.72) is its possibility to describe different models, like the Fateev-Zamolodchikov model (linked to the discretised sine Gordon model at root of unity) or the XXZ spin s " pp ´1q{2 at root of unity, simply by tuning the different parameters p n .

Eventually, let us emphasise that important results are known for this model, from the characterisation of the spectrum (using separation of variables) up to the computation of some form factors [START_REF] Grosjean | On form factors of local operators in the lattice sine-Gordon model[END_REF][START_REF] Grosjean | On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models[END_REF][START_REF] Grosjean | The τ 2 -model and the chiral Potts model revisited : Completeness of Bethe equations originated from Sklyanin SOV[END_REF] for the periodic model. Now we aim to start to describe the model with general integrable boundaries.

Chapter 2

Quantum integrable models with boundaries

As briefly stated in the introduction, the study of models with boundaries has attracted a large research enthusiasm as they can describe both equilibrium and out of equilibrium physics. Some interesting applications concern the description of classical stochastic relaxation processes, like ASEP [START_REF] Derrida | An exactly soluble non-equilibrium system: the asymmetric simple exclusion process[END_REF][START_REF] Schutz | Phase transitions and critical phenomena[END_REF], [START_REF] Shiroishi | Bethe Ansatz equation for the Hubbard model with boundary fields[END_REF][START_REF] Shiroishi | Integrable boundary conditions for the one-dimensional Hubbard model[END_REF][START_REF] Alcaraz | Reaction-diffusion processes, critical dynamics, and quantum chains[END_REF][170][START_REF] De Gier | Bethe Ansatz solution of the asymmetric exclusion process with open boundaries[END_REF][START_REF] De Gier | Exact spectral gaps of the asymmetric exclusion process with open boundaries[END_REF] and quantum transport properties in spin systems [START_REF] Sirker | Diffusion and ballistic transport in one-dimensional quantum systems[END_REF][START_REF] Prosen | Open XXZ spin chain: nonequilibrium steady state and a strict bound on ballistic transport[END_REF].

In the previous chapter, we recalled that Hamiltonians of quantum integrable systems can be obtained from the transfer matrix. However, with the previous definition for this transfer matrix as a trace of the monodromy matrix, it turns out that for all the considered models the reconstructed Hamiltonians have periodic boundary conditions. The aim of this chapter is to introduce more general transfer matrices, in order to describe more general boundaries.

A first generalisation is presented considering twisted transfer matrices. In paragraph 2.1, we show that one can indeed use scalar representations of the Yang-Baxter algebra to generate twisted transfer matrices, leading to quasi-periodic integrable Hamiltonians.

However, the main goal of this chapter is to outline the reflection algebras introduced by Sklyanin [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF]. This one allows in particular to define a boundary transfer matrix (the trace of the boundary monodromy), which can be used to generate quantum integrable Hamiltonians with general integrable boundaries.

An important feature that emerges to effectively define local Hamiltonians is the notion of fundamental and non fundamental models, regarding to the relative dimensions of the auxiliary and quantum spaces. Thus, the last paragraph is dedicated to the recall of the fusion procedure, which allows to define spin chain Hamiltonians for non fundamental models. Let us mention that all along this chapter, the XXZ spin chain is used as the main example to explain the various concepts and tools.

Twisted boundary conditions

As mentioned earlier, a first generalisation and a first step to take into account boundaries in the framework of the Quantum Inverse Scattering Method, is given by the twisted transfer matrix. In the general setting of section 1.2, instead of considering the transfer matrix (1.16), one can define a twisted transfer matrix:

T pΣq pλq " tr a tΣ a pλqM aQ pλqu (2.1)
with Σpλq a scalar matrix, i.e. an element of M m pCq (We recall that m denotes the dimension of the auxiliary space V a ). One of the most interesting feature in considering the twisted transfer matrix is the possibility to describe local Hamiltonians with twisted boundary conditions. Thanks to the trace identities, the quasiperiodic boundary conditions are linked to the choice of Σpλq. An example is given by the XXZ-spin 1/2 chain in the next paragraph.

The XXZ-1/2 quasi-periodic chain

Let us consider the XXZ-1/2 spin chain, whose periodic case has been studied in section 1.4.1.

One can find the following solutions for the scalar representations of the 6-vertex Yang-Baxter algebra:

@pα, aq P C ˆt0, 1u , Σpλq " Σ pα,aq " pσ x q a ˆeα 0 0 e ´α˙(

2.4)

where σ x is the Pauli matrix. Let us denote then T pα,aq pλq the associated twisted transfer matrix, given by (2.1). By taking the logarithmic derivative of this matrix, one can show that similarly to equation (1.51), the quantity H pα,aq " pq ´q´1 q d ln `T pα,aq pλq dλ ˇˇˇˇλ

"1 ´pq `q´1 q N 2 (2.5)
is the Hamiltonian of the XXZ spin 1/2 chain with twisted boundary conditions. Explicitly,

H pα,aq " N ÿ n"1 " σ x n σ x n`1 `σy n σ y n`1 `1 2 pq `q´1 qσ z n σ z n`1  (2.6)
with the following quasi-periodic boundary conditions:

σ s N `1 " Σ pα,aq 1 σ s 1 Σ pα,aq 1
for s " x, y or z (2.7)

In particular, to consider the twisted transfer matrix T p0,1q pλq " Bpλq`Cpλq leads to the anti-periodic chain. Let us simply mention here that at the thermodynamic limit, i.e. for an infinite number of sites, the results concerning the form factors and the correlation functions should not depend on the twist function Σ.

The 6-vertex reflection algebra and integrability

The question we address now is to describe chains with general integrable boundary conditions. For example, we aim to describe the action of two magnetic fields at the boundaries of the spin XXZ 1/2 chain. To answer this question, we report here the work of Sklyanin [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF], who introduced the notion of reflection algebra for a boundary version of the monodromy matrix. We present this algebra on the example of the 6-vertex R-matrix, and give the definition of the boundary transfer matrix, which will be used to derive integrable local Hamiltonians with integrable boundaries.

The 6-vertex reflection algebra

Let us consider Rpλq the 6-vertex R-matrix (1.20). Then following [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF] let us introduce the algebra U ´, the associative algebra defined by the generators Upλq α,β , with pα, βq P t1, 2u, considered as the elements of a square matrix Upλq, and by the relation: The algebra U `is defined in the same way with a matrix U `pλq and the relation:

R
R 12 pµ{λqU t1 `1pλqR 12 p1{pλµqqqU t2 `2pµq " U t2 `2pµqR 12 p1{pλµqqqU t1 `1pλqR 12 pµ{λq (2.9)
The equations (2.8) and (2.9) are called the reflection equations.

Moreover, Sklyanin gives a way to find representations of these algebras. In fact, his construction allows to use representations of the Yang-Baxter algebra to generate representations of the reflection algebra thanks to scalar solutions of the reflection equations. Explicitly, a solution of the reflection equation (2.8) in the total quantum space H can be obtained by the following boundary monodromy matrix:

U 0Q pλq " M 0Q pλqK 0 pλq M0Q pλq with M0Q pλq " σ y 0 M t0 0Q p1{λqσ y 0 (2.10)
where M 0Q pλq is the monodromy matrix introduced for the Yang- in the space V 1 ˆV2 (we recall that the auxiliary spaces are of dimension 2,

V i » C 2 ).
The expression for M0Q pλq is reminiscent of the inversion formula for the bulk monodromy matrix (1.46), and indeed it is linked to the inverse of the bulk monodromy thanks to the quantum determinant:

M0Q pλq " q-det M 0Q pq{λq ¨M ´1 0Q pq{λq (2.12)
As for the Yang-Baxter algebra, we will use a standard notation:

U 0Q pλq " ˆApλq Bpλq Cpλq Dpλq ˙(2.13)
to denote the operators of the boundary monodromy matrix. These ones are thus the generators of the reflection algebra, let us write explicitly some of the relations they satisfy: We emphasise on these relations as they will be used in the next; let us in particular note the commutativity of the families Bpλq and Cpλq.

CHAPTER 2. QUANTUM INTEGRABLE MODELS WITH BOUNDARIES

The same way as Kpλq is defined, let K `pλq be a scalar solution of the reflection equation (2.9): R 12 pµ{λqK t1 `1pλqR 12 p1{pλµqqqK t2 `2pµq " K t2 `2pµqR 12 p1{pλµqqqK t1 `1pλqR 12 pµ{λq (2.24)

The most general expressions for the boundary matrices Kpλq and K `pλq are known [START_REF] Cherednik | Factorizing particles on a half-line and root systems[END_REF][START_REF] Vega | The highest weight property for the SUq(n) invariant spin chains[END_REF]. Let us recall here:

K 0 pλq " K 0 pλ|ξ ´, τ ´, κ ´q " 1 ξ ´´1 ξ´ˆλ ξ ´q´1{2 ´1{pλξ ´q´1{2 q κ ´eτ´`λ2 {q ´q{λ 2 κ´e ´τ´`λ2 {q ´q{λ 2 ˘q1{2 ξ ´{λ ´λ{pq 1{2 ξ ´q˙r 0s (2.25)
It is easy to remark that the general scalar solution of the reflection equation (2.24) admits the following form in terms of the solution of the reflection equation (2.11):

K t0 `0pλq " K 0 pqλq (2.26)
Hence the following expression:

K `0pλq " K `0pλ|ξ `, τ `, κ `q " 1 ξ `´1
ξ`ˆλ ξ `q1{2 ´1{pλξ `q1{2 q κ `eτ``q λ 2 ´1{pqλ 2 q κ`e ´τ``q λ 2 ´1{pqλ 2 q ˘q´1{2 ξ `{λ ´λ{pq ´1{2 ξ `q˙r 0s (2.27) The parameters ξ ˘, τ ˘and κ ˘are generic, and associated to the boundaries. A standard notation to refer to these matrices is given by:

K 0 pλ|ξ ´, τ ´, κ ´q " ˆa´p λq b ´pλq c ´pλq d ´pλq ˙and K `0pλ|ξ `, τ `, κ `q " ˆa`p λq b `pλq c `pλq d `pλq ˙(2.28)
As for the 6-vertex Yang-Baxter algebra, there is the notion of quantum determinant. Indeed, using once again the degeneracy of the 6-vertex R-matrix at the point λ " 1{q, Sklyanin [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF] shows that the element q-det U 0Q pλq " tr 12

! P 12 U 1Q pq ´1{2 λqR 12 pλ 2 {qqU 2Q pq 1{2 λq ) (2.29)
is central for any representation of the reflection algebra (2.8):

@pλ, µq P C 2 , rq-det U 0Q pλq, U 0 1 Q pµqs " 0 (2.30)

The proof of this centrality is very similar to the one for the Yang-Baxter algebra.

Boundary transfer matrix

In a way, the reflection algebras generalise the Yang-Baxter algebra as one can define a one parameter family of commuting matrices, the so-called boundary transfer matrices, which are used to describe Hamiltonians with general integrable boundaries. The procedure is the following. Let K `pλq be a solution of the reflection equation (2.24) and let us consider the boundary monodromy matrix given by (2.10). The boundary transfer matrix is then defined as:

T pλq " tr 0 tK `0pλqU 0Q pλqu (2.31)
Thanks to the reflection equations, this is a one parameter family of commuting operators [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF]:

@pλ, µq P C 2 , rT pλq, T pµqs " 0 (2.32)
This relation is, as for the Yang-Baxter algebra, at the heart of the integrability of a system. Indeed, using trace identities, Hamiltonians can be expressed via the boundary transfer matrix (the example of the XXZ spin chain is given right after, cf (2.38)), so one exhibits a one-parameter family of commuting operators commuting also with the Hamiltonian. What is more, if the spectrum of the boundary transfer 2.3. LOCAL HAMILTONIANS ASSOCIATED TO REPRESENTATIONS OF THE REFLECTION ALGEBRA matrix is simple, i.e. if there is only one eigenvector associated to an eigenvalue, the set of conserved operators is complete as the eigenvectors are univocally labelled by the eigenvalues.

We end this section highlighting a symmetry of the boundary transfer matrix, and making explicit the generators of the reflection algebra in terms of the generators of the Yang-Baxter algebra. Following again [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF], it is indeed more convenient to work with the next shifted definition of the monodromy matrix for the Yang-Baxter algebra, compared with (1.49):

M 0Q pλq " L 0N pλ{pξ N q 1{2 qq...L 01 pλ{pξ 1 q 1{2 qq " ˆApλq Bpλq Cpλq Dpλq ˙(2.33)
This shift on λ ensures the symmetry:

T pλq " T p1{λq (2.34)
We shall come back to that issue later, which concerns both the symmetry of the eigenvalues of the boundary transfer matrix and the symmetry of the boundary terms of the Hamiltonian.

Eventually, let us write explicitly the boundary transfer matrix, which is the object at the heart of the integrability of a system, in terms of the generators of the reflection algebra:

T pλq " a `pλqApλq `d`p λqDpλq `c`p λqBpλq `b`p λqCpλq (2.35)
Moreover, we recall that the generators of the reflection algebra are quadratic in terms of the generators of the Yang-Baxter algebra (2.10), for instance:

Bpλq " ´a´p λqApλqBp1{λq `b´p λqApλqAp1{λq ´c´p λqBpλqBp1{λq `d´p λqBpλqAp1{λq (2.36) This expression for the boundary transfer matrix has to be compared with T pλq " Apλq `Dpλq for the periodic case. The study of general boundaries appears thus a priori much more involved than for periodic systems.

Remark Originally, these reflection algebras were introduced in [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF] for general R-matrices Rpλq verifying the following assumptions:

(i) The symmetries R 21 pλq " R 12 pλq and R t1 12 pλq " R t2 12 pλq

(ii) The unitarity R 12 pλqR 12 p1{λq " ρpλq for ρpλq a scalar function (iii) The crossing unitarity R t1 12 pλqR t1 12 p1{pq 2 λqq " ρpλq for ρpλq a scalar function and q a characteristic constant of the R-matrix.

The 6-vertex R-matrix is indeed satisfying these properties, with ρpλq " ´pλq ´1{pλqqq pλ{q ´q{λq and ρpλq " ρpλqq (2.37)

We aim to generalise these algebras in chapter 5. To this purpose, we will turn back to the original Cherednik picture of reflection equations [START_REF] Cherednik | Factorizing particles on a half-line and root systems[END_REF], namely the description of the reflection of two particles on a wall, but for different types (i. e. representations) of particles. But for now, we recall in the next paragraph how to compute integrable Hamiltonians from the boundary transfer matrix.

Local Hamiltonians associated to representations of the reflection algebra

As this is the case for the Yang-Baxter algebra, one has to show that the boundary transfer matrix allows to construct the local Hamiltonians one is interested in. In this section we start by giving the CHAPTER 2. QUANTUM INTEGRABLE MODELS WITH BOUNDARIES representative example of the XXZ spin chain, and then we derive a more general expression, valid for more general fundamental models.

The XXZ 1/2 spin chain with general integrable boundary conditions

In Then, up to an additive constant, the following derivative of the boundary transfer matrix in the homogeneous limit (ξ n " 1 on every site) reproduces the Hamiltonian of the XXZ 1/2 spin chain with integrable boundaries [START_REF] Niccoli | Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Form factors and complete spectrum[END_REF]:

2pq ´q´1 q 1´2N tr 0 tK `0pq 1{2 qu tr 0 tK 0 pq 1{2 qu dT pλq dλ ˇˇˇλ "q 1{2 `cste " H open XXZ (2.38)
where H open XXZ denotes the Hamiltonian:

H open XXZ " N ´1 ÿ n"1 " σ x n σ x n`1 `σy n σ y n`1 `1 2 pq `q´1 qσ z n σ z n`1  `B1 `BN (2.39)
for wich the boundary terms on sites 1 and N are given by: The main idea of the proof is given in a slightly more general case in the next paragraph. Let us just comment that contrary to the quasi-periodic Hamiltonian, the boundary Hamiltonians are obtained thanks to the derivative of the boundary transfer matrix with respect to the spectral parameter, and not as the logarithmic derivative. Moreover, with the expressions (2.39), (2.40) and (2.41) we can note the symmetry between the boundaries on sites 1 and N. They have indeed the same algebraic form, only the boundary parameters ξ ˘, τ ˘and κ ˘are different.

B 1 " q ´1{q ξ ´´1{ξ ´ˆ1

Local Hamiltonians with boundaries for fundamental models

Actually, in his original paper [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF], Sklyanin could give the expression of local Hamiltonians with integrable boundaries for fundamental models (for which the quantum spaces and the auxiliary space are isomorphic) associated to R-matrices satisfying the properties (i)-(iii) of paragraph 2.2.2, and for models whose Lax operators coincide with the R-matrix. The XXZ 1/2 spin chain is a fundamental model of this kind.

Sklyanin used a variant of Baxter argument (trace identities) to derive local Hamiltonians. Explicitly, if it holds the following properties:

• The auxiliary space V a and the local quantum spaces H n are isomorphic (fundamental models)

• There exists a point λ 0 such that Kpλ 0 q " 1 2.3. LOCAL HAMILTONIANS ASSOCIATED TO REPRESENTATIONS OF THE REFLECTION ALGEBRA

• The Lax operator (which is an R-matrix) reduces to the permutation between the spaces V a and H n in the same point λ 0 : R an pλ 0 q " P an then one can show that the first derivative of the boundary transfer matrix, at the homogeneous limit (ξ n " 1 on every site), can be put under the form:

dT pλq dλ ˇˇˇλ "λ0
" 2 H tr a tK `apλ 0 qu `tr a # dK `apλq dλ ˇˇˇλ "λ0

+ (2.42)
with the following Hamiltonian with boundaries:

H " N ´1 ÿ n"1 H n,n`1 `1 2 d K 1 pλq d λ ˇˇˇλ "λ0
`tr a tK `apλ 0 qH N a u tr a tK `apλ 0 qu (2.43)

where the bulk interactions are given by: Let us eventually emphasise that the symmetry R n,n`1 pλq " R n`1,n pλq is sufficient to have the same bulk interactions H n,n`1 for general integrable boundaries as for the bulk interactions of the quasi-periodic systems obtained from Yang-Baxter algebra 1 . Indeed, for fundamental models with the Lax operators coinciding with the R-matrix, the transfer matrix is given by: T pλq " tr a tR aN pλ{ξ N q...R a1 pλ{ξ 1 qu (2. 46) and assuming that there exists a point λ 0 such that R an pλ 0 q " P an , the first derivative leads to:

H n,
dT pλq dλ ˇˇˇλ "λ0 " N ÿ n"1 P n,n`1 dR n,n`1 pλq dλ ˇˇˇλ "λ0 (2.47)
We shall come back to that issue later, when we will derive the Hamiltonians for non fundamental models (in chapter 5).

In a nutshell, the reflection algebras mainly result in the possibility to associate to any closed integrable quantum model, characterised by a solution of the Yang-Baxter equation, its integrable quantum model with boundaries, characterised by the associated Sklyanin's solutions of the reflection equation.

The fusion procedure

As we have just seen, we know a way to define local integrable Hamiltonians (quasi-periodic or with general boundaries) for fundamental systems. In this paragraph, we briefly present the fusion procedure, which allows to consider the Hamiltonians for higher spin representations while keeping for example the 6-vertex 1 We shall come back to that issue in section 5.5.4. Let us comment that more generally, it seems only necessary to have

dR n,n`1 pλq dλ ˇˇλ "λ 0 9 dR n`1,n pλq dλ ˇˇλ "λ 0
, with a proportionality coefficient different from ´1, in order to have the same bulk interactions between the chain with quasi-periodic boundary conditions and the chain with general integrable boundaries.

CHAPTER 2. QUANTUM INTEGRABLE MODELS WITH BOUNDARIES

R-matrix. Indeed, the XXZ 1/2 spin chain is described by the 6-vertex R-matrix, and is a fundamental model having isomorphic auxiliary and quantum spaces (of dimension 2). If one wants to consider the XXZ 1 spin chain, the quantum spaces become of dimension 3 (and thus the Lax operators cannot be reduced to the permutation, as the involved spaces cannot be isomorphic).

Let us consider a fundamental R-matrix satisfying the Yang-Baxter equation (1.13). The fusion procedure, introduced in [START_REF] Kulish | Yang-Baxter equation and representation theory : I[END_REF], is a method to construct solutions of (1.13) in higher dimensional spaces.

Here we present the fusion procedure on the example of the XXZ spin chain, which allows the construction of a solution of the Yang Baxter algebra with an auxiliary space of dimension 3, starting from the 6-vertex R-matrix (solution of the Yang-Baxter algebra with an auxiliary space of dimension 2). Let us recall the degeneracy (1.39) of the 6-vertex R-matrix in the point λ " 1{q, which reduces (up to a scalar) to the rank one projector P ´, and let us consider

P `" 1 ´P ´(2.48)
which is also a projector. Using the Yang-Baxter algebra, and the property P ´P `" 0, one can show that the so-called fused R-matrices: in the spaces V ă12ą ˆV3 ˆV4 and V 1 ˆV2 ˆVă34ą respectively, where as usual the spaces V i are 2 dimensional. Regarding the definition of the spaces V ă12ą and V ă34ą , obtained as the projections of V 1 ˆV2 and V 3 ˆV4 respectively, they are of dimension 3. Similarly, in [START_REF] Mezincescu | Fusion procedure for open chains[END_REF] the authors tackle the fusion procedure for open chains. Using the degeneracy of the R-matrix, the projectors P ´and P `and the reflection equation (2.11) Hamiltonians for representations with arbitrary spin s, so we are able to give the Hamiltonians associated to XXZ spin s chains2 . As already mentioned, we aim to generalise these results, in particular to be able to compute the Hamiltonians for non fundamental models. It will be done in the chapter 5, for models associated to cyclic representations of the 6-vertex reflection algebra.

R
We henceforth have an algebraic framework to deal with general integrable boundaries and moreover, thanks to Sklyanin's construction, we can associate to any representation of the Yang-Baxter algebra, and a boundary matrix, a representation of the reflection algebra. However, the generators of the latter are quadratic in terms of the generators of the Yang-Baxter algebra, and the boundary transfer matrix appears much more cumbersome. In particular, the Algebraic Bethe Ansatz seems hardly applicable for general boundaries. To nevertheless study this boundary transfer matrix, the next chapter introduces the quantum separation of variables, which is another approach to the characterisation of the transfer matrix spectrum in the Quantum Inverse Scattering Method framework.

Chapter 3

Quantum separation of variables

In this chapter, we introduce the important notion of separation of variables for integrable systems. Starting from the classical point of view, we recall the usual definition of separate variables. Then we show the implementation of this concept in quantum models, emphasising this way on the analogy between the classical and quantum descriptions. The quantum separation of variables (SoV) allows for the characterisation of the transfer matrix spectrum in the framework of the Quantum Inverse Scattering Method. Notably, one can describe models for which the reference state |0y used in Algebraic Bethe Ansatz cannot be found directly. As an example, we construct the explicit separate basis for the anti-periodic XXZ 1/2 spin chain and highlight on this example, the main motivations to work with this tool.

Classical separation of variables

In order to understand where the concept of quantum separation of variables comes from, we start recalling briefly the classical point of view. Thus, let us consider a classical system described by 2N canonical variables pq i q 1ďiďN and pp i q 1ďiďN , and let us moreover suppose that it is Liouville integrable [16], thanks to N independent conserved quantities pH i q 1ďiďN in involution in the sense of the Poisson bracket t., .u. This way we have: @pi, jq P 1, N 2 , tq i , q j u " tp i , p j u " 0 and tq i , p j u " δ ij (3.1) as well as @pi, jq P 1, N 2 , tH i , H j u " 0 (3.2)

Then a set of 2N canonical variables px i q 1ďiďN and pv i q 1ďiďN is called separate if there exists N separate relations of the form:

@i P 1, N , f i px i , v i , H 1 , ..., H N q " 0 (3.3)
linking, through some functions f i , any couple px i , v i q with the conserved charges.

The separation of variables is one of the most powerful method to deal with integrable systems. Its prime motivation is to transform the original problem in 2N variables (q i 's and p i 's) into a set of N independent problems involving only one variable and its conjugate (x i and v i ). Indeed, the H i 's commuting with the Hamiltonian (in the sense of the Poisson bracket), they are constant and their value can be fixed. Equations (3.3) give then N relations, each one involving only one couple px i , v i q.

The standard procedure follows in general three steps to actually solve a system. First, one has to get a set of action-angle variables, defined such that their evolution equation is linear in time. Thus one can easily perform the time evolution of this set, and lastly one has to go back to the original variables at the time t, via an inverse transformation. Such a resolution by quadrature defines the Liouville integrability [16,18].

The first part of this procedure can be solved by the Liouville-Arnold theorem [16]. In particular, if the phase space of dimension 2N is compact and connected, it is diffeormophic to a N -torus. The action variables are the conserved quantities pH i q 1ďiďN , and it is possible to define angle variables thanks to a generating function SrpH i q 1ďiďN , pq i q 1ďiďN s. The latter is defined as an integral on the level manifold where the pH i q 1ďiďN are fixed, but hardly handleable in general as for a given i P 1, N , p i depends on the whole set pq i q 1ďiďN . However, if the set of variables pq i q 1ďiďN and pp i q 1ďiďN is separate, then one can use the separate relations (3.3) to simplify its expression. In particular, if it is possible to express p i as a function of q i (the quadrature), the integral splits into N one variable integrals, leading to:

SrpH i q 1ďiďN , pq i q 1ďiďN s " N ÿ k"1 S k rpH i q 1ďiďN , q k s (3.4)
Each S k , k P 1, N , is now computable or at least handleable, and the angle variables can be computed.

A simple example of separate variables is given by the classical Hamiltonian describing the motion of a particle in a central potential. The use of the spherical coordinates r,θ, φ is indeed rather natural, given the symmetry of the system under rotations. But there is more in fact, as it decouples the original problem in the three variables x,y and z (orthonormal basis) in three independent problem in r,θ, φ. The generating function and the effective quadrature of this problem can be read for example in [18], in particular in paragraph 2.7.

Construction of separate variables

For a given integrable system there is in general no universal method to construct a set of canonical separate variables. However, when there exists a Lax pair ensuring the integrability of the system, a general construction allows to define such a set. As we will highlight here, the key relation to the construction of separate variables is the quadratic Poisson bracket for the Lax operators, involving the r-matrix (cf paragraph 1.1).

To make a clear statement, let Lpλq be a Lax matrix of dimension 2 ˆ2, associated to a given system with phase space of dimension 2N. It depends on a spectral parameter λ, and of course on the canonical dynamical variables tq i , p i u 1ďiďN (we will omit writing them for simplicity):

Lpλq " ˆApλq Bpλq Cpλq Dpλq ˙(3.5)
By definition, the conserved quantities H i are obtained from the spectral invariants of Lpλq. Let us assume that the coefficients of the Lax matrix are polynomials in the spectral parameter, and that they satisfy the Sklyanin quadratic Poisson bracket:

tL 1 pλq, L 2 pµqu " rr 12 pλ, µq, L 1 pλqL 2 pµqs (3.6)
with the following r-matrix:

r 12 pλ, µq " P 12 λ ´µ (3.7)
where P 12 is the permutation operator between the spaces V 1 and V 2 . This represents already a large class of models, as for instance the Toda chain [18,[START_REF] Toda | Wave propagation in anharmonic lattices[END_REF] satisfies these assumptions.

The main line of the method is to consider the eigenvalues of the Lax matrix. Let us denote this way zpλq an eigenvalue of this matrix and Ωpλq the associated eigenvector, such that it holds:

LpλqΩpλq " zpλqΩpλq (3.8)

QUANTUM SEPARATION OF VARIABLES

The characteristic equation for the eigenvalue problem (3.8) is thus:

det pLpλq ´zpλq1q " 0 (3.9)
It defines an algebraic curve Γ, the so-called spectral curve, which is the locus of the points λ and zpλq such that it holds (3.9). We shall come back to that notion later, but we can as of now claim that this object is at the heart of the separation of variables methods: it indeed provides separate relations. The cornerstone to understand this property is the following decomposition of the determinant entering in (3.9): det pLpλq ´zpλqq " z 2 pλq ´zpλq trLpλq `det Lpλq (3.10)

The trace and the determinant of the Lax matrix being spectral invariants, they can be expressed via the conserved quantities H i . This way, the equation (3.9) only involves 2 variables, λ and zpλq, and the conserved quantities.

In this context of 2-dimensional Lax matrices, it is easy to find eigenvalues. Indeed, let x i be a zero of Bpλq. This variable, which non trivially depends on the dynamical variables of the system, is of prime interest as in such a point the Lax matrix is lower triangular:

Lpx i q " ˆApx i q 0 Cpx i q Dpx i q ˙(3.11)
Thus the eigenvalues are immediately known, and in particular the couple px i , Apx i qq belongs to the spectral curve (3.9). Let us denote γ i " Apx i q.

The main feature with these variables is that we can moreover show that the couples px i , γ i q can be easily used to construct separate variables, namely canonical variables. The quadratic Poisson bracket (3.6)-(3.7), i. e. the integrability in some sense, is the cornerstone of the method. Indeed, from these commutation relations it follows [18,135]: tx i , x j u " 0 , tγ i , γ j u " 0 and tγ i , x j u " γ i δ ij (3.12) We can mention that the first Poisson commutation relation comes from the commutativity of the polynomial family Bpλq, and that the two others can be computed using for instance from the Poisson bracket tBpλq, Apµqu.

Thus, the couples px i , v i " logpγ i qq form a set of canonical coordinates, and thus of separate variables with respect to the spectral curves (3.9).

Let us mention that a more general framework has been developed to deal with Lax matrices of arbitrary size m ˆm. This makes use of the Baker-Akhiezer function, a properly normalised eigenvector of the Lax matrix (see [135] and references therein). This way the separate relations are still given by spectral curves. The variables x i are obtained as the zeros of a function defined by a combination of the entries of the Lax matrix (which reduces to Bpλq in dimension 2), while the v i variables are obtained from the value in x i of a function defined by another combination of the entries of the Lax matrix (which reduces to Apλq in dimension 2).

As we will see in the next paragraph, a very similar structure can be used to compute separate variables in quantum systems. In particular, the use of the operators Bpλq and Apλq for 2 dimensional Lax operators is an efficient way to construct separate variables.

Quantum separation of variables

Let us now move to the quantum version of separation of variables, and let us consider a quantum integrable model having N degrees of freedom, described by an (operatorial) Hamiltonian H.

The main idea remains the same as for the classical case [133,135,[START_REF] Babelon | Quantization of solitons and the restricted sine-Gordon model[END_REF][START_REF] Babelon | Null-vectors in integrable field theory[END_REF][START_REF] Kuznetsov | On Bäcklund transformations for many body systems[END_REF][START_REF] Sklyanin | Quantum inverse scattering method. Selected topics. Quantum groups and quantum integrable systems[END_REF][START_REF] Sklyanin | Separation of variables in the quantum integrable models related to the Yangian Y rslp3qs[END_REF][START_REF] Gutzwiller | The quantum mechanical Toda lattice II[END_REF][START_REF] Komarov | Goryachev-Chaplygin top in quantum mechanics[END_REF], namely to map the coupled N -variables spectral problem to a set of N one-variable (thus decoupled) soluble spectral problems, involving one operator and its canonical conjugate. Here, the notion of canonical operators has to be understood in the sense of the usual commutator, that is a set of 2N canonical operators pX i q 1ďiďN and pP i q 1ďiďN satisfies by definition the commutation relations: @pi, jq P 1, N 2 , rX i , X j s " rP i , P j s " 0 and rX i , P j s " i δ ij (3.13) Let us suppose that there exists a complete set of N commuting observables pH i q 1ďiďN , ensuring the integrability of the system: @pi, jq P 1, N 2 , rH i , H j s " rH i , Hs " 0 (3.14)

The aim of the method is then to find a set of canonical operators pX i , P i q 1ďiďN satisfying N separate relations of the form:

@i P 1, N , F i pX i , P i , H 1 , ..., H N q |Ψy " 0 (3.15)
where |Ψy is the wave function. Moreover, when the operators X i are diagonalisable and with simple spectrum, the common eigenbasis is called the separate basis. In this basis xx| " xx 1 , ..., x N |, the wave function Ψpxq " xx|Ψy=Ψpx 1 , ..., x N q factorises in a product of Ψ i px i q.

The example of the Hydrogen atom

Let us briefly illustrate the notion of quantum separation of variables with the quantum model of Hydrogen atom. In spherical coordinates, and in position representation, the Hamiltonian is given by:

xr, θ, φ| H " xr, θ, φ| " ´ 2 2m B 2 B 2 r `1 2mr 2 L 2 ´e2 r  (3.16)
where L is the angular momentum, depending only on θ and φ:

xr, θ, φ| L 2 " xr, θ, φ| " 1 tanpθq B Bθ `B2 B 2 θ `1 sin 2 pθq B 2 B 2 φ  (3.17)
A complete set of commuting observables is obtained with: (3.20). We emphasise that the wave functions are univocally described by the triplet pk, l, mq of quantum numbers which describe all the eigenvalues of the Hamiltonian, which is the essence of the integrability.

H 1 " L z ,
To generalise this notion to integrable systems with N degrees of freedom is the aim of the next paragraph. In particular, we show how to define the separate operators in the framework of the 6-vertex Yang-Baxter algebra.

Construction of separate variables from the 6-vertex Yang-Baxter algebra

As for classical systems, there is in general no standard method to get quantum separation of variables for a given quantum system. However, as already mentioned, the Lax operators in the framework of Quantum Inverse Scattering Method can be wisely used, and provide the necessary ingredients.

The first one to obtain results this way has been Sklyanin for the Toda chain [133], relying on the works of Gutzwiller [START_REF] Gutzwiller | The quantum mechanical Toda lattice II[END_REF] and Komarov [START_REF] Komarov | Goryachev-Chaplygin top in quantum mechanics[END_REF]. It led to the so called functional Bethe ansatz. The method has thereafter been well developed and many models were characterised, one can cite among others the XXX [137,138] and XXZ quasi-periodic spin chain [START_REF] Niccoli | Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Form factors and complete spectrum[END_REF][START_REF] Niekamp | The XXZ model with anti-periodic twisted boundary conditions[END_REF], the sinh Gordon model [START_REF] Sklyanin | Exact quantization of the sinh-Gordon model[END_REF], the sine Gordon model [START_REF] Niccoli | The sine-Gordon model revisited I[END_REF] or even the more general τ 2 chain [START_REF] Grosjean | The τ 2 -model and the chiral Potts model revisited : Completeness of Bethe equations originated from Sklyanin SOV[END_REF]. See also the works [136,[START_REF] Amico | Separation of variables for integrable spin-boson models[END_REF][START_REF] Amico | Integrable spin-boson models descending from rational six-vertex models[END_REF].

In the framework of the 6-vertex Yang-Baxter algebra, Sklyanin's approach to the separation of variables is the following. The aim being to find a separate basis for the transfer matrix spectral problem, i.e. a basis where this problem separates in independent one variable problems, let us consider the standard monodromy matrix (1.21). If the operator Bpλq is diagonalisable and with simple spectrum, the commutativity (1.22) of this family shows the existence of a common eigenbasis. Then we can consider the zero operators ηi of Bpλq, diagonal operators in the eigenbasis, such that (for example for Bpλq polynomial of degree N in λ):

Bpλq " B 0 N ź i"1 pλ ´η i q (3.24)
The ηi being diagonals in the same basis, they commute between themselves. Then, as in the classical case, we use the integrability to construct the canonical operator associated to ηi . And very similarly to the classical case, using the 6-vertex Yang-Baxter algebra one can show that the operators Apη i q, where η i are the eigenvalues of ηi , are of prime interest. Indeed, let us parametrise the eigenbasis of Bpλq by the eigenvalues η i , i.e. let us consider the covectors xη 1 , ..., η N | such that it holds:

xη 1 , ..., η N | Bpλq " b η pλq xη 1 , ..., η N | with b η pλq " b 0 N ź i"1 pλ ´ηi q (3.25)
Using the simplicity of spectrum of the operator Bpλq, and the Yang-Baxter algebra, one can show that the action of Apη i q and Dpη i q on the elements xη 1 , . The action of the operators Apη i q is known only on the basis elements containing η i , namely xη 1 , ..., η i , ..., η N |. However, up to some assumptions (e. g. the polynomiality of Apλq), the action on H of Apλq can in general be obtained (e. g. thanks to an interpolation formula). Thus in this basis, the spectral problem for T pλq " Apλq`Dpλq can be written in terms of one variable equations only, the so-called Baxter equations. And the diagonalisation basis of Bpλq is a separate basis.

Let us now see an explicit example to deal with concrete operators, in the case of the antiperiodic CHAPTER 3. QUANTUM SEPARATION OF VARIABLES XXZ 1/2 spin chain.

The example of the XXZ 1/2 antiperiodic spin chain

Here we present the method on the simple and representative example of the XXZ 1/2 antiperiodic spin chain, the generalisation to other representations of the 6-vertex Yang-Baxter algebra being discussed just after. This way we consider the chain described in paragraph 2.1.1, in particular the one associated to the twist Σ 0 " Σ p0,1q 0

" σ x 0 . The corresponding monodromy will be denoted:

M0Q pλq " M p0,1q 0Q pλq " ˆĀpλq Bpλq Cpλq Dpλq ˙(3.28)
and we will consider the associated antiperiodic transfer matrix:

T p0,1q pλq " T pλq " tr 0 M0Q pλq ( (3.29)
The distinction with a bar notation emphasises on the difference between these operators and the operators associated to the periodic transfer matrix, which will be denoted here by: Let us now sketch the proof that the operator Bpλq is diagonalisable and with simple spectrum, and that its diagonalisation basis is a separate basis for the transfer matrix spectral problem. Here we reproduce the results given in [START_REF] Niccoli | Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Form factors and complete spectrum[END_REF], where the proof follows from the explicit construction of the diagonalisation basis. Let us also remark that Bpλq " Dpλq, and that the eigenbasis of Dpλq is the so-called F -basis of [START_REF] Maillet | Drinfel'd twists and algebraic Bethe ansatz[END_REF][START_REF] Terras | Drinfel'd Twists and Functional Bethe Ansatz[END_REF].

M 0Q pλq " M p0,
The key observation is that the operators Āpλq and Dpλq are Laurent polynomials of degree N ´1 in λ, respectively even for N odd and odd for N even. Thus, we only need to know their action in N different points to know their action on the whole space, by a Lagrange interpolation formula. The same holds thus for the antiperiodic transfer matrix T pλq " Āpλq `Dpλq. Diagonalisation Let x0| be the reference state with all spins up. By its very definition, it satisfies:

x0| Bpλq " dpλq x0| (3.36)
The key point to remark is that, due to the Yang-Baxter algebra (1.24), we have: Let us sketch the proof of (3.40). First, using the commutativity of the Apλq family, one can write:

@
@a P 1, N , xh| " 1 N x0| ˆĀpξ a q āpξ a q ˙ha N ź n"1 n‰a ˆĀpξ n q āpξ n q ˙hn (3.41)
Applying thus the commutation (3.37) with µ " ξ a , it holds:

xh| Bpλq " dpλq N ź a"1 f pλ, ξ a q ha xh| Ǹ ÿ a"1 1 N dpξ a qgpλ, ξ a qδ ha,1 N ź n"1 n‰a f pλ, ξ n q hn x0| Āpλq āpξ a q N ź n"1 n‰a ˆĀpξ n q āpξ n q ˙hn (3.42)
Let us comment that the first term of (3.42) is obtained using the first term of the right hand side of equation (3.37) to commute all the Āpξ n q with Bpλq, while the generic term (a) in the sum comes from the use of the first term of the right hand side of equation (3.37) to commute all the Āpξ n q, n ‰ a, with Bpλq but the second one to commute the Āpξ a q with Bpλq. Using the fact that dpξ a q " 0, one eventually gets (3.40). Thus, for generic inhomogeneities satisfying:

@pn, mq P 1, N , @j P t´1, 0, 1u , ξ n ‰ q j ξ m for m ‰ n (3.43) the states xh| form a set of 2 N independent vectors and so they are the eigenbasis of Bpλq which is therefore diagonalisable and with simple spectrum (to each eigenvalue is associated a unique eigenvector). Let us comment that the eigenvalues η n of the zeros of Bpλq are given by:

η n " ξ n {q hn for h n P t0, 1u (3.44)
In the parametrisation (3.39) of the eigenbasis, we use the constants h n to distinguish the different states, but the basis is genuinely parametrised by the eigenvalues of the zeros of Bpλq, as h n and η n are in one-to-one correspondence. Qtph n q (3.50)

Separate relations

It is worth emphasising once again on the analogy with the classical case, notably as the Baxter equations can be seen as the analogue of the spectral curve (3.9).

We would like to end this paragraph using the example of the XXZ antiperiodic spin chain to highlight the main features of the separation of variables method.

In particular, the spectrum of T pλq is simple (i.e. to any eigenvalue is associated a unique eigenvector), and the eigenvalues are characterised, in a class of function, by the solutions to a discrete system of equations.

Indeed, the eigenvalues of the transfer matrix inherit the symmetries of the transfer matrix itself. Here, this operator is a Laurent polynomial in λ of degree N ´1, with the opposite parity with respect to the parity of N . And so are the eigenvalues.

Then, it is worth rewriting the Baxter equations in a matrix form:

@n P 1, N , ˆtpξ n q ´āpξ n q ´dpξ n {qq tpξ n {qq ˙ˆΨtph 1 , ..., h n " 0, ...h N q Ψtph 1 , ..., h n " 1, ...h N q ˙" ˆ0 0 ˙(3.51)
Thus, if tpλq is an eigenvalue of T pλq, the previous system has to have a non trivial solution, and so the 3.2. QUANTUM SEPARATION OF VARIABLES eigenvalues are characterised by functions satisfying the previous symmetries, and such that det Dtpξ n q " 0 (3.52)

where Dtpξ n q are the 2 ˆ2 matrices entering in (3.51). On top of that, the rank of this matrix being one, the solution is unique (up to normalisation), and T -spectrum is simple. Let us comment that the condition (3.52) is written:

tpξ n q tpξ n {qq ´q-det M0Q pξ n q " 0 (3.53)

It is interesting to note that the characterisation (3.52) appears thus as the fusion relation [START_REF] Mezincescu | Bethe ansatz solution of the Fateev-Zamolodchikov quantum spin chain with boundary terms[END_REF][START_REF] Mezincescu | Fusion procedure for open chains[END_REF] for the transfer matrix T pλq in the point λ " ξ n .

Moreover, the provided description of the spectrum is complete in the following sense: not only any eigenvector of the transfer matrix is characterised by the Baxter equations (3.49), but also any solution of the Baxter equations characterises an eigenvector.

Indeed, let tpλq be a function in the class previously defined (i.e. a Laurent polynomial of degree N ´1 in λ, with the opposite parity with respect to the parity of N ), let Qtph n q denote solutions to the associated Baxter system (3.49), and let us consider:

tD

" 1 ÿ h1,...,h N "0 N ź n"1 Qtph n q ˜ź 1ďbăaďN η a {η b ´ηb {η a ¸´1 |h 1 , ..., h N y (3.54)
This is equivalent to the factorisation property (3.50) when one uses the following decomposition of the identity:

1 " 1 ÿ h1,...,h N "0 ˜ź 1ďbăaďN η a {η b ´ηb {η a ¸´1 |h 1 , ..., h N y xh 1 , ..., h N | (3.55)
Then, it holds:

@n P 1, N , xh 1 , ..., h N | T pη n q tD " tpη n q @ h 1 , ..., h N ˇˇt D (3.56)
Thus, being T pλq a polynomial of degree N ´1 in λ, it holds:

xh 1 , ..., h N | T pλq tD " tpλq @ h 1 , ..., h N ˇˇt D (3.57)
that is tpλq is an eigenvalue and tD given by (3.54) the associated eigenvector.

A powerful and promising tool

The quantum separation of variables exposed here can be generalised to different representations. This method is a priori more generally applicable than the Algebraic Bethe Ansatz, as no reference state is needed. In the example of the XXZ 1/2 anti-periodic spin chain, such a state is used to give an explicit basis of diagonalisation of the operator Bpλq. And here lies maybe the only limitation: in order for the method to be applicable, the operator Bpλq of a generic transfer matrix must be diagonalisable and with simple spectrum. For instance, the Bpλq operator associated to the periodic transfer matrix of the XXZ 1/2 spin chain is not diagonalisable (it is nilpotent) whereas the ones associated to all the other quasiperiodic transfer matrices are. Still there are some conditions for the separation of variables method to be applicable, but simplicity of spectrum for Bpλq is related to the inhomogeneity parameters being in generic position, see (3.43).

Eventually, let us emphasise two points that emerged in all the models studied so far with the quantum separation of variables [START_REF] Niccoli | Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Form factors and complete spectrum[END_REF][START_REF] Grosjean | The τ 2 -model and the chiral Potts model revisited : Completeness of Bethe equations originated from Sklyanin SOV[END_REF][START_REF] Niccoli | Antiperiodic XXZ chains with arbitrary spins: Complete eigenstate construction by functional equations in separation of variables[END_REF]. First, one can show, almost by construction, the completeness of the description of the eigenvectors, i.e. all the eigenvectors of the transfer matrix are described this way. Second, this technique leads to the simplicity of spectrum of the transfer matrix, hence characterising the integrability (up to the fact that the Hamiltonian commutes with this family).

In a nutshell, the quantum separation of variables seems very promising as compared to other methods to analyse the spectral problem (Algebraic Bethe Ansatz, Baxter Q-operator, analytic Bethe ansatz) as it works in a large class of models for which the others do not apply. It characterises both the spectrum and the eigenvectors, the completeness of the description is almost straightforward and the same way it is not a hard task to prove the simplicity of spectrum of the transfer matrix.

Within the 6-vertex reflection algebra

The question of quantum separation of variables henceforth naturally arises for models with general integrable boundary conditions. For some models associated to the 6-vertex reflection algebra for example, with triangular boundary matrices, the operator Bpλq can be used to construct a separate basis.

But to deal with the general case is more involved, this problem is at the heart of the next chapter, considering cyclic representations of the 6-vertex reflection algebra.

Chapter 4

Boundary transfer matrix spectrum by quantum separation of variables

In this chapter we study the boundary transfer matrix spectral problem for cyclic representations of the 6-vertex reflection algebra associated to the Bazahnaov-Stroganov Lax operator. As already mentioned in the introduction, particular cases include the spin s chains with the anisotropy coupling at root of unity, the chiral Potts model or the discretised sine Gordon model at root of unity.

Implementing the quantum separation of variables for these representations with the most general integrable boundaries, we will characterise in two different ways the spectrum of the transfer matrix: a discrete characterisation, as the set of solutions to a discrete system of polynomial equations in a given class of functions, and an equivalent characterisation as the set of solutions to a functional Baxter like TQ equation.

In order to be more understandable, we present our method in two steps. The first one is to consider one general Kpλq matrix (left boundary) and one triangular K `pλq matrix (right boundary), such that in this case a separate basis is obtained as the diagonalisation basis of Bpλq. Then, to allow the description of completely general integrable boundaries, we will introduce a gauge transformation. Wisely choosing the gauge parameters, we will be able to deal with a gauged 6-vertex reflection algebra and we will put the gauge right boundary matrix triangular, leading to the framework of the case treated before.

The results described in this chapter have been published in the two articles I and II.

The model in the framework of the Quantum Inverse Scattering Method

To begin with, we make explicit some key properties and symmetries of the cyclic representations of the 6-vertex reflection algebra. Thanks to Sklyanin's construction of the boundary transfer matrix [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF], we consider cyclic representations of the 6-vertex Yang-Baxter algebra, i.e. the Bazhanov-Stroganov Lax operators (1.72), and define the monodromy matrix with a shift (cf paragraph 2.2.2 ):

M 0Q pλ, p Q q " L 0N pλ{q 1{2 , p N q...L 01 pλ{q 1{2 , p 1 q " ˆApλq Bpλq Cpλq Dpλq ˙(4.1)
We recall that p Q " pp 1 , ..., p N q denotes the parameters in the whole quantum space. The expression for the quantum determinant is thus the following shifted one:

q-det M 0Q pλ, p Q q " Apλq 1{2 qDpλq ´1{2 q ´Bpλq 1{2 qCpλq ´1{2 q " apλqdpλ{qq (4.2)
where the functions apλq and dpλq have been introduced in (1.86). Let K 0 pλq and K `0pλq be the general scalar solutions of the reflection equations (2.25) and (2.27).

CHAPTER 4. BOUNDARY TRANSFER MATRIX SPECTRUM BY QUANTUM SEPARATION OF VARIABLES

The boundary monodromy matrix is defined in the standard way:

U 0Q pλ, p Q q " M 0Q pλ, p Q qK 0 pλq M0Q pλ, p Q q with M0Q pλ, p Q q " p´1q N σ y 0 M t0 0Q p1{λ, p Q qσ y 0 (4.3)
as well as the boundary transfer matrix:

T pλq " tr 0 tK `0pλqU 0Q pλ, p Q qu (4.4)
Following the same proof given in Sklyanin's paper [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF], one shows that this is a one parameter family of commuting operators and our aim here is to characterise its spectrum. Let us recall here some notations and properties of these representations.

First, we can compute the quantum determinant (2.29) associated to the boundary matrices, it holds:

q-det K 0 pλq " `λ2 {q 2 ´q2 {λ 2 ˘g´p λq 1{2 qg ´pq 1{2 {λq (4.5)
and q-det K `0pλq " ´`λ 2 q 2 ´1{pλ 2 q 2 q ˘g`p λq 1{2 qg `pq 1{2 {λq (4.6)

with the functions:

g ˘pλq " pλα ˘{q 1{2 ´q1{2 {pλα ˘qqpλβ ˘{q 1{2 `q1{2 {pλβ ˘qq pα ˘´1{α ˘q pβ ˘`1{β ˘q (4.7)
where we used a more convenient parametrisation for the boundary parameters:

pα ˘´1{α ˘q pβ ˘`1{β ˘q " ξ ˘´1{ξ κ˘a nd pα ˘`1{α ˘q pβ ˘´1{β ˘q " ξ ˘`1{ξ κ˘( 4.8)
Let Apλq, Bpλq, Cpλq and Dpλq be the elements of the boundary monodromy matrix, namely the generators of the 6-vertex reflection algebra. Then, following Sklyanin's paper [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF], one can show the symmetries:

Dpλq " pλ 2 {q ´q{λ 2 q pλ 2 ´1{λ 2 q Apλ ´1q `pq ´1{qq pλ 2 ´1{λ 2 q Apλq (4.9)

as well as Bpλ ´1q " ´pλ 2 q ´1{ `qλ 2 ˘q pλ 2 {q ´q{λ 2 q Bpλq and Cpλ ´1q " ´pλ 2 q ´1{ `qλ 2 ˘q pλ 2 {q ´q{λ 2 q Cpλq (4.10) Moreover, by using the previous symmetries, and the definition (2.29), the boundary quantum determinant can be expressed:

q-det U 0Q pλ, p Q q " `λ2 {q 2 ´q2 {λ 2 ˘rApλq 1{2 qApq 1{2 {λq `Bpλq 1{2 qCpq 1{2 {λqs (4.11)
or equivalently:

q-det U 0Q pλ, p Q q " `λ2 {q 2 ´q2 {λ 2 ˘rDpλq 1{2 qDpq 1{2 {λq `Cpλq 1{2 qBpq 1{2 {λqs (4.12)
Moreover, we have the factorisation:

q-det U 0Q pλ, p Q q " q-det K 0 pλq q-det M 0Q pλ, p Q q q-det M 0Q p1{λ, p Q q (4.13)
leading to the expression: which is such that the boundary quantum determinant can also be written:

q-det U 0Q pλ, p Q q " pλ
q-det U 0Q pλ, p Q q " pλ 2 {q 2 ´q2 {λ 2 qD ´pλq 1{2 qD ´pq 1{2 {λq (4.17)
Furthermore, let us introduce the coefficients:

a `pλq " pλ 2 q ´1{ `qλ 2 ˘qpλξ `{q 1{2 ´q1{2 {pλξ `qq pλ 2 ´1{λ 2 qpξ `´1{ξ `q (4.18) and d `pλq " pλ 2 q ´1{ `qλ 2 ˘qpξ `q1{2 {λ ´λ{pq 1{2 ξ `qq pλ 2 ´1{λ 2 qpξ `´1{ξ `q (4.19)
It follows two important rewritings of the boundary transfer matrix (2.35):

T pλq " a `pλqApλq `a`p 1{λqAp1{λq `c`p λqBpλq `b`p λqCpλq (4.20)

" d `pλqDpλq `d`p 1{λqDp1{λq `c`p λqBpλq `b`p λqCpλq (4.21)
Lastly, using the symmetries (4.10) and the coefficients of the boundary transfer matrix, we can easily show the following proposition.

Proposition 4.1.1. The most general boundary transfer matrix admits the following symmetries:

T pλq " T p1{λq and T p´λq " T pλq (

This symmetry is thus satisfied for all the parameters and it will be used in the spectrum characterisation. Now that the main notations have been introduced, and some symmetries characterised, we can tackle the spectral problem of this operator family.

The case of a triangular right boundary matrix

As stated previously, we start the characterisation of the spectrum for models with the special boundary condition associated to a general Kpλq (left boundary) and a lower triangular K `pλq (right boundary) to give a clear example. It lays the foundations of the quantum separation of variables method which will be then generalised to the case of general left and right boundary matrices.

The idea developed here is a straightforward generalisation of the concept of separation of variables developed for the Yang-Baxter algebra: we use the operator Bpλq of the boundary monodromy matrix to construct a separate basis. We first show that this operator is indeed diagonalisable and with simple spectrum, then we show that the diagonalisation basis is indeed a separate basis for this particular transfer matrix spectral problem, and we finally give a discrete characterisation of the spectrum.

The procedure to show that Bpλq is diagonalisable is as follows. First, we consider a subclass of parameters for which we can show the proposition thanks to an explicit construction, via an explicit reference state. This is done in the proposition 4.2.1. Then for the most general parameters, we use a general argument to state the existence of a reference state, and from the explicit case treated before we use a continuity argument to show that the proposition holds for almost any value of the parameters.

Prelude: an explicit construction of Bpλq diagonalisation basis for a subclass of parameters

In this paragraph, we show the diagonalisability of the operator Bpλq for a general boundary matrix Kpλq but with the following extra constraint on the quantum parameters:

@n P 1, N , b p n `ap n " 0 (4.23)
This constraint allows us to explicitly construct a diagonalisation basis in this subclass of parameters, thanks to an explicit reference state. Indeed, from (4.23) we can construct, by a tensor product of simple local states1 , a covector xΩ| and a vector ˇˇΩ D which are eigenstates of the operators Apλq and Dpλq, and such that the action of Bpλq is zero:

xΩ| Apλq 1{2 q " apλq xΩ| , xΩ| Dpλq 1{2 q " dpλq xΩ| , xΩ| Bpλq " 0 ¯, xΩ| Cpλq ‰ 0 ¯(4.24) Apλq 1{2 q ˇˇΩ D " ˇˇΩ D apλqq, Dpλq 1{2 q ˇˇΩ D " ˇˇΩ D dpλ{qq, Bpλq ˇˇΩ D " 0 ¯, Cpλq ˇˇΩ D ‰ 0 ¯(4.25)
Thereafter, we use the states xΩ| and ˇˇΩ D to construct explicitly the elements of the basis in which the operator Bpλq is diagonal. Using the explicit form of Bpλq:

Bpλq " p´1q N p´a ´pλqApλqBp1{λq `b´p λqApλqAp1{λq ´c´p λqBpλqBp1{λq `d´p λqBpλqAp1{λqq From these first eigenstates, we can in fact construct new eigenvectors of Bpλq thanks to the repeated action of the operator Apλq in some particular points, up to the determination of the Bpλq-eigenstates basis of the Hilbert space (and of its dual). These points are zeros of the quantum determinant shifted by powers of the parameter q. We introduce them here, in order to state the proposition 4.2.1.

Let pµ n,`q 1ďnďN and pµ n,´q 1ďnďN be respectively the zeros of the functions apλq and dpλ{qq, they read:

µ n,`" iq 1{2 pa n β n {α n b n q 1{2 and µ n,´" iq 1{2 pc n β n {α n d n q 1{2 (4.29)
We define the following functions parametrised thanks to N-tuples h " ph 1 , ..., h N q P 0, ..., p ´1 N :

b h pλq " b ´pλqa h pλqa h p1{λq (4.30)
with

a h pλq " p´1q N N ź n"1 pα n β n q 1{2 p λ ξ phnq n ´ξphnq n λ q (4.31)
The variables ξ phq n are simply the zeros µ n,`, shifted by a power q h`1{2 :

@n P 1, N , ξ phq n " µ n,`q h`1{2
b ´pλq ‰ 0, µ p n,`‰ µ p m,`@ n ‰ m P 1, N (4.33)

THE CASE OF A TRIANGULAR RIGHT BOUNDARY MATRIX

as well as @h P 1, p ´1 , @pn, mq P 1, N 2 , @ P t´1, 1u,

µ 2p n,`‰ ˘1, µ 2 n,`‰ q ´2h α 2 ´, µ 2 n,`‰ ´q´2h β 2 ´, µ 2 n,`‰ q ´2h´2 µ 2 m,´( 4.34)
the states:

xh 1 , ..., h N | " xh| " 1 n xΩ| N ź n"1 hn ź kn"1 Ap1{ξ pkn´1q n q A ´p1{ξ pkn´1q n q (4.35) |h 1 , ..., h N y " |hy " 1 n N ź n"1 p´2 ź kn"hn Dpξ pkn`1q n q D ´pξ pkn`1q n q | Ωy (4.36)
where h n P 0, p´1 and n is a free normalisation, define a Bpλq-eigenstates basis of H ˚and H respectively:

xh|Bpλq " b h pλqxh| (4.37)

Bpλq|hy " |hyb h pλq (4.38)
In particular, this result states that for almost all the values of the parameters, when the constraints (4.23) are satisfied, the operator Bpλq is diagonalisable and with simple spectrum.

In order to understand the role of the constraints (4.33)-(4.34), let us highlight the main steps of the proof. The idea is to efficiently use the reflection algebra to construct, as mentioned before, new eigenvectors from xΩ| and ˇˇΩ D , the key elements being the commutation relations (2.14)-(2.19) of the 6-vertex reflection algebra generators. In more details, for the left case it holds (2.16):

c 1 pλ, µq BpλqApµq " ApµqBpλq ´Bpµq `c2 pλ, µq Apλq `c3 pλ, µq Dpλq ˘(4.39)
This way, one can repeat the procedure given for the diagonalisation of the operator Bpλq in the paragraph 3.2.3: starting from (4.27), the action of Apµq for µ a root of b 0 pλq gives that the covector xΩ| Apµq is another eigenvector of Bpλq, and so on for repeated actions, up to (4.37). The same is true for (4.38), using the commutation relation (2.17).

As by condition p4.33q each state xh| (respectively |hy) is associated to a different eigenvalue of Bpλq, the only thing that we need to prove to get their linear independence, and thus the fact that they form a Bpλq-eigenstate basis of H ˚(respectively H), is that each such state is non-zero. This can be proven using the action of the operator Apλq (respectively Dpλq) in some points on the states xh| (respectively |hy). Then this non-zero condition becomes equivalent to the non annihilation of the quantum determinant, which hold thanks to conditions (4.34).

In this framework where the bulk parameters satisfy the extra constraints (4.23), we can explicitly construct the eigenbasis of Bpλq. In the next, we extend this procedure to unconstrained parameters, leading to a non explicit basis. However, as we will emphasise, the exact knowledge of the basis elements is not necessary for the computation of the dynamics of the model.

For generic parameters Bpλq is diagonalisable and with simple spectrum: the general proof

In this paragraph, we give a procedure leading to the existence, for general parameters, of at least one eigenvectors of Bpλq on the right |Ω R y and on the left xΩ L |. These vectors play the same role as the vectors ΩD and xΩ| introduced in the previous proposition, thus from these states one can prove the diagonalisability of Bpλq, using the procedure previously described.

Proposition 4.2.2. There exists at least one eigenstate on the right |Ω R y and one eigenstate on the left VARIABLES xΩ L | of the one parameter family of commuting operators

Bpλq: Bpλq|Ω R y " |Ω R yb ´p λ 2 q ´q λ 2 q N ź a"1 p λ b´,a
´b ´,a λ qpλ b´,a ´1 λ b´,a q (4.40)

xΩ L |Bpλq " b ´p λ 2 q ´q λ 2 q N ź a"1 p λ b´,a ´b ´,a λ qpλ b´,a ´1 λ b´,a qxΩ L | (4.41)
where b ´, b´,a and b´,a for a P 1, N are non-zero complex numbers.

To understand these formulas, let us sketch the proof of the existence of |Ω R y. First, from the expression (4.26) of the generator Bpλq in terms of the generators of the Yang-Baxter algebra, Bpλq is a Laurent polynomial in the spectral parameter λ (of degree 2N `2) if the left boundary matrix is non-diagonal (b ´pλq ‰ 0) and the bulk parameters are generals, i.e. if the leading coefficient is non-zero:

b ´" p´1q N κ ´eτ´N ź a"1 α n β n ‰ 0 (4.42)
Then, we can deduce thanks to the symmetry (4.10), the following functional form:

Bpλq " b ´p λ 2 q ´q λ 2 q N ź a"1 p λ B ´,a ´B´,a λ qpλB ´,a ´1 λB ´,a q (4.43)
where the B ´,a are invertible commuting operators. After, reasoning with the Jordan normal form of the operators B ´,1 , B ´,2 and so on, we show that there exists at least one simultaneous right eigenstate of all the operators B ´,a , which is thus an eigenvector of Bpλq.

At this point, we can state the following proposition 4.2.3, which is the generalisation of 4.2.1 to almost any value of the parameters.

Proposition 4.2.3. Under the condition p4.42q, for almost all the values of the bulk and boundary parameters, the operator family Bpλq is diagonalisable and it has simple spectrum.

In particular it holds

@n ‰ m P 1, N , bp ´,m ‰ bp ´,n (4.44)
as well as @h P 1, p ´1 , @pn, mq P 1, N 2 , @ε P t´1, 1u, b2p

´,n ‰ ˘1, b2

´,n ‰ q ´2h`1 α 2ε

´, b2

´,n ‰ ´q´2h`1 β 2ε

´, b2

´,n ‰ q ´2ε`1´2h µ 2ε m,´, b2

´,n ‰ q ´2ε`1´2h µ 2ε m,( 4.45) and the states:

xh 1 , ..., h N | " xh| " 1 n xΩ L | N ź n"1 hn ź kn"1 Ap1{ξ pkn´1q n q A ´p1{ξ pkn´1q n q (4.46) |h 1 , ..., h N y " |hy " 1 n N ź n"1 p´2 ź kn"hn Dpξ pkn`1q n q D ´pξ pkn`1q n q |Ω R y (4.47)
where h n P 0, p ´1 , ξ phq n " b´,n q h and n is a free normalisation, define a Bpλq-eigenstates basis of H As already mentioned, the proof of this proposition uses a continuity argument: we know that it holds for one particular subclass of parameters, we can extend it to almost every value of the parameters. Let us give some more details.

First, we observe that by definition the operator family B ´pλq is a polynomial in the bulk and boundary parameters, so the same must be true for its spectrum. This implies in particular that the quantities bp

´,n ´b p ´,m , for any n ‰ m P 1, N are either identically zero or they can be zero only over subspaces of non-zero co-dimension in the space of the bulk parameters. So here we have just to prove that it is not identically zero to derive that p4.44q holds for almost all the values of the parameters. To do so we can just recall that from the results derived in the previous paragraph it holds:

@m P 1, N , bp
´,m " q p{2 µ p m,`( 4.50)

under the condition Bpλq nilpotent, i.e. (4.23). Then, the condition p4.44q holds as soon as we impose:

@n ‰ m P 1, N , β p n {α p n ‰ β p m {α p m (4.51)
Thereafter, one can follow step by step the proof of the proposition 4.2.1 to show that the states (4.46) and (4.47) indeed lead to the relations (4.48) and (4.49). The simplicity of spectrum comes from the condition (4.44).

Eventually, let us note that the repeated action of the operator Apλq in the zeros of Bpλq can generate a null vector only if some of the zeros of Bpλq coincides with the zeros of the quantum determinant. Anyhow, as discussed in the previous paragraph, under the condition p4.34q we are always able to chose an appropriate set of p N zeros of Bpλq which do not coincide with those of the quantum determinant, so that we generate exactly p N independent states for almost all the parameters.

Remark. Propositions similar to the previous two hold also for Cpλq so that we can say that for general values of the boundary and bulk parameters the standard Algebraic Bethe Ansatz does not applies, being

Cpλq and Bpλq not nilpotent operators.

The separate basis for a lower triangular right boundary

In this paragraph, we show that the eigenbasis of Bpλq defined previously in proposition 4.2.3 are separate basis for the boundary transfer matrix spectral problem associated to a lower triangular K `pλq matrix. Here and in the next of this section, we consider the general ξ phnq n defined in the proposition 4.2.3, i.e. we do not suppose the extra condition (4.23).

We start by giving the needed expressions of the actions of the generators of the reflection algebra on the elements of the basis, and then we show thanks to these expressions, that the spectral problem indeed separates for this special boundary.

But first, let us introduce some more notation. Let h n P 0, p ´1 be associated to the site n of the chain, n P 1, N . We define:

ζ phnq n " $ ' & ' % ξ phnq n @n P 1, N ´ξph n´N q n´N ¯´1 @n P N `1, 2N (4.52)
which is just a convenient way to deal with ξ phnq n and its inverse for n ą N . Moreover, in the next the following notations will give more compact expressions: Λ " pλ 2 `1{λ 2 q and X " q `1{q (4.53)

@b P 1, 2N , X ph b q b " pζ ph b q b q 2 `1{pζ ph b q b q 2 (4.
q 2 {q ´q{pζ phaq a q 2 qppζ phaq a q 2 ´1{pζ phaq a q 2 q N ź b"1 b‰a modN Λ ´Xph b q b X phaq a ´Xph b q b A ´pζ phaq a q ˆxh|T ´ϕa a `p´1q N q-det M 0Q p1q pλ{q 1{2 `q1{2 {λq 2 N ź b"1 Λ ´Xph b q b X ´Xph b q b xh| `p´1q N pξ ´`1{ξ ´q pξ ´´1{ξ ´q q-det M 0Q piq pλ{q 1{2 ´q1{2 {λq 2 N ź b"1 Λ ´Xph b q b X `Xph b q b xh| (4.55)
whereas the action on the right of the operator Dpλq on a generic state |hy reads:

Dpλq|hy " 2N ÿ a"1 T ´ϕa a |hy pλ 2 {q ´q{λ 2 qpλζ phaq a ´1{ζ phaq a λq ppζ phaq a q 2 {q ´q{pζ phaq a q 2 qppζ phaq a q 2 ´1{pζ phaq a q 2 q N ź b"1 b‰a modN Λ ´Xph b q b X phaq a ´Xph b q b ˆD´p ζ phaq a q `|hyp´1q N q-det M 0Q p1q pλ{q 1{2 `q1{2 {λq 2 N ź b"1 Λ ´Xph b q b X ´Xph b q b `p´1q N`1 |hy pξ ´`1{ξ ´q pξ ´´1{ξ ´q q-det M 0Q piq pλ{q 1{2 ´q1{2 {λq 2 N ź b"1 Λ ´Xph b q b X `Xph b q b (4.56)
In these expressions, the operators T ȃ are simply shifts:

xh 1 , ..., h a , ..., h N |T ȃ " xh 1 , ..., h a ˘1, ..., h N | (4.57) T ȃ |h 1 , ..., h a , ..., h N y " |h 1 , ..., h a ˘1, ..., h N y (4.58)
while ϕ a is defined by:

@a P 1, 2N , ϕ a " $ & % 1 @a P 1, N ´1 @a P N `1, 2N (4.59)
Proof. This proposition can be derived without difficulty from the known form of the vectors xh| and |hy. Indeed, one easily computes the action of Apζ ph b q b q for b P 1, 2N thanks to the commutation (4.39) for the 6-vertex reflection algebra generators and the simplicity of spectrum of Bpλq. This is the standard computation, we presented it in paragraph 3.2.3 for the Yang-Baxter algebra. Up to simple modifications to deal with the reflection algebra, and using the expression for the boundary quantum determinant (4.11), we have:

xh|Apζ ph b q b q " A ´pζ ph b q b qxh|T ´ϕb b (4.60)
Moreover, Apλq is a Laurent polynomial of degree 2N+1 in λ. Thus, remarking that we can compute the action of the operators in the points q 1{2 and iq 1{2 :

Upq 1{2 q " p´1q N q-det M 0Q p1q 1 and Upiq 1{2 q " ip´1q N`1 ξ ´`1{ξ ξ´´1 {ξ ´q-det M 0Q piq σ z 0 (4.61)
we get the interpolation formula for its action on xh|. We proceed using the same steps for representations on the right.

The previous proposition gives in fact the action of the four generators of the reflection algebra on the left and on the right. Indeed, the representation of Dpλq on the left and of Apλq on the right follow from the symmetry (4.9), while the ones of Cpλq follow from by the quantum determinant.

THE CASE OF A TRIANGULAR RIGHT BOUNDARY MATRIX

As mentioned before, we restrict our attention in this paragraph to the special boundary condition b `pλq " 0 (see the notation (2.28)). Indeed, we aim now to prove that a separate basis is given by the Bpλq-eigenstates basis. That is we consider the spectral problem of the boundary transfer matrix under the following conditions on the boundary parameters: b `pλq " 0 and b ´pλq ‰ 0.

(4.62)

Note that the condition b `pλq " 0, keeping instead if desired a c `pλq ‰ 0, can be simply realised by the following renormalisation of the boundary parameters κ `" e ´γ κ 1 `and e τ`" e τ 1 `´γ by sending γ Ñ `8. So, by using the symmetry (4.20), we are left with:

T pλq " a `pλqA ´pλq `a`p 1{λqA ´p1{λq `c`p λq Bpλq where we introduced the coefficients and the wave functions:

apλq " a `pλqA ´pλq and Ψ τ phq " xh|τ y (4.66)

as well as the shifted states:

T n phq " ph 1 , . . . , h n ˘1, . . . , h N q (4.67)
The spectrum, both the eigenvalues and eigenstates, of the transfer matrix T pλq is thus characterised by the discrete system of equations (4.65) for any n P 1, N and h P 0, p ´1 N , i.e. a system of p N Baxter-like equations in the wave functions. These are separate equations as for any fixed n the coefficients are only function of the spectrum of one separate variable as well as the shifts on the wave functions are only along the same variable.

The same kind of Baxter equations are obtained for the left eigenvectors of T pλq, using this time the expression (4.21) for the boundary transfer matrix, and the right eigenbasis of Bpλq.

So the eigenbasis of Bpλq are indeed separate basis for this special boundary. In the next section, we refer to it as the left separate basis (for xh|) and right separate basis (for |hy), and use it to completely characterise both the eigenvalues and eigenstates of the boundary transfer matrix.

Discrete characterisation of the spectrum

In this paragraph we present the complete characterisation of the spectrum of the boundary transfer matrix T pλq associated to cyclic representations of the 6-vertex reflection algebra with the constraint K `pλq lower triangular. In the next, we will denote by Σ τ the transfer matrix spectrum, namely the set of all the eigenvalues of T pλq.

But first, let us present a preliminary property satisfied by all the eigenvalue functions of all the boundary transfer matrices T pλq associated to cyclic representations of the 6-vertex reflection. This one, given in the following proposition, states the functional form of the eigenvalues, which is the same for all the parameters, even if the boundary matrix K `pλq is not triangular. Proposition 4.2.5. Any function τ pλq P Σ τ is an even function of λ invariant under the transformation VARIABLES λ Ñ 1{λ which admits the following interpolation formula:

τ pλq " N ÿ a"1 Λ 2 ´X2 pX p0q a q 2 ´X2 N ź b"1 b‰a Λ ´Xp0q b X p0q a ´Xp0q b τ pζ p0q a q `p´1q N pΛ `Xq 2 N ź b"1 Λ ´Xp0q b X ´Xp0q b q-det M 0Q p1q `p´1q N pΛ ´Xq 2 N ź b"1 Λ ´Xp0q b X `Xp0q b pξ ``1{ξ `q pξ `´1{ξ `q pξ ´`1{ξ ´q pξ ´´1{ξ ´q q-det M 0Q piq `pΛ 2 ´X2 qτ 8 N ź b"1 pΛ ´Xp0q b q (4.68)
where:

τ 8 " κ `κ´p e τ`´τ´ś N b"1 δ b γ b `eτ´´τ`ś N b"1 α b β b q pξ `´1{ξ `q pξ ´´1{ξ ´q (4.69)
This proposition is simply an interpolation formula. Indeed, we know the symmetries of T pλq, and thus the ones of τ pλq P Σ τ , and we use the polynomiality in the spectral parameter. The asymptotic behaviour τ 8 " lim

log λÑ˘8 λ ¯2pN`2q T pλq (4.70)
is also known after some simple computations.

Let us now go back to a lower triangular right boundary. Under the reparametrisation κ `" e ´γ κ 1 ànd e τ`" e τ 1 `´γ , with γ Ñ `8, the asymptotic of the transfer matrix reads:

τ 8 " p´1q N κ 1 `κ´e τ´´τ 1 `śN b"1 α b β b pξ `´1{ξ `q pξ ´´1{ξ ´q (4.71)
In the following we will suppress the unnecessary prime in κ `and τ `when dealing with the constraint b `pλq " 0 but c `pλq ‰ 0.

The previous proposition defines the set of polynomials to which belong the transfer matrix eigenvalues. In order to completely characterise the eigenvalues, we introduce now the following one-parameter family D t pλq of p ˆp matrices, depending on a generic function tpλq:

D t pλq " ¨tpλq ´ap1{λq 0 ¨¨¨0 ´apλq ´apqλq tpqλq ´ap1{ pqλqq 0 ¨¨¨0 0 . . . . . . . . . ¨¨¨. . . . . . ¨¨¨. . . . . . . . . 0 0 . . . 0 ´apq 2l´1 λq tpq 2l´1 λq ´ap1{ `q2l´1 λ ˘q ´ap1{ `q2l λ ˘q 0 . . . 0 ´apq 2l λq tpq 2l λq ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' (4.72)
It is clear from the equation (4.65) that the matrix D τ pλq will allow to write the Baxter system, for τ pλq P Σ T , in a matrix form. Moreover, with some simple computations on the rows and columns, one can prove that if tpλq is a function of λ invariant under the transformation λ Ñ 1{λ then det D t pλq is a function of λ p invariant under the transformation λ p Ñ 1{λ p . This property will be used in the following.

The last point we need before to state the result of the characterisation is to compute the scalar product between states of the left and right separate basis: Proposition 4.2.6. Using an appropriate choice for the norm n in the basis elements (4.46) and (4.47),

THE CASE OF A TRIANGULAR RIGHT BOUNDARY MATRIX it holds:

@ h, k P 0, p ´1 N , xh|ky " δ hk ź 1ďbăaďN 1 X phaq a ´Xph b q b (4.73)
In particular, this implies the spectral decomposition of the identity in the separate basis:

p´1 ÿ h1,...,h N "0 ź 1ďbăaďN pX phaq a ´Xphaq a q|h 1 , ..., h N yxh 1 , ..., h N | " 1 (4.74)
The fact that the states are orthogonal comes from the fact that all the eigenvalues of Bpλq are different. Then, to compute the value of the scalar product, we compute the matrix element θ a " xh 1 , ..., h a , ..., h N |Apξ pha`1q a q|h 1 , ..., h a `1, ..., h N y, for a P 1, N , using the left action and the right action of Apλq from the proposition 4.2.4.

We are now able to give the complete characterisation of the spectrum in the separate basis, it is stated in the next proposition: Proposition 4.2.7 (Characterisation of the spectrum). If the conditions:

@h P 1, p ´1 , @n P 1, N , b2
´,n ‰ q ´2h`1 ξ ˘2 `(4.75)

and p4.44q-p4.45q are satisfied, then T pλq has simple spectrum and Σ T coincides with the set of polynomials τ pλq of the form p4.68q with p4.71q which satisfy the following discrete system of equations:

@a P 1, N , detD τ pζ p0q a q " 0 (4.76)
• The right T -eigenstate corresponding to τ pλq P Σ τ is defined by the following decomposition in the right separate basis:

|τ y " p´1 ÿ h1,...,h N "0 N ź a"1 Q phaq τ,a ź 1ďbăaďN pX phaq a ´Xph b q b q|h 1 , ..., h N y (4.77)
where the Q phaq τ,a are the unique nontrivial solution up to normalisation of the linear homogeneous system:

D τ pζ p0q a q ¨Qp0q τ,a . . . Q pp´1q τ,a ‹ ‹ ' " ¨0 . . . 0 ‹ ' (4.78)
• The left T -eigenstate corresponding to τ pλq P Σ τ is defined by the following decomposition in the left separate basis:

xτ | " p´1 ÿ h1,...,h N "0 N ź a"1 Qphaq τ,a ź 1ďbăaďN pX phaq a ´Xph b q b qxh 1 , ..., h N | (4.79)
where the Qphaq τ,a are the unique nontrivial solution up to normalisation of the linear homogeneous system:

´Q p0q τ,a . . . Qpp´1q τ,a ¯´D τ pζ p0q a q ¯t0 " `0 . . . 0 ˘(4.80)
and Dτ pλq is the family of p ˆp matrices defined substituting in D τ pλq the coefficient apλq with dpλq " d `pλqD ´pλq (4.81) VARIABLES

In the followig, we just aim to highlight the main points of the proof. The key is the use of the matrix D τ pλq to rewrite the Baxter equations. Details are given in article I.

Key elements of the proof

The Baxter system admits the following equivalent representation as a p-linear system of homogeneous equations:

D τ pξ p0q n q ¨Ψτ ph 1 , ..., h n " 0, ..., h N q Ψ τ ph 1 , ..., h n " 1, ..., h N q . . . Ψ τ ph 1 , ..., h n " p ´1, ..., h N q ‹ ‹ ‹ ' " ¨0 0 . . . 0 ‹ ‹ ‹ ' (4.82)
for any n P 1, N and for any h m‰n in 0, p ´1 . Then the condition τ pλq P Σ τ implies the compatibility equations for these linear systems, i.e. it must hold: @a P 1, N , detD τ pξ p0q a q " 0 (4.83)

The rank of the matrices D τ pξ p0q n q in p4.82q is p ´1, being: @h P 1, p ´1 , @n P 1, N , a ´pξ phq n q ‰ 0 and a ´p1{ξ ph´1q

n q ‰ 0 (4.84)
from the conditions p4.44q-p4.45q and p4.75q. Then, up to an overall normalisation, the solution is unique and independent from the h m‰n P 0, p ´1 for any n P 1, N . So fixing τ pλq P Σ τ there exists, up to normalisation, one and only one corresponding T pλq eigenstate |τ y, with coefficients of the factorised form given by p4.77q and p4.78q. Hence T pλq has simple spectrum.

Vice versa, if τ pλq is in the set of functions (4.68) with (4.71) and satisfies (4.76), then the state |τ y defined by p4.77q-p4.78q satisfies:

@n P 1, N , xh| T pζ phnq n q|τ y " τ pζ phnq n qxh|τ y (4.85)
for any eigenstate xh| of Bpλq and this implies:

@λ P C , xh| T pλq|τ y " τ pλqxh|τ y (4.86)
which is equivalent to τ pλq P Σ τ and |τ y is the corresponding eigenstate.

For the left T pλq eigenstates the proof follows as above, we just remark in this case that the matrix elements xτ |T pζ phnq n q|hy are computed in the right separate basis, so it holds:

@n P 1, N , τ pζ phnq n q Ψτ phq " dpζ phnq n q Ψτ pT ń phqq `dp1{ζ phnq n q Ψτ pT ǹ phqq (4.87)
where we denote: Ψτ phq " xτ |hy and dpλq " d `pλqD ´pλq (4.88)

After some simple computations, one can prove that it holds det Dτ pλq " detD τ pλq.

A link between the right and left eigenvectors

For future applications it is interesting to show that we can obtain the coefficients of a left eigenstate of the boundary transfer matrix in terms of those of the right one. In particular, we can show the next proposition:

Proposition 4.2.8. Let τ pλq P Σ τ then we have:

Qphq τ,a Qph´1q τ,a " ap1{ζ ph´1q a q dp1{ζ ph´1q a q Q phq τ,a Q ph´1q τ,a (4.89)

THE MOST GENERAL INTEGRABLE BOUNDARIES

This property will be used later in the functional characterisation of the spectrum.

The most general integrable boundaries

The aim of this section is now to tackle the spectral problem of the boundary transfer matrix associated to cyclic representations of the 6-vertex reflection algebra, for the most general integrable boundaries. To characterise the eigenfunctions, we aim to find a separate basis, in which the spectral problem separates as in the previous section.

The main idea is to use a gauge transformation on the generators of the 6-vertex reflection algebra and on the boundary matrices, and to use this freedom to put lower triangular the gauged boundary matrix K `pλq, to be able to do the same steps of computation that we did in the previous section. As we will see, the separate basis are given by the pseudo-diagonalisation basis of the gauged Bpλq operator, in which this operator is pseudo-diagonal (in a precise sense that we are going to define).

We first introduce the gauge transformation in question, and then give the sketch of the proof that the gauged Bpλq is indeed pseudo-diagonalisable. Then we give the discrete characterisation of the spectrum, as it was done in the previous section, and eventually provide another characterisation via Baxter's type functional equation, which leads to a Bethe ansatz like form for the eigenvectors.

The gauge transformation

Keeping in mind that the boundary monodromy matrix is expressed in terms of the Yang-Baxter monodromy matrix, we first use a gauge transformation on the generators of the Yang-Baxter algebra. We follow the same approach used in the spin 1/2 case, namely the use of the gauge transformation introduced by Baxter's series of paper [52][53][54] (and its use to simplify the boundary matrix [START_REF] Fan | Algebraic Bethe ansatz for eight vertex model with general open-boundary conditions[END_REF][START_REF] Cao | Exact solution of XXZ spin chain with unparallel boundary fields[END_REF]), but with different gauge matrices, adapted to the cyclic case. This results in a gauge modification of the 6-vertex Yang-Baxter algebra. Then we define a gauge transformation on the generators of the 6-vertex reflection algebra, which satisfy a gauge modification of the reflection algebra commutation relations. The modified generators define a modified algebra which remains closed and allow for the introduction of the separation of variables in a way similar to the ungauged case.

Gauge transformation of the Yang-Baxter algebra

For arbitrary non zero complex parameters α, β and γ, let us introduce the following two matrices:

Gpλ|α, βq " ˆ1{pαβλq β{pαλq 1 1 ˙and Ḡpλ|γq " ˆ1{pλγq 0 1 1 ˙(4.90)
as well as their inverses:

G ´1pλ|α, βq " αλ β ´1{β ˆ´1 β{pαλq 1 ´1{pαβλq ˙and Ḡ´1 pλ|γq " ˆλγ 0 ´λγ 1 ˙(4.91)
The gauge transformed bulk monodromy matrix is then defined by: Let us recall that M0Q pλq, defined in (4.3), is a modification of the monodromy matrix used to express the boundary monodromy matrix. VARIABLES

M 0Q pλ|α, β, γq " G ´1pλq 1{2 |α, βq M 0Q pλq Ḡpλq 1{2 |γq " ˆApλ|α, β,
The definition here chosen of these gauge transformations differs with respect to that used previously in the literature [START_REF] Fan | Algebraic Bethe ansatz for eight vertex model with general open-boundary conditions[END_REF][START_REF] Cao | Exact solution of XXZ spin chain with unparallel boundary fields[END_REF]. On the one hand for the particular choice of the gauge matrix Ḡpλ|γq, which can be seen as lim aÑ`8 Gpλ|aγ 1{2 , γ 1{2 {aq and, on the other hand, as the parameters on the left and the right transformations are a priori independent.

These gauge transformed Yang-Baxter generators are of special interest as they define a closed set of commutation relations. Here we will not write all of them, let us simply highlight how the commutation between the generators are modified (with respect to (1.22)

We can prove by direct computation these relations, using the action of the gauge matrices on the Yang-Baxter equation.

As far as the quantum determinant is concerned, it is simple to prove that it holds:

M0Q pλ|α, β, γq " f pα, β, γqp´1q N σ y 0 M t0 0Q pλ|α, β, γqσ y 0 (4.100)
where we define the following function of the gauge parameters:

f pα, β, γq " γp1 ´β2 q αβ (4.101)
This way, it implies that we can express the quantum determinant thanks to the gauge transformed generators. The same way as in paragraph 1.3, it holds four different re-writings, let us here mention two of them:

q-det M 0Q pλq " f pα, qβ, qγq " Apλq 1{2 |α, β, γqDpλ{q 1{2 |α, βqq ´Bpλq 1{2 |α, βqCpλ{q 1{2 |α, βq, γqq ı (4.102)
and q-det M 0Q pλq " f pα, β{q, qγq

" Dpλq 1{2 |α, βqApλ{q 1{2 |α, β{q, γqq ´Cpλq 1{2 |α, β, γqBpλ{q 1{2 |α, β{qq ı (4.103)
The results here presented highlight that this modified Yang-Baxter algebra is very similar to the original one allowing for the extension of the quantum separation of variables to this framework.

THE MOST GENERAL INTEGRABLE BOUNDARIES

Gauge transformation of the reflection algebra

In this paragraph, we now construct the gauge boundary monodromy matrix. Using the previous gauge matrices, we define:

Upλ|α, βq " q 1{2 λ G ´1pλq 1{2 |α, βq Upλq Gpq 3{2 {λ|α, βq " ˆApλ|α, βq 2 q Bpλ|α, βq Cpλ|α, βq 2 q Dpλ|α, βq ˙(4.104)
Note the definition of the gauge generators, with a shift of q 2 on β for further convenience. Thus we consider the following linear combination of operators: where the gauged left boundary matrix reads:

Apλ|α,
Kpλ|γq " q 1{2 λ Ḡ´1 pλq 1{2 |γq Kpλq Ḡpq 1{2 {λ|γqq (4.111)
In these gauge transformed boundary matrix extra degrees of freedom are added that can be used wisely to eventually characterise the boundary transfer matrix spectrum.

Here, we can remark that Upλ|α, βq does not depend on the internal gauge parameter γ, so we are free to choose it at will. In our separation of variables construction, we will exclude the following values of γ:

γ " γ ˘" ´ξ´˘b ξ 2 ´`4κ 2 2qe τ´κ ´(4.112)
which leads to the vanishing of the coefficient (2,1) of this matrix:

@λ P C , Kpλ|γ ˘q21 " 0 (4.113)
(An explicit expression of the coefficients can be read in article II, formulas (B.6),(B.7),(B.8) and (B.9)).

In fact, one can argue that the function Kpλ|γq 21 plays the same role as the function b ´pλq for the ungauged case and, as we have already seen, we had to suppose b ´pλq ‰ 0 in order to implement the separation of variables.

The two next propositions state that the gauged generators satisfy similar commutation relations to the 6-vertex reflection algebra, and similar symmetry properties compared to the ungauged ones. In this proposition, we emphasise only some of these commutation relations. They are similar to the ones used in the ungauged case and they allow to extend to this gauged case the quantum separation of variables.

@pλ 1 , λ 2 q P C 2 , Bpλ 2 |α, βqBpλ 1 |α, β{q 2 q " Bpλ 1 |α, βqBpλ 2 |α, β{q 2 q (4.114) Apλ 2 |α, βq 2 qBpλ 1 |α, βq " pλ 1 q{λ 2 ´λ2 {qλ 1 qpλ 1 λ 2 {q ´q{λ 1 λ 2 q pλ 1 {λ 2 ´λ2 {λ 1 qpλ 1 λ 2 ´1{λ 1 λ 2 q Bpλ 1 |α,

Proposition 4.3.2 (Symmetries). Similar commutation relations of proposition 4.3.1 involving Cpλ|α, βq can be written by using the following β-symmetries:

Bpλ|α, βq " Cpλ|α, q 2 {βq and Apλ|α, βq " Dpλ|α, q 2 {βq (4.118)

Moreover, these gauge transformed operators satisfy the following parity properties:

Dpλ|α, βq " pq ´1{qqpλ 2 β{q ´q{λ 2 βq pβ{q 2 ´q2 {βqpλ 2 ´1{λ 2 q Apλ|α, βq `pβ{q ´q{βqpλ 2 {q ´q{λ 2 q pβ ´1{βqpλ 2 ´1{λ 2 q Ap1{λ|α, βq (4.119)

Bp1{λ|α, βq " ´pλ 2 q ´1{qλ 2 q pλ 2 {q ´q{λ 2 q Bpλ|α, βq and Cp1{λ|α, βq " ´pλ 2 q ´1{qλ 2 q pλ 2 {q ´q{λ 2 q C ´pλ|α, βq (4.120)

Lastly, the boundary quantum determinant can be written in terms of the gauge transformed boundary generators as it follows:

det q Upλq pλ 2 {q 2 ´q2 {λ 2 q " Apq 1{2 λ |α, βq 2 qApq 1{2 {λ |α, βq 2 q `B´p q 1{2 λ |α, βqCpq 1{2 {λ |α, βq 2 q (4.121) " Dpq 1{2 λ |α, βqDpq 1{2 {λ |α, βq `Cpq 1{2 λ |α, βq 2 qBpq 1{2 {λ |α, βq (4.122)
for any P t´1, 1u.

Once again, one can remark very similar relations compared to the ungauged case.

Pseudo-diagonalisability of the gauged operator Bpλ|α, βq

In this paragraph, we show that the gauged operator Bpλ|α, βq is pseudo-diagonalisable with simple pseudo-spectrum. Here, the term pseudo has to be understood as the fact that we can exhibit states which satisfy a similar relation to the one defining an eigenstate, only the value of the gauge parameter β is modified by the action of Bpλ|α, βq. However, this pseudo-diagonalisation basis of Bpλ|α, βq is extremely important as, for a wise choice of the gauge parameter α, it defines a separate basis for T pλq.

Statement of result

As usual, we start with some notation. Let b ´,a pβq be a set of complex numbers, for 1 ď a ď N , and let b ´pβq P C. We introduce the following function of the N-tuple h " ph 1 , ..., h n q P 0, p ´1 N :

b h pλ|βq " b ´pβqp λ 2 q ´q λ 2 q N ź a"1 p λ b ´,
b 2p ´,n pβq ‰ ˘1, b p ´,m pβq ‰ b p ´,n pβq, @n ‰ m P 1, N (4.124) b 2 ´,n pβq ‰ q 1´2h α 2 ´, b 2 ´,n pβq ‰ ´q1´2h β 2
´(4.125)

and b 2 ´,n pβq ‰ q 1´2h´2 µ 2 m,`, b 2 ´,n pβq ‰ q 1´2h´2 µ 2 m,´( 4.126)
and such that the following set of states: and with simple pseudo-spectrum, namely there is only one pseudo-eigenvector associated to a pseudoeigenvalue.

xβ, h| " xβ, h 1 , ..., h N | " 1 n β xΩ β | N ź n"

Sketch of proof

The proof of this theorem, which paves the way to one of the main result of this thesis, follows the same line as the one of the proof of diagonalisability of the operator Bpλq. However, the proof is much more technical, we simply highlight in the following the main ideas.

As for the ungauged case, the existence of the states xΩ β | and |Ω β y can be proven by an explicit construction for a subclass of parameters. The same holds for the coefficients pb ´,a pβqq 1ďaďN and b ´pβq, and so we explicitly show the Theorem 4.3.3 for this subclass of parameters. VARIABLES Then, a general argument states the existence of the vectors xΩ β | and |Ω β y for any parameters, i.e. the existence of at least one pseudo-eigenvector on the left and on the right. Thereafter, the theorem is proven using a continuity argument based on the fact that the quantities in question are polynomials in the bulk, boundary and gauge parameters, and the fact that an explicit construction under special values of these parameters is known. Let us give some more details:

• Special representation In the following, we want to study the conditions for which a nonzero state identically annihilated by the action of the operator family Apλ|α, β, γq exists:

xΩ, α, β,
µ 2p n, n ‰ ˘1, µ 2p n, n ‰ α 2p ´, µ 2p n, n ‰ ´β2p ´, µ 2p n,`‰ µ 2p m,´, µ p n, n ‰ µ p m, n (4.138)
Then the operator family Bpλ|α, βq is pseudo-diagonalised: As for the previous propositions, the conditions (4.138) implies the simplicity. Indeed, each state xβ, h| is associated to a different pseudo-eigenvalue of Bpλ|α, βq, then the only thing that we need to prove to get their linear independence is that each such state is nonzero. This is done as for the ungauged case, namely we know by construction that xΩ β | is nonzero and we prove by induction, thanks to the action of Apλ|α, βq 2 q on the left in some points, that all the states are non zero as it holds (4.138). Thus, they define a left basis of linear space of the representation.

xβ, h 1 , ..., h N |Bpλ|α, βq " b h pλ|βqxβ{q 2 ,
One can repeat the same procedure for the elements on the right, and it holds a very similar proposition for the right pseudo-eigenbasis. Thus, we have a subclass of parameters for which we have an explicit construction of the elements of theorem 4.3.3.

• General parameters As pointed out in the beginning, the proof that the operator Bpλ|α, βq is pseudo-diagonalisable and with simple pseudo-spectrum for general parameters follows from two steps. The first one is the existence of the states |Ω β y and xΩ β | in general, which can be done using the same kind of reasoning as in the ungauged case, and the second one relies on the special case of proposition 4.3.4, using a continuity argument. Let us give some more details.

To begin with, we can show that there exists at least one left and right pseudo-eigenstate |Ω β y and xΩ β | of the one parameter family of pseudo-commuting operators Bpλ|α, βq, with pseudo-eigenvalue b 0 pλ|βq satisfying the conditions p4.124q, p4.125q and p4.126q. VARIABLES Indeed, the operator family Bpλ|α, βq admits the following representation:

Bpλ|α, βq " p λ 2 q ´q λ 2 q N ÿ a"0 Λ a Ba pβqT ´2 β (4.146)
where the following commutation relations hold: @a, b P 1, N , Ba pβqT β " T β Ba pβqq and " Ba pβq, Bb pβq ‰ " 0 (4.147)

as a consequence of the commutation relations p4.114q. The result of the proposition 4.3.4 implies that for the special choice of the boundary, bulk and gauge parameters introduced, all the operators B´,a,β are invertible and Bpλ|α, βq admits the following representation:

Bpλ|α, βq " b ´pβqp λ 2 q ´q λ 2 q N ź a"1 p λ B a pβq ´Ba pβq λ qpλB a pβq ´1 λB a pβq qT ´2 β (4.148)
where the B a pβq are commuting and invertible operators. Then the fact that these operators depend continuously on the parameters implies that this statement is true for almost any values of these parameters. This also implies that for almost all the values of the boundary, bulk and gauge parameters we can use the above representation for Bpλ|α, βq.

We can now recall that, repeating the reasoning of proposition 4.2.2, we can always find a nonzero simultaneous eigenstate of commuting operators such as the B a pβq for any a P 1, N . This is a pseudo-eigenstate of the operator family Bpλ|α, βq.

Now, for the same set of representations considered in the proposition 4.3.4 we know that the pseudo-eigenvalues of Bpλ|α, βq satisfy the conditions p4.125q and p4.126q. Then, we can use once again the continuity argument to argue that the eigenvalues on the common eigenstate still satisfy p4.125q and p4.126q. We can thus prove the theorem 4.3.3, namely the pseudo-diagonalisability and simplicity of pseudospectrum of Bpλ|α, βq. Indeed, under the conditions p4.125q and p4.126q we can prove that all the left and right states are well defined and nonzero states which are pseudo-eigenstates of Bpλ|α, βq associated to different pseudo-eigenvalues as a consequence of the gauge transformed commutation relations. The proof of the fact that the states p4.127q and p4.128q are all nonzero is done reproducing the standard argument.

The separate basis

In this paragraph lies the heart of the quantum separation of variables. In a first part, we show a rewriting of the boundary transfer matrix for general parameters in terms of a gauged K `pλq right boundary matrix. Then, for an appropriate choice of the gauge parameter, we show that its spectral problem separates in the pseudo-eigenbasis of Bpλ|α, βq. To this end, we give the action of the generators of the gauged reflection algebra on the elements of the pseudo-eigenbasis.

A gauge transformation on the right boundary matrix K `pλq

Let us denote by Xpλ|α, βq and Y pλ|α, βq the respectively first and second columns of the gauge matrix Gpλ|α, βq, and similarly let us define Ỹ pλ|α, βq and Xpλ|α, βq the respectively first and second lines of the gauge matrix G ´1pλ|α, βq. Moreover, we define some modifications of these quantities, namely: Xpλ|α, q 2 βq " q β{q ´q{β β ´1{β Xpλ|α, q 2 βq and Ŷ pλ|α, β{q 2 q " q qβ ´1{pqβq β ´1{β Y pλ|α, β{q 2 q (4.149) ˆY pq 1{2 {λ|α, β{q 2 qK `pλq Xpλ{q 1{2 |α, q 2 βq Y pq 1{2 {λ|α, βqK `pλq Ŷ pλ{q 1{2 |α, β{q 2 q Xpq 1{2 {λ|α, βqK `pλq Xpλ{q 1{2 |α, q 2 βq Xpq 1{2 {λ|α, q 2 βqK `pλq Ŷ pλ{q 1{2 |α, β{q 2 q ˙(4.151)

Xpλ|α
One can thus compute the explicit expressions for the elements of the matrix K`p λ|α, βq. For example, it holds: K`p λ|α, β{qq 12 " 1{pξ `´1{ξ `q1{pβ ´1{βqα{q `qλ 2 ´1{pqλ 2 q ˘`qκ `e´τ`p β{αq 2 ´q2 {ξ `β{α ´κ`e T pλq " K`p λ|α, β{qq 11 Apλ|α, βq `K `pλ|α, β{qq 22 Dpλ|α, βqK `pλ|α, β{qq 21 Bpλ|α, β{q 2 q `K `pλ|α, β{qq 12 Cpλ|α, q 2 βq (4.153)

The interest in these gauge transformation is that for a judicious choice of the gauge, one gets that K`p λ|α, βq is triangular. Then, we are in the same framework as what has been done for the constraint boundary. More precisely, we can state: Proposition 4.3.6. If we fix the gauge parameter α as it follows:

α " ´ββ `{q 2 α `eτ`( 4.154) then K`p λ|α, βq 12 " 0 and the boundary transfer matrix can be equivalently written:

T pλq " a `pλqApλ|α, βq `a`p 1{λqAp1{λ|α, βq `q c `pλ|βqBpλ|α, β{q 2 q (4.155)

or T pλq " d `pλqDpλ|α, βq `d`p 1{λqDp1{λ|α, βq `q´1 c `pλ|βqBpλ|α, βq (4.156)

where we have defined:

a `pλq " ´λ2 q ´1{qλ 2 λ 2 ´1{λ 2 g `pλq , d `pλq " λ 2 q ´1{qλ 2 λ 2 ´1{λ 2 g `pq{λq (4.157)
c `pλ|βq " ´qpλ 2 q ´1{qλ 2 q pββ `{qα `´qα `{ββ `q β pα `´1{α `q pβ ``1{β `q (4.158)

(we recall that the function g `pλq has been introduced in (4.7)).

Note that (4.155) and (4.156) are direct consequences of the symmetry (4.120). Let us emphasise on the fact that these are the same kind of formulas than the symmetries (4.20) and (4.21) proven in paragraph 4.1 for the ungauged case.

The spectral problem separates

As it was expected, the pseudo-eigenbasis of the operator Bpλ|α, βq is a separate basis for the boundary transfer matrix with general parameters. Indeed, we show here that the action of the generators Apλ|α, βq and Dpλ|α, βq are simply shifts on the elements of this basis, leading to a system of Baxter equations.

First, we introduce the notation:

ζ phnq n " $ & % b ´,n pβqq hn @n P 1, N `b´,n´N pβqq h n´N ˘´1 @n P N `1, 2N (4
@ h, k P 0, p ´1 N , xβ, h|βq 2 , ky " δ hk ź 1ďbăaďN 1 X phaq a ´Xph b q b (4.168)
In particular, this implies the spectral decomposition of the identity in the separate basis:

p´1 ÿ h1,...,h N "0 ź 1ďbăaďN pX phaq a ´Xphaq a q|βq 2 , h 1 , ..., h N yxβ, h 1 , ..., h N | " 1 (4.169)
where we recall the definition

@b P 1, N , X ph b q b " pb ´,b pβqq h b q 2 `1{pb ´,b pβqq h b q 2 " pζ ph b q b q 2 `1{pζ ph b q b q 2 (4.170)
Therefore, we have the following characterisation of the boundary transfer matrix:

Proposition 4.3.8.
For almost all the values of the boundary and bulk parameters, T pλq has simple spectrum and Σ τ , the set of all the eigenvalues, coincides with the set of polynomials τ pλq of the form p4.68q with (4.69) which satisfy the following discrete system of equations: @a P 1, N , detD τ pζ p0q a q " 0 (4.171)

• The right T pλq eigenstate corresponding to τ pλq P Σ τ is defined by the following decomposition in the right separate basis:

|τ y " p´1 ÿ h1,...,h N "0 N ź a"1 q phaq τ,a ź 1ďbăaďN pX phaq a ´Xph b q b q|β, h 1 , ..., h N y (4.172)
where the gauge parameters α and β satisfy the condition p4.154q and the q phaq τ,a are the unique nontrivial solutions up to normalisation of the linear homogeneous system:

D τ pζ p0q a q ¨qp0q τ,a . . . q pp´1q τ,a ‹ ‹ ' " ¨0 . . . 0 ‹ ' (4.173)
• The left T pλq eigenstate corresponding to τ pλq P Σ τ is defined by the following decomposition in the left separate basis:

xτ | " p´1 ÿ h1,...,h N "0 N ź a"1 qphaq τ,a ź 1ďbăaďN pX phaq a ´Xph b q b qxβ{q 2 , h 1 , ..., h N | (4.174)
where the gauge parameters α and β satisfy the condition p4.154q and the qphaq τ,a are the unique nontrivial solutions up to normalisation of the linear homogeneous system:

´q p0q τ,a . . . qpp´1q τ,a ¯´D τ pζ p0q a q ¯t0 " `0 . . . 0 ˘(4.175)
with Dτ pλq is the family of p ˆp matrices defined substituting in D τ pλq the coefficient apλq with dpλq.

Lastly, there is the following link: let τ pλq P Σ τ then it holds:

qphq τ,a qph´1q τ,a " ap1{ζ ph´1q a q dp1{ζ ph´1q a q q phq τ,a q ph´1q τ,a (4.176)
The proof of this proposition follows the same lines as the one dedicated to the characterisation of the spectrum under the constraint that the right boundary is triangular, cf proposition 4.2.7.

This proposition is one of the main result of this thesis. Before to move to an equivalent characterisation in terms of functional equations, we would like to state the following proposition: and where the boundary matrices are diagonal, K 0 pλ|ξ ´, 0, 0q and K `0pλ|ξ `, 0, 0q, with the associated boundary parameters satisfying moreover |ξ ´| " |ξ `| " 1. The ˚operation is the complex conjugation. A simple direct calculation made for example in [START_REF] Grosjean | The τ 2 -model and the chiral Potts model revisited : Completeness of Bethe equations originated from Sklyanin SOV[END_REF] leads to the following Hermitian conjugate of the monodromy matrix:

M : 0Q pλ|P Q |P C q " σ y 0 M 0Q pλ ˚|P Q |P C qσ y 0 (4.178)
where σ y 0 denotes the Pauli matrix. From this relation, and using the specific left boundary matrix introduced, one can compute the Hermitian conjugate of the boundary monodromy matrix:

U : 0Q pλq " U t0 0Q p1{λ ˚q (4.179)
Then, from the definition of the boundary transfer matrix, and for the special choice of representation here chosen, we can show:

T : pλq " T p1{λ ˚q (4.180)
Thus for this special representation the boundary transfer matrix is normal. Then it follows that the determinant of the matrix p N ˆpN of elements xe i |τ j y, where xe i | is the generic element of a left orthogonal basis and |τ j y is the generic transfer matrix eigenvector, is non zero.

Noting that this determinant is a fractional function of the bulk and boundary parameters, non zero for the special choice of the parameters above defined, it follows that it is non zero for almost every choice of the parameters. Which concludes the proof. This property will be used later, notably in chapter 5.

Functional characterisation of T pλq spectrum

In this section, we give an equivalent characterisation of T pλq spectrum as the set of solutions to a Baxter like T-Q functional equation, and rewrite the eigenstates in an Algebraic Bethe Ansatz form. This makes an interesting link with the Bethe ansatz characterisation, but as we are going to see, for generic parameters the equation has an inhomogeneous term which has to be tackled. The literature on the analysis of the inhomogeneous Baxter equations is still at a preliminary stage. Let us mention that they were first introduced in [START_REF] Cao | Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fields[END_REF] for the XXZ spin 1/2 chain and the work of Nepomechie [START_REF] Nepomechie | Inhomogeneous T-Q equation for the open XXX chain with general boundary terms: completeness and arbitrary spin[END_REF] has given, for the XXX case, some first numerical evidence that they define a complete characterisation of the transfer matrix spectrum while a first rigorous proof of the completeness is appeared in [START_REF] Kitanine | Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables[END_REF].

In order to get a shorter presentation, we here introduce the functional characterisation for the most general integrable boundaries. The case of a triangular right boundary matrix can be explicitly read in paper I, or can be traced back from the following.

Here we mainly state the result, the fairly technical proofs can be found in paper II. The propositions 4.3.10 and 4.3.11 are preliminary results in order to understand the main result, the theorems 4.3.12 and 4.3.13. i.e. it is invariant under the transformations λ p Ñ 1{λ p and λ Ñ ´λ. Moreover, if tpλq is a Laurent polynomial of degree N `2 in Λ then det D t pλq is a Laurent polynomial of degree N `2 in Z.

The first symmetry is the same one as the one already seen for the ungauged case. The proof of the second statement consists in showing that det D t pq 1{2 q and det D t pi q 1{2 q are finite when tpλq is a Laurent

det D τ pλq " F pλq ˆλ2p ´1 λ 2p ˙2 p´1 ź k"0 pτ 8 ´pq k a 8 `q´k a 0 qq (4.184)
The proof consists in the following. From the previous proposition, we know that det D τ pλq is a polynomial of degree N+2 in Z. We can explicitly compute the asymptotic behaviour, and N zeros are known when τ pλq is an eigenvalue, thanks to the discrete characterisation (4.171). Moreover, we show that in fact det D τ pq 1{2 q and det D τ pi q 1{2 q are vanishing (for τ pλq an eigenvalue).

Let us now turn to the main results of this section, Baxter's TQ-functional equations. In the following, we denote with Qpλq a polynomial in Λ " λ 2 `1{λ 2 of degree N Q of the form:

Qpλq " N Q ź b"1 pΛ ´Λb q (4.185)
Theorem 4.3.12. For almost all the values of the boundary and bulk parameters such that:

@k P 0, p ´1 , τ 8 ‰ q ´ka 8 `qk a 0 (4.186)
it holds τ pλq P Σ τ if and only if τ pλq is an entire function and there exists and is unique a polynomial Qpλq of the form p4.185q with N Q " pp ´1q N, satisfying the following functional equation:

τ pλqQpλq " apλqQpλ{qq `ap1{λqQpλqq `"τ 8 ´pq ´NQ a 8 `qN Q a 0 q ‰ `Λ2 ´X2 ˘F pλq (4.187)
and the conditions:

@a P 1, N , pQpζ p0q a q, ..., Qpζ pp´1q a qq ‰ p0, ..., 0q (4.188)
The difficult task is to prove the existence of such a polynomial Qpλq, the other implication being quite straightforward thanks to the transfer matrix discrete characterisation (4.171). Thus, the trick is to consider a more general equation than (4.187) (namely (II, 5.30)), and to show that the solution X 0 pλq which can be computed thanks to the Cramer's rule, is in fact the searched Qpλq. Using similar techniques as for the propositions 4.3.10 and 4.3.11 to show that apparently diverging quantities are in fact finite, we show each statements: the equation (4.187) is satisfied, Qpλq has the form (4.185), it is unique and satisfies (4.188). For this last point, we use the polynomiality in the parameters, and show that the statement holds for a particular set of parameters, so it holds almost everywhere. VARIABLES In the previous theorem we have excluded the constraint on the boundary and bulk parameters leading to an identically zero det D τ pλq for any τ pλq P Σ τ . These specific cases are considered in the next theorem. Theorem 4.3.13. Let us assume that there exists a k P 0, p ´1 such that it holds:

τ 8 " q ´ka 8
`qk a 0 (4.189)

Then for almost all the values of the boundary and bulk parameters, τ pλq P Σ τ if and only if τ pλq is an entire function and there exists and is unique a polynomial Qpλq of the form p4.185q with N Q ď pp ´1q pN `1q and N Q " k mod p, satisfying the following homogeneous Baxter equation:

τ pλqQpλq " apλqQpλ{qq `ap1{λqQpλqq (4.190)
and the conditions:

@a P 1, N , pQpζ p0q a q, ..., Qpζ pp´1q a qq ‰ p0, ..., 0q (4.191)
Let us simply comment that in this case, the polynomial Qpλq can be explicitly constructed via the cofactor of the element p1, 1q of the matrix D τ pλq, the proof being based on some symmetries of the cofactors.

We end this section by giving a Bethe ansatz type formulation for the eigenvectors. In more details, let us introduce the states:

xβ, ω| " p´1 ÿ h1,...,h N "0 N ź a"1 ha´1 ź ka"0 ap1{ζ pkaq a q dp1{ζ pkaq a q ź 1ďbăaďN pX phaq a ´Xph b q b q xβ, h 1 , ..., h N | (4.192) |β, ωy " p´1 ÿ h1,...,h N "0 ź 1ďbăaďN pX phaq a ´Xph b q b q|β, h 1 , ..., h N y (4.193)
and a renormalisation of the operator Bpλ|α, βq " Bpλ|α, βqT 2 β pλ 2 {q ´q{λ 2 qb ´pβq (4.194) which is a degree N polynomial in Λ. As first remarked in paper [136], from the functional (polynomial) characterisation of the Q-function and from the discrete characterisation of the boundary transfer matrix eigenvectors, it follows the Bethe-like rewriting of these eigenstates:

A first step toward the dynamic

As already mention, the characterisation of the spectrum is the very first step in the computation of physical quantities, aiming to characterise the dynamic of the system. In this short section, we give a little step toward the computation of the form factors and correlation functions for the class of models under consideration, namely we compute the scalar products of the so-called separate states. where the elements of the size N matrix M pα,βq are given by:

@pa, bq P 1, N 2 , M pα,βq a,b " p´1 ÿ h"0 α phq a β phq a pX phq a q pb´1q (4.199)
The proof is quite straightforward, it is based on the fact that Vandermonde determinants appear when computing the scalar product. One of the main corollary is the orthogonality of two eigenstates xτ | and |τ 1 y of the boundary transfer matrix associated to two different eigenvalues τ pλq and τ 1 pλq:

xτ |τ 1 y " 0 (4.200)
The computation of such scalar products is the very first step toward the dynamic, several further steps are required to reach this characterisation for the models associated to cyclic representations of the 6-vertex reflection algebra (see paragraph 1.5): the reconstruction of the local operators in separate variables, the identification of the ground state, the homogeneous and the thermodynamic limit. For example a rewriting of the determinant representations for the form factors obtained from separation of variables will be necessary to overcome the standard problems related to the homogeneous limit. This problem has been addressed and solved for the XXX spin 1/2 chain, linking the separation of variables type determinants with Izergin's, Slavnov's and Gaudin's type determinants [START_REF] Kitanine | The open XXX spin chain in the SoV framework: scalar product of separate states[END_REF][START_REF] Kitanine | On determinant representations of scalar products and form factors in the SoV approach: the XXX case[END_REF].

Let us comment that we have chosen, for both the triangular and general K `boundary matrices, to present the existence of the separate basis avoiding a recursive construction, which was instead done for example in the sine Gordon model [START_REF] Niccoli | The sine-Gordon model revisited I[END_REF]. Here, we first considered some special representations in which we could construct explicitly the separate basis (in both triangular and general cases), while the existence of the separate basis for general representations is proven by some general arguments. Let us argue that this does not lead to severe limitations per se. For the spectrum indeed, we are still able to prove that the functional equation characterises the eigenvalues for general parameters. Moreover, for the future computations of the dynamics, the relevant point is that we can still characterise Sklyanin's measure (4.73) and (4.168) by a Vandermonde formula, leading to formula of the kind (4.198)-(4.199) for the scalar products of separate states. Then, the rewriting of the scalar products and form factors by determinant formulas adapted to the homogeneous limit should allow to absorbe the separate variable eigenvalue spectrum in Laurent polynomials whose homogeneous limit is finite and known; as it happens Chapter 5

Local Hamiltonians associated to cyclic representations of the reflection algebra

In this chapter, we solve the problem to define local Hamiltonians with integrable boundaries, commuting with the boundary transfer matrix associated to cyclic representations of the 6-vertex reflection algebra. The search of an Hamiltonian is an interesting problem, as this results in the modelling of the interactions at a microscopic level, emphasising on the physics of the system at hand.

To that end, we follow the subsequent steps. First we consider the fundamental R-matrix, solution of the so-called mixed Yang-Baxter equation intertwining two Bazhanov-Stroganov Lax operators acting in two different quantum spaces. We define then a mixed reflection equation intertwining these Lax operators and two scalar boundary matrices, respectively defined on the auxiliary space (of dimension 2) and quantum space (of dimension p). By explicitly solving this mixed reflection equation, we find diagonal scalar solutions for the boundary matrices on the quantum space, associated to the known diagonal scalar matrices on the auxiliary space (the matrices Kpλq and K `pλq of the previous chapter, but taken here diagonal). Then we can define a multi spectral parameter family of fundamental transfer matrices, commuting between themselves and with the transfer matrix associated to the original cyclic reflection algebra. Moreover, we prove that these fundamental transfer matrices generate multi parameter families of commuting local Hamiltonians with integrable boundaries.

We end the chapter making totally explicit the Hamiltonians for some models of interest, for 3dimensional local quantum spaces (i.e. p=3): the XXZ spin 1 chain at root of unity, the super integrable chiral Potts model, the sine-Gordon model at root of unity and a more general model. They all can be obtained from the Bazhanov-Stroganov Lax operator, by an appropriate choice of the parameters.

The results of this chapter have been published in article III.

Fundamental R-matrix

For fundamental quantum integrable models, e.g. the XXZ spin 1/2 chain, the construction of the local Hamiltonians is given by standard techniques, see paragraph 2.3.2, relying on the fact that the associated Lax operator reduces to the permutation operator in some special value of the spectral parameter. However, the Bazhanov-Stroganov models are not of this type, as the Lax operator (1.72) does not reduce in general to the permutation operator (its bi-dimensional auxiliary space and its p-dimensional quantum space are non isomorphic). Hence, in order to define local Hamiltonians for Bazhanov-Stroganov models, one has to consider the fundamental R-matrix [START_REF] Bazhanov | Chiral Potts model as a descendant of the six vertex model[END_REF][START_REF] Tarasov | Cyclic monodromy matrices for slpnq trigonometric R-matrices[END_REF], denoted S, solution of the so-called mixed Yang-Baxter equation which intertwines two Lax operators acting in two different quantum spaces V a and V b , while acting on the same THE REFLECTION ALGEBRA auxiliary space V 0 : @ λ P C , S ba pp b , p a qL 0a pλ, p a qL 0b pλ, p b q " L 0b pλ, p b qL 0a pλ, p a qS ba pp b , p a q (5.1) Such a fundamental R-matrix acts in the tensor product of two cyclic representations.

In [START_REF] Bazhanov | Chiral Potts model as a descendant of the six vertex model[END_REF], this Yang-Baxter equation has first appeared and it has been solved for the subclass of cyclic representations associated to the chiral-Potts model. In [START_REF] Tarasov | Cyclic monodromy matrices for slpnq trigonometric R-matrices[END_REF] the fundamental R-matrix has been determined for the most general cyclic representations. Here, we reproduce these results and fundamental steps in their derivations, following our convention making use of the untwisted 6-vertex R-matrix unlike what it is done in the literature [START_REF] Bazhanov | Chiral Potts model as a descendant of the six vertex model[END_REF][START_REF] Tarasov | Cyclic monodromy matrices for slpnq trigonometric R-matrices[END_REF]. Thus in this section we first use the chiral Potts model as a starting point to present the fundamental R-matrix. Then, we extend to more general representations the computation of this fundamental R-matrix and we eventually give two important properties, namely its unitarity and crossing-unitarity.

Parametrisation by points in C 4

Let us first give a different parametrisation for the Lax operator L 0n pλ, p n q (1.72). Following [START_REF] Bazhanov | Chiral Potts model as a descendant of the six vertex model[END_REF], let us thus consider a point P P C 4 . Generally, we will refer to its canonical coordinates by the notation P " pa P , b P , c P , d P q or equivalently we will use the parameters:

x P " a P d P y P " b P c P s P " d P c P t P " x P y P (5.2)
Then a more convenient parametrisation of the Lax operator (1.72), for future computations, can be given by the triplet pP, Q n , R n q thanks to the correspondence:

λα n " ´t´1{2 P b Qn b Rn q ´1{2 b n " 1 q 2 a Qn d Rn (5.3)
β n λ " ´t1{2 P d Qn d Rn q 1{2 a n " ´cQn b Rn (5.4) λδ n " 1 q 2 t ´1{2 P
a Qn a Rn q 1{2 d n " ´dQn a Rn (5.5)

γ n λ " t 1{2 P c Qn c Rn q ´1{2 c n " b Qn c Rn . (5.6)
We recall here that q is a parameter entering in the R-matrix (1.18), and that it is assumed to be a p th -root of unity: q " e 2ikπ{p for some k P N ˚, with k and p coprime integers. We also recall that p is odd:

p " 2l `1 with l P N ˚.
With this parametrisation, the dependence in the point P is related to the spectral parameter λ by choosing λ 9 t ´1{2 P , and we can use the notation

L 0n pλ|Q n , R n q " γ n ˜´λy Qn y Rn vn `1 λ σ n v´1 n ûn p´1 q y Rn vn `1 q x Qn σ n v´1 n q û´1 n pqy Qn vn ´1 q σ n x Rn v´1 n q 1 λ vn ´1 q 2 λσ n x Qn x Rn v´1 n ¸(5.7)
where we introduced the parameter σ n " s Qn s Rn and where we used the parameter γ n as a global normalisation.

Let us emphasise that each Bazhanov-Stroganov Lax operator depends on the spectral parameter λ and on a bunch of 5 independent parameters x Qn , x Rn , y Qn , y Rn and σ n associated to the site n. The total number of independent parameters in the monodromy matrix M 0Q pλq is thus 5N (indeed, one can always absorb for example σ 1 in λ).

In the following, this parametrisation with points in C 4 is almost exclusively used to deal with the fundamental R-matrix and the computation of the local Hamiltonians.

The chiral Potts representation

In [START_REF] Bazhanov | Chiral Potts model as a descendant of the six vertex model[END_REF], the authors have shown that the chiral Potts model can be described and studied through the Yang-Baxter algebra associated to the 6-vertex R-matrix. Indeed, the Bazhanov-Stroganov Lax operators are general solutions of the Yang-Baxter equation, and to particularise the points Q n and R n of (5.7) to a specific curve leads to the description of the chiral Potts model.

In this short section, we recall the expression of the so-called chiral Potts curves and of the intertwiner S ba pp b , p a q (5.1) in this context. The main tool to express the intertwiners is the dilogarithm function [START_REF] Au-Yang | Commuting transfer matrices in the chiral Potts models: Solutions of star-triangle equations with genus>1[END_REF][START_REF] Au-Yang | Solvable models in statistichal mechanics and Riemann surfaces of genus greater than one[END_REF][START_REF] Baxter | New solutions of the star-triangle relations for the chiral Potts model[END_REF], which for the chiral Potts representations reduces to the Boltzmann weights. The next two propositions 5.1.1 and 5.1.2 recall respectively its definition and how the fundamental R-matrix is computed for the chiral Potts representations: Proposition 5.1.1 (Dilogarithm functions). Let us denote by W QP and W QP the dilogarithm functions associated to a couple pQ, P q P C 4 ˆC4 . By definition, these functions are given, up to normalisation, by:

@n P N , W QP pnq W QP p0q " ˆsQ s P ˙n n ź k"1 y P ´q´2k x Q y Q ´q´2k x P and W QP pnq W QP p0q " ps Q s P q n n ź k"1 q ´2x Q ´q´2k x P y P ´q´2k y Q
(5.9) The dilogarithms are geometric quasi-periodic functions of quasi-period p:

W QP pn `pq W QP pnq " k ´p QP W QP pn `pq W QP pnq " k´p QP (5.10)
where we have defined k QP and kQP the constants, defined up to a p th -root of unity, such that:

k p QP " s p P s p Q y p Q ´xp P y p P ´xp Q and kp QP " 1 s p P s p Q y p P ´yp Q x p Q ´xp P (5.11)
The key point with these functions is the fact that they are solutions of the following recursion relations:

@n P N , W QP pnq W QP pn `1q " ´sP s Q
x P y P q ´2pn`1q 1 ´yQ x P q 2pn`1q 1 ´xQ y P q ´2pn`1q

(5.12) and W QP pnq W QP pn `1q " ´q2pn`1q 1 s P s Q y P x P 1 ´yQ y P q ´2pn`1q 1 ´xQ

x P q 2n

(5.13)

Moreover, if the points P and Q belong to a given chiral Potts curve, then these dilogarithm functions are periodic of period p, i.e. it holds k p QP " kp QP " 1 Then we can recall the following proposition, from the work of Bazhanov and Stroganov [START_REF] Bazhanov | Chiral Potts model as a descendant of the six vertex model[END_REF]: Proposition 5.1.2 ( [158]). Let pQ a , R a , Q b , R b q P `Ck chP ˘4, and let us consider two Lax operators L 0a pλ|Q a , R a q and L 0b pλ|Q b , R b q on two different quantum spaces V a and V b . THE REFLECTION ALGEBRA Let us consider the following basis for the tensor product of the quantum spaces: @ph 1 , h 2 q P 0, p ´1 and let S ba pQ b , R b |Q a , R a q be the operator defined, on the tensor product of the two p-dimensional spaces, by its elements:

@ph 1 , h 2 , h 2 1 , h 2 2 q P 0, p ´1 2 , xh 1 , h 2 | S ba pQ b , R b |Q a , R a q h 2 1 , h 2 2 D " W Q b Qa ph 1 ´h2 2 qW R b Qa ph 2 1 ´h2 2 qW R b Ra ph 2 ´h2 1 qW Q b Ra ph 2 ´h1 q (5.15) Then S ba pQ b , R b |Q a , R a q is well defined, in the sense that xh 1 , h 2 | S ba pQ b , R b |Q a , R a q |h 2 1 , h 2 2 y is p-periodic for each of the variables h 1 , h 2 , h 2
1 and h 2 2 , and is the following intertwiner in the quantum spaces:

@λ P C , S ba pQ b , R b |Q a , R a qL 0a pλ|Q a , R a qL 0b pλ|Q b , R b q " L 0b pλ|Q b , R b qL 0a pλ|Q a , R a qS ba pQ b , R b |Q a , R a q (5.16)
Proof. The fact that S ba pQ b , R b |Q a , R a q is p-periodic is clear since each of the Boltzmann weight is pperiodic for points on the chiral Potts curve. The proof of the fact that the relation (5.16) holds follows from the recursion relations satisfied by the dilogarithm functions. This equation can be seen as an equality of matrices in the auxiliary space, with elements in EndpH a b H b q, and we show this equality for any matrix element pi 1 , i 2 q P t1, 2u 2 by direct computation.

The key observation, at the basis of our work, is that this proof only requires points on the chiral Potts curves to get a well defined intertwiner, i.e. p-periodic in the four variables. The relation (5.16) holds independently of the curves (5.8), it is only based on the recursion relations satisfied by the dilogarithm functions.

More general representations

Here we present an expression of the fundamental R-matrix for Bazhanov-Stroganov Lax operators beyond the chiral Potts model (giving the expression of each of its components), i.e. we compute an intertwiner for the Lax operators (5.7) when the points Q n and R n are not necessarily on a chiral Potts curve.

To this end, thanks to the observation of the previous paragraph, rather than imposing the points Q n and R n to live on the chiral Potts curves to have p periodic dilogarithm functions, we choose here to use a bunch of four scalars to make p periodic the whole product of the four terms in (5.15).

Explicitly, let S ba pQ b , R b |Q a , R a q be the matrix, acting on the tensor product of the two p-dimensional spaces V a and V b , whose matrix elements read:

@ph 1 , h 2 , h 2 1 , h 2 2 q P 0, p ´1 2 , xh 1 , h 2 | S ba pQ b , R b |Q a , R a q h 2 1 , h 2 2 D " a h1 1 a h2 2 a h"1 3 a h"2 4 W Q b Qa ph 1 ´h2 2 qW R b Qa ph 2 1 ´h2 2 qW R b Ra ph 2 ´h2 1 qW Q b Ra ph 2 ´h1 q (5.17)
with the following scalars:

a 1 " kQ b Qa k Q b Ra ; a 2 " kR b Ra k Q b Ra ; a 3 " k R b Qa kR b Ra and a 4 " 1 kQ b Qa k R b Qa (5.18)
By construction, these matrix elements are p periodic in each of the variables h 1 , h 2 , h

y p Q b ‰ y p Qa ; y p R b ‰ y p Ra ; x p Q b ‰ x p Qa ; x p R b ‰ x p Ra (5.20) and y p Q b ‰ x p Ra ; x p Q b ‰ y p Ra ; y p R b ‰ x p Qa ; x p R b ‰ y p Qa (5.21)
Thus, the operator family S ba pQ b , R b |Q a , R a q PEndpH a bH b q is well defined for any choice of pQ a , R a , Q b , R b q in this set of parameters.

Existence condition

The use of the expression (5.17) seems promising as the matrix elements are p-periodic without any requirement. However the next proposition defines some restriction on the parameters:

Proposition 5.1.3. Let S ba pQ b , R b |Q a , R a q be defined by (5.17). It satisfies the mixed Yang-Baxter equation for any λ P C:

S ba pQ b , R b |Q a , R a qL 0a pλ|Q a , R a qL 0b pλ|Q b , R b q " L 0b pλ|Q b , R b qL 0a pλ|Q a , R a qS ba pQ b , R b |Q a , R a q (5.22)
if and only if the points pQ a , R a , Q b , R b q P `C4 ˘4 are such that the following three independent conditions are satisfied: @i P t1, 2, 3u , a i " 1.

(5.23)

Proof. To show the equation (5.22) from (5.23) is trivial thanks to proposition 5.1.2 and the remark on the fact that points on chiral Potts curves are only necessary to have p periodic functions. The other way of the proposition is done using the same lines as for the proof of proposition 5.1.2, namely to consider (5.22) as four equations in the quantum space V a ˆVb , but now the quantities are dressed with the coefficients a i . At the end, to require (5.22) leads to the condition (5.23). A more explicit proof can be read in appendix of paper III.

Thus, the existence of an intertwiner of the form (5.17) is guaranteed if the pre factors satisfy the condition (5.23), which can be put under the form:

$ ' & ' % kQ b Qa " k Q b Ra k R b Qa " kR b Ra kR b Ra k Q b Ra " 1 (5.24)
The pre factors being defined up to a p th -root of unity, one can always choose these roots such that (5.24) is equivalent to:

$ ' ' & ' ' % kp Q b Qa " k p Q b Ra k p R b Qa " kp R b Ra kp R b Ra k p Q b Ra " 1 (5.25)
Let us introduce the following notation:

@i P ta, bu , X Qi " x p Qi ; X Ri " x p Ri ; Y Qi " y p Qi ; Y Ri " y p Ri ; S i " σ p i " s p Qi s p Ri (5.26)
Then, if the quantum parameters on sites a and b satisfy the following system (5.27), the intertwinner THE REFLECTION ALGEBRA S ba pQ b , R b |Q a , R a q exists and is given by (5.17).

$ & % pY Qa ´YQ b q{pX Q b ´XQa q " S a pY Q b ´XRa q{pY Ra ´XQ b q pY Ra ´YR b q{pX R b ´XRa q " S b pY Ra ´XQ b q{pY Q b ´XRa q pY Ra ´YR b q{pX R b ´XRa q " S a pY R b ´XQa q{pY Qa ´XR b q
(5.27)

Explicit parametrisations

In this paragraph, we find explicit solutions of the system (5.27), i.e. we find explicit conditions on the parameters for the fundamental R-matrix to exist. Let us first introduce the following notation:

L i " S i X Qi ´YRi 1 ´Si M i " S i X Qi ´YRi S i X Qi X Ri ´YRi Y Qi N i " Y Qi ´Si X Ri 1 ´Si (5.28) and R i " X Qi X Ri ´YQi Y Ri X Qi ´YRi K i " Y Qi ´XRi X Qi ´YRi T i " Y Qi Y Ri S i (5.29)
or equivalently in the original parametrisation of the Bazhanov-Stroganov Lax operator:

L i " q p{2 a p i `bp i γ p i `βp i M i " q p{2 a p i `bp i α p i `δp i N i " q p{2 c p i `dp i γ p i `βp i (5.30) and R i " q p{2 α p i `δp i a p i `bp i K i " d p i `cp i a p i `bp i T i " α p i β p i (5.31)
Then the following proposition holds: Proposition 5.1.4. Let us assume that the conditions (5.23) are satisfied. The solutions to (5.23) can be parametrised by the following four sets of conditions: A) If the points pQ a , R a , Q b , R b q P `C4 ˘4 satisfy the conditions:

@i P ta, bu, S i ‰ 1 and Y Qi Y Ri ´Si X Qi X Ri ‰ 0 (5.32)
as well as Y Qi ´Si X Ri ‰ 0 and Y Ri ´Si X Qi ‰ 0 (5.33) then (5.23) is equivalent to the following three conditions of homogeneities between the sites a and b:

L a " L b " L, M a " M b " M, N a " N b " N (5.34)
where L, M and N are three free parameters. B) If the points pQ a , R a , Q b , R b q P `C4 ˘4 satisfy the conditions:

S i " 1, X Qi X Ri ´YQi Y Ri ‰ 0, @i P ta, bu then (5.23
) is equivalent to the following two conditions of homogeneities between the sites a and b:

R a " R b " R, K a " K b " K (5.35)
where R and K are two free parameters. C) If the points pQ a , R a , Q b , R b q P `C4 ˘4 satisfy the conditions: pS a , S b q ‰ p1, 1q and @i P ta, bu , Y Ri " S i X Qi and Y Qi " S i X Ri (5.36) then (5.23) is equivalent to the following condition of homogeneity between the sites a and b:

T a " T b " T (5. 37
)
where T is a free parameter. D) If the points pQ a , R a , Q b , R b q P `C4 ˘4 satisfy the conditions: @i P ta, bu , S i " 1 ; Y Ri " X Qi and Y Qi " X Ri (5.38) then (5.23) is satisfied without any more requirement.

Proof. The proof of this proposition is based on the following equivalent formulation for (5.27):

$ & % Y Qa Y Ra ´Sa X Qa X Ra " pY Qa ´Sa X Ra qX Q b ´pS a X Qa ´YRa qY Q b `pS a ´1qY Q b X Q b Y Q b Y R b ´Sb X Q b X R b " pY R b ´Sb X Q b qX Ra ´pS b X R b ´YQ b qY Ra `pS b ´1qY Ra X Ra pS a X Qa ´YRa qpY Q b ´XR b q " pY Qa ´Sa X Ra qpX Q b ´YR b q `pS a ´1qpX Q b Y Q b ´XR b Y R b q (5.39)

Remark. Points C) and D) of the proposition correspond to the special representations with nilpotent off-diagonal elements of the Bazhanov-Stroganov Lax operators.

Remark. Following the idea given by [START_REF] Tarasov | Cyclic monodromy matrices for slpnq trigonometric R-matrices[END_REF], one of the simplest necessary condition for the intertwiner S ba to exist is probably the equality of the mean values2 , i.e. it must hold:

@λ P C , ă L 0a pλ|Q a , R a q ąă L 0b pλ|Q b , R b q ą"ă L 0b pλ|Q b , R b q ąă L 0a pλ|Q a , R a q ą (5.40)
We can show that the generic solution of this equation is the generic solution given by point A) of proposition 5.1.4, thus this proposition states that for generic parameters, the equality of mean values is in fact equivalent to the existence of S ba . However, this result does not hold for example for the point C), where the system (5.40) is trivially satisfied whereas the extra condition (5.37) is needed for the existence of the intertwiner.

Let us resume the results we have obtained so far. In each of the four points of proposition 5.1.4, we state the existence of the intertwiner S ba pQ b , R b |Q a , R a q of the form (5.17) when imposing that three parameters have to be the same on the both spaces V a and V b :

• pL, M, N q for point A)

• pS " 1, R, Kq for point B)

• pY R ´SX Q " 0, Y Q ´SX R " 0, T q for point C) • and pS " 1, Y R ´XQ " 0, Y Q ´XR " 0q for point D)
As a consequence, these homogeneous constant parameters along the chain can be seen as characterising the model. We will denote by P C this set of three parameters in a general way.

Then, as on each site n of the chain are associated 5 independent parameters, let us denote P n the two remaining quantum parameters on each site. They can be seen for example, as the remaining 6-vertex inhomogeneity plus a quantum inhomogeneity. Eventually, we will denote the set of all the quantum parameters of the chain (2N parameters) by:

P Q " tP n u 1ďnďN
(5.41)

Accordingly with this notation, to emphasise the fact that the parameters belong to certain sets for which the intertwiner exists, in all the next we will denote the fundamental R-matrix by S ba pP b |P a |P C q and we will denote the Bazhanov-Stroganov Lax operators by L 0n pλ|P n |P C q. THE REFLECTION ALGEBRA Let us comment here that the fundamental R-matrix (5.17) has formally the same expression as for the chiral Potts representations (5.15). The difference lies in the fact that each Boltzmann weight is now quasi-periodic, keeping periodic however the product of the four weights with respect to the variables h 1 , h 2 , h" 1 and h" 2 .

Remark. The chiral Potts model is obtained as a particular case of our study, as it holds for it for this model:

kp Q b Qa " 1 ; k p R b Qa " 1 ; kp R b Ra " 1 and k p Q b Ra " 1 (5.42)
so the system (5.27) is trivially satisfied. Moreover, it is included for a generic parameter k ‰ 0 of C k chP , in the point A) of proposition 5.1.4. The constants of the model read:

L " ´1 k M " 1 k and N " 1 k (5.43)
For a chiral Potts model on a curve with k " 0, the point B) applies and the constants of the models read: R " 0 and K " 1 (5.44)

Unitarity and crossing-unitarity

In this paragraph, we give some properties satisfied by the fundamental R-matrix (5.17). In particular, we state its unitarity and crossing-unitarity, two properties that will be of interest when dealing with reflection equations. Let us consider two quantum spaces V a and V b isomorphic to C p , and parameters such that it exists a set P C entering in one of the points of proposition 5.1.4, leading to the existence of S ba pP b |P a |P C q. We recall that the parameters are given by sets px Qa , x Ra , y Qa , y Ra , σ a q and px Q

b , x R b , y Q b , y R b , σ b q with σ i " s Qi s Ri , associated to couples of points Q a , R a and Q b , R b .
We first show by direct computation the next lemma, which is at the basis of many of the properties: Lemma 5.1.5. Let ph 2 , h" 2 q P 0, p ´1 2 . It holds the following identities:

p´1 ÿ h"0 W X b Xa ph ´h2 qW XaX b ph" 2 ´hq " δ h2h"2 Σ
pXq ab (5.45) where the functions Σ pXq ab , for X " R or Q, are independent of h 2 and given by:

Σ pXq ab " p´1 ÿ h"0 χ pXq ab,h with χ pXq ab,h "
px Xa ´xX b q py X b ´yXa q px Xa q h´1 ´xX b q 1´h q py X b q h´1 ´yXa q 1´h q (5.46)

Unitarity

The next proposition states the unitarity of the fundamental R-matrix, a property which is known for the 6-vertex R-matrix, cf [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF].

Crossing-unitarity

In order to make explicit another symmetry of the fundamental R-matrix, we introduce an operation Υ on the parameters. On the original parameters, it reads:

Υ : pα, β, γ, δ, a, b, c, dq Ñ p´δ{q, ´qγ, ´qβ, ´α{q, ´b, ´a, ´d, ´cq (5.50)

while, using the parametrisation with points in C 4 , we have:

Υ : px Q , x R , y Q , y R , σq Ñ py R , y Q , x R {q 2 , x Q {q 2 , 1{σq (5.51)
This non involutive operation is useful to state the crossing-unitarity of the fundamental transfer matrix.

But first, we give this important proposition: The transpositions are given in the û-basis.

Proof. The first relation that we use is given thanks to the operation Υ, which acts as the following on Bazhanov-Stroganov Lax operators:

L t b 0b pλ|P Υ b b q 9 L ´1 0b pλ|P b q (5.54)
Then, one can show the proposition thanks to simple computations, similar to the proof of the unitarity, and using the symmetries:

@h P 0, p ´1 , W Υ b QaR b phq " W Q b Qa phq and W Υ b QaQ b phq " W R b Qa p´hq (5.55)
Remark. Moreover, one easily shows by direct computation:

S ab pP ΥaΥa a |P Υ b Υ b b |P C q " S ab pP a |P b |P C q and L 0b pλ|P Υ b Υ b b q " qL 0b pqλ|P b q (5.56)
So far, we have obtained the existence and the expression of the fundamental R-matrix for models beyond the chiral Potts case, as well as the unitarity and crossing-unitarity for this matrix. As introduced previously, one of the role of this matrix is to overcome the fact that the auxiliary and quantum spaces of the Bazhanov-Stroganov Lax operators are not isomorphic. In the following, we use this fundamental operator to compute local Hamiltonians. Moreover, we aim to describe integrable boundaries, for systems associated to cyclic representations of the 6-vertex reflection algebra. In order to implement this idea, we now turn to the definition of the reflection equation involving cyclic representations.

Reflection equations

In this short paragraph, we give a general point of view on the reflection equations, introducing the fact that an automorphism can describe the reflection process at the boundaries. We emphasise the role of these equations in integrability, as they can be used to construct a family of commuting operators, and we make a link with (ABCD)-type quantum algebras [START_REF] Freidel | Quadratic algebras and integrable systems[END_REF]. This point of view will be useful to tackle reflection equations involving one or more cyclic representations. As an example, we particularise this general frame to the 6-vertex case and find back the Sklyanin's description of the reflection algebras.

General considerations

Introduced by Cherednik [START_REF] Cherednik | Factorizing particles on a half-line and root systems[END_REF], and at the heart of Sklyanin's construction of the one parameter family of boundary transfer matrices in the 6-vertex case [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF], the reflection equations can be used to describe the behaviour of particles reflecting on a wall while they have factorised scattering in the bulk. Considering this idea of two particles labelled 1 and 2, we can give this general expression for the reflection equation:

R φ2φ1

21 pP 2 , P 1 qK 1 pP 1 qR φ2 12 pP 1 , P 2 qK 2 pP 2 q " K 2 pP 2 qR φ1 21 pP 2 , P 1 qK 1 pP 1 qR 12 pP 1 , P 2 q (5.57)

Here, the matrix R 12 pP 1 , P 2 q encodes the collision of the two particles, P i stands for the spectral parameters associated to space V i describing the "colour" of the particle, K i describes the reflection of the particle i on the boundary and the automorphism φ i describes the reflected particle i. (cf. figure 5.1) 5.1: On the left, particle 1 and particle 2 are reflecting on a wall. On the right, the same situation but with particle 1 translated. By conservation laws, the equation (5.57) holds. The operations φ i caracterise the reflection. The order R 12 or R 21 is determined thanks to the relative position of the particles 1 and 2 at the asymptotic times.

• • • • K 1 K 2 2 1 • K 1 1 • • • K 2 2 Figure
In parallel, let us recall that an (ABCD)-type quantum quadratic algebra [START_REF] Freidel | Quadratic algebras and integrable systems[END_REF] is defined as the associative algebra generated by the elements of a matrix T ´, which satisfy the relations

A 12 T ´,1 B 12 T ´,2 " T ´,2 C 12 T ´,1 D 12 (5.58)
for four given operators A 12 , B 12 , C 12 and D 12 . To be a consistent algebra, the latter have to satisfy eight compatibility conditions, explicitly given by the relations (15) of [START_REF] Freidel | Quadratic algebras and integrable systems[END_REF]. These ones are involving a third space V 3 , labelled 3. As an example, it must hold: then the boundary matrix K is a scalar representation of an (ABCD)-type quantum quadratic algebra, associated to the generators A 12 " R φ2φ1 21 pP 2 , P 1 q, B 12 " R φ2 12 pP 1 , P 2 q, C 12 " R φ1 21 pP 2 , P 1 q and D 12 " R 12 pP 1 , P 2 q (5.61)

A 12 A
There are two main interests in the (ABCD)-quantum type algebra.

• First, the compatibility conditions ensure the following property [START_REF] Freidel | Quadratic algebras and integrable systems[END_REF]: In particular, starting from a scalar representation of the algebra (5.58) (so a solution to the reflection equation (5.57)), one can dress it up to a representation in the total quantum space H.

• Second, this algebra is useful to define a commuting family of operators, making thus a link with integrable systems. Indeed, following [START_REF] Freidel | Quadratic algebras and integrable systems[END_REF] we can consider the dual algebra of (5.58), defined as the associative algebra generated by the elements of a matrix T `which satisfy the relations:

`At1t2 12 ˘´1 T `,1 ´`B t1 12 ˘´1 ¯t2 T `,2 " T `,2 ´`C t2 12 ˘´1 ¯t1 T `,1 `Dt1t2 12
˘´1
(5.62)

This algebra has been introduced in order to define the following boundary transfer matrices:

T pP 1 q " tr 1 tK `,1 pP 1 qT ´,1Q pP 1 qu (5.63)

with T ´,1Q a representation of the reflection algebra in the total quantum space, and K t1 `,1 a scalar representation of the dual algebra. Indeed, this defines a family of commuting operators: rT pP 1 q, T pP 1 qs " 0 (5.64)

for P 1 and P 1 two different sets of spectral parameters associated to the (auxiliary) space V 1 .

To understand where this commutativity and this dual equation come from, let us give some details of the procedure explained in [START_REF] Freidel | Quadratic algebras and integrable systems[END_REF]. The very efficient way used by the authors is a rewriting of the reflection algebra (5.58) using bi-vector indices: the first step is to consider the matrices T ´,i as elements of V i b V i with the notation T ´,i " T

t i 1
´,ii 1 . The convention is that any bold index is a vector index. It holds thus:

A 12 T t 1 1 ´,11 1 B 1 1 2 T t 2 1 ´,22 1 " T t 2 1 ´,22 1 C 12 1 T t 1 1 ´,11 1 D 1 1 2 1 (5.65)
This can be rewritten: R 11 1 ,22 1 T ´,11 1 T ´,22 1 " T ´,22 1 T ´,11 1 (5.66)

with the following matrix (assuming the invertibility of the involved operators): As by definition of the bi-vector index notations it holds K

R 11 1 ,22 1 " ´Ct 2 1 12 1 ¯´1 ´Dt 1 1 t 2 1 1 1 2 1 ¯´1 A 12 B t 1
t 11 1 `,11 1 T ´,11 1 " tr 1 K t1 `,1 T ´,1 (
, it is exactly the commutation (5.64) of (5.63).

The origin of the dual algebra (5.62) is thus clear, as in matrix notation the equation (5.68) simply reads (5.62).

Let us comment that Sklyanin was the first to introduce and prove the commutativity of the boundary transfer matrices using two algebras, in [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF], for the (fundamental) 6-vertex representation of the Yang-Baxter algebra. His proof is equivalent as the one in [START_REF] Freidel | Quadratic algebras and integrable systems[END_REF], but more technical as it remains in matrix notations. Note that also in the case of different spaces V 1 and V 2 a proof à-la Sklyanin of the transfer matrix commutativity can be still applied.

This set up lays the foundation of the next of the chapter, namely the use of this framework for mixed representations, i.e. spaces V 1 and V 2 non isomorphic. But before, let us end this section considering again the 6-vertex reflection algebra (2.8) of Sklyanin, under the light of this general reflection equation framework.

Reflection algebra associated to the 6-vertex R-matrix

In this paragraph we have a fresh look at the reflection algebra associated to the 6-vertex R-matrix, cf paragraph 2.2.1. In particular, we show that the reflection equations (2.8) and (2.9) define indeed (ABCD)-type quantum quadratic algebras for an appropriate choice of the automorphism φ describing the reflection.

Here, let 1 and 2 denote two quantum spaces of dimension 2. Following the work of Sklyanin [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF], we know an automorphism φ for which the reflection equation (5.57), associated to the 6-vertex Rmatrix, admits a non-trivial solution (and which leads, at the end of the day, to the fact that the algebra (5.58)-(5.61) and its dual (5.62) effectively describe the quantum XXZ spin chain with open boundaries conditions.). Denoting σ this automorphism, it consists in the following action on the associated spectral parameter λ :

σ : λ Ñ q λ (5.70)
The action will be indifferently denoted by its action on the operator or on the parameters: is an (ABCD)-quantum type algebra. We can also use the (ABCD)-quantum type algebra to understand the second algebra introduced by Sklyanin in [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF]. Indeed, taking the dual algebra T `from (5.62) Thus it is the second reflection algebra introduced by Sklyanin, cf (2.24) (up to the definition of T `with a transposition). Let K t0 `0pλq be a scalar representation of this dual reflection algebra (5.81). Then we immediately know that the boundary transfer matrix studied in chapter 4, given by (5.63), is a commuting family. Here, we add the upper-script p6V q to emphasise that this matrix is constructed from the 6-vertex R-matrix:

L σ0 0a pλ|P a |P C q " L 0a pλ σ0 |P a |P C q (5.
T p6V q pλ|P Q |P C q " tr 0 tK `0pλqU 0Q pλ|P Q |P C qu (

Mixed reflection equation

In this section, we start the study of mixed representations, i.e. the reflection equation (5.57) for two non isomorphic spaces 1 and 2.

We consider thus the reflection equation (5.57) for mixed representations, with a space V a » C p (quantum space) and a space V 0 » C 2 (auxiliary space). The Bazhanov-Stroganov Lax operator mixing both the representations, our starting point is thus the following mixed reflection equation, associated to THE REFLECTION ALGEBRA the known scalar matrix K 0 pλq (2.25) in the auxiliary space:

L θaσ0 a0 pP a |λ|P C qK 0 pλqL θa 0a pλ|P a |P C qK a pP a |P C q " K a pP a |P C qL σ0 a0 pP a |λ|P C qK 0 pλqL 0a pλ|P a |P C q (5.83)
Similarly to σ 0 , the operation θ a is a transformation on the parameters associated to the quantum space:

L θa 0a pλ|P a |P C q " L 0a pλ|P θa a |P C q, S θa ab pP a |P b |P C q " S ab pP θa a |P b |P C q (5.84)
which leaves λ and P C unchanged, and which is consistent with the Yang-Baxter equations. That is we can apply θ a to the Yang-Baxter equation (1.12) and (5.1) leaving them satisfied.

A solution to this equation, i.e. both an automorphism θ a and a matrix K a pP a |P C q in the quantum space, is a first step toward the computation of the fundamental boundary transfer matrix. This one will serve to derive the local Hamiltonian, and the matrix K a pP a |P C q will take part in the description of the boundaries of the system.

An interesting point of view is to consider that the matrix K a pP a |P C q can be seen as a scalar representation of the following algebra, generated by the elements of a matrix T ´satisfying:

L θaσ0 a0 pP a |λ|P C qT ´0pλqL θa 0a pλ|P a |P C qT ´apP a q " T ´apP a qL σ0 a0 pP a |λ|P C qT ´0pλqL 0a pλ|P a |P C q (5.85)
In order to interpret (5.85) as an (ABCD)-quantum type algebra, the operators L 0a and L a0 must satisfy the Freidel-Maillet compatibility, cf relations (15) of [START_REF] Freidel | Quadratic algebras and integrable systems[END_REF].

Proposition 5.3.1. Let us take parameters such that a set P C ensures the existence of an intertwiner (5.1). Then, once we do the identification:

L a0 pP a |λ|P C q " L ´1 0a pλ|P a |P C q (5.86)
the algebra (5.85) is an (ABCD)-quantum type algebra.

Proof. To show the proposition, only the compatibility conditions have to be checked. So one needs to verify the eight relations (15) of [START_REF] Freidel | Quadratic algebras and integrable systems[END_REF], as (5.59), when the third space (labelled 3) is of dimension 2 and also when the third space is of dimension p. Let 0 and 0 1 be two indexes denoting two spaces of dimension 2, V 0 and V 0 1 , and let a and b be two indexes denoting two spaces of dimension p, V a and V b .

The proof relies on the following identifications:

A 00 1 " R σ 0 1 σ0 0 1 0 ; A 0a " L σ0θa a0
and A ab " S θ b θa ba (5.87)

B 00 1 " R σ 0 1 00 1 ; B 0a " L θa 0a and B ab " S θ b ab (5.88) C 00 1 " R σ0 0 1 0 ; C 0a " L σ0 a0
and C ab " S θa ba (5.89)

D 00 1 " R 00 1 ; D 0a " L 0a and D ab " S ab (5.90)
where R, L and S are of course respectively the 6-vertex R-matrix, the Bazhanov-Stroganov Lax operator and the fundamental R-matrix. Accordingly to the spaces, the automorphisms are φ 0 " σ 0 on twodimensional spaces and φ a " θ a on p-dimensional spaces. With this identification, when the third space is taken bi-dimensional (and denoted 0 1 ), the equation (5.59) can be written:

L θaσ0 a0 R σ 0 1 σ0 0 1 0 L σ 0 1 θa 0 1 a " L σ 0 1 θa 0 1 a R σ 0 1 σ0 0 1 0 L θaσ0 a0 (5.91)
Using the fact that σ and θ are automorphisms, it must hold:

L a0 R 0 1 0 L 0 1 a " L 0 1 a R 0 1 0 L a0 (5.92)
so that the identification of the proposition is well-posed, using the mixed Yang-Baxter equation (1.12).

MIXED REFLECTION EQUATION

The same holds for all the other relations: when the third space is taken bi-dimensional, the equation (1.12) ensures the compatibility.

When the third space is taken p-dimensional (and denoted b), the equation (5.59) is written:

L θaσ0 a0 L θ b σ0 b0 S θ b θa ba " L θaσ0 b0 L θaσ0 b0 S θ b θa ba (5.93)
Then using the identifications introduced, this equation is equivalent to the intertwining relation (5.1).

The same holds for all the other conditions: when the third space is taken p-dimensional, the equation (5.1) ensures the compatibility.

Let us note that this compatibility condition is also imposed to be able to dress scalar solutions of (5.83) by Yang-Baxter solutions, in order to get new solutions of (5.83), i.e. non scalar representations of (5.85).

Choice of the automorphism θ and symmetries

In this paragraph, we define an automorphism θ describing the reflection in the cyclic space.

According to the fact that P C must be invariant under the action of θ, we restrict ourself to look for solutions to the above problem once we impose one condition on the parameters P C .

In particular, we impose:

L " ´M for parameters of point A) of proposition 5. In this context we can define the local transformation θ a , acting in space V a , as it follows:

θ a : pα a , β a , γ a , δ a , a a , b a , c a , d a q Ñ p´γ a , ´δa , ´αa , ´βa , a a , b a , c a , d a q (5.97)
or equivalently, using the parametrisation with points in C 4 :

θ a : px Qa , x Ra , y Qa , y Ra , σ a q Ñ ˆq2 x Ra , q 2 x Qa , 1 y Ra , 1 y Qa , σ a x Qa x Ra q 2 y Qa y Ra ˙(5.98)
This acts only on the local parameters P a of the space V a while leaving the parameters λ and P C unchanged. In particular S θa ba , S θ b ba and S θ b θa ba exist when S ba exists. Moreover, the 6-vertex consistency conditions:

α a γ a " a a c a and β a δ a " b a d a (5.99) are invariant under θ a so that the action of θ a on the Yang-Baxter equations leaves all them satisfied.

Remark. There is a point that could be investigated further: the existence of an automorphism θ which does not require (5.94)-(5.96) and which reduces to our definition of θ when these conditions are satisfied.

If there exists such an automorphism, the results we will obtain should remain true in a wider class of parameters. It is worth to mention anyhow that the conditions (5.94)-(5.96) correspond to still quite general 6-vertex cyclic representations which contains as specific representations those associated to the chiral Potts, the sine-Gordon model and the XXZ spin chain at the root of units.

Symmetries and rewritings

Following from the definitions of σ 0 and θ a , by direct computation one can show:

L θaσ0 a0 pP a |λ|P C q " ´L0a pλ|P a |P C q (5.100)

Let us comment that, using the identifications (5.87) and (5.88), the dressing of a scalar representation K 0 pλq of the reflection algebra (5.58) by the operators A 0n " ´L0n pλ|P n |P C q and B 0n " L ´1 0n pq{λ|P n |P C q THE REFLECTION ALGEBRA (for V 0 a bi-dimensional space and V n a local quantum space of dimension p) leads to the construction of the boundary transfer matrix U 0Q pλ|P Q |P C q. Moreover, we have the following identity:

Proposition 5.3.2. When it exists, the fundamental R-matrix satisfies:

S θ b θa ba pP b |P a |P C q " S ab pP a |P b |P C q (5.101)
Proof. Looking to the mixed Yang-Baxter equation (5.1), and using (5.100), we can prove that it holds:

S θ b θa ba pP b |P a |P C q 9 S ab pP a |P b |P C q (5.102)
In fact, looking to the definition (5.17) for the intertwiner S ba , one can prove by calculations that the proportionality coefficient is one.

Let us use now the symmetry outlined above to rewrite the mixed reflection equation (5.83). This can be put under the form:

L 0a pλ|P a |P C qK 0 pλqL σ0 a0 pP a |λ|P C qK a pP a |P C q " K a pP a |P C qL σ0 a0 pP a |λ|P C qK 0 pλqL 0a pλ|P a |P C q (5.103)
or equivalently:

L 0a pλ|P a |P C qK 0 pλqL θa 0a pλ|P a |P C qK a pP a |P C q " K a pP a |P C qL θa 0a pλ|P a |P C qK 0 pλqL 0a pλ|P a |P C q (5.104)
These symmetries are of prime interest, as we are now able to show that these equations contain the equations obtained by the fusion procedure (cf paragraph 2.3.3):

The fusion procedure as a particular case

Let us rewrite the mixed equation proposed by Nepomechie et al for the fusion case [START_REF] Mezincescu | Fusion procedure for open chains[END_REF]. Using the symmetry of the 6-vertex R-matrix, it can be rewritten in the form (x12y denotes the fused space, of dimension 3): R 0x12y pλ|P x12y qK 0 pλqR σ0 x12y0 pP x12y |λqK x12y pP x12y q " K x12y pP x12y qR σ0 x12y0 pP x12y |λqK 0 pλ|ξqR 0x12y pλ|P x12y q (5.105)

Here we have done the identifications: P x12y " µ and R 0x12y pλ|P x12y q " R 0x12y pλ{µq (5.106) and σ0 is defined, on the space 0, by: σ0 : λ Ñ 1{λ (5.107)

Let us observe that the 6-vertex cyclic representations corresponding to these fusion procedure are contained in point D) of proposition 5.1.4. It is now easy to remark that a part the shifted definition of the σ0 with respect to σ 0 , the equation proposed by Nepomechie et al [START_REF] Mezincescu | Fusion procedure for open chains[END_REF] is a particular case of the equation p5.103q and so it is equivalent to the reflection equation p5.83q.

Let us comment that the shift in question is due to a different choice of convention. Indeed, in [START_REF] Mezincescu | Fusion procedure for open chains[END_REF] they base on the reflection algebra T of Sklyanin (equation ( 12) of [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF]) while in this thesis we consider the shifted version, the algebra U (equation (35) of [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF]).

Solutions of the mixed reflection equation

The aim of this paragraph is to compute solutions of the mixed reflection equations. In a first part, we show how to dress a known scalar solution K a pP a |P C q to get a solution in the whole quantum space.

MIXED REFLECTION EQUATION

Then, we explicitly find a scalar solution K a pP a |P C q, associated to a diagonal boundary matrix in the space V 0 . As we have already seen, the knowledge of such matrices are of prime interest, as they will allow to describe integrable boundaries for the models associated to the 6-vertex cyclic representations.

Dressing of solutions

Let us suppose that one knows a matrix K a pP a |P C q solution of the mixed reflection equation. In this short part, we dress this boundary matrix to get a solution of the mixed reflection equation in the whole quantum space. Later on, subscripts n P 1, N will refer to the local quantum spaces V n » C p . Proposition 5.3.3. Let us define the fundamental boundary monodromy matrix:

V aQ pP a |P Q |P C q " S θ Q θa Qa pP Q |P a |P C qK a pP a |P C qS θ Q aQ pP a |P Q |P C q (5.108)
where we have defined:

S θ Q θa Qa pP Q |P a |P C q " S θ N θa N a pP N |P a |P C q ¨¨¨S θ1θa 1a pP 1 |P a |P C q (5.109)
and

S θ Q aQ pP a |P Q |P C q " S θ1 a1 pP a |P 1 |P C q ¨¨¨S θ N aN pP a |P N |P C q (5.110)
Then, it satisfies the following mixed reflection equation:

L θaσ0 a0 pP a |λ|P C qU 0Q pλ|P Q |P C qL θa 0a pλ|P a |P C qV aQ pP a |P Q |P C q " V aQ pP a |P Q |P C qL σ0 a0 pP a |λ|P C qU 0Q pλ|P Q |P C qL 0a pλ|P a |P C q (5.111)
where U 0Q pλ|P Q |P C q is the boundary monodromy matrix (studied in chapter 4).

Proof. The proof is a direct consequence of proposition 5.2.1, using the identifications (5.87) and (5.88).

The next proposition states the unicity of the solutions: Proposition 5.3.4. Up to a normalisation the following statements are true: i) there exists at most one K a pP a |P C q solution to the corresponding mixed reflection equation (5.83) once K 0 pλq is fixed. ii) if the K a pP a |P C q associated to K 0 pλq exists, then V aQ pP a |P Q |P C q is the only solution of (5.111) once U 0Q pλ|P Q |P C q is fixed. Moreover, if the normalisation can be fixed such that:

K a pP θa a |P C q " K ´1 ´a pP a |P C q (5.112)
then it follows :

V θa aQ pP a |P Q |P C q " ρ aQ pP a |P Q |P C qρ aQ pP a |P θ Q Q |P C qV ´1 aQ pP a |P Q |P C q (5.113)
with the scalar function:

ρ aQ pP a |P Q |P C q " N ź k"1 ρ ak pP a |P k |P C q (5.114)
We recall that the function ρ ak pP a |P k |P C q have been introduced to deal with the unitarity and crossingunitarity of the fundamental R-matrix.

Proof. The proof of the proposition is based on the following lemma:

Lemma 5.3.5 (Center of the Reflection Algebra). The centre of the reflection algebra is proportional to the identity.

This one is easily proven using the basis (4.127) for the Hilbert space H and the simplicity of pseudospectrum of the operator Bpλ|α, βq. Then, from the reflection equation (5.83) (respectively (5.111)), THE REFLECTION ALGEBRA one can show that K a pP a |P C qK a pP θa a |P C q (respectively V aQ pP a |P Q |P C qV θa aQ pP a |P Q |P C q) is central with respect to the reflection algebra associated to U 0taQu pλ|P a , P Q |P C q, the dressed boundary monodromy matrix on a chain with N `1 quantum sites, the last site coinciding with V a . The relation (5.113) follow from the appropriate choice of normalisation.

Scalar solutions to the mixed reflection equations

Let us compute now an explicit scalar solution of the mixed reflection equation. In this paragraph, we restrict to the case of a diagonal K 0 pλq solution of the original 6-vertex reflection equation, i.e. a matrix K 0 pλ|ξ, 0, 0q given by (2.25). For this type of boundaries, we could construct the unique associated scalar solution in the quantum space of the mixed reflection equation. The following proposition holds: Proposition 5.3.6. Let P C be a set of constant parameters such that the fundamental R-matrix S aQ pP a |P Q |P C q exists and let us denote P a the arbitrary parameters of the cyclic representation in the space V a . Then the mixed reflection equation (5.83) admits the following non-trivial solution:

• In the framework of points A) and B) of proposition 5.1.4, the boundary matrix K a pξ|P a |P C q is diagonal in the v-basis (cf. (1.77)), and its elements satisfy:

xh `1| vK a pξ|P a |P C q|h `1y v xh| vK a pξ|P a |P C q|hy v " ´γa q h`1{2 `δa q k´ph`1{2q α a q h`1{2 `βa q k´ph`1{2q @h P t0, ..., p ´2u (5.115)

This solution exists if and only if the boundary parameter ξ satisfies:

δ p a {ξ 2p ´γp a α p a ´βp a {ξ 2p " 1
(5.116)

• In the framework of points C) and D) of proposition 5.1.4, satisfying moreover the identities:

b a " ´q2j´1 a a , d a " ´q2j´1 c a (5.117)

for a given j P t0, ..., p ´1u , it holds for any value of ξ that the boundary matrix K a pξ|P a |P C q is diagonal in the v-basis and its element satisfy:

xh `1| vK a pξ|P a |P C q|h `1y v xh| vK a pξ|P a |P C q|hy v " ´γa q h`1{2 ´δa q ´ph`1{2q {ξ 2 α a q h`1{2 ´βa q ´ph`1{2q {ξ 2 @h P tj, ..., p `j ´2u (5.118)

Proof. The idea is to consider the reflection equation (5.83) as four equations in the auxiliary space of dimension 2, with elements in the quantum space V a . The fact that K a pP a |P C q is diagonal (in the v-basis) follows from the fact that K 0 pλq is diagonal, and from the equations associated to the elements p1, 1q and p2, 2q of the auxiliary space. Then, looking for diagonal solutions, we find the result of the proposition. Let us comment that the condition (5.116) is a compatibility condition, needed to ensure the periodicity of the solution. For nilpotent models, no condition is required.

Remark. Let us mention that the scalar solutions of the reflection equation hold when the set P C is general, namely the extra constraints (5.94)-(5.96) are not necessary. For further applications, we will nevertheless have to impose these constraints, leading to the following simplification of the compatibility condition (5.116): the boundary parameter ξ must satisfy ξ 2 " ´q´k , for some k P t0, ..., p ´1u.

Thanks to the previous proposition, we can compute diagonal boundary matrices in the v-basis. Let us mention that the existence and the derivation of non diagonal solutions is currently under examination as it can lead to describe more general boundaries for these classes of representations.

The dual mixed reflection equation

In this paragraph, we consider the dual mixed reflection equation. As we have already seen, it allows to define a family of commuting operators. First we rewrite this equation in a more handleable way, and 5.3. MIXED REFLECTION EQUATION then we find diagonal scalar solutions similarly to the original equation, i.e. associated to a diagonal scalar boundary matrix K `0pλq.

For a general cyclic representation, the mixed dual reflection equation (5.62) associated to the most general scalar solution K `0pλq (2.27) of the 6-vertex dual reflection equation reads:

pL ´1 0a pλ|P a |P C qq t0ta K `0pλq " `Lt0 0a pλ|P θa a |P C q ˘´1 ı ta K `apP a |P C q " K `apP a |P C q " `Lt0 0a pλ|P θa a |P C q ˘´1 ı ta K `0pλqpL ´1 0a pλ|P a |P C qq t0ta (5.119)
In order to solve the problem to compute the scalar solution K `apP a |P C q, we show the following rewriting: Proposition 5.3.7. The dual mixed reflection equation (5.119) is equivalent to the equation:

K ta `apP a |P C qL 0a p1{qλ|P a |P C qK t0 `0pλqL ´1 0a pλ|P a |P C q " L ´1 0a pλ|P a |P C qK t0 `0pλqL 0a p1{qλ|P a |P C qK ta `apP a |P C q (5.120)
Proof. The equivalence is shown with some simple computations, using the transposition on the different spaces.

The key is that we can show

" `Lt0 0a pλ|P θa a |P C q ˘´1 ı t0 9 L 0a p1{qλ|P a |P C q (5.121)
which is proven also using the definition of the quantum determinant (1.46).

Scalar solutions to the dual mixed reflection equations

As before, in this part we restrict to the case of a diagonal K `0pλq solution of the dual 6-vertex reflection equation, i.e. a matrix K `0pλ|ξ, 0, 0q given by (2.27). For this type of boundaries, we construct the associated scalar solution in the quantum space of the dual mixed reflection equation. The following proposition holds: Proposition 5.3.8. Let P C be a set of constant parameters such that the fundamental R-matrix S aQ pP a |P Q |P C q exists and let us denote P a the arbitrary parameters of the cyclic representation in the space V a . Then the dual mixed reflection equation (5.119) admits the following non-trivial solution:

• In the framework of points A) and B) of proposition 5.1.4, the matrix K `apξ|P a |P C q is diagonal in the v-basis and its elements satisfy:

xh `1| vK `apξ|P a |P C q|h `1y v xh| vK `apξ|P a |P C q|hy v "
´αa q h´1{2 `βa q k´ph´1{2q γ a q h`3{2 `δa q k´ph`3{2q @h P t0, ..., p ´2u (5.122)

The solution exists if and only if the boundary parameter ξ satisfies:

β p a ξ 2p ´αp a γ p a ´δp a ξ 2p " 1 (5.123)

• In the framework of points C) and D) of proposition 5.1.4, satisfying moreover the identities:

b " ´q2j´1 a, d " ´q2j´1 c (5.124)

for a given j P t0, ..., p ´1u , it holds for any value of ξ that the boundary matrix K `apξ|P a |P C q is diagonal in the v-basis and its element satisfy:

xh `1| vK `apξ|P a |P C q|h `1y v xh| vK `apξ|P a |P C q|hy v " ´αa q h´1{2 ´βa q ´ph´1{2q ξ 2 γ a q h`3{2 ´δa q ´ph`3{2q ξ 2 @h P tj, ..., p `j ´2u (5.125) THE REFLECTION ALGEBRA Proof. The proof follows the same line as the one dedicated to the proof of the mixed reflection equation, using the form (5.121) for the equation. A similar condition on the boundary parameter ξ is needed to ensure the periodicity.

Remark. As for the reflection equation, let us mention that the scalar solutions of the dual reflection equation hold when the set P C

is general, namely the extra constraints (5.94)-(5.96) are not necessary. However, it is important to note that when these ones are imposed, the condition (5.123) reduces to the same condition as for the reflection equation, namely ξ has to satisfy: ξ 2 " ´q´k , for some k P t0, ..., p ´1u.

Remark. In the same way as for the original mixed reflection equation, let us comment that the question of the existence and expression of more general (non-diagonal in the v-basis) boundaries is still under examination. Nevertheless in the next, K a pP a |P C q and K `apP a |P C q will denote as before, general scalar solutions of the mixed reflection equation and its dual, while K a pξ|P a |P C q and K `apξ|P a |P C q will stand for the explicit diagonal scalar solutions given in propositions 5.3.6 and 5.3.8. We keep these general matrices just to emphasise that the results that we will present in the next for the fundamental transfer matrix are not related to the specific form of the boundary matrices but just to the fact that they are solutions of the reflection equations.

The fundamental transfer matrix

We can now define the fundamental transfer matrix (5.63) for cyclic representations of the 6-vertex Yang-Baxter algebra associated to Bazhanov-Stroganov Lax operators:

T pf undq pP a |P Q |P C q " tr a tK `apP a |P C qV aQ pP a |P Q |P C qu (5.126)
Here K ta `apP a |P C q is a scalar solution of the mixed dual equation, and V aQ pP a |P Q |P C q is the fundamental boundary monodromy matrix defined by (5.108). In this definition, the space V a can be considered as an auxiliary space, but this time of dimension p. Thus the set P a of two parameters can be considered as a set of spectral parameters for the fundamental transfer matrix.

Using the fact that the mixed reflection algebra and its dual are (ABCD)-quantum type algebras, it immediately holds the following proposition which is at the heart of the computation of local Hamiltonians commuting with the 6-vertex boundary transfer matrix: Proposition 5.3.9. Let P C be a set of constant parameters ensuring the existence of the fundamental R-matrix S aQ pP a |P Q |P C q and satisfying the associated constraints (5.94), (5.95) or (5.96), and let P a be the set of two spectral parameters on the space V a . Then it holds:

rT pf undq pP a |P Q |P C q, T p6V q pλ|P Q |P C qs " 0 (5.127)
Let us comment that this commutativity can also be proved by a direct computation, using an approach à-la Sklyanin. Moreover, to compute local Hamiltonians which commute between themselves, we will use the essential next proposition: Proposition 5.3.10. Let P C be a set of constant parameters ensuring the existence of the fundamental R-matrix S aQ pP a |P Q |P C q and satisfying the associated constraints (5.94), (5.95) or (5.96), and let P a and P 1 a be two sets of spectral parameters on the space V a . Then it holds:

rT pf undq pP a |P Q |P C q, T pf undq pP 1 a |P Q |P C qs " 0 (5.128)
Proof. The proof is based on the simplicity of the spectrum of T p6V q pλ|P Q |P C q, which has been derived in chapter 4. Indeed, let |τ y be an eigenvector of T p6V q pλ|P Q |P C q:

T p6V q pλ|P Q |P C q |τ y " τ pλ|P Q |P C q |τ y (5.129)
with τ pλ|P Q |P C q the associated eigenvalue. Thanks to proposition 5.3.9 and the mentioned simplicity, we can set

T pf undq pP a |P Q |P C q |τ y " τ pP a |P Q |P C q |τ y (5.130)
with τ a scalar function. Then, using the proposition 4.3.9, we know that for almost all the parameters the set t|τ yu of all the eigenvectors forms a basis of the Hilbert space. Thus the equation (5.128) holds for almost all the values of the parameters. Observing that the fundamental transfer matrix T pf undq pP a |P Q |P C q is a rational fraction in all the parameters, then the left hand side of (5.128) is a continuous functions of the parameters zero almost for every values of the parameters and so it is identically zero and (5.128) is proven. This fundamental transfer matrix family plays a crucial role in the determination of local Hamiltonians for models associated to cyclic representations, it is the purpose of paragraph 5.4 to explain how. Let us early mention that the main point to understand this is the fact that we could compute a fundamental transfer matrix which is similar to the standard one T p6V q , but which has an auxiliary space of dimension p. Thus the auxiliary and quantum spaces are isomorphic for this matrix and the hope, to compute local Hamiltonians, is to be able to use a similar standard computation than the one dedicated to fundamental models (cf paragraph 2.3.2). Before to end this section, we emphasise on the link between the mixed reflection equation and its dual (for the particular diagonal boundary matrices), and we eventually give a reflection equation at a cycliccyclic level. As we will see, this latter point has to be investigated more (this work is under consideration) to be able to deal with an (ABCD)-quantum type algebra at a cyclic-cyclic level.

The dual mixed reflection equation from the mixed reflection equation

In this paragraph, we make explicit the link between the mixed dual reflection equation and its dual. First, from the expression for the K a pξ|P a |P C q and K `apξ|P a |P C q boundary matrices, cf. propositions 5.3.6 and 5.3.8, it follows the corollary: Corollary 5.3.11. The scalar solution K `apξ|P a |P C q of the dual reflection equation can be written in terms of the original K a pξ|P a |P C q as it follows:

K `apξ|P a |P C q " K a p1{ξ|P θa a |P C q (5.131)
where we make use of the automorphism θa acting on the parameters as:

θa : pα a , β a , γ a , δ a , a a , b a , c a , d a q Ñ p´qγ a , ´δa {q, ´αa {q, ´qβ a , a a , b a , c a , d a q (5.132)
Using the parametrisation with points in C 4 , we have:

θa : px Qa , x Ra , y Qa , y Ra , σ a q Ñ ˆq3 x Ra , q 3 x Qa , q y Ra , q y Qa , σ a x Qa x Ra q 2 y Qa y Ra ˙(5.133)
This property is the equivalent of the link (2.26) between the boundary matrices on the auxiliary space V 0 of dimension 2. As a consequence, we can show the next proposition: Proposition 5.3.12. The original mixed reflection equation (5.83) for the diagonal scalar boundary matrix K a pξ|P a |P C q reduces to the dual one (5.119) for the diagonal scalar boundary matrix K `ap1{ξ|P a |P C q under the transformation σ0 θa , where σ0 is the following involutive automorphism on the auxiliary space:

σ0 : λ Ñ 1 λ (5.134) THE REFLECTION ALGEBRA
The proof directly follows from the action of σ0 on K 0 pλ|ξq:

K σ0 0 pλ|ξq " K `0pλ|1{ξq (5.135)
and the following two identities:

L 0a pλ|P θa a |P C q " L 0a pλq|P θa a |P C q " ´L´1 0a p1{λ|P a |P C q (5.136)
and L 0a pλ| `P θa a ˘θa |P C q " L 0a pλ{q|P a |P C q (5.137)

We end this section by giving some properties related to the operation θ, allowing to write differently the equations. By a proof similar to the one of proposition 5.3.2, one easily shows:

S θa θb ab pP a |P b |P C q " S ba pP b |P a |P C q (5.138)
Moreover, the three operations θ, θ and Υ (introduced in (5.50)) are linked via the obvious identity:

θ θ " θθΥ ´1Υ ´1 (5.139)
Then, in order to write the dual mixed reflection equation in a form similar to the original one, we can introduce the operations: φ 0 " σ0 σ 0 σ0 and Ψ a " θa θ a θa (5.140)

The dual mixed reflection equation can be written:

L φ0Ψa 0a pλ|P a |P C qK `0pλ|ξqL Ψa a0 pλ|P a |P C qK `apξ|P a |P C q " K `apξ|P a |P C qL φ0 0a pλ|P a |P C qK `0pλ|ξqL a0 pλ|P a |P C q (5.141)
Thanks to this form, we can state the following:

Proposition 5.3.13 (Equations at a cyclic-cyclic level). Let K a pξ|P a |P C q and K b pξ|P b |P C q be the two diagonal solutions of the mixed reflection equation (5.83). Then they satisfy the following cyclic-cyclic reflection equation:

cpP a , P b , P C q S θ b θa ba pP b |P a |P C qK a pξ|P a |P C qS θ b ab pP a |P b |P C qK b pξ|P b |P C q " K b pξ|P b |P C qS θa ba pP b |P a |P C qK a pξ|P a |P C qS ab pP a |P b |P C q (5.142)
The dual matrices K `apξ|P a |P C q and K `bpξ|P b |P C q, solutions of the dual mixed reflection equation (5.141), satisfy:

cpP θa a , P θb b , P C qS ΨaΨ b ab pP a |P b |P C qK `apξ|P a |P C qS Ψ b ba pP b |P a |P C qK `bpξ|P b |P C q " K `bpξ|P b |P C qS Ψa ab pP a |P b |P C qK `apξ|P a |P C qS ba pP b |P a |P C q (5.143)
The scalar cpP a , P b , P C q satisfies, for any sets of spectral parameters P a and P b , the following properties: cpP a , P b , P C q " 1 cpP b , P a , P C q " 1 cpP θa a , P b , P C q " 1 cpP a , P θ b b , P C q and cpP a , P a , P C q " 1 (5.144)

The equation (5.143) can be equivalently written:

`S´1 ab pP a |P b |P C q ˘tatb K `apξ|P a |P C q " `Sta ab pP θa a |P b |P C q ˘´1 ı t b K `bpξ|P b |P C q " cpP θa a , P θb b , P C qK `bpξ|P b |P C q " `Sta ab pP θa a |P b |P C q ˘´1 ı t b K `apξ|P a |P C q `S´1 ab pP a |P b |P C q ˘tatb (5.145)
where t a and t b are the transposition on spaces a and b respectively, given in the û-basis.

As it will be shown in paragraph 5.5, we can explicitly compute this scalar for p " 3, and it indeed reduces to one. The proof that this is the case for any value of p odd is currently under consideration.

Remark. The interest in this proposition is that, with dressed solutions of these cyclic-cyclic reflection equations, one could provide a direct proof of the commutativity of the fundamental transfer matrices between themselves, with the standard procedure induced by the (ABCD)-quantum type algebras (see (5.69)).

In this way it would avoid the indirect argument that we have used, based on the simplicity of 6-vertex fundamental transfer matrix.

Expression of local Hamiltonians

Now that the reflection equations have been studied, and that we explicitly know some boundary matrices, we can use them and tackle the problem to define integrable local Hamiltonians associated to cyclic representations of the 6-vertex reflection equation.

Let us consider the fundamental transfer matrix associated to the diagonal boundary scalar matrices, with boundary parameters denoted ξ ´and ξ `:

T pf undq pP a |P Q |P C q " tr a tK `apξ `|P a |P C qV aQ pP a |P Q |P C qu (5.146) with V aQ pP a |P Q |P C q " S θ Q θa Qa pP Q |P a |P C qK a pξ ´|P a |P C qS θ Q aQ pP a |P Q |P C q (5.147)
Making use of (5.146), the standard procedure in order to define local Hamiltonians consists in taking its (first order) derivative with respect to the spectral parameters, in a particular value (see paragraph 2.3.2). Indeed, for the fundamental transfer matrix, the auxiliary space V a and the local quantum spaces V n are isomorphic so that one can try to derive local Hamiltonians following the procedure introduced by Sklyanin in [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF].

In this section we show that for cyclic representations, the first order derivative does not produce non-trivial local Hamiltonians. However, one can compute the second order derivative and obtain from it the local Hamiltonians. At the end, we get a model of interactions where the two boundary matrices K a pξ ´|P a |P C q and K `apξ `|P a |P C q effectively describe the left and right boundaries of a chain, and where the nearest neighbour interactions are given thanks to derivatives of the fundamental R-matrix.

First order derivative of the fundamental transfer matrix

Following Sklyanin's procedure [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF], an important property is needed to construct local Hamiltonians from the fundamental transfer matrix: there must exist at least one set of spectral parameters such that the boundary matrix K a pP a |P C q reduces to the identity and such that the fundamental R-matrix reduces to the permutation operator. This property is satisfied for the models under consideration:

• The matrices Kpξ|P a |P C q and K `pξ|P a |P C q reduce to the identity when the spectral parameters P a belong to certain hypersurfaces. Indeed, let us introduce the sets:

P

´" P a " px Qa , x Ra , y Qa , y Ra , σ a q | x Qa x Ra " q 2 and y Qa y Ra " 1 ( (5.148) and P `" P a " px Qa , x Ra , y Qa , y Ra , σ a q | x Qa x Ra " 1{q 2 and y Qa y Ra " q 2 ( (5.149)

Then we have, by choosing the appropriate normalisation for the boundary transfer matrices, that for any complex parameter ξ it holds: @P á P P ´, Kpξ|P á |P C q " 1 and @P à P P `, K `pξ|P à |P C q " 1 (5.150)

• From the Yang-Baxter equation, we directly have that for any set P of spectral parameters:

S ab pP |P |P C q " P ab (5.151) THE REFLECTION ALGEBRA where P ab denotes the permutation operator between the spaces V a and V b .

Moreover, from the definition of the action of θ on the parameters, it is clear that: @P ´P P ´, S θa ab pP ´|P ´|P C q " P ab (5.152) Definition 5.4.1 (Homogeneous chain). Let P be a set of spectral parameters. A chain is said homogeneous when, on every site n, the quantum parameters P n are given by P n " P . For brevity, we will simply denote this by P Q " P , where the meaning is @n P 1, N , P n " P .

Let P ´P P ´, and let us denote px Q , x R , y Q , y R , σq its (non-independent) components. The standard computation to get local Hamiltonians is to calculate the first derivative of the fundamental transfer matrix in the homogeneous limit, i.e. where the chain is homogeneous (P Q " P ´) and where the spectral parameters are evaluated in the same values (P a " P ´). Indeed, one can then extract from the obtained formula the expression of Hamiltonians which commute with the fundamental transfer matrix (see paragraph 2.3.2).

So let x a be a component of the set of spectral parameters P a . We aim to compute: The first derivative in an homogeneous limit P ´P P ´of the fundamental transfer matrix has the following expression:

d T pf undq pP a |P Q |P C q dx a
d T pf undq pP a |P Q |P C q dx a ˇˇˇP a"P ṔQ "P ´" tr a # dK `apξ `|P a |P C q dx a ˇˇˇP a "P ´+ `tr a K `apξ `|P ´|P C q ( N ´1 ÿ k"1 H pxq k,k`1 `tr a K `apξ `|P ´|P C q ( dK 1 pξ ´|P 1 |P C q dx 1 ˇˇˇP 1 "P ´`tr a ! K `apξ `|P ´|P C qH pxq N a ) (5.154)
The local interactions are given by: The upper-script pxq emphasises on the fact that we derive in the direction x a .

H pxq k,k`1 " dS k`1,k pP k`1 |P k |P C q dx k`1 ˇˇˇP k`1 "P Ṕk "P ´Pk,k`1 `Pk,k`1 dS k`1,k pP k`1 |P k |P C q dx k`1 ˇˇˇP k`
Proof. The proof is a direct computation. We have: Then using the properties (5.151) and (5.152), we have V aQ pP ´|P ´|P C q " 1. Indeed, the unitarity in the point P ´gives ρpP ´|P ´|P C q " 1.

d T pf undq pP a |P Q |P C q dx a ˇˇˇP a "P ṔQ "P ´" tr a # dK `apξ `|P a |P C q dx a ˇˇˇP a "P ´VaQ pP ´|P ´|P C q + `tr a $ ' ' & ' ' % K `apξ `|P ´|P C q dV aQ pP a |P Q |P C q dx a

EXPRESSION OF LOCAL HAMILTONIANS

For the other term, it holds:

dV aQ pP a |P Q |P C q dx a ˇˇˇP a "P ṔQ "P ´" dS θaθ Q Qa pP Q |P a |P C q dx a ˇˇˇˇP a"P ṔQ "P ´Ka pξ ´|P ´|P C q S θ Q aQ pP ´|P ´|P C q (5.158) `Sθaθ Q Qa pP ´|P ´|P C q dK a pξ ´|P ´|P C q dx a ˇˇˇP a "P ´S θ Q aQ pP ´|P ´|P C q (5.159) `Sθaθ Q Qa pP ´|P ´|P C q K a pξ ´|P ´|P C q dS θ Q aQ pP a |P Q |P C q dx a ˇˇˇˇP a "P ṔQ "P ´(5.160)
Each one is now calculated using the reduction to the permutation operator in the special value P ćonsidered here.

Two comments are required. The first one concerns the interactions H pxq k,k`1 . Indeed, let us introduce the following notation : (5.161)

h pxq k,k`1 " dS k`1,k pP k`1 |P k |P C q dx k`1 ˇˇˇP k`
Then the local interactions can be written:

H pxq k,k`1 " h pxq k,k`1 `hpxq k`1,k and H pxq N a " h pxq N a `hpxq aN (5.162)
Let us early mention that within this formalism, the bulk Hamiltonian for a chain described by the reflection algebra (so with integrable boundaries) is a symmetrised version of the bulk Hamiltonian obtained for the same chain described by the Yang-Baxter algebra (with quasi-periodic boundary conditions):

´Hreflection bulk ¯pxq k,k`1 " h pxq k,k`1 `hpxq k`1,k while ´HY ang´Baxter bulk ¯pxq k,k`1 " h pxq k,k`1 (5.163)
We shall come back later to the impact of the boundaries on the bulk interactions, for some specific models in paragraph 5.5.

The second comment concerns the Hamiltonian itself. Indeed, one is tempted to divide by the quantity tr a tK `apξ `|P ´|P C qu and thus to obtain an Hamiltonian which expression is extremely similar to the one derived in (2.43), paragraph 2.3.2. However, as it is explained in the following paragraph, this cannot be done when one considers cyclic representations.

Peculiarities for cyclic representations

As already mentioned, for cyclic representations some technical difficulties emerge due to the fact that q is a p th root of unity. In particular, the following proposition holds: Proposition 5.4.2. For cyclic representations (associated to diagonal boundary matrices), the first order derivative of the fundamental transfer matrix reduces to a scalar multiple of the identity:

@P ´P P ´, d T pf undq pP a |P Q |P C q dx a ˇˇˇP a"P ṔQ "P ´9 1 (5.164)
As a consequence, the proof that the local Hamiltonians commute with the boundary transfer matrix cannot be done using this derivative. THE REFLECTION ALGEBRA The aim of this paragraph is to give the main steps of the proof of the proposition. It relies on the propositions 5.4.3 and 5.4.5. One important property is given by the corollary 5.4.4, stating that the trace of the boundary matrix tr a tK `apξ `|P ´|P C qu is zero. In particular, it explains that we cannot define local Hamiltonians of the form (2.43). The proofs of the propositions 5.4.3 and 5.4.5 are based on the fact that in the special homogeneous limit P ´considered, we can give an explicit and handleable formula of the right and left boundary matrices, in terms of the operators v. Indeed, in this point, the recursion formulas of the propositions 5.3.6 and 5.3.8 can be solved explicitly.

Proposition 5.4.3. In the û-basis, the diagonal elements of the matrices K `apξ|P ´|P C q and K a pξ|P `|P C q are vanishing: @pP ´, ξq P P ´b C , @h P 0, p ´1 , xh| û K `apξ|P ´|P C q |hy û " 0 (5.165) @pP `, ξq P P `b C , @h P 0, p ´1 , xh| û K a pξ|P `|P C q |hy û " 0 (5.166)

We chose to give here the explicit proof, as it introduces an important reconstruction of the boundaries matrices that will be used later.

Proof. We show the first relation (5.165). From proposition 5.3.8, we know the ratios defining the diagonal coefficients of the matrix K `apξ|P a |P C q in the v-basis. At the special point P a " P ´, this ratio R p`q h (5.122) or (5.125) can be written:

R p`q h " q 2 f phq f ph `2q (5.167)
with f phq " q 2h ´s, where s " σqξ 2 is a scalar depending on the parameters at the homogeneous limit P ´and on the boundary parameter ξ.

Thanks to the specific form (5.167) of these ratios, we can compute:

xh| v K `apξ|P ´|P C q |hy v " N ˆ1 q 2ph`1q ´s ´1 q 2h ´s ẇhere the pre-factor N is explicitly given by N " x0| v K `apξ|P ´|P C q |0y v f p0qf p1q 1 ´q2 . (5.168)

The key observation to note is that we can then give an expression of the matrix K `apξ|P ´|P C q in terms of the v operators:

K `apξ|P ´|P C q " N q 2 v2 ´s ´N v2 ´s (5.169)
Now, the cornerstone of the proof is the following inversion formula (see (5.12) of [START_REF] Grosjean | On form factors of local operators in the lattice sine-Gordon model[END_REF]):

@y P C , 1 `yp y v2 `1 " p´1 ÿ j"0
p´1q j y j v2j (5.170) which leads, with a simple computation, to:

K `apξ|P ´|P C q " p´1 ÿ j"0 C j v2j with C j " N s p´1 1 ´sp s ´j `q2j ´1˘( 5.171)
It follows the proposition as:

xh| û K `apξ|P ´|P C q |hy û " p´1 ÿ j"0 C j xh| û |h ´jy û " C 0 " 0 (5.172)
Using a very similar method, one can prove (5.166). Let us simply comment that the reconstruction of

EXPRESSION OF LOCAL HAMILTONIANS

the matrix is different, as it holds:

K a pξ|P `|P C q " N `v p´2 ´t vp´4 ˘(5.173)
with N and t are scalars depending on the parameters at the homogeneous limit P ´and on the boundary parameter ξ.

Corollary 5.4.4. In the points where K `apξ|P a |P C q (respectively K a pξ|P a |P C q) reduces to a scalar, the trace of K a pξ|P a |P C q (respectively K `apξ|P a |P C q) is vanishing: @pP ´, ξq P P ´b C , tr a K `apξ|P ´|P C q ( " 0 (5.174) @pP `, ξq P P `b C , tr a K a pξ|P `|P C q ( " 0 (5.175)

As already mentioned, this is an important obstacle in our way to the computation of local Hamiltonians. Moreover, using this corollary, it remains only the term tr a tK `apξ `|P ´|P C qH N a u in (5.154). The following lemma is then at the heart of the proof of proposition 5.4.2: Proposition 5.4.5. In any direction x for the derivative, it holds:

@pP á , ξq P P ´b C tr a ! K `apξ|P ´|P C qh pxq N a
)

" κ pxq 1 and tr a ! K `apξ|P ´|P C qh pxq aN

) " κpxq 1 (5.176)

The two scalars are explicitly known, and given by: W RQ ph a `∆a qW QR ph a q (5.178)

κ pxq " p´1 ÿ ∆a"0 C ∆a d WRaR N p∆ a q d x a ˇˇˇP N "P Ṕa "P ´p´1 ÿ ha"0 W RQ ph a qW QR ph a `∆a q (5.
In these formulas, the coefficient C j is the one introduced in (5.171) for the decomposition of the boundary matrix in terms of the generators of the Weyl algebra, W and W denote the dilogarithm functions entering in the expression of the fundamental R-matrix, and the points denoted R and Q are the homogenous limits of the points R n and Q n of the chain.

The proof of these equalities is rather technical (see paper III). We simply highlight that the heart of the proof lies in some particular relations satisfied by the dilogarithm functions, for example WQQ ph ´lq " δ h,l (the Kronecker symbol). Computing the trace in the û-basis allows to use the proposition 5.4.3, which simplifies the computation. Lastly, we use the reconstruction (5.171) of K `apξ|P ´|P C q in terms of the generators of the Weyl algebra, which permits to complete the computations. This last proposition ends the proof of proposition 5. Thus the first order derivative reduces to a scalar in the considered limit, and the standard procedure to obtain local Hamiltonians cannot be used.

Expression of local Hamiltonians

Here, we prove that the second order derivative with respect to the spectral parameters can be used to express local Hamiltonians and to show their commutativity with the fundamental transfer matrix. The key point of the computation relies on the fact that the non-trivial expression for the second order derivative simplifies thanks to proposition 5.4.2. In particular, we show that the highly non local interactions are in fact cancelled. THE REFLECTION ALGEBRA

Second derivative of the fundamental transfer matrix

Thus we tackle the computation of the second order derivative with respect to the spectral parameters, in the homogeneous limit, of the fundamental transfer matrix:

d 2 T pf undq pP a |P Q |P C q dx 2 a ˇˇˇP a "P ṔQ "P ´(5.179)
After a rather long computation, with more involved steps than for the first order derivative but of the same kind (explicit computation of the different terms in the derivative, reduction of the fundamental R-matrix to the permutation operator, reduction of the boundary matrix to the identity and so on), we find:

d 2 T pf undq pP a |P Q |P C q dx 2 a ˇˇˇP a "P ṔQ "P ´" tr a # d 2 K `apξ `|P a |P C q dx 2 a ˇˇˇP a "P ´+ `tr a K `apξ `|P ´|P C q ( H pxq nonlocal `2 ˜tr a # dK `apξ `|P a |P C q dx a ˇˇˇP a "P ´+ `tr a ! K `apξ `|P ´|P C qH pxq N a ) ¸N´1 ÿ k"1 H pxq k,k`1 `2 ˜tr a # dK `apξ `|P a |P C q dx a ˇˇˇP a "P ´+ `tr a ! K `apξ `|P ´|P C qH pxq N a ) ¸dK 1 pξ ´|P 1 |P C q dx 1 ˇˇˇP 1 "P 2 tr a # dK `apξ `|P a |P C q dx a ˇˇˇP a "P ´H pxq N a + `tr a ! K `apξ `|P ´|P C qH p2qpxq N a ) `2 tr a ! K `apP á |ξ `qh pxq N a h pxq aN ) `2 tr a ! K `apξ `|P ´|P C q " H pxq N ´1,N , h pxq aN ı) (5.180)
In this formula, H pxq nonlocal denotes a highly non local and non trivial term, which is however irrelevant in our calculations as it is multiplied by 0 (proposition 5.4.4). We also introduced: Moreover, we can rewrite the previous equality (5.180) as:

H p2qpxq N a " d 2 S
d 2 T pf undq pP a |P Q |P C q dx 2 a ˇˇˇP a "P ṔQ "P ´" tr a # d 2 K `apξ `|P a |P C q dx 2 a ˇˇˇP a "P ´+ `αpxq N ´1 ÿ k"1 H pxq k,k`1 `αpxq dK 1 pξ ´|P 1 |P C q dx 1 ˇˇˇP 1 "P ´`2 tr a # dK `apξ `|P a |P C q dx a ˇˇˇP a "P ´H pxq N a + `tr a ! K `apξ `|P ´|P C qH p2qpxq N a ) `2 tr a ! K `apξ `|P ´|P C qh pxq N a h pxq aN ) (5.182)
with the scalar (proposition 5.4.5) α pxq given by: Proof. The proof uses a continuity argument, similar to the one used previously many times in chapter 4. From the expression (5.183), one can see that α pxq is a continuous function of the parameters. Then, it is sufficient to show that there is at least one set of parameters for which α pxq ‰ 0, and the result will hold for almost any value of the parameters. So let us consider the following set of parameters:

α pxq " 2 ˜tr a # dK `apξ `|P a |P C q dx a
P a " px Qa , x Ra , y Qa , y Ra , σ a q " pq χ a , q χ a , χ a , χ a , Σq (5.185)

and

P N " px Q N , x R N , y Q N , y R N , σ N q " pq χ N , q χ N , χ N , χ N , Σq (5.186)
with χ a , χ N and Σ free parameters. The homogeneous limit is given by χ a " χ N " χ " 1. The idea is to check the behaviour of α pχq under the limit Σ Ñ `8. After some computations, using for instance the reconstruction (5.171), we show that α pχq ‰ 0 for this model, which concludes the proof.

This proposition leads to another main result of this thesis, exposed in the following conclusion.

Expression of local Hamiltonians associated to cyclic representations of the 6-vertex reflection algebra

For α pxq ‰ 0, we can define the following integrable local Hamiltonians with boundaries:

H pxq " N ´1 ÿ k"1 H pxq k,k`1 `dK 1 pξ ´|P 1 |P C q dx 1 ˇˇˇP 1 "P ´`B pxq N pξ `|P C q (5.187)
where the right boundary (on site N) is given by:

B pxq N pξ `|P C q " 1 α pxq ˜2 tr a # dK `apξ `|P a |P C q dx a ˇˇˇP á H pxq N a + `tr a ! K `apξ `|P á |P C qH p2qpxq N a ) `2tr a ! K `apξ `|P á |P C qh pxq N a h pxq aN ) ¸(5.188)
Thanks to the commutativity exposed in propositions 5.3.10 and 5.3.9, these Hamiltonians commute with the fundamental transfer matrix and with the 6-vertex transfer matrix for any direction x of derivation:

@pP a , P ´q P P b P

´,

"

T pf undq pP a |P ´|P C q, H pxq ı " 0 (5.189) @pλ, P ´q P C b P

´, " T p6V q pλ|P ´|P C q, H pxq ı " 0 (5.190)

The left and right boundaries are encoded via the parameters ξ ´and ξ `respectively. Let us comment that similarly to Sklyanin's procedure [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF] for the expression of the local Hamiltonian, cf (2.43), the boundaries have different expressions, with the one for the right boundary seeming much more involved. We shall come back to that issue very soon, in section 5.5.

Remark.

Let us comment that in the more general case, two Hamiltonians H pxq and H pyq of the form (5.187) can be obtained thanks to the derivative in two different direction x and y. Then, one can combine these two expressions with a free parameter z to obtain H " H pxq `z H pyq , another commuting family which commutes with the fundamental transfer matrix and with the 6-vertex transfer matrix. Indeed, from the commutativity of proposition 5.3.10, it holds " H pxq , H pyq ‰ " 0. Thus the Hamiltonian H depends on

Some explicit (new) models for 3-dimensional local quantum spaces

In the last section of this thesis, we explicitly study four models of interest with finite cyclic representation of dimension 3, i.e. at the value p " 3. The fact to take a small numerical value for p allows to get explicit expressions and thus to develop an intuition for what happens with generic representations.

In particular a symmetry on the boundaries of the Hamiltonians appears, as well as some properties for the bulk interactions. Moreover, we have results concerning the impact of the boundaries on the bulk Hamiltonians, and we check the reflection equations at a cyclic-cyclic level.

The models we are considering are the followings. For each of them, we give the parameters px Qn , x Rn , y Qn , y Rn , σ n q in the quantuml spaces V n (of dimension 3) of the chain. The same sets of parameters hold in the auxiliary space V a (of dimension 3), simply replacing the subscript n by a. Moreover, we will give the homogeneous limit of these models, i.e. for which the parameters P ´" px Q , x R , y Q , y R , σq P P

´.

In addition to this, we give an explicit expression for the generators of the Weyl algebra. For example in the so-called v-basis, it holds: û " ¨0 1 0 0 0 1 1 0 0 ' and v " ¨1 0 0 0 q 0 0 0 q2 ' , with q 3 " 1.

(

• The XXZ spin chain at root of unity. In our study, the XXZ spin s " 1 chain can be obtained with the following set of parameters:

@n P 1, N , σ n " q , x Qn " i q 2 1 ξ n , x Rn " ´qi 1 ξ n , y Qn " ´i 1 ξ n and y Rn " qi 1 ξ n (5.192) where " ˘1 and ξ n are free parameters. Indeed, the Lax operator (5.7) reduces to the Rmatrix obtained by fusion [START_REF] Frappat | A complete Bethe ansatz solution for the open spin-s XXZ chain with general integrable boundary terms[END_REF][START_REF] Mezincescu | Fusion procedure for open chains[END_REF] from the 6-vertex R-matrix under the following similarity transformation P.û.P ´1 and P.v.P ´1 on the generators of the Weyl algebra represented by (5.191). The matrix P is explicitly given by:

P " ¨´1 0 0 0 0 1 0 1 0 ' (5.193)
The anisotropy ∆ of the model is related to q by ∆ " pq `1{qq{2. For this model, which corresponds to the point D) of proposition 5.1.4, the homogeneous limit in a point P ´P P ´is obtain when the parameters satisfy ξ n " ξ with ξ 2 " q.

This model is used, in this section, mainly as a verification tool. Indeed, as we shall see our method allows to find back the known Hamiltonian of the spin 1 XXZ chain.

• The super integrable chiral Potts chain. With the following set of parameters:

@n P 1, N , σ n " 1 , x Qn " q 2´i y Rn and x Rn " q i y Qn (5.194) where the parameters y Qn and y Rn are free and i is a free integer, one can describe the super integrable chiral Potts model. This one corresponds also to the point D) of proposition 5.1.4. The homogeneous limit in a point P ´P P ´is obtain when the parameters y Q and y R are satisfying y Q y R " 1. @n P 1, N , σ n " 1 , x Qn " q 2 y Qn , x Rn " y Rn , y Qn " 1 iq 1{2 κ n ξ n and y Rn " iq 1{2 κ n ξ n (5.195) where ξ n and κ n are free, one can describe the so-called Fateev-Zamolodchikov model [START_REF] Fateev | Self-dual solutions of the star-triangle relations in ZNmodels[END_REF]. This model is interesting per se, but also for its link with the discretised sine-Gordon model at q root of unity. Indeed, the Lax operator (5.7) satisfies: L 0n pλ|Q n , R n q " L psGq 0n pλq.σ x 0 (5.196) where L psGq 0n pλq is the Lax operator of the sine-Gordon model, see [START_REF] Niccoli | The sine-Gordon model revisited I[END_REF] for instance, and σ x 0 denotes the Pauli matrix in the space V 0 . It has been shown that, for an even number of sites (N=2M), there is a mapping between the transfer matrices of these two models. As far as an odd N is concerned, the two transfer matrices have different eigenvalues (see appendix D of [START_REF] Niccoli | The sine-Gordon model revisited I[END_REF] for an explicit comparison between these two models). This model enters in the framework of point B) of proposition 5.1.4 with the constant parameters R " 0 and K " 1. Let us mention that the homogeneous limit in a point P ´P P ´is obtain when the parameter ξ n and κ n satisfy ξ n " ξ and κ n " κ , with ξ 2 " 1 and κ free.

• A general model. We will also consider a model which generalises the Fateev-Zamolodchikov model, parametrised as follow:

@n P 1, N , σ n " q k , x Qn " q i y Qn K and x Rn " q 2´i K y Rn (5.197) with y Qn , y Rn and K free, and free integers k and i. It enters in the framework of point B) of proposition 5.1.4, where the constants parameters read R " 0 and K. The homogeneous limit in a point P ´P P ´can be easily realised, the parameters y Q and y R have to satisfy y Q y R " 1.

Let us simply note that for the XXZ model, there is only one direction of derivation, ξ a . For the other models, pxq can take two different values: ξ a or κ a for the sine Gordon model, and y Qa or y Ra for the two others.

These models being now characterised, let us compute their local Hamiltonians and analyse some of their properties.

Symmetry of the boundary Hamiltonians

For all the four considered models, we check that when the coefficient α pxq given by (5.183) is non zero, the following symmetry holds:

B pxq N pξ `|P C q " dK N pξ `|P N |P C q dx N ˇˇˇP N "P
´`cst (5.198) where cst is a scalar whose exact value is irrelevant, as the Hamiltonians are anyhow defined up to an additive constant. Thus, the left and right boundaries have a symmetric expression on the two boundary sites 1 and N with just different parameters ξ ´and ξ `.

The Hamiltonians are given (up to an additive constant) by:

H pxq " N ´1 ÿ k"1 H pxq k,k`1 `dK 1 pξ ´|P 1 |P C q dx 1 ˇˇˇP 1 "P ´`dK N pξ `|P N |P C q dx N ˇˇˇP N "P
´ (5.199) This is a nice compact form of the boundaries terms, which should probably hold for generic values of p. For the boundary transfer matrix associated to the spin 1/2 XXZ spin chain, the symmetry with respect to λ going in 1{λ allows to show the same kind of symmetrical rewriting of the boundaries for the Hamiltonians (2.43).

Here, we have to investigate the symmetry of the fundamental transfer matrix with respect to the set of parameters P a . A symmetry of T pf undq pP a |P Q |P C q may be responsible for the symmetry of the boundaries.

Reflection equations at a cyclic-cyclic level

For these particular models with a cyclic representation of dimension 3, we can check by direct computation that for all the considered models, the cyclic-cyclic reflection equation (5.142) and its dual (5.145) are satisfied, with the scalar:

cpP a |P b |P C q " 1 (5.200)
Thus for the considered models, it holds:

S θ b θa ba pP b |P a |P C qK a pξ ´|P a |P C qS θ b ab pP a |P b |P C qK b pξ ´|P b |P C q " K b pξ ´|P b |P C qS θa ba pP b |P a |P C qK a pξ ´|P a |P C qS ab pP a |P b |P C q (5.201)
as well as

pS ba pP b |P a |P C qq tat b K `apξ `|P a |P C q " `Sta ab pP θa a |P b |P C q ˘´1 ı t b K `bpξ `|P b |P C q " K `bpξ `|P b |P C q " `Sta ab pP θa a |P b |P C q ˘´1 ı t b K `apξ `|P a |P C q pS ba pP b |P a |P C qq tat b (5.202)
As already mentioned, these equations allow to show the commutativity of the fundamental transfer matrix between themselves by an elegant and simple algebraic proof, using the framework of the (ABCD)quantum type algebras. The needed compatibility conditions are satisfied as we can check the Yang-Baxter equation for the fundamental R-matrix:

S ab pP a |P b |P C qS ac pP a |P c |P C qS bc pP b |P c |P C q " S bc pP b |P c |P C qS ac pP a |P b |P C qS ab pP a |P b |P C q (5.203)
We expect that these results hold for every dimension p of the representation, and for every scalar solution (not only diagonal) for the boundaries.

In the two last paragraphs of this chapter, we explicitly study the Hamiltonians of the four considered models. To that end, we first give the explicit expressions of the bulk Hamiltonians for the quasi-periodic models, i.e. associated to the Yang-Baxter algebra, in 5.5.3. Then in 5.5.4 we give the expressions of the boundary matrices, and see how the bulk interactions in the reflection algebra case are modified compared with the Yang-Baxter algebra framework.

A part the XXZ model which is considered as a verification of our approach, we are generating new local Hamiltonians, both for the quasi-periodic conditions and for integrable boundary conditions.

On the bulk Hamiltonians associated to the Yang-Baxter algebra

Following the standard procedure presented in chapter 1 (trace identity), we easily show that the Hamiltonians: (5.204) where we recall that h pxq k,k`1 is defined in (5.161), are commuting with the bulk transfer matrix: acting on V k b V k`1 of this quasi-periodic chain, in terms of the generators of the Weyl algebra.

H pxq " N ÿ k"1 h pxq k,k`1
T pP a |P Q |P C q " tr a tS aN pP a |P N |P C q...S

The XXZ spin 1 model at root of unity

Computing the derivative in (5.161) for the set of parameters (5.192), the bulk Hamiltonians have the following expression in terms of the generators û and v of the Weyl algebra:

h pξq k,k`1 " Γ 0 2 ÿ α k "0 2 ÿ β k "0 2 ÿ α k`1 "0 2 ÿ β k`1 "0 hpα k , β k , α k`1 , β k`1 q ûα k k vβ k k ûα k`1 k`1 vβ k`1
k`1 (5.206) In this formula, the parameter Γ 0 denotes a global normalisation and hp0, 0, 0, 0q is free, the Hamiltonian being defined up to an additive constant. All the coefficients hpα k , β k , α k`1 , β k`1 q are zero, except: hp0, 0, 0, 1q " ´1{2 ; hp0, 0, 0, 2q " ´q{2 ; hp0, 1, 0, 0q " ´1{2 ; hp0, 2, 0, 0q " ´q{2 hp1, 0, 2, 0q " ´q2 ; hp1, 0, 2, 1q " q 2 ; hp1, 0, 2, 2q " q 2 ; hp1, 1, 2, 0q " q hp1, 2, 2, 0q " 1 ; hp2, 0, 1, 0q " ´q2 ; hp2, 0, 1, 1q " q ; hp2, 0, 1, 2q " 1 (5.207)

hp2, 1, 1, 0q " q 2 ; hp2, 2, 1, 0q " q 2

As mentioned before, we can check that this Hamiltonian is indeed the bulk one expected of the XXZ spin 1 chain. This spin 1 Hamiltonian appeared with the work of Fateev and Zamolodchikov [START_REF] Fateev | Model factorized S matrix and an integrable Heisenberg chain with spin 1[END_REF], it has the following expression:

h XXZ k,k`1 " τ k,k`1 ´τ 2 k,k`1 `2shpηq 2 `τ z k,k`1 ´pτ z k,k`1 q 2 `pS z k q 2 `pS z k`1 q 2 4sh 2 pη{2q `τ K k,k`1 τ z k,k`1 `τ z k,k`1 τ K k,k`1 ˘(5.208)
where:

τ k,k`1 " Ý Ñ S k . Ý Ñ S k`1 ; τ K k,k`1 " S x k S x k`1 `Sy k S y k`1 and τ z k,k`1 " S z k S z k`1
(5.209) and Ý Ñ S are the standard spin 1 generators of sup2q. Here, the parameter η is related to the anisotropy of the chain via ∆ " chpηq, and linked to our notation thanks to q " e η . For the particular choice Γ 0 " ´q{3 for the normalisation, and for q a 3 rd -root of unity, it holds:

h XXZ k,k`1 " P k P k`1 .h pξq k,k`1 .P ´1 k P ´1 k`1 `cst (5.210)
for the representation (5.191) of the generators û and v.

The super integrable chiral Potts model

When the parameters satisfy (5.194), the Hamiltonians, for example in the case i " 1, have the following expressions:

h py Q q k,k`1 " Γ pQq 0 2 ÿ α k "0 2 ÿ β k "0 2 ÿ α k`1 "0 2 ÿ β k`1 "0 h pQq pα k , β k , α k`1 , β k`1 q ûα k k vβ k k ûα k`1 k`1 vβ k`1 k`1
(5.211) denote a global normalisation, h pQq p0, 0, 0, 0q and h pRq p0, 0, 0, 0q are free, and all the h pQq pα k , β k , α k`1 , β k`1 q and h pRq pα k , β k , α k`1 , β k`1 q are zero, except: h pQq p0, 0, 0, 1q " ´2 ; h pQq p0, 0, 0, 2q " 1 ; h pQq p0, 1, 0, 0q " 1 ; h pQq p0, 2, 0, 0q " ´2 h pQq p1, 0, 2, 0q " ´1 ; h pQq p1, 0, 2, 1q " q 2 ; h pQq p1, 0, 2, 2q " q ; h pQq p1, 1, 2, 0q " q (5.213) h pQq p1, 2, 2, 0q " q 2 ; h pQq p2, 0, 1, 0q " ´1 ; h pQq p2, 0, 1, 1q " q ; h pQq p2, 0, 1, 2q " q 2 h pQq p2, 1, 1, 0q " q 2 ; h pQq p2, 2, 1, 0q " q and h pRq p0, 0, 0, 1q " 1 ; h pRq p0, 0, 0, 2q " ´2 ; h pRq p0, 1, 0, 0q " ´2 ; h pRq p0, 2, 0, 0q " 1 h pRq p1, 0, 2, 0q " ´1 ; h pRq p1, 0, 2, 1q " q 2 ; h pRq p1, 0, 2, 2q " q ; h pRq p1, 1, 2, 0q " q (5.214)

h py R q k,k`1 " Γ pRq 0 2 ÿ α k "0 2 ÿ β k "0 2 ÿ α k`1 "0 2 ÿ β k`1 "0 h pRq pα k , β k , α k`1 , β k`1 q ûα k k vβ k k ûα k`1 k`1 vβ k`1 k`1
h pRq p1, 2, 2, 0q " q 2 ; h pRq p2, 0, 1, 0q " ´1 ; h pRq p2, 0, 1, 1q " q ; h pRq p2, 0, 1, 2q " q 2 h pRq p2, 1, 1, 0q " q 2 ; h pRq p2, 2, 1, 0q " q Let us mention that these Hamiltonians are different from the Hamiltonians introduced by Baxter [260] and then generalised by Tarasov [START_REF] Tarasov | Cyclic monodromy matrices for the R-matrix of the six-vertex model and the chiral Potts model with fixed spin boundary conditions[END_REF][START_REF] Tarasov | Cyclic monodromy matrices for slpnq trigonometric R-matrices[END_REF], for the so-called chiral Potts model with fixed spin boundary conditions. Indeed, they could give an expression for the Hamiltonian of the chiral Potts model under a special boundary condition, for which only the operator Apλq of the Lax matrix is used to construct the Hamiltonian. Moreover, this Hamiltonian is obtained from a completely different method than the one presented here, as it is defined as the coefficient in λ N ´1 of the operator Apλq (for a chain composed of N sites). Our Hamiltonian is obtained from the full transfer matrix.

The Fateev-Zamolodchikov model

For the parameters describing the Fateev-Zamolodchikov model, i.e. when they satisfy (5.195), we can also have two different directions of derivation.

• In the direction ξ a , the Hamiltonian is given by:

h pξq k,k`1 " Γ 0 2 ÿ α k "0 2 ÿ β k "0 2 ÿ α k`1 "0 2 ÿ β k`1 "0 hpα k , β k , α k`1 , β k`1 q ûα k k vβ k k ûα k`1 k`1 vβ k`1 k`1 (5.215)
The parameter Γ 0 denotes a global normalisation, hp0, 0, 0, 0q is once again free and all the hpα k , β k , α k`1 , β k`1 q are zero, except: hp0, 0, 0, 1q " ´q2 κ 2 ; hp0, 0, 0, 2q " ´q2 κ 2 ; hp0, 1, 0, 0q " ´q2 κ 2 hp0, 2, 0, 0q " ´q2 κ 2 ; hp1, 0, 2, 0q " ´2q 2 κ 2 ; hp1, 0, 2, 1q " ´qκ 2 pκ 2 ´1q hp1, 0, 2, 2q " κ 2 ´1 ; hp1, 1, 2, 0q " κ 2 ´1 ; hp1, 2, 2, 0q " ´qκ 2 pκ 2 ´1q

(5.216) hp2, 0, 1, 0q " ´2q 2 κ 2 ; hp2, 0, 1, 1q " κ 2 ´1 ; hp2, 0, 1, 2q " ´qκ 2 pκ 2 ´1q hp2, 1, 1, 0q " ´qκ 2 pκ 2 ´1q ; hp2, 2, 1, 0q " κ 2 ´1

• In the direction κ a , the Hamiltonian is given by:

h pκq k,k`1 " Γ0 2 ÿ α k "0 2 ÿ β k "0 2 ÿ α k`1 "0 2 ÿ β k`1 "0 h pκq pα k , β k , α k`1 , β k`1 q ûα k k vβ k k ûα k`1 k`1 vβ k`1 k`1
(5.217)

The parameter Γ0 stands for a global normalisation, h pκq p0, 0, 0, 0q is free and all the 5.5. SOME EXPLICIT (NEW) MODELS FOR 3-DIMENSIONAL LOCAL QUANTUM SPACES

h pκq pα k , β k , α k`1 , β k`1 q are zero, except:
h pκq p0, 0, 0, 1q " ´qκ ; h pκq p0, 0, 0, 2q " ´qκ ; h pκq p0, 1, 0, 0q " qκ h pκq p0, 2, 0, 0q " qκ ; h pκq p1, 0, 2, 1q " ´κpκ 2 ´1q ; h pκq p1, 0, 2, 2q " q 2 pκ 2 ´1q{κ (5.218)

h pκq p1, 1, 2, 0q " ´q2 pκ 2 ´1q{κ ; h pκq p1, 2, 2, 0q " κpκ 2 ´1q ; h pκq p2, 0, 1, 1q " q 2 pκ 2 ´1q{κ h pκq p2, 0, 1, 2q " ´κpκ 2 ´1q ; h pκq p2, 1, 1, 0q " κpκ 2 ´1q ; h pκq p2, 2, 1, 0q " ´q2 pκ 2 ´1q{κ

The general model Lastly, for the parameters satisfying (5.197) with k " 1 and i " 2 for instance, the bulk Hamiltonians have the following expressions:

h py Q q k,k`1 " Γ pQq 0 2 ÿ α k "0 2 ÿ β k "0 2 ÿ α k`1 "0 2 ÿ β k`1 "0 h pQq pα k , β k , α k`1 , β k`1 q ûα k k vβ k k ûα k`1 k`1 vβ k`1 k`1
(5.219)

h py R q k,k`1 " Γ pRq 0 2 ÿ α k "0 2 ÿ β k "0 2 ÿ α k`1 "0 2 ÿ β k`1 "0 h pRq pα k , β k , α k`1 , β k`1 q ûα k k vβ k k ûα k`1 k`1 vβ k`1 k`1
(5.220)

The parameters Γ pQq 0 and Γ pRq 0 denote a global normalisation, h pQq p0, 0, 0, 0q and h pRq p0, 0, 0, 0q are free and all the h pQq pα k , β k , α k`1 , β k`1 q and h pRq pα k , β k , α k`1 , β k`1 q are zero, except: h pQq p0, 1, 0, 0q " q 2 ; h pQq p0, 2, 0, 0q " 1 ; h pQq p1, 0, 2, 0q " q ; h pQq p1, 1, 2, 0q " ´p1 `q{Gq h pQq p1, 2, 2, 0q " ´qpG `qq ; h pQq p2, 0, 1, 0q " q ; h pQq p2, 1, 1, 0q " ´pG `qq (5.221)

h pQq p2, 2, 1, 0q " ´qp1 `q{Gq and h pRq p0, 0, 0, 1q " q 2 ; h pRq p0, 0, 0, 2q " 1 ; h pRq p1, 0, 2, 0q " q ; h pRq p1, 0, 2, 1q " ´pq `Gq h pRq p1, 0, 2, 2q " ´q G pq `Gq ; h pRq p2, 0, 1, 0q " q ; h pRq p2, 0, 1, 1q " ´1 G pq `Gq (5.222)

h pRq p2, 0, 1, 2q " ´q pq `Gq

We denoted G " K{y 2 Q the re-scaled free parameter K.

On the bulk Hamiltonians associated to the reflection algebra

In this final paragraph we analyse the impact of the presence of integrable boundaries on the bulk Hamiltonians previously derived, i.e. the local interactions H pxq k,k`1 of the four models (5.192)- (5.197). We recall that it holds

H pxq k,k`1 " h pxq k,k`1

`hpxq

k`1,k for the bulk interactions of the chain with boundaries. In a way we are thus considering a symmetrised version of the chain.

Using symmetries specific to each of the models, and to each direction x a of derivation, we show that the behaviour of the bulk of the chain with integrable boundaries really depends on the considered model.

The XXZ spin 1 model at root of unity

When the parameters are tuned to describe the XXZ spin 1 chain, the fundamental R-matrix only depends on the ratio of the spectral parameters. It is of the form:

S k`1,k pP k`1 |P k |P C q " S k`1,k ˆξk`1 ξ k ˙(5.223) THE REFLECTION ALGEBRA
Moreover, we can check the symmetry:

S k`1,k ˆξk`1 ξ k ˙" S k,k`1 ˆξk`1 ξ k ˙(5.224)
This way, it holds:

h pξq k,k`1 " h pξq k`1,k
and thus

N ´1 ÿ k"1 H pξq k,k`1 " 2 N ´1 ÿ k"1 h pξq k,k`1 (5.225)
We find thus that up to a factor 2, the bulk interactions of the chain with general integrable boundary conditions are the same as the bulk interactions of the periodic chain.

We can also explicitly see the relation

h pξq k,k`1 " h pξq k`1
,k as the coefficients (5.207) satisfy:

@pα k , β k , α k`1 , β k`1 q P 0, 2 4 , hpα k , β k , α k`1 , β k`1 q " hpα k`1 , β k`1 , α k , β k q (5.226)
Let us write the diagonal boundary matrices for this model. Thanks to propositions 5.3.6 and 5.3.8, we know the coefficients of these matrices in the v-basis:

K a pξ ´|ξ a |P C q " ¨1´qξ 2 a ξ 2 ξ2 a ´ξ2 ´1´ξ 2 a ξ 2 qξ 2 a ´ξ2 ´0 0 0 1 0 0 0 q 1´qξ 2 a ξ 2 ξ2 a ´ξ2 ´‹ ‹ ' (5.227)
and

K `apξ `|ξ a |P C q " ¨q2 q´ξ 2 a ξ 2 ξ2 a ´ξ2 `1´ξ 2 a ξ 2 q2 ξ 2 a ´ξ2 `0 0 0 1 0 0 0 q q´ξ 2 a ξ 2 ξ2 a ´ξ2 `‹ ‹ ' (5.228)
Thus, we can equivalently write them in terms of the generators. It holds:

K a pξ ´|ξ a |P C q " 1{N pξ a , ξ ´q " α 0 pξ a , ξ ´q1 `α1 pξ a , ξ ´qv a `α2 pξ a , ξ ´qv 2 a ‰ (5.229)
The factor N pξ a , ξ ´q is simply a normalisation, and the three coefficients α i pξ a , ξ ´q are polynomials in the variables ξ a and ξ ´. Explicitly, for the choice of normalisation (5.227), it holds:

N pξ a , ξ ´q " 3pξ 2 a ´ξ2 ´qpqξ 2 a ´ξ2 ´q (5.230)
and the coefficients α i pξ a , ξ ´q read:

α i pξ a , ξ ´q " α i0 pξ ´q `αi2 pξ ´qξ 2 a `αi4 pξ ´qξ 4 a (5.231)
The polynomials α ij pξ ´q are given by:

α 00 pξ ´q " 1 ´qξ 2 ´`ξ 4
´; α 02 pξ ´q " q 2 `2q 2 ξ 2

´`q 2 ξ 4

´; α 04 pξ ´q " q ´ξ2

´`qξ 4

´(5.232)

α 10 pξ ´q " 1 ´q2 ξ 2
´`q 2 ξ 4

´; α 12 pξ ´q " 1 ´ξ2

´`ξ

´; α 14 pξ ´q " 1 ´qξ 2

´`qξ 4

´(5.233)

α 20 pξ ´q " 1 ´ξ2 ´`qξ 4
´; α 22 pξ ´q " q ´qξ 2

´`qξ

4

´; α 24 pξ ´q " q 2 ´q2 ξ 2

´`qξ 4

´(5.234)

Exploiting the symmetry (5.131), we can state:

K `apξ `|ξ a |P C q " K a p1{ξ `|1{ξ a |P C q (5.235)
Thus, the integrable boundaries on the sites j " 1 and j " N explicitly read:

B pξq j pξ ˘|P C q " dK j pξ ˘|ξ j |P C q dξ j ˇˇˇξ j "q 1{2 " 1{ N pξ ˘q " γ 0 pξ ˘q1 `γ1 pξ ˘qv a `γ2 pξ ˘qv 2 a ‰ (5.236)
where the coefficients γ i pξq read: γ 0 pξq " 3q 2 `pq ´1qξ 2 ´3q 2 ξ 4 ; γ 1 pξq " ´qpq ´1qp1 `ξ2 ´ξ4 q ; γ 2 pξq " q 2 pq ´1qp1 ´ξ2 ´ξ4 q (5.237)

and the normalisation N pξq is given by: N pξq " ´3{2 qp1 `ξ2 `ξ4 q (5.238)

As mentioned before, this model serves as a tool for the verification of the good working of our approach. Indeed, we find back the known results for the integrable boundaries of the XXZ spin 1 chain at root of unity. For example, the boundary matrices K a pξ ´|ξ a |P C q and K `apξ `|ξ a |P C q are effectively the diagonal matrices reduction of the more general ones derived in [START_REF] Frappat | A complete Bethe ansatz solution for the open spin-s XXZ chain with general integrable boundary terms[END_REF]: up to an additive constant, the diagonal boundary term H b of [START_REF] Frappat | A complete Bethe ansatz solution for the open spin-s XXZ chain with general integrable boundary terms[END_REF] on site 1, that we will note H bound,1 pξ ´q is indeed recovered via:

P 1 .B pξq 1 pξ ´|P C q.P ´1 1 " γ H bound,1 pξ ´q (5.239)
when we use the representation (5.191) for the generators û and v. (γ is a numerical factor)

The super integrable chiral Potts model (at p " 3)

For the super integrable chiral Potts model, the fundamental R-matrix is of the form

S k`1,k pP k`1 |P k |P C q " S k`1,k p y Q k`1 y Q k , y R k`1 y R k q (5.240)
When taking the derivatives with respect to y Q k or y R k , the Hamiltonians satisfy:

h py Q q k,k`1 " h py Q q k`1,k `γpy Q q k`1 ´γpy Q q k (5.241)
and

h py R q k,k`1 " h py R q k`1,k `γpy R q k`1 ´γpy R q k (5.242)
where γ py Q q i and γ

py R q i are matrices acting only in space i. These symmetries come from symmetries on the fundamental R-matrix:

@x P C , S k`1,k px, 1q " S k,k`1 px, 1q `Pk,k`1 ´Γpy Q q k`1 pxq ´Γpy Q q k pxq `Mpy Q q k`1,k pxq ¯(5.243) @x P C , S k`1,k p1, xq " S k,k`1 p1, xq `Pk,k`1 ´Γpy R q k`1 pxq ´Γpy R q k pxq `Mpy R q k`1,k pxq ¯(5.244)
where Γ py Q q i and Γ

py R q i are matrices acting only in space V i , and where matrices M

py Q q k`1,k and M py R q
k`1, satisfy:

dM py Q q k`1,k pxq dx ˇˇˇˇx "1
" 0 and

dM py R q k`1,k pxq dx ˇˇˇˇx "1 " 0 (5.245)
Then it holds:

N ´1 ÿ k"1 H py Q q k,k`1 " 2 N ´1 ÿ k"1 h py Q q k,k`1 `γpy Q q 1 ´γpy Q q N and N ´1 ÿ k"1 H py R q k,k`1 " 2 N ´1 ÿ k"1 h py R q k,k`1 `γpy R q 1 ´γpy R q N (5.246)
This way, as previously, the bulk interactions for the chain with general integrable boundary conditions are the same as for the quasi-periodic chain up to a factor 2, but moreover here there is an extra modification of the boundaries via the matrices γ 1 and γ N . One can read the matrices γ from the coefficients (5.213)-(5.214): (5.252)

γ py Q q " Γ pQq 0 `v ´v 2 ˘`cst and γ py R q " Γ pRq 0 `v 2 ´v ˘`cst
The polynomials α ij pξ ´q are given by:

α 00 pξ ´q " 1 ´q2 ξ 2 ´`ξ 4
´; α 01 pξ ´q " 1 `2ξ 2

´`ξ

4

´; α 02 pξ ´q " 1 ´qξ 2

´`ξ 4

´(5.253) α 10 pξ ´q " q ´q2 ξ 2

´`q 2 ξ 4

´; α 11 pξ ´q " 1 ´ξ2

´`ξ

4

´; α 12 pξ ´q " q 2 ´qξ 2

´`qξ 4

´(5.254) α 20 pξ ´q " q 2 ´q2 ξ 2

´`qξ

4

´; α 21 pξ ´q " 1 ´ξ2

´`ξ

4

´; α 22 pξ ´q " q ´qξ 2

´`q 2 ξ 4

´(5.255)

Exploiting the symmetry (5.131), we can state:

K `apξ `|y Qa , y Ra |P C q " K a p1{ξ `|q{y Ra , q{y Qa |P C q (5.256)
Thus, the integrable boundaries on the sites j " 1 and j " N explicitly read:

B py Q q j pξ ˘|P C q " dK j pξ ˘|y Qj , y Rj |P C q dy Qj ˇˇˇy Q j y R j "y Q y R "1
(5.257)

" y R { N py Qa , y Ra , ξ ˘q " γ 0 py Qa , y Ra , ξ ˘q1 `γ1 py Qa , y Ra , ξ ˘qv a `γ2 py Qa , y Ra , ξ ˘qv 2 a

‰

The normalisation N py Qa , y Ra , ξq is given by: N py Qa , y Ra , ξq " 3p1 `ξ2 `ξ4 q (5.258) while the three other coefficients are written: γ 0 py Qa , y Ra , ξq " 3 `qpq ´1qξ 2 ´3ξ 4 ; γ 1 py Qa , y Ra , ξq " qpq ´1qp1 `ξ2 ´ξ4 q (5.259) γ 2 py Qa , y Ra , ξq " ´qpq ´1qp1 ´ξ2 ´ξ4 q (5.260)

As the boundary matrices depend on the product y Qa y Ra , then it follows:

B py R q j pξ ˘|P C q " dK j pξ ˘|y Qj , y Rj |P C q dy Rj ˇˇˇy Q j y R j "y Q y R "1 " 1{y 2 R B py Q q j
pξ ˘|P C q (5.261)

SOME EXPLICIT (NEW) MODELS FOR 3-DIMENSIONAL LOCAL QUANTUM SPACES

The Fateev-Zamolodchikov model

As far as the Fateev-Zamolodchikov model is concerned, the fundamental R-matrix is written:

S k`1,k pP k`1 |P k |P C q " S k`1,k p ξ k`1 ξ k , κ k`1 , κ k q (5.262)
The boundary matrices are:

K a pξ ´|ξ a , κ a |P C q " ¨1 0 0 0 q 2 ´ξ2 a ξ 2 q2 ξ 2 a ´ξ2 ´0 0 0 p´1`ξ 2 a ξ 2 ´qp´q 2 `ξ2 a ξ 2 ´q pξ 2 a ´ξ2 ´qpq 2 ξ 2 a ´ξ2 ´q ‹ ‹ ' (5.263) and K `apξ `|ξ a , κ a |P C q " ¨1 0 0 0 1´qξ 2 a ξ 2 q2 pξ 2 a ´ξ2 `q 0 0 0 p´q`ξ 2 a ξ 2 `qp´1`qξ 2 a ξ 2 `q pξ 2 a ´ξ2 `qpq 2 ξ 2 a ´ξ2 `q ‹ ‹ ' (5.264)
In order for the mixed reflection equation and its dual to be satisfied, it must hold ξ 2p

´" 1 and ξ 2p `" 1. We can equivalently write them in terms of the generators. It holds:

K a pξ ´|ξ a , κ a |P C q " 1{N pξ a , ξ ´q " α 0 pξ a , ξ ´q1 `α1 pξ a , ξ ´qv a `α2 pξ a , ξ ´qv 2 a ‰ (5.265)
The factor N pξ a , ξ ´q is simply a normalisation, and the three coefficients α i pξ a , ξ ´q are polynomials in the variables ξ a and ξ ´. Explicitly, for the choice of normalisation (5.263) , it holds: The polynomials α ij pξ ´q are given by: α 00 pξ ´q " 1 ´ξ2

N pξ a , ξ ´q " 3pξ
´`qξ

4

´; α 02 pξ ´q " 1 `2q 2 ξ 2

´`qξ

4

´; α 04 pξ ´q " 1 ´qξ 2

´`qξ 4

´(5.268) α 10 pξ ´q " q ´q2 ξ 2

´`qξ

4

´; α 12 pξ ´q " q 2 ´qξ 2

´`ξ

4

´; α 14 pξ ´q " 1 ´ξ2

´`q 2 ξ 4

´(5.269) α 20 pξ ´q " q 2 ´qξ 2

´`qξ

4

´; α 22 pξ ´q " q ´ξ2

´`q 2 ξ 4

´; α 24 pξ ´q " 1 ´q2 ξ 2

´`ξ 4

´(5.270)

Exploiting the symmetry (5.131), we can state:

K `apξ `|ξ a , κ a |P C q " K a p1{ξ `|q 2 {ξ a , κ a |P C q (5.271)
For this model, the two directions of derivation lead to different behaviours.

• In the direction ξ a , we use the next symmetry which is similar to the one of the XXZ model:

S k`1,k ˆξk`1 ξ k , κ, κ ˙" S k,k`1 ˆξk`1 ξ k , κ, κ ˙(5.272)
Then, it holds

h pξq k,k`1 " h pξq k`1,k leading to N ´1 ÿ k"1 H pξq k,k`1 " 2 N ´1 ÿ k"1 h pξq k,k
@pα k , β k , α k`1 , β k`1 q P 0, 2 4 , hpα k , β k , α k`1 , β k`1 q " hpα k`1 , β k`1 , α k , β k q (5.274)
As far as the integrable boundaries are concerned, it holds:

B pξq j pξ ˘|P C q " dK j pξ ˘|ξ j , κ j |P C q dξ j ˇˇˇξ j "1; κj "κ " 1{ N pξ ˘q " γ 0 pξ ˘q1 `γ1 pξ ˘qv a `γ2 pξ ˘qv 2 a ‰ (5.275)
where the coefficients γ i pξq read: γ 0 pξq " 3q 2 `q2 pq´1qξ 2 ´3ξ 4 ; γ 1 pξq " pq´1qpq 2 `qξ 2 ´ξ4 q ; γ 2 pξq " pq´1qpq´ξ 2 ´q2 ξ 4 q (5.276) and the normalisation N pξq is given by: N pξq " 3{2pξ 2 ´1qpq 2 ´ξ2 q (5.277)

• In the direction κ a however, we have α pκq " 0. So the formula (5.187) does not apply.

Thus, one may try to compute for example a higher derivative of the fundamental transfer matrix to get an expression. However, let us just remark here that the symmetry:

S k`1,k p1, κ k`1 , κ k q " S k,k`1 p1, κ k , κ k`1 q (5.278)
and the expression:

S k`1,k p1, κ k`1 , κ k q " P k,k`1 `S k`1,k p κ k`1 κ k q `pκ k`1 ´κk q Šk`1,k pκ k`1 , κ k q (5.279)
where S and Š are regular functions, allow us to write:

h pκq k,k`1 " ´hpκq k`1,k (5.280)
This way, even if in the case of periodic boundary conditions the κ a direction of derivation generates a non trivial local hamiltonian, the previous symmetry implies that the integrable boundary conditions lead to a trivial local hamiltonian in this direction:

N ´1 ÿ k"1 H pκq k,k`1 " 0 (5.281)
One can moreover check that the K j pξ|P a |P C q matrix does not depend on κ: dK j pξ ˘|ξ j , κ j |P C q dκ j ˇˇˇξ j "1; κj "κ " 0 (5.282)

In the direction κ a , the reflection makes to vanish all the interactions. pq´y Qa y Ra ξ 2 `qp1´qy Qa y Ra ξ 2

`q ‹ ‹ ' (5.284)
In order for the mixed reflection equation and its dual to be satisfied, it must hold ξ 2p

´" 1 and ξ 2p `" 1 for the boundary parameters.

We can equivalently write them in terms of the generators. It holds: (5.287)

K a pξ ´|y Qa , y Ra |P C q " 1{N
The polynomials α ij pξ ´q are given by: α 00 pξ ´q " 1 ´ξ2

´`q 2 ξ 4

´; α 01 pξ ´q " 1 `2qξ 2

´`q 2 ξ 4

´; α 02 pξ ´q " 1 ´q2 ξ 2

´`q 2 ξ 4

´(5.288)

α 10 pξ ´q " 1 ´q2 ξ 2 ´`ξ 4
´; α 11 pξ ´q " q 2 ´ξ2

´`qξ

4

´; α 12 pξ ´q " q ´qξ 2

´`q 2 ξ 4

´(5.289)

α 20 pξ ´q " 1 ´qξ 2 ´`qξ 4
´; α 21 pξ ´q " q ´q2 ξ 2

´`ξ

4

´; α 22 pξ ´q " q 2 ´ξ2

´`q 2 ξ 4

´(5.290)

Exploiting the symmetry (5.131), we can state:

K `apξ `|y Qa , y Ra |P C q " K a p1{ξ `|q{y Ra , q{y Qa |P C q (5.291)
Thus, the integrable boundaries on the sites j " 1 and j " N explicitly read:

B py Q q j pξ ˘|P C q " dK j pξ ˘|y Qj , y Rj |P C q dy Qj ˇˇˇy Q j y R j "y Q y R "1
(5.292)

" y R { N py Qa , y Ra , ξ ˘q " γ 0 py Qa , y Ra , ξ ˘q1 `γ1 py Qa , y Ra , ξ ˘qv a `γ2 py Qa , y Ra , ξ ˘qv 2 a

‰

The normalisation N py Qa , y Ra , ξq is given by: N py Qa , y Ra , ξq " 3q pξ 2 ´1qpq 2 ξ 2 ´1q (5.293) while the three other coefficients are written: γ 0 py Qa , y Ra , ξq " 3q `pq ´1qξ 2 ´3q 2 ξ 4 ; γ 1 py Qa , y Ra , ξq " qpq ´1qp1 `qξ 2 ´q2 ξ 4 q (5.294) γ 2 py Qa , y Ra , ξq " ´pq ´1qp1 ´qξ 2 ´q2 ξ 4 q (5.295) THE REFLECTION ALGEBRA

As the boundary matrices depend on the product y Qa y Ra , then it follows:

B py R q j pξ ˘|P C q " dK j pξ ˘|y Qj , y Rj |P C q dy Rj ˇˇˇy Q j y R j "y Q y R "1 " 1{y 2 R B py Q q j pξ ˘|P C q (5.296)
For this model, the bulk Hamiltonians of the chain with integrable boundary conditions seem not to satisfy any particular symmetry with the bulk Hamiltonians for the Yang-Baxter chain. The boundaries completely change the behaviour of the bulk of the chain.

With these four models, we have presented a representative sample of the different behaviours of a chain with integrable boundaries. Depending on the symmetries of the fundamental transfer matrix, the fact to consider integrable boundaries can lead to the same bulk interactions, or to completely different bulk interactions, even possibly leading to the vanishing of the interactions.

Conclusion

The main theoretical steps to understand the macroscopic behaviour of quantum systems are certainly the determination of their Hamiltonian spectrum and the computation of their correlation functions, where all the information is encoded. This thesis takes place in the development of such a research program to study quantum integrable models with general integrable boundary conditions, the long-range goal being to be able to exactly describe both equilibrium and out of equilibrium physics.

Precisely, we started the analysis of the class of integrable quantum models associated to cyclic representations of the 6-vertex reflection algebra, including as particular cases the XXZ spin chain at root of unity, the lattice sine-Gordon model at root of unity and the chiral Potts model.

A large part of the work has been devoted to the development of the quantum separation of variables method to solve the transfer matrix spectral problem for the models associated to general integrable boundary conditions, described by boundary K-matrices scalar solutions of the 6-vertex reflection equation, and to the general Bazhanov-Stroganov cyclic Lax operator, local solution of the 6-vertex Yang-Baxter algebra. In a first time, we have proven that the quantum separation of variables was applicable for the general cyclic representation under one constraint on the boundary parameters. In particular, we have considered one completely general and one triangular boundary matrix. Thereafter, we have extended this method to the most general integrable boundaries. This has been done solving the problem to generalise Baxter's gauge transformations to these cyclic reflection algebras. Indeed, without any loss of generality, we have used these gauge transformations to transform one general boundary matrix in a triangular form, leading to the separate basis construction in a framework similar to the triangular case.

The boundary transfer matrix spectrum, both eigenvalues and eigenstates, is completely characterised in terms of the set of solutions to a discrete system of polynomial equations. Moreover, we have proven the equivalence of this characterisation with the set of solutions, in a given class of functions, to a Baxter like functional equation, showing also the rewriting of the boundary transfer matrix eigenstates in an Algebraic Bethe Ansatz form. It is worth remarking that this functional equation has in general an inhomogeneous term, leading to a new type of equation for which new methods have to be developed. However, let us mention that this inhomogeneous term vanishes when just one constraint is imposed, either on the boundary or on the bulk parameters, resulting this way in a homogeneous Baxter like equation. In that particular case, the boundary transfer matrix spectral problem is reduced to the standard analysis of an equivalent system of Bethe equations.

In the continuity of this work, we have solved the problem to define cyclic local Hamiltonians with integrable boundaries commuting with the above boundary transfer matrix, i.e. associated to general cyclic representations of the reflection algebra. The main difficulty has been to overcome the fact that the auxiliary and quantum spaces of the Bazahanov-Stroganov Lax operators are not isomorphic. Hence, we have considered first the fundamental cyclic R-matrix, solution of the so-called mixed Yang-Baxter equation intertwining two Lax operators acting in two different quantum spaces. Such an R-matrix acts in the tensor product of two cyclic representations. Then, we defined a mixed reflection equation intertwining Lax operators and two scalar boundary matrices, one defined on the auxiliary space and one defined on the quantum space. We found diagonal scalar solutions (boundary matrices) on the quantum space, associated to known scalar diagonal solutions on the auxiliary space. This achievement allowed us to define a multi spectral parameter family of fundamental transfer matrices commuting between themselves and with the boundary transfer matrix associated to the original cyclic reflection algebra.

In addition to that, we have proven that these fundamental transfer matrices generate multi parameter families of commuting local Hamiltonians with boundaries. It is worth mentioning that these cyclic local Hamiltonians are obtained as second derivatives with respect to the spectral parameters of the fundamental transfer matrices, as the first order derivatives reduce to the identity. It is worth mentioning also that the Hamiltonians we have constructed are new. In particular, they differ from the one considered by Baxter and Tarasov

The natural continuation of this work, already partly under consideration, is to clarify the role of the fundamental R-matrix for building up reflection equations at a cyclic-cyclic level and their compatibility conditions. This should also lead to new type of boundary conditions and new integrable local Hamiltonians, generalising the ones we have obtained to non diagonal boundary matrices.

Moreover, it should be interesting to investigate the Baxter T-Q equations in the context of general integrable boundaries and in particular the link between the Q-operator [51] and the fundamental transfer matrix. This last connection is done, for periodic boundaries and non compact representations of general quantum affine algebras, in [START_REF] Meneghelli | Integrable light-cone lattice discretizations from the universal Rmatrix[END_REF]. What is more, it must be investigated wether the fundamental transfer matrix introduced in this thesis is related to the fused transfer matrix obtained in [START_REF] Von Gehlen | The Baxter-Bazhanov-Stroganov model: separation of variables and the Baxter equation[END_REF]. In both cases, this would allow to compute (or at least to characterise) the eigenvalues of the fundamental transfer matrix, and thus the ones of the corresponding local Hamiltonians.

Another continuation of this work is to tackle the study of the dynamics. The first step is to consider the matrix elements, in the eigenstate basis, of local operators for these models. These form factors are an essential ingredient for computing correlation functions. At this point, the approach being through the quantum separation of variables, the determinants one will obtain should suffer from the standard problem regarding their non-trivial homogeneous limits, i.e. the limit in which all inhomogeneity parameters are put to the same value. It should be interesting to investigate, in this cyclic representation framework, the equivalence between these types of determinants for scalar products, essentially related to dressings of Vandermonde determinants, with Izergin's and Slavnov's type determinants, as was demonstrated recently for the XXX and XXZ boundary integrable models [START_REF] Kitanine | The open XXX spin chain in the SoV framework: scalar product of separate states[END_REF][START_REF] Kitanine | On determinant representations of scalar products and form factors in the SoV approach: the XXX case[END_REF]. This should make possible to take their homogeneous limit and then their thermodynamic limit (where the size N of the chain becomes large). Let us comment that the impact of boundaries in the thermodynamic limit is non trivial for a half-infinite chain only, where solely one boundary is sent away. In addition to that, it is worth emphasising that due to the boundary, the 1-point correlation functions are already non trivial.

Even if there is still a lot to be done, for example concerning the characterisation of the ground state, if successful these techniques should also allow for the computation of the order parameters of the chiral Potts model in a rather direct way along the preliminary results described in [START_REF] Grosjean | On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models[END_REF]. Another application would be the computation of form factors for the lattice sine-Gordon model, in particular at its q-root of unity reduction points, linked to minimal models of Conformal Field Theory. and also giving access to first computations of matrix elements of local operators [33] in the eigenstates basis.

The aim of the paper is to generalize this type of results for the 6-vertex cyclic representations. Here we solve this problem in the case of one triangular and one general boundary matrix, so that our current results define also the setup for the solution of the most general boundary case as well as the paper [33] has introduced the tools to solve the case with the most general boundary in [35].

The paper is organized as it follows. In section 2, we recall the cyclic representations of the 6-vertex Yang-Baxter algebra associated to the Bazhanov-Stroganov Lax-operator. In section Here, we characterize the spectral problem and we start the analysis of the dynamical problem (by determining scalar products of separate states) for the class of models in the Sklyanin's construction associated to general scalar solutions of the 6-vertex reflection equation [11,13] and the general Bazhanov-Stroganov cyclic solution of the 6-vertex Yang-Baxter algebra [113].

It may be instructive to recall some main literature on open integrable quantum chains, even for different representations with regards to those studied here. Indeed, this allows us to point out the difficulties that arise in their analysis and that we have also encountered for the models studied in this paper and to give the motivations for the approach that we have followed to overcome them. The spectrum of the open XXZ spin 1/2 quantum chain with parallel z-oriented magnetic fields on the boundaries has been characterized in [3], in the framework of the algebraic Bethe ansatz. While its dynamics has been studied by the exact computation of correlation functions first in [14,15] and then in [16], there generalizing in the ABA framework the method established in [114,115] for the periodic chains. These open quantum spin chains with z-oriented boundary magnetic fields correspond in the Sklyanin's construction to the diagonal scalar solution of the reflection equation. However, the most general scalar solution of the 6-vertex/8-vertex reflection equation is non-diagonal [11,13] producing unparalleled and not z-oriented boundaries magnetic fields. Under this general setting the analysis of the spectrum and dynamics has shown to be much more involved. The ABA method cannot be directly applied to these open chains with general boundary. In fact, it was first understood in [37], for the XYZ spin 1/2 open chain, that the use of the Baxter's gauge transformations [116] allows to generalize the ABA method limitedly to non-diagonal boundary matrices which satisfy one special constraint. After that, the same approach, based now on the trigonometric version of the Baxter's gauge transformations, was used in [38,39] to describe the XXZ spectrum by ABA under similar constraints. Let us comment that for the XXZ spectrum the same constraint was derived independently by a pure functional method based on the use of the fusion of transfer matrices and truncations identities for the roots of unit case in [22]- [24]. This has given access to the study of the spectrum leaving however the study of the dynamics for these open models still unsolved.

Results on the spectrum for the most general unconstrained boundary conditions have been achieved only more recently and they have required the introduction of methods different from the ABA. Pure eigenvalue analysis has been implemented in [25] by a functional method leading to nested Bethe ansatz type equations similar to those previously introduced in [26].

Moreover, an ansatz for polynomial T-Q functional equations with an inhomogeneous term has been recently argued in [27]. Eigenstate construction has been first considered under these general boundary in [21] by the q-Onsager algebra formalism. A different approach, based on the generalization of the Sklyanin's separation of variables (SoV) method to the reflection algebra framework, has then lead to the complete eigenvalues and eigenstates characterization [29,30,[33][34][35][36], proving its equivalence to an inhomogeneous TQ functional equation [36],

where the local quantum determinants read:

det q L a,n (λ) ≡ L a,n (λ) 11 L a,n (λq -1 ) 22 -L a,n (λ) 12 L a,n (λq -1 ) 21 (2.17)
= L a,n (λ) L a,n (λq -1 ) -L a,n (λ) L a,n (λq -1 )
.

(2.18)

They admit the following explicit form:

det q M a (λ) = N n=1 k n ( λ µ n,+ - µ n,+ λ )( λ µ n,- - µ n,- λ ) (2.19) = (-q) N N n=1 β n a n c n α n ( 1 λ + q -1 b n α n a n β n λ)( 1 λ + q -1 d n α n c n β n λ) (2.20) = a(λ)d(λ/q), (2.21)
where:

k n ≡ (a n b n c n d n ) 1/2 , µ n,h ≡ iq 1/2 (a n β n /α n b n ) 1/2 h = +, iq 1/2 (c n β n /α n d n ) 1/2 h = -.
(2.22)

a(λ) ≡ a 0 N n=1 ( β n λ + q -1 b n α n a n λ), d(λ) ≡ (-1) N a 0 N n=1 a n c n α n ( 1 λ + q d n α n c n β n λ), (2.23)
and a 0 is a free non zero parameter.

Cyclic representations of the 6-vertex reflection algebra

In this section we define the most general cyclic representations of the 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax-operator. This is done by following the general procedure introduced by Sklyanin

(λ/µ) 1 (λ) R 12 (λµ/q) 2 (µ) = 2 (µ) R 12 (λµ/q) 1 (λ) R 12 (λ/µ). (3.1)
Here, we have defined:

a (λ) = M a (λ)K a (λ) Ma (λ), (3.2)
where K a (λ; ζ, κ, τ) is the most general scalar (boundary matrix) solution of the 6-vertex reflection equation [11,12]:

K a (λ; ζ, κ, τ) = 1 ζ -1 ζ λζ q 1/2 - q 1/2 λζ κe τ λ 2 q - q λ 2 κe -τ λ 2 q - q λ 2 q 1/2 ζ λ -λ ζq 1/2 a , (3.3)
and we have defined:

Ma (λ) = (-1) N σ y a M t a a (1/λ) σ y a . (3.4)
Using this correspondence, the most general cyclic representations of the 6-vertex Yang-Baxter algebra, associated to the bulk monodromy matrix

(2.11), define the most general cyclic representations of the 6-vertex reflection algebra, corresponding to the boundary monodromy matrices: The local generators of the cyclic 6-vertex Yang-Baxter algebra can be now defined as the elements of the following Bazhanov-Stroganov Lax operator:

a,- (λ) = M a (λ)K a,- (λ) Ma (λ) = -(λ) -(λ) -(λ) - (λ) a , (3.5)
t a a,+ (λ) = M t a a (λ)K t a a,+ (λ) M t a a (λ) = + (λ) + (λ) + (λ) + (λ) a , (3.6
L a,n (λ) ≡ λα n v n - β n λ -1 v -1 n u n q -1/2 a n v n + q 1/2 b n v -1 n u -1 n q 1/2 c n v n + q -1/2 d n v -1 n γ n v n /λ - δ n λ/v n a ∈ End( 2 ⊗ n ), (2.7)
where a denote the so-called auxiliary space V a 2 . Indeed, under the condition

γ n = a n c n /α n , δ n = b n d n /β n , (2.8)
L a,n (λ) is a solution of the 6-vertex Yang-Baxter equation:

R 12 (λ/µ)L 1,n (λ)L 2,n (µ) = L 2,n (µ)L 1,n (λ)R 12 (λ/µ), (2.9)
with regards to the standard 6-vertex R-matrix: ). Then, the following monodromy matrix:

R a b (λ) =   qλ -q -1 λ -1 λ - λ -1 q -q -1 q -q -1 λ - λ -1 qλ -q -1 λ -1   , ( 2
M a (λ) = A(λ) B(λ) C(λ) D(λ) a ≡ L a,N (λq -1/2 ) • • • L a,1 (λq -1/2 ) ∈ End( 2 ⊗ ), (2.11)
is also a solution of the Yang-Baxter equation:

R 12 (λ/µ)M 1 (λ)M 2 (µ) = M 2 (µ)M 1 (λ)R 12 (λ/µ), (2.12)
and its elements define a representation of the Yang-Baxter algebra on the tensor product of the local representation spaces, i.e.

=

⊗ N n=1 n . Note that one can also consider cyclic representations of the 6-vertex Yang-Baxter algebra associated to q, an even root of unit, these have been studied in [117].

Bulk transfer matrix and quantum determinant

The Yang-Baxter equation implies that the bulk transfer matrix

τ 2 (λ)
≡tr a M a (λ) defines a one parameter family of commuting operators. Note that we have:

[τ 2 (λ), Θ] = 0, where Θ ≡ N n=1 v n .
(2.13)

In [113,118] it was related to the analysis of the chP-model [119][120][121][122][123][124][125][126] and characterized by SoV in [92][93][94][95][96][97]. The Yang-Baxter equation also implies that the so-called quantum determinant is a central element and it has the following factorized form:

det q M a (λ) ≡ A(λq 1/2 )D(λq -1/2 ) -B(λq 1/2 )C(λq -1/2 ) (2.14) = D(λq 1/2 )A(λq -1/2 ) -C(λq 1/2 )B(λq -1/2 ) (2.15) = N n=1 det q L a,n (λ), (2.16)
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We omit the proof of this proposition as it can be derived repeating the main steps of the original Sklyanin's paper, where similar statements were proven for the case of spin 1/2 representations of the 6-vertex reflection algebra. Let us introduce now the following notation:

\ (λ) ≡ a + (λ) -(λ) + d + (λ) -(λ), (3.20)
for the diagonal part of the transfer matrix, i.e. the one associated to the diagonal elements of the matrix a,-(λ), and the coefficients:

a + (λ) ≡ (λ 2 q -1/ qλ 2 )(λζ + /q 1/2 -q 1/2 /(λζ + )) (λ 2 -1/λ 2 )(ζ + -1/ζ + ) , (3.21) d + (λ) ≡ (λ 2 q -1/ qλ 2 )(ζ + q 1/2 /λ -λ/(q 1/2 ζ + )) (λ 2 -1/λ 2 )(ζ + -1/ζ + ) . (3.22)
then we can prove the following:

Corollary 3.1. The most general transfer matrix admits the following symmetries:

(λ) = (1/λ), (-λ) = (λ), (3.23)
and the diagonal part \ (λ) has the following explicitly symmetric forms:

\ (λ) ≡ a + (λ) -(λ) + a + (1/λ) -(1/λ) (3.24)
= d

+ (λ) -(λ) + d + (1/λ) -(1/λ). (3.25
This is done in order to make completely explicit the construction of this basis; however, the proof of the diagonalizability of -(λ) can be done without these constraints and under completly general values of the inner boundary matrix and of the bulk parameters and it will be presented in appendix.

Pseudo-vacuum states

We implement the above constraints by imposing:

b n = -q 2 j n -1 a n , (4.2)
where for any n ∈ {1, ..., N} we have fixed j n ∈ {0, ..., p -1}, then we have:

〈 j n -1, n| L a,n 12 = 0 ¯, L a,n 12 | j n , n〉 = 0 ¯, (4.3)
as well as:

〈 j n -1, n| L a,n (λ) 11 = a n (λq j n -1 ) 〈 j n -1, n| (4.4)
〈 j n -1, n| L a,n (λ) 22 = d n (λq 1-j n ) 〈 j n -1, n| , (4.5
K a,± (λ) = K a (λq (1±1)/2 ; ζ ± , κ ± , τ ± ) = a ± (λ) b ± (λ) c ± (λ) d ± (λ) a , (3.7)
and ζ ± , δ ± , τ ± are arbitrary complex parameters.

Boundary transfer matrix and quantum determinant

These boundary monodromy matrices define a one parameter family of commuting transfer matrices:

(λ) ≡ tr a {K a,+ (λ) M a (λ) K a,- (λ) Ma (λ)} (3.8) = tr a {K a,- (λ) a,+ (λ)} = tr a {K a,+ (λ) a,- (λ)}
(3.9)

= a + (λ) -(λ) + d + (λ) -(λ) + b + (λ) -(λ) + c + (λ) - (λ). (3.10)
This statement follows by using the reflection equation as Sklyanin has proven in [3].

The characterization of the spectrum (eigenvalues and eigenstates) of this class of transfer matrices is the main subject of this paper. In particular, we will restrict our attention to the special boundary condition b + (λ) = 0, which can be analyzed by implementing the SoV approach once is proven the diagonalizability of the -(λ) family of commuting operators. In order to introduce this spectral analysis we start pointing out some important properties satisfied by the generators of the reflection algebra

-(λ), -(λ), - (λ)
and -(λ).

Let us start with the following re-parametrization of the boundary parameters [22]:

(α --1/α -) (β -+ 1/β -) ≡ ζ --1/ζ - κ - , (α -+ 1/α -) (β --1/β -) ≡ ζ -+ 1/ζ - κ - . (3.11)
Then we define the following functions:

A -(λ) ≡ g -(λ)a(λq -1/2 )d(1/(q 1/2 λ)), (3.12)
where:

g -(λ) ≡ (λα -/q 1/2 -q 1/2 /(λα -))(λβ -/q 1/2 + q 1/2 /(λβ -)) (α --1/α -) (β -+ 1/β -) . ( 3
det q a,- (λ) ≡ ((λ/q) 2 -(q/λ) 2 )[ -(λq 1/2 ) -(q 1/2 /λ) + -(λq 1/2 ) -(q 1/2 /λ)] (3.14) = ((λ/q) 2 -(q/λ) 2 )[ -(λq 1/2 ) -(q 1/2 /λ) + -(λq 1/2 ) -(q 1/2 /λ)], (3.15)
is a central element in the reflection algebra, i.e.

[det q a,- (λ), a,-

(µ)] = 0, (3.16)
and its explicit expression reads:

det q a,- (λ) = (λ 2 /q 2 -q 2 /λ 2 )A -(λq 1/2 )A -(q 1/2 /λ). (3.17)
Moreover, the generators of the reflection algebra satisfy the following properties:

-(λ) = (λ 2 /q -q/λ 2 ) (λ 2 -1/λ 2 ) -(λ -1
) + (q

-1/q) (λ 2 -1/λ 2 ) -(λ), (3.18) and -(λ -1 ) = - (λ 2 q -1/ qλ 2 ) (λ 2 /q -q/λ 2 ) -(λ) , -(λ -1 ) = - (λ 2 q -1/ qλ 2 ) (λ 2 /q -q/λ 2 ) -(λ). (3.19)
where:

ξ (h) n = µ n,+ q h+1/2 , ξ (h) n+N ≡ ξ (h) n ∀n ∈ {1, ..., N}, a 0 (λ) = a(λ/q 1/2
).

(

Moreover, next, we will need also the following notations:

Λ = (λ 2 + 1/λ 2 ), X (h b ) b = (ζ (h b ) b ) 2 + 1/(ζ (h b ) b ) 2 , X = q + 1/q (4.20) ζ (h) n = ξ (h) n ϕ n for h ∈ {0, ..., p -1} and ∀n ∈ {1, ..., 2N}, (4.21)
ϕ a = 1 -2θ (a -N) with θ (x) = {0 for x ≤ 0, 1 for x > 0}.
and

µ 2p n,+ = ±1, µ 2 n,+ = q -2h α 2ε -, µ 2 n,+ = -q -2h β 2ε -, µ 2 n,+ = q -2ε-2h µ 2ε m,- (4.24)
for any

ε = ±1, h ∈ {1, ..., p -1} and n, m ∈ {1, ...,
N}, then the states: Proof. Let us write explicitly the decomposition of the reflection algebra generator:

〈h 1 , ..., h N | ≡ 1 N 〈Ω| N n=1 h n k n =1 -(1/ξ (k n -1) n ) A -(1/ξ (k n -1) n ) , (4.25
〈h| -(λ) = 2N a=1 (λ 2 /q -q/λ 2 )(λζ (h a ) a -1/ζ (h a ) a λ) ((ζ (h a ) a ) 2 /q -q/(ζ (h a ) a ) 2 )((ζ (h a ) a ) 2 -1/(ζ (h a ) a ) 2 ) N b=1 b =a modN Λ -X (h b ) b X (h a ) a -X (h b ) b A -(ζ (h a ) a ) × 〈h|T -ϕ a a + (-1) N det q M (1) (λ/q 1/2 + q 1/2 /λ) 2 N b=1 Λ -X (h b ) b X -X (h b ) b 〈h| + (-1) N (ζ -+ 1/ζ -) (ζ - -1/ζ -) det q M (i) (λ/q 1/2 -q 1/2 /λ) 2 N b=1 Λ -X (h b ) b X + X (h b ) b 〈h|,
-(λ) = (-1) N [-a -(λ)A(λ)B(1/λ) + b -(λ)A(λ)A(1/λ) -c -(λ)B(λ)B(1/λ) + d -(λ)B(λ)A(1/λ)] , (4.29)
in terms of the generators of the Yang-Baxter algebra. Then, by using the identities

11 | j n , n〉 = | j n , n〉 a n (λq j n ), L a,n (λ) 22 | j n , n〉 = | j n , n〉 d n (λq -j n ), (4.6)
where:

a n (λ) = λα n - β n /λ, d n (λ) = γ n /λ - λδ n , (4.7)
which is of course compatible with the local quantum determinant at site n:

〈 j n - 1, n| det q L a,n (λ) = 〈 j n - 1, n| L a,n (λ) 11 L a,n (λ/q) 22 -L a,n 12 L a,n 21 (4.8) = a n (λq j n -1 )d n (λq -j n ) 〈 j n - 1, n| (4.9) det q L a,n (λ) | j n , n〉 = L a,n (λ) 22 L a,n (λ/q) 11 -L a,n 21 L a,n 12 | j n , n〉 (4.10) = | j n , n〉 a n (λq j n -1 )d n (λq -j n ), (4.11)
being:

a n (λq j n -1 )d n (λq -j n ) = -q β n a n c n α n ( 1 λ -q 2( j n -1) α n β n λ)( 1 λ + q -1 d n α n c n β n λ).
(4.12)

Then we can define the following left and right "reference states":

〈Ω| = ⊗ N n=1 〈 j n - 1, n| , Ω = ⊗ N n=1 | j n , n〉 . (4.13)
The following properties are satisfied:

〈Ω| A(λq 1/2 ) = a(λ) 〈Ω| , 〈Ω| D(λq 1/2 ) = d(λ) 〈Ω| , 〈Ω| B(λ) = 0 ¯, 〈Ω| C(λ) = 0 ¯, (4.14) A(λq 1/2 ) Ω = Ω a(λq), D(λq 1/2 ) Ω = Ω d(λ/q), B(λ) Ω = 0 ¯, C(λ) Ω = 0 ¯, (4.15)
where it is simple to verify that as it should:

a(λ) = N n=1 a n (λq j n -1 ), d(λ) = N n=1 d n (λq 1-j n ), (4.16)
once we have fixed the free parameter:

a 0 = (-q) N N n=1 q -j n .
Of course, the coefficients a(λ)

and d(λ) as well as the reference states depend on the choice of the N-tuple { j 1 , ..., j N } but for simplicity we do not write it explicitly. we need to introduce some notations. We define the following functions parametrized by the

Representation of the reflection algebra in

N -tuples h ≡ (h 1 , ..., h N ) ∈ {0, ..., p - 1} N : B h (λ) ≡ κ -e τ -(λ 2 /q -q/λ 2 ) (ζ -- 1/ζ -) a h (λ)a h (1/λ), (4.17) with a h (λ) ≡ (-1) N N n=1 (α n β n ) 1/2 ( λ ξ (h n ) n - ξ (h n ) n λ ), (4.18)
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On the generic state |h〉, the action of the remaining reflection algebra generators follows by:

-(λ)|h〉 = 2N a=1 T -ϕ a a |h〉 (λ 2 /q -q/λ 2 )(λζ (h a ) a -1/ζ (h a ) a λ) ((ζ (h a ) a ) 2 /q -q/(ζ (h a ) a ) 2 )((ζ (h a ) a ) 2 -1/(ζ (h a ) a ) 2 ) N b=1 b =a modN Λ -X (h b ) b X (h a ) a -X (h b ) b × D -(ζ (h a ) a ) + |h〉(-1) N det q M (1) (λ/q 1/2 + q 1/2 /λ) 2 N b=1 Λ -X (h b ) b X -X (h b ) b + (-1) N+1 |h〉 (ζ -+ 1/ζ -) (ζ - -1/ζ -) det q M (i) (λ/q 1/2 -q 1/2 /λ) 2 N b=1 Λ -X (h b ) b X + X (h b ) b , (4.39)
where: Proof. The proof is given along the same steps used in the previous theorem, we just need to make the following remarks. First of all by using the identities (4.15) it follows that Ω is a -(λ)-eigenstate with non-zero eigenvalue:

D -(λ) = k(λ)A -(q/λ), T ± a |h 1 , ..., h a , ..., h N 〉 = |h 1 , ..., h a ± 1, ..., h N 〉.
-(λ) Ω ≡ Ω B p-1 (λ). (4.41)
Now all we need are the following reflection algebra commutation relations:

-(λ 1 ) -(λ 2 ) = (λ 1 q/λ 2 -λ 2 /(λ 1 q))(λ 1 λ 2 /q -q/(λ 1 λ 2 )) (λ 1 /λ 2 -λ 2 /λ 1 )(λ 1 λ 2 -1/(λ 1 λ 2 )) -(λ 2 ) -(λ 1 ) - (q -1/q)(λ 1 λ 2 /q -q/(λ 1 λ 2 )) (λ 1 /λ 2 -λ 2 /λ 1 )(λ 1 λ 2 -1/(λ 1 λ 2 )) -(λ 1 ) -(λ 2 ) - q -1/q (λ 1 λ 2 -1/(λ 1 λ 2 )) -(λ 1 ) -(λ 2
).

D -(ζ (h) a ) = κ (h) a A -(q/ζ (h) a ) (4.43)
for any h ∈ {0, ..., p -1} and a ∈ {1, ..., 2N} and Now by using the reflection algebra commutation relations:

det q -(ξ (h+1/2) a ) ( (ξ (h+3/2) a ) 2 -1/(ξ (h+3/2) a ) 2 ) = D -(ξ (h+1) a )D -(1/ξ (h) a ) = A -(ξ (h+1) a )A -(1/ξ (h) a ),
- (λ 2 ) - (λ 1 ) = (λ 1 q/λ 2 - λ 2 /(λ 1 q))(λ 1 λ 2 /q - q/(λ 1 λ 2 )) (λ 1 /λ 2 - λ 2 /λ 1 )(λ 1 λ 2 -1/(λ 1 λ 2 )) - (λ 1 ) - (λ 2 ) + (λ 2 1 /q - q/λ 2 1 )(q -1/q) (λ 2 /λ 1 - λ 1 /λ 2 )(λ 2 1 -1/λ 2 1 ) - (λ 2 ) - (λ 1 ) - q -1/q (λ 2 1 -1/λ 2 1 )(λ 1 λ 2 -1/(λ 1 λ 2 )) - (λ 2 ) ˜ - ( λ 1 ) (4.31)
we can follow step by step the proof given in the only thing that we need to prove to get their linear independence is that each such state is nonzero. We know by construction that the state 〈Ω| is nonzero so let us assume by induction that the same is true for the state

〈h (0) | = 〈h (0) 1 , ..., h (0) N | with h (0) j ∈ {0, ..., p -2} and let us show that 〈h (0) j | = 〈h (0) 1 , ..., h (0) j + 1, ..., h (0)
N | is nonzero. We have that:

〈h (0) j | - (ξ (h (0) j +1) j ) = A - (ξ (h (0) j +1) j )〈h (0) | = 0 ¯j ∈ {1, ..., N} (4.32)
so that 〈h (0) j | is nonzero. Using this we can prove that all states 〈h Finally, by using the identities:

| = 〈h (0) 1 + x 1 , ..., h (1)
-

(q 1/2 ) = (-1) N det q M (1) I 0 , - (iq 1/2 ) = i(-1) N+1 ζ - + 1/ζ - ζ --1/ζ - det q M (i) σ z 0 , (4.33)
and remarking that -(λ) has the following functional dependence with regards to λ:

- (λ) = 2N+1 a=0 λ (2a-2N+1) -,a , (4.34)
where -,a ∈End( ) are some fixed operators, we get our interpolation formula for its action on 〈h|.

Similarly, defining:

κ (h) a = k(ζ (h) a ), for h ∈ {0, ..., p -1}, a ∈ {1, ..., 2N}, (4.35)
and the function:

k(λ) = λ 2 -1/λ 2 /(λ 2 /q 2 - q 2 /λ 2 ), (4.36)
we have similar properties for the right representations: and so:

|h 1 , ..., h N 〉 ≡ 1 N N n=1 p-2 k n =h n - (ξ (k n +1) n ) κ (k n +1) n A - (1/ξ (k n ) n ) | Ω〉, (4.37
θ a = (q -1/q)A -(1/ξ (h a ) a ) ((ξ (h a ) a ) 2 -1/(ξ (h a ) a ) 2 ) 〈h 1 , ..., h a , ..., h N |h 1 , ..., h a , ..., h N 〉. (4.56)
These results lead to the identity:

〈h 1 , ..., h a + 1, ..., h N |h 1 , ..., h a + 1, ..., h N 〉 〈h 1 , ..., h a , ..., h N |h 1 , ..., h a , ..., h N 〉 = N b=1 b =a X (h a ) a -X (h b ) b X (h a +1) a -X (h b ) b , (4.57)
from which one can prove:

〈h 1 , ..., h N |h 1 , ..., h N 〉 〈p -1, ..., p -1|p -1, ..., p -1〉 = 1≤b<a≤N X (p-1) a -X (p-1) b X (h a
= 1≤b<a≤N X (p-1) a -X (p-1) b 〈Ω| N n=1 p-2 k n =0 -(1/ξ (k n ) n ) A -(1/ξ (k n ) n ) | Ω〉 1/2 . (4.60)
The previous theorem implies the following spectral decomposition of the identity in the SoV basis:

≡ p-1 h 1 ,...,h N =0 1≤b<a≤N (X (h a

Separate states and their scalar products

Let us introduce a class of left and right states, the so-called separate states, characterized by the following type of decompositions in the left and right SoV-basis:

〈α| = p-1 h 1 ,...,h N =0 N a=1 α (h a ) a 1≤b<a≤N (X (h a ) a -X (h b ) b )〈h 1 , ..., h N |, (4.62)
|β〉 = p-1 h 1 ,...,h N =0 N a=1 β (h a ) a 1≤b<a≤N (X (h a ) a -X (h b ) b )|h 1 , ..., h N 〉, (4.63)
where the coefficients

α (h a
〈α|β〉 = det N || (α,β) a,b
|| with

(α,β) a,b ≡ p-1 h=0 α (h) a β (h) a (X (h) a ) (b-1)
.

(4.64)

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Change of basis and SoV spectral decomposition of the identity

In this section we present the main properties of the

〈h| = 〈h|U (L) = p N i=1 U (L) (h),i 〈 -1 (i) | and |h〉 = U (R) |h〉 = p N i=1 U (R) i, (h) | -1 (i)〉, (4.47)
where is the isomorphism between the sets {0, ..., p -1} N and {1, ..., p N } defined by:

: h ∈ {0, ..., p - 1} N → (h) ≡ 1 + N a=1 p (a-1) h a ∈ {1, ..., p N }. (4.48)
From the diagonalizability of

-(λ) it follows that U (L)
and U (R) are invertible matrices for which it holds:

U (L) -(λ) = ∆ -(λ)U (L) , -(λ)U (R) = U (R) ∆ -(λ), (4.49)
where ∆ -(λ) is the p N × p N diagonal matrix defined by:

∆ -(λ) i, j ≡ δ i, j B -1 (i) (λ) ∀i, j ∈ {1, ..., p N }. (4.50)
We can prove that it holds: Proposition 4.1. The p

N × p N matrix M ≡ U (L) U (R)
consisting of scalar products of left and right --eigenstates is diagonal and it is characterized by the following diagonal entries: ) we get:

M (h) (h) = 〈h|h〉 = 1≤b<a≤N 1 X (h a ) a -X (h b ) b . ( 4
θ a = (q - 1/q)A -(1/ξ (h a ) a ) ((ξ (h a ) a ) 2 - 1/(ξ (h a ) a ) 2 ) N b=1 b =a X (h a +1) a -X (h b ) b X (h a ) a -X (h b ) b × 〈h 1 , ..., h a + 1, ..., h N |h 1 , ..., h a + 1, ..., h N 〉 (4.53)
while using the decomposition (3.18) and the fact that:

〈h 1 , ..., h a , ..., h N | -(1/ξ (h a +1) a )|h 1 , ..., h a + 1, ..., h N 〉 = 0 (4.54)
it holds:

θ a = k (h a +1) a (q - 1/q)A -(1/ξ (h a ) a ) ((ξ (h a +1) a ) 2 - 1/(ξ (h a +1) a ) 2 ) 〈h 1 , ..., h a , ..., h N |h 1 , ..., h a , ..., h N 〉, (4.55)
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The previous lemma defines the set of polynomials to which belong the transfer matrix eigenvalues; in order to completely characterize the eigenvalues we introduce now the following one-parameter family D τ (λ) of p × p matrices:

D τ (λ) ≡                τ(λ) -A(1/λ) 0 • • • 0 -A(λ) -A(qλ) τ(qλ) -A(1/ (qλ)) 0 • • • 0 0 . . . . . . . . . • • • . . . . . . • • • . . . . . . . . . 0 0 . . . 0 -A(q 2l-1 λ) τ(q 2l-1 λ) -A(1/ q 2l-1 λ ) -A(1/ q 2l λ ) 0 . . . 0 -A(q 2l λ) τ(q 2l λ)                , (5.5)
where for now τ(λ) is a generic function and we have defined:

A(λ) = a + (λ)A - (λ).
(

Note that the coefficient A(λ) satisfies the quantum determinant condition: (5.9)

A(λq 1/2 )A(q 1/2 /λ) = a + (λq 1/2 )a + (q 1/2 /λ)det q -(λ) (λ/q) 2 -(q/λ) 2 . ( 5
Let us now observe that:

D τ (λq) = C p→1 R p→1 (D τ (λ)), (5.10)
where R p→1 is the operation on a p × p matrix which move the last row in the first row leaving the order of the others unchanged and similarly C p→1 is the operation on a p × p matrix which move the last column in the first column leaving the order of the others unchanged. This clearly implies that:

det p D τ (qλ) = det p D τ (λ), (5.11)
which completes the proof of the lemma.

As mentioned before, we will restrict our attention to the special boundary condition b + (λ) = 0, for which the SoV-basis coincides with the --eigenstates basis. That is we consider the spectral problem of the transfer matrix:

(λ) ≡ \ (λ) + c + (λ) -(λ), (5.12)
under the following conditions on the boundary parameters:

b + (λ) = 0 and b -(λ) = 0.
(5.13)

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Proof. The proof follows the same method as in [97]. The formula (4.51)

implies, using the representation of the states 〈α| and |β〉, the following:

〈α|β〉 = p-1 h 1 ,...,h N =0 V (X (h 1 )
1 , ...,

X (h N ) N ) N a=1 α (h a ) a β (h a ) a , (4.65)
where we have denoted by

V (x 1 , ..., x N ) ≡ 1≤b<a≤N (x a -x b ) the Vandermonde determinant.
Finally, using the multilinearity of the determinant we get our result.

-spectrum characterization in the SoV basis

In this section we present the complete characterization of the spectrum of the transfer matrix (λ) associated to the cyclic representations of the 6-vertex reflection algebra. We first present some preliminary properties satisfied by all the eigenvalue functions of the transfer matrix (λ): Lemma 5.1.

Denote by

Σ the transfer matrix spectrum, then any τ(λ) ∈ Σ is an even function of λ invariant under the transformation λ → 1/λ which admits the following interpolation formula:

τ(λ) = N a=1 Λ 2 -X 2 (X (0) a ) 2 -X 2 N b=1 b =a Λ -X (0) b X (0) a -X (0) b τ(ζ (0) a ) + (-1) N (Λ + X ) 2 N b=1 Λ -X (0) b X -X (0) b det q M (1) -(-1) N (Λ -X ) 2 N b=1 Λ -X (0) b X + X (0) b (ζ + + 1/ζ + ) (ζ + -1/ζ + ) (ζ -+ 1/ζ -) (ζ - -1/ζ -) det q M (i) + (Λ 2 -X 2 )τ ∞ N b=1 (Λ -X (0) b ), (5.1)
where:

τ ∞ ≡ κ + κ -(e τ + -τ - N b=1 δ b γ b + e τ - -τ + N b=1 α b β b ) (ζ + -1/ζ + ) (ζ - -1/ζ -)
.

(5.2)

Proof. In the previous section, we have shown that the transformations λ → -λ and λ → 1/λ are symmetries of the transfer matrix (λ)

so if τ(λ) ∈ Σ then τ(λ)
is left unchanged under these transformations. Moreover, the asymptotic of the transfer matrix can be easily derived by direct computations, it is central and it holds:

τ ∞ = lim log λ→±∞ λ ∓2(N+2) (λ).
(5.3)

The identities (4.33) imply after some simple computation that the transfer matrix is central in q ±1/2 and iq ±1/2 and that it holds:

(q ±1/2 ) = (-1) N X det q M (1), (iq ±1/2 ) = (-1) N X (ζ + + 1/ζ + ) (ζ + -1/ζ + ) (ζ -+ 1/ζ -) (ζ - -1/ζ -)
det q M (i).

(5.4)

The known functional form of (λ)

with regards to λ together with this identities imply the interpolation formula in the lemma.

Proof. The spectrum (eigenvalues and eigenstates) of the transfer matrix (λ) in the SoV-basis is characterized by the following discrete system of equations:

τ(ξ (h n ) n )Ψ τ (h) = A(ξ (h n ) n )Ψ τ (T - n (h)) + A(1/ξ (h n ) n )Ψ τ (T + n (h)), (5.24
T ± n (h) ≡ (h 1 , . . . , h n ± 1, . . . , h N ).
(5.26)

This system admits the following equivalent representation as N linear systems of homogeneous equations:

D τ (ξ (0) n )     Ψ τ (h 1 , ..., h n = 0, ..., h N ) Ψ τ (h 1 , ..., h n = 1, ..., h N ) . . . Ψ τ (h 1 , ..., h n = p -1, ..., h N )     =     0 0 . . . 0     , (5.27
Note that for the previous lemma this condition is verified also in the points

ζ (0) a = 1/ξ (0)
a-N for any a ∈ {N + 1, ..., 2N}. The rank of the matrices in

(5.27) is p -1 being A(ξ (h) n ) = 0, A(1/ξ (h-1) n ) = 0 ∀h ∈ {1, ..., p -1}, n ∈ {1, ..., N}, (5.29)
for the conditions

〈h 1 , ..., h N | (ζ (h n ) n )|τ〉 = τ(ζ (h n ) n )〈h 1 , ..., h N |τ〉 ∀n ∈ {1, ..., N} (5.30)
for any --eigenstate 〈h 1 , ..., h N | and this implies:

〈h 1 , ..., h N | (λ)|τ〉 = τ(λ)〈h 1 , ..., h N |τ〉 ∀λ ∈ , (5.31)
i.e.

τ(λ)

∈ Σ and |τ〉 is the corresponding -eigenstate.

For the left -eigenstates the proof follows as above, we just remark in this case that the matrix elements:

〈τ| (ζ (h n ) n )|h 1 , ..., h N 〉, (5.32)
are computed in the right --representation:

τ(ζ (h n ) n ) Ψτ (h) = D(ζ (h n ) n ) Ψτ (T - n (h)) + D(1/ζ (h n ) n ) Ψτ (T + n (h)), ∀n ∈ {1, ..., N} (5.33)
where:

Ψτ

(h) ≡ 〈τ|h 1 , ..., h N 〉, D(λ) ≡ d + (λ)D - (λ).
(

τ ∞ = (-1) N κ + κ -e τ --τ + N b=1 α b β b (ζ + -1/ζ + ) (ζ --1/ζ -)
.

(5.14)

In the following we will suppress the unnecessary prime in κ + and τ + .

Theorem 5.1. If the conditions:

µ 2 n,+ = q -2h ζ ±2 + , ∀h ∈ {1, ...
|τ〉 = p-1 h 1 ,...,h N =0 N a=1 Q (h a ) τ,a 1≤b<a≤N (X (h a ) a -X (h b ) b )|h 1 , ..., h N 〉, (5.17)
where the Q

(h a )
τ,a are the unique nontrivial solution up to normalization of the linear homogeneous system:

D τ (ζ (0) a )    Q (0) τ,a . . . Q (p-1) τ,a    =   0 . . . 0   . (5.18)
II) The left -eigenstate corresponding to τ(λ) ∈ Σ is defined by the following decomposition in the left SoV-basis:

〈τ| = p-1 h 1 ,...,h N =0 N a=1 Q(h a ) τ,a 1≤b<a≤N (X (h a ) a -X (h b ) b )〈h 1 , ..., h N |, (5.19)
where the Q(h a ) τ,a are the unique nontrivial solution up to normalization of the linear homogeneous system: (5.21)

Q(0) τ,a . . . Q(p-1)
Finally, using ideas from [97], let us note that if

τ(λ) = τ (λ) ∈ Σ : N b=1 (τ,τ )
τ(λ) -τ (λ) ≡ Λ 2 -X 2 N b=1 x (τ,τ ) b Λ b-1 , ( 5
Q (x+1) τ,a Q (x-1) τ,a = τ(ζ (x) a ) A(1/ζ (x) a ) Q (x) τ,a Q (x-1) τ,a - A(ζ (x) a ) A(1/ζ (x) a ) (5.44)
and so:

Q (x+1) τ,a Q (0) τ,a = τ(ζ (x) a ) A(1/ζ (x) a ) Q (x) τ,a Q (0) τ,a - A(ζ (x) a ) A(1/ζ (x) a ) Q (x-1) τ,a Q (0) τ,a (5.45)
which by using the formula (5.40) for h = x -1 and h = x reads: (5.46) which just proves the formulae (5.40) and (5.41) for h = x + 1 once we recall that:

Q (x+1) τ,a Q (0) τ,a = τ(ζ (x) a )t (x) τ,a -A(ζ (x) a )A(1/ζ (x-1) a )t (x-1) τ,a x b=0 A(1/ζ (b) a ) ,
A(ζ (x) a )A(1/ζ (x-1) a ) = det q K a,+ (ξ (x-1/2) a )det q a,- (ξ (x-1/2) a ) (ξ (h+1/2) a ) 2 -1/(ξ (h+1/2) a ) 2 .
(5.47)

The proof of the first identity in (5.40) can be done in the same way, we have just to use now that:

D(ζ (x) a )D(1/ζ (x-1) a ) = det q K a,+ (ξ (x-1/2) a )det q a,- (ξ (x-1/2) a ) (ξ (h+1/2) a ) 2 -1/(ξ (h+1/2) a ) 2 .
(5.48)

Functional equation characterizing the -spectrum

In this section we assume that at any quantum site the following constraints are satisfied:

+ (λ) = 0.
Here, we show that the SoV characterization of the (λ)-spectrum can be reformulated by functional equations. Before this, let us just observe that the central value of the asymptotic of (λ) is given by:

τ ∞ = (-1) N κ + κ -e ε(τ - -τ + ) N b=1 α b β b (ζ + -1/ζ + ) (ζ - -1/ζ -) , (6.3)
A(λ) = λq -1/2 A(λ), (6.4)
A ∞ = lim λ→+∞ λ -2(N+2) A(λ) = (-1) N α -β -ζ + κ - N n=1 b n c n q 1+N (ζ + -1/ζ + ) (ζ - -1/ζ -) , (6.5
)

20 SciPost Phys. 2, 009 (2017)
It is simple to observe that it holds: detD τ (λ) = det Dτ (λ), (5.35) as a consequence of the following identities: D(λ) = α(λ)A(q/λ) (5.36) and:

α(1/λ)α(qλ) = 1, p-1 a=0 α(λq a ) = 1
(5.37)

where:

α(λ) = s(λ) s(q/λ) k(λ), s(λ) = λ 2 q -1/ qλ 2 λ 2 -1/λ 2 .
(5.38)

Finally, the identity (5.22) is quite general in the SoV framework and can be proven also in our case of cyclic representations of the 6-vertex reflection algebra by following the same proof first given in the case of a periodic lattice [97].

For next applications it is interesting to show that we can obtain the coefficients of a left transfer matrix eigenstates in terms of those of the right one by introducing a recursion formula that produces both coefficients in terms of the transfer matrix eigenvalues. The following lemma holds: Lemma 5.

Let τ(λ)

∈ Σ then we have:

Q(h) τ,a Q(h-1) τ,a = A(1/ζ (h-1) a ) D(1/ζ (h-1) a ) Q (h) τ,a Q (h-1)
τ,a , (5.39) being:

Q(h) τ,a Q(0) τ,a = t (h) τ,a h-1 b=0 D(1/ζ (b) a ) , Q (h) τ,a Q (0) τ,a = t (h) τ,a h-1 b=0 A(1/ζ (b) a )
, (5.40) where the t (h) τ,a are defined by the following recursion formula:

t (h) τ,a = τ(ζ (h-1) a )t (h-1) τ,a - det q K + (ξ (h-3/2) a )det q -(ξ (h-3/2) a ) (ξ (h-1/2) a ) 2 -1/(ξ (h-1/2) a ) 2 t (h-2)
τ,a for h ∈ {1, ..., p -1} (5.41)

with the following initial conditions t (-1)

τ,a = 0, t (0) τ,a = 1.
Proof. We just need to prove the last two identities in this lemma as the first ones are simple consequences of them. Let us prove the second identity in (5.40).

For h = 1 we have:

Q (1)
τ,a

Q (0) τ,a = τ(ζ (0) a ) A(1/ζ (0) a ) , ( 5
= det p          τ ∞ 0 0 • • • 0 -A ∞ -x A ∞ xτ ∞ 0 0 • • • 0 0 -x 2 A ∞ x 2
-x 2l-1 A ∞ x 2l-1 τ ∞ 0 0 0 . . . 0 -x 2l A ∞ x 2l τ ∞          , ( 6.16)
where we have denoted with t the transpose of the matrix and x = q 2(N+2) , so that it holds:

lim log λ→±∞ λ ∓2p(N+2) det p Dτ (λ) = τ p ∞ -A p ∞ .
(6.17) These results fix completely the Laurent's polynomial det p Dτ (λ) to satisfy (6.9).

Let us introduce now the following function:

G(λ|x, y)

=F (λ) τ ∞ -q -2(p-1)N A ∞ x 1 a=0 λ i a q 1/2 - i a q 1/2 λ i a q 1/2 λ - 1 i a q 1/2 λ +(i -1) y A(iq 1/2 ) 4F (iq 1/2
)

1 a=0 λq (1-2a)/2 - 1 λq (1-2a)/2 , (6.18)
and the states:

〈ω| = p-1 h 1 ,...,h N =0 N a=1 h a -1 k a =0 A(1/ζ (k a ) a ) D(1/ζ (k a ) a ) 1≤b<a≤N (X (h a ) a -X (h b ) b ) 〈h 1 , ..., h N | , (6.19)
| ω〉 = p-1 h 1 ,...,h N =0 1≤b<a≤N (X (h a ) a -X (h b ) b )|h 1 , ..., h N 〉, (6.20)
and a renormalization of the operator

ˆ -(λ) = (ζ - -1/ζ -) κ -e τ -(λ 2 /q -q/λ 2 ) N n=1 α n β n -(λ), (6.21)
which is a degree

N polynomial in Λ.
In the following we denote with Q(λ) a polynomial in det p Dτ In order to do so let us observe that the Q(λ) which is solution of (6.26) is defined up to a constant factor so that we are free to fix it by writing:

Λ = λ 2 + 1/λ 2 of degree N Q ≤ (p -
(i a q 1/2 ) (λ p + (-1) a /λ p ) 2 4 (-1) a F (i a q 1/2 ) = τ p ∞ - A p ∞ F (λ) λ 2p -1 λ 2p 2 . ( 6
Q(λ) = N Q b=1 (Λ -Λ b ) (6.30)
where

Λ b = λ 2 b + 1/λ 2 b
, then we have that it holds:

N Q b=1 ˆ -(λ b )| ω〉 = p-1 h 1 ,...,h N =0 N Q b=1 N a=1 (X (h a ) a -Λ b ) 1≤b<a≤N (X (h a ) a -X (h b ) b )|h 1 , ..., h N 〉 = p-1 h 1 ,...,h N =0 N a=1 Q(X (h a ) a ) 1≤b<a≤N (X (h a ) a -X (h b ) b )|h 1 , ..., h N 〉, (6.31)
where in the last line appears the SoV characterization of the right eigenstate associated to τ(λ) ∈ Σ . The same steps are used to prove (6.27) for the left eigenstate. Let us comment that the polynomiality of Q(λ) is the central property that we have used to prove these rewriting of the SoV characterizations of the transfer matrix eigenstates. This type of rewriting was first observed in the case of the noncompact XXX chains [90]. It is quite general and it has been used already for different models in the SoV framework, see for example [101][102][103].

Let us now assume that τ(λ) ∈ Σ and let us prove that it exists Q(λ) of the form (6.22).

In order to do so it is enough to show that the Baxter's equation holds in pN

+ 2 different
values of Λ, that we chose to be Λ = ±(q + 1/q) and the

Λ = X (h a ) a
for any a ∈ {1, ..., N} and h a ∈ {0, ..., p -1}. This set of equations can be organized in the following form:

Dτ (ξ (0) a )         Q(ξ (h=0) a ) Q(ξ (h=1) a ) . . . . . . Q(ξ (h=p-1) a )         p×1 =         0 . . . . . . . . . 0         p×1 ∀a ∈ {1, ...,
N}.

(6.32)

They are equivalent to the following system of equations:

Q(ξ (0) a )τ(ξ (0) a )/A(1/ξ (0) a ) =Q(ξ (1)
a ) (6.33)

τ(ξ (h) a ) A(1/ξ (h) a ) Q(ξ (h) a ) - A(ξ (h) a ) A(1/ξ (h) a ) Q(ξ (h-1)
q ∞ ≡ (p-1)N+1 a=1 (p-1)N+1 b=1,b =a Q(ξ a ) w a - w b , q 0 ≡ Q(iq 1/2
).

(6.24)

Here, and the following functional equation: Proof. The proof of this type of reformulation of the spectrum is by now quite standard and it has been proven for several models once they admit SoV description, see for example [36,[94][95][96][101][102][103]. So we will try to point out just some main features of the proof. Let us start proving the first part of the statement. It is simple to remark that the r.h.s. of the equation (6.26) is a Laurent polynomial in λ, indeed we can write: (6.28) so that the limits λ → ±1, λ → ±i are all finite. Moreover, it is simple to observe that the r.h.s. of (6.26) is invariant under the transformations λ → -λ and λ → 1/λ, so that the r.h.s. of (6.26) is in fact a polynomial of degree N

q ∞ is the coefficient in Λ (p-
τ(λ)Q(λ) = A(λ)Q(λ/q) + A(1/λ)Q(λq) + G(λ|q ∞ , q 0 ),
A(λ)Q(λ/q) + A(1/λ)Q(λq) = X(λ)Q(q/λ) -X(1/λ)Q(λq) λ 2 - 1/λ 2
Q + N + 2 ≤ pN + 2 in Λ.
Then, the fact that τ(λ) is entire in λ implies by the equation (6. ) = 0, for any choice of q ∞ , q 1 . Then, the previous theorem directly implies our corollary. Finally, let us remark that in this case we do not have any restriction on the degree of the polynomial Q(λ) imposed by the Baxter's equation (6.26), so that we are only left with

N Q ≤ (p -1)N.

Conclusions

In this paper we have studied the transfer matrix spectrum of the class of cyclic 6-vertex representations of the reflection algebra in the case of one completely general and one triangular boundary matrices and for bulk parameters satisfying some specific constraints. Our result is the complete characterization of the spectrum (eigenvalues and eigenstates) of this class of models both by a discrete system of Baxter's like second order difference equations and by a single inhomogeneous TQ-functional equation within a class of polynomial Q-functions. The present paper represents a natural starting point to solve the spectral problem in the most general setting. In order to do so we need to generalize to the 6-vertex cyclic representations the Baxter's gauge transformations used in the 6-vertex spin 1/2 highest weight representations and prove then the pseudo-diagonalizability of the associated family of gauge transformed --operators. These points are currently under analysis.

An interesting point to which we would like to come back in future investigations is the explicit construction of the Q-operator for the cyclic representations of the 6-vertex reflection algebra. This can lead to new connections with transfer matrices of exactly solvable class of models of non 6-vertex type. Indeed, let us recall that for the special class of 6-vertex cyclic representations of the Yang-Baxter algebra, parametrized by points on the algebraic curve (A.37), the associated integrable models have some remarkable connections with the inhomogeneous p-state chiral Potts models. Indeed, the chiral Potts transfer matrices play the role of the Q-operators for the transfer matrix associated to the cyclic representations of the 6-vertex Yang-Baxter algebra [113].

where we have defined:

x a,0,h = τ(ξ (h) a ) A(1/ξ (h) a ) , x a,-,h = - A(ξ (h) a ) A(1/ξ (h) a )
, ∀h ∈ {1, ..., p -2}, ∀a ∈ {1, ..., N}. (6.37) One can rewrite the previous system as it follows:

Q(ξ (h) a ) =
Q(λ) = (i -1)A(iq 1/2 )q 0 4 τ ∞ -q 2-2(p-1)N A ∞ F (iq 1/2 ) , ( 6
τ(λ)Q(λ) = A(λ)Q(λ/q) + A(1/λ)Q(λq).

A.2 The lattice sine-Gordon model with integrable boundaries

In this appendix we point out that the results on the transfer matrix spectrum for general cyclic representations developed in this paper apply also to characterize the spectrum of the transfer matrix of the lattice sine-Gordon model with integrable open boundary conditions. Let us recall that the Lax operator defining the lattice sine-Gordon model has the following form: admits the following functional form:

L sG a,n (λ|U n , V n , κ n , r n , s n ) ≡   U n q -1/2 κ 2 n r n s n V n + q 1/2 s n V n r n κ n i λr n V n -1 λr n V n κ n i λ r n V n - r n V n λ U -1 n q 1/2 r n V n s n + q -1/2 κ 2 n s n r n V n   a , ( A
M sG a (λ) = L a,N (λq -1/2 /ξ N ) • • • L a,1 (λq -1/2 /ξ 1 ) ∈ End( 2 ⊗ ), (A.14)
-(λ) = B -( λ 2 q - q λ 2 ) N a=1 ( λ -,a --,a λ )(λ -,a - 1 λ -,a
),

(A.3)
where the -,a are invertible commuting operators. The above functional form implies that under the condition (A.

-(λ)|Ω R 〉 = |Ω R 〉B -( λ 2 q - q λ 2 ) N a=1 ( λ b-,a - b-,a λ )(λ b-,a - 1 λ b-,a ) (A.5) 〈Ω L | -(λ) = B -( λ 2 q - q λ 2 ) N a=1 ( λ b-,a - b-,a λ )(λ b-,a -

N}.

Then the following identity holds: Moreover, under the above conditions and imposing the following identifications:

M sG a (λ) = M a (λ) σ
τ ε = ε x τ sG ε , κ ε = ε x κ sG ε , ζ ε = ζ sG ε ε x for ε = ±, (A.22)
of the boundary parameters of the τ Proof. It is simple to observe that the following identities hold:

L sG a,n (λ/ξ n |U n , V n , κ n , r n , s n ) = L a,n (λ)σ
d n = κ 2 n /(r n s n ), α n = -iκ n r n /ξ n , β n = -iκ n ξ n /r n . (A.27)
Similarly by direct computations it is easy to show that defined:

LsG a,n (λ) ≡ σ x a L sG a,n (λ|U n , V n , κ n , r n , s n )σ x a (A.28) it holds: LsG a,n (λ) = L sG a,n (λ|U -1 n , V -1 n , κ n , r -1 n , s -1 n ).
(A.29)

Let us now observe that for

x = 0, we can write: with p even and p odd and coprime, which is equivalent to the following identities among the generators of the local algebras:

M sG a (λ) = [L a,2M ( λq -1/2 ξ 2M )σ x a ][ La,2M-1 ( λq -1/2 ξ 2M-1 )σ x a ] • • • [L a,2 ( λq -1/2 ξ 2 )σ x a ][ La,1 ( λq -1/2 ξ 1 )σ x a ] (A.30) 29
S + n = u -1 n (v n -1/v n ) /2i, S - n = u n (v n /q -q/v n ) /2i
• • • 1 • • • 0 t 0 , a ∈ {1, ..., 2s + 1} (A.54)
the element a of the canonical basis given by the column vector with all elements zero except that in row a which is 1. In this basis we have the following representations for the generators .55) where:

S + n =      0 f (1) . . . . . . f (2s) 0      , S - n =      0 f (1) . . . . . . . . . f (2s) 0      , ( A
f ( j) = sinh jη sinh(p -j)η = i(q j -q -j
)/2, (A.56) and the second identity holds for q p = 1, and: Similarly we have:

S z n =      2s
S + n |a, n〉 = |a + 1, n〉 f (a) = |a + 1, n〉(q -a -q a )/2i (A.61) = u -1 n (v n -1/v n ) /2i
|p -a, n〉 ∀a ∈ {1, ..., p}

(A.62)
and

S - n |a, n〉 = |a -1, n〉 f (a -1) = |a -1, n〉(q 1-a -q a-1
)/2i where the q n are free elements of k and ∆ is the following discrete automorphism of the curve:

(A.63) = [u n (v n /q -q/v n ) /2i] |p + 2 -a,
∆ : x = (a x , b x , c x , d x ) ∈ k → ∆(x) = (b x , a x , d x , c x ) ∈ k , (A.43)
which implies:

x p n = y q n , y p n = x q n , s p n = s -1 q n . (A.44)
Finally, this class of representations reduce to the superintegrable chiral Potts model under the following special homogeneous limits: In particular, we have that our functional equation characterization of the spectrum under a special homogeneous limit defines the spectrum of the following local Hamiltonian given by fusion and the Sklyanin formula:

x p q n → 1 + k /k,
α n = β n = 1/2, a n = q -1/2 /2i, (A.48) b n = iq 1/2 /2, c n = q -1/2
/2i, be the Lax operator of the spin s XXZ chain with anisotropy cosh η = (q + 1/q) /2, then it holds:

d n = iq 1/2 /2
L X X Z a,n (λ) = L a,n (λ/q) (A.51)
form factors) was considered in previous works [106][107][108][109][110][111], generalizing in particular [126,[START_REF] Boos | Hidden Grassmann Structure in the XXZ Model[END_REF]. The interest in such a problem is due to the fact that special cases include the Sine-Gordon lattice model at roots of unity and the Chiral Potts model [128][129][130][131][132][133][134][135][136][137]. In [1] we started the analysis considering the special case where one of the boundary K-matrices has triangular form (which is equivalent to one constraint on the boundary parameters). For that situation we have been able to apply successfully the SoV method by identifying the separate basis as the eigenstate basis of a special diagonalizable B-operator with simple spectrum which can be constructed from the boundary monodromy matrix entries. Then using this separate basis, the spectrum (eigenvalues and eigenstates) for the boundary transfer matrix was completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions.

The purpose of the present article is to address this spectral problem for the most general boundary conditions preserving integrability, namely for the most general K-matrices solution of the reflection equation. The method to reach this goal is to design a gauge transformation that enable us to put this general situation into correspondence with the previous one, namely with a model having one triangular K-matrix. For that purpose, the standard idea of Baxter's gauge transformations, see e.g. [83,95] and references therein, has to be adapted in a way similar to [68,70] and generalized to these cyclic representations of the 6-vertex Yang-Baxter algebra. Then using this correspondence, the method and tools obtained in our first paper [1] can be used, leading to the complete characterization of the spectrum (again eigenvalues and eigenstates) of the general boundary transfer matrix. We also give determinant formula for the scalar products of the separate states. Further, we show that the spectrum characterization admits a representation in terms of functional equations of Baxter TQ-equation type. Let us remark that an analogous inhomogeneous Baxter's like equation has been already proposed in [60] on the basis of pure functional arguments on the fusion of transfer matrices. Thanks to our SoV construction, we prove in the present article that our inhomogeneous Baxter's like equation does characterize the full transfer matrix spectrum. A direct comparison with [60] is complicated as the Q-function proposed there has 4p more zeros than the Q-function we consider here (see section 5). This article is organized as follows. In section 2 we just recall the basics of the cyclic representations associated to the Bazhanov-Stroganov quantum Lax operator. In section 3 we define the gauged transformed reflection algebra that put into correspondence the most general boundary condition K-matrix with a triangular one. It enables us to adapt the SoV method that we already described in our first article [1] to this more general context, leading in section 4 to the transfer matrix spectrum characterization in this SoV basis. There we also present the scalar product formulae for the so-called separate states containing the transfer matrix eigenstates. In section 5 we show that the spectrum characterization admit a representation in terms of functional equations of Baxter TQ-equation type. Details about the construction of the gauge transformation are given in Appendices A and B together with determinant identities used in the spectrum characterization in Appendix C.

2 Cyclic representations of 6-vertex reflection algebra.

Following Sklyanin's paper [20], we consider the most general cyclic solutions of the 6-vertex reflection equation associated to the Bazhanov-Stroganov Lax operator [121]: (2.13) where:

R 12 (λ/µ) U 1,-(λ) R 21 (λµ/q) U 2,-(µ) = U 2,-(µ) R 21 (λµ/q) U 1,-(λ) R 12 (λ/µ) ( 2
A -(λ) ≡ g -(λ)a(λq -1/2 )d(1/(q 1/2 λ)), D -(λ) = k(λ)A -(q/λ),
a(λ) ≡ a 0 N n=1 ( β n λ + q -1 b n α n a n λ), k(λ) = λ 2 -1/λ 2 (λ 2 /q 2 -q 2 /λ 2 ) , (2.14)
d(λ) ≡ (-1) N a 0 N n=1 a n c n α n ( 1 λ + q d n α n c n β n λ), (2.15)
a 0 is a free nonzero parameter and

g ǫ (λ) ≡ (λα ǫ /q 1/2 -q 1/2 /(λα ǫ ))(λβ -ǫ ǫ /q 1/2 + q 1/2 /(λβ -ǫ ǫ )) (α ǫ -1/α ǫ ) (β ǫ + 1/β ǫ ) , (2.16)
where ǫ = ±1 and we have defined:

(α ǫ -1/α ǫ ) (β ǫ + 1/β ǫ ) ≡ ζ ǫ -1/ζ ǫ κ ǫ , (α ǫ + 1/α ǫ ) (β ǫ -1/β ǫ ) ≡ ζ ǫ + 1/ζ ǫ κ ǫ , (2.17)
Moreover, later we will use:

µ n,h ≡ iq 1/2 (a n β n /α n b n ) 1/2 h = +, iq 1/2 (c n β n /α n d n ) 1/2 h = -.
(2.18)

Following the Sklyanin's paper [20] the next proposition holds: Proposition 2.1. The most general boundary transfer matrix associated to the Bazhanov-Stroganov Lax operator in the cyclic representations of the reflection algebra is defined by:

T (λ) ≡ tr a {K a,+ (λ)U a,-(λ)} (2.19) = a + (λ)A -(λ) + d + (λ)D -(λ) + b + (λ)C -(λ) + c + (λ)B -(λ), (2.20)
It is a one parameter family of commuting operators satisfying the following symmetries proprieties:

T (λ) = T (1/λ), T (-λ) = T (λ).
(2.21)

The boundary quantum determinant:

det q U a,-(λ) ≡ ((λ/q) 2 -(q/λ) 2 )[A -(λq 1/2 )A -(q 1/2 /λ) + B -(λq 1/2 )C -(q 1/2 /λ)] (2.22) = ((λ/q) 2 -(q/λ) 2 )[D -(λq 1/2 )D -(q 1/2 /λ) + C -(λq 1/2 )B -(q 1/2 /λ)], (2.23)
is a central element in the reflection algebra, i.e.

[det q U a,-(λ), U a,-(µ)] = 0, (2.24) and its explicit expression reads:

det q U a,-(λ) = (λ 2 /q 2 -q 2 /λ 2 )A -(λq 1/2 )A -(q 1/2 /λ).
(2.25)

6
where the two sides of the equation belong to End(V 1 ⊗ V 2 ⊗ H) and are defined by the following boundary monodromy matrices:

U a,-(λ) = M a (λ)K a,-(λ) Ma (λ) = A -(λ) B -(λ) C -(λ) D -(λ) a ∈ End(V a ⊗ H), (2.2)
where:

Ma (λ) = (-1) N σ y a M ta a (1/λ) σ y a , (2.3)
and

V a ≃ C 2
is the so-called auxiliary space. Here,

M a (λ) = A(λ) B(λ) C(λ) D(λ) a ≡ L a,N (λq -1/2 ) • • • L a,1 (λq -1/2 ) ∈ End(V a ⊗ H), (2.4)
is the cyclic solution of the 6-vertex Yang-Baxter equation:

R 12 (λ/µ)M 1 (λ)M 2 (µ) = M 2 (µ)M 1 (λ)R 12 (λ/µ) ∈ End(V 1 ⊗ V 2 ⊗ H), (2.5) associated to the R-matrix R ab (λ) =    qλ -q -1 λ -1 0 0 0 0 λ -λ -1 q -q -1 0 0 q -q -1 λ -λ -1 0 0 0 0 qλ -q -1 λ -1    ∈ End(V a ⊗ V b ), (2.6)
and defined in terms of the Bazhanov-Stroganov's Lax operators [121]:

L a,n (λ) ≡ λα n v n -β n λ -1 v -1 n u n q -1/2 a n v n + q 1/2 b n v -1 n u -1 n q 1/2 c n v n + q -1/2 d n v -1 n γ n v n /λ -δ n λ/v n a ∈ End(V a ⊗ R n ), (2.7)
where:

γ n = a n c n /α n , δ n = b n d n /β n .
(

u n v m = q δn,m v m u n with u p n = v p m = 1 ∀n, m ∈ {1, ..., N}, (2.9)
and:

q = e -iπβ 2
K a,± (λ) = a ± (λ) b ± (λ) c ± (λ) d ± (λ) a ≡ K a (λq (1±1)/2 ; ζ ± , κ ± , τ ± ), (2.11)
where:

K a (λ; ζ, κ, τ ) = 1 ζ -1 ζ   λζ q 1/2 -q 1/2 λζ κe τ λ 2 q -q λ 2 κe -τ λ 2 q -q λ 2 q 1/2 ζ λ -λ ζq 1/2   a ∈ End(V a ).
(2.12)

form a left and a right basis of the representation space defining the following decomposition of the identity:

I ≡ p-1 h1,...,hN=0 1≤b<a≤N (X (ha) a -X (ha) a )|βq 2 , h 1 , ..., h N β, h 1 , ..., h N |, (3.11)
with

β, h 1 , ..., h N |βq 2 , k 1 , ..., k N = 1≤a≤N δ ha,ka 1≤b<a≤N 1 X (ha) a -X (hb) b , (3.12)
where

X (hb) b = (b -,b (β)q hb ) 2 + 1/(b -,b (β)q hb ) 2 (3.13)
for the non-zero normalization fixed by

n β =   1≤b<a≤N X (p-1) a -X (p-1) b Ω β |Ω βq 2   1/2 . ( 3

.14)

In this basis the operator family

B -(λ|β) is pseudo-diagonalized: β, h 1 , ..., h N |B -(λ|β) = b h (λ|β) β/q 2 , h 1 , ..., h N |, (3.15)
B -(λ|β)|β, h 1 , ..., h N = |q 2 β, h 1 , ..., h N b h (λ|β), (3.16
)

with simple pseudo-spectrum b 2p -,n (β) = ±1, b p -,m (β) = b p -,n (β), ∀n = m ∈ {1, ..., N}, (3.17
β, h 1 , ..., h N |A -(ζ (ha) a |βq 2 ) = A -(ζ (ha) a ) β, h 1 , ..., h N |T -ϕa a , (3.18)
D -(ζ (ha) a |β)|β, h 1 , ..., h N = T -ϕa a |β, h 1 , ..., h N D -(ζ (ha) a ), (3.19)
where

β, h 1 , ..., h a , ..., h N |T ± a = β, h 1 , ..., h a ± 1, ..., h N |, (3.20)
T ± a |β, h 1 , ..., h a , ..., h N = |β, h 1 , ..., h a ± 1, ..., h N , (3.21)
and: In this paper we want to solve the same type of spectral problem but for the most general boundary conditions. In order to do so we can follow the same approach used in the case of the transfer matrix associated to the spin-1/2 reflection algebra [68]. That is, we introduce the following linear combinations of the original reflection algebra generators:

ζ (h) n = b -,n (β)q h ϕn for h ∈ {0, ..., p -1} and ∀n ∈ {1, ..., 2N}, (3.22)
ϕ a = 1 -2θ(a -N) with θ(x) = {0 for x ≤ 0, 1 for x > 0}. ( 3
A - (λ|β) = -λq 3/2 /β A - (λ) -αqB - (λ) + C - (λ)/(αq) + βD - (λ)/ λq 3/2 /(β/q 2 -q 2 /β) (3.1) B - (λ|β) = -λβ/q 1/2 A - (λ) -αqB - (λ) + β 2 /αq C - (λ) + βq 1/2 /λ D - (λ) /(β - 1/β) (3.2) C - (λ|β) = λq 3/2 /β A - (λ) + αqB - (λ) -q 3 /αβ 2 C - (λ) -q 5/2 /λβ D - (λ) /(β/q 2 -q 2 /β) (3.3) D - (λ|β) = λβ/q 1/2 A - (λ) + αqB - (λ) -C - (λ)/αq -q 1/2 /λβ D - (λ) /(β - 1/β), (3.4)
where β = ±1, ±q 2 and α are arbitrary complex values; to simplify the notation, we won't explicit the dependance in α.

As it is discussed in the appendices A and B, these operators families still satisfy a set of commutation relations which are gauged versions of the reflection algebra commutation relations. In the following we will refer to these families as the gauge transformed reflection algebra generators. In the same appendices, we prove the following theorem, characterizing the representation of these generators:

:

Ω β |B - (λ|β) = b 0 (λ|β) Ω β/q 2 |, B - (λ|β)|Ω β = |Ω q 2 β b 0 (λ|β), (3.5)
where: b

h (λ|β) = b - (β)( λ 2 q -q λ 2 ) N a=1 ( λ b -,a (β)q ha - b -,a (β)q ha λ )(λq ha b -,a (β)
β) = q 1-2h α 2ǫ -, b 2 -,n (β) = -q 1-2h β 2ǫ -, (
) b 2 -,n (β) = q 1-2h µ 2ǫ m,+ , b 2 -,n (β) = q 1-2h µ 2ǫ m,-, (3.7
for any ǫ = ±1, n, m ∈ {1, ..., N} and h ∈ {1, ..., p -1}. Then, the following set of states:

β, h 1 , ..., h N | = 1 n β Ω β | N n=1 hn kn=1 A - (q 1-kn /b -,n (β)|βq 2 ) A - (q 1-kn /b -,n (β)) , (3.9
)

|β, h 1 , ..., h N ≡ 1 n β/q 2 N n=1 hn kn=1 D - (q 1-kn /b -,n (β)|β) D - (q 1-kn /b -,n (β)) |Ω β , (3.10)
where:

Λ ≡ (λ 2 + 1/λ 2 ) and X ≡ q + 1/q (4.2)
and

τ ∞ ≡ κ + κ -(e τ+-τ-N b=1 δ b γ b + e τ--τ+ N b=1 α b β b ) (ζ + -1/ζ + ) (ζ - -1/ζ -) . (4.3)
Proof. This lemma coincides with Lemma 5.1 of our previous paper.

We introduce the following one-parameter family D τ (λ) of p × p matrices:

D τ (λ) ≡                 τ (λ) -a(1/λ) 0 • • • 0 -a(λ) -a(qλ) τ (qλ) -a(1/ (qλ)) 0 • • • 0 0 . . . . . . . . . • • • . . . . . . • • • . . . . . . . . . 0 0 . . . 0 -a(q 2l-1 λ) τ (q 2l-1 λ) -a(1/ q 2l-1 λ ) -a(1/ q 2l λ ) 0 . . . 0 -a(q 2l λ) τ (q 2l λ)                 , (4.4)
where for now τ (λ) is a generic function and we have defined:

a(λ) = a + (λ)A -(λ), (4.5)
where the coefficient a(λ) satisfies the quantum determinant condition:

a(λq 1/2 )a(q 1/2 /λ) = a + (λq 1/2 )a + (q 1/2 /λ)det q U -(λ) (λ/q) 2 -(q/λ) 2 . ( 4.6)
The separation of variables lead to the following discrete characterization of the transfer matrix spectrum.

|τ = p-1 h1,...,hN=0 N a=1 q (ha) τ,a 1≤b<a≤N (X (ha) a -X (hb) b )|β, h 1 , ..., h N , (4.8)
where the gauge parameters α and β satisfy the condition (3.26) and the q (ha) τ,a are the unique nontrivial solutions up to normalization of the linear homogeneous system:

D τ (ζ (0) a )    q (0) τ,a . . . q (p-1) τ,a    =    0 . . . 0    .
(4.9) 10 Proposition 3.1. The quantum determinant can be written in terms of the gauge transformed boundary generators as: the transfer matrix can be written as

det q U - (λ) (λ 2 /q 2 -q 2 /λ 2 ) = A - (q 1/2 λ ǫ |βq 2 )A - (q 1/2 /λ ǫ |βq 2 ) + B - (q 1/2 λ ǫ |β)C - (q 1/2 /λ ǫ |βq 2 ) (3.24) = D - (q 1/2 λ ǫ |β)D - (q 1/2 /λ ǫ |β) + C - (q 1/2 λ ǫ |βq 2 )B - (q 1/2 /λ ǫ |β),
T (λ) = a + (λ)A - (λ|β) + a + (1/λ)A - (1/λ|β) + qc + (λ|β)B - (λ|β/q 2 ) (3.27) T (λ) = d + (λ)D - (λ|β) + d + (1/λ)D - (1/λ|β) + c + (λ|β)B - (λ|β)/q, (3.28)
where we have defined:

a + (λ) = -λ 2 q - 1/qλ 2 λ 2 - 1/λ 2 g + (λ), d + (λ) = λ 2 q - 1/qλ 2 λ 2 - 1/λ 2 g + (q/λ), (3.29) c + (λ|β) = -q(λ 2 q - 1/qλ 2 ) (ββ + /qα + -qα + /ββ + ) β (α + - 1/α + ) (β + + 1/β + ) . (3.30)
Proof. The proof of this statement coincides with the one given in [68] for the XXZ spin 1/2 quantum chain with general integrable boundaries; in fact, this statement is representation independent.

The only difference is that here we have used a Laurent polynomial form while in the XXZ case it was a trigonometric form.

T -spectrum characterization in SoV basis and scalar products

In this section we present the complete characterization of the spectrum of the transfer matrix T (λ) associated to the cyclic representations of the 6-vertex reflection algebra. We first present some preliminary properties satisfied by all the eigenvalue functions of the transfer matrix T (λ):

τ (λ) = N a=1 Λ 2 -X 2 (X (0) a ) 2 -X 2 N b=1 b =a Λ -X (0) b X (0) a -X (0) b τ (ζ (0) a ) + (-1) N (Λ + X) 2 N b=1 Λ -X (0) b X -X (0) b det q M (1) - (-1) N (Λ -X) 2 N b=1 Λ -X (0) b X + X (0) b (ζ + + 1/ζ + ) (ζ + - 1/ζ + ) (ζ - + 1/ζ - ) (ζ -- 1/ζ - ) det q M (i) + (Λ 2 -X 2 )τ ∞ N b=1 (Λ -X (0) b ), (4.1)
where σ y 0 denotes the Pauli matrix. From this relation, and using the specific inner boundary matrix introduced, one can compute the Hermitian conjugate of the boundary monodromy matrix:

U † 0Q (λ) = U t0 0Q (1/λ * ) (4.17)
Then, from the definition of the boundary transfer matrix, and for the special choice of representation here chosen, we can show: ∈ Σ T then it holds:

T † (λ) = T (1/λ * ) ( 4
q(h) τ,a q(h-1) τ,a = a(1/ζ (h-1) a ) d(1/ζ (h-1) a ) q (h) τ,a q (h-1) τ,a . (4.19)
Let us introduce a class of left and right states, the so-called separate states, characterised by the following type of decompositions in the left and right separate basis:

α| ≡ p-1 h1,...,hN=0 N a=1 α (ha) a 1≤b<a≤N (X (ha) a -X (hb) b ) β/q 2 , h 1 , ..., h N | (4.20) |β = p-1 h1,...,hN=0 N a=1 β (ha) a 1≤b<a≤N (X (ha) a -X (
α|β = det M (α,β) (4.22)
where the elements of the size N matrix M (α,β) are given by:

∀(a, b) ∈ [1, N ] 2 , M (α,β) a,b ≡ p-1 h=0 α (h) a β (h) a (X (h) a ) (b-1) (4.23) 12
II) The left T -eigenstate corresponding to τ (λ) ∈ Σ T is defined by the following decomposition in the left SoV-basis:

τ | = p-1 h1,...,hN=0 N a=1 q(ha) τ,a 1≤b<a≤N (X (ha) a -X (hb) b ) h 1 , ..., h N , β/q 2 |, (4.10)
where the gauge parameters α and β satisfy the condition (3.26) and the q(ha) τ,a are the unique nontrivial solutions up to normalization of the linear homogeneous system: q(0) τ,a . . . A simple direct calculation made for example in [110] leads to the following Hermitian conjugate of the monodromy matrix:

q(p-1)
β) = q 1-2h α ±2 + , b 2 -,n (β) = -q 1-2h β ±2 + , ∀h ∈ {1, ..., (
M † 0Q (λ|P Q |P C ) = σ y 0 M 0Q (λ * |P Q |P C )σ y 0 (4.16)
where

x(λ) = λ 2 - 1 λ 2 a(λ), (5.3)
and D τ,i,j (λ) denotes the (p

× (p -1) matrix obtained from D τ (λ) removing the row i and the column j. From the identity:

det p-1 D τ,(p+1)/2,(p+1)/2+1 (λq 1/2 ) = det p-1 D τ,(p+1)/2,(p+1)/2-1 (q 1/2 /λ), (5.4)
and the regularity of these two determinants for λ → ±1 and λ → ±i, it follows that det p D τ (i a q 1/2

) are finites too for a ∈ {0, 1}. Now, our statement about the Laurent polynomiality of degree N+2 of det p D τ (λ) w.r.t. Z follows from the symmetries and from the fact that τ (λ) and x(λ) are Laurent polynomials in λ of degree 2N + 4.

Let us introduce the following notations:

a ∞ = lim λ→+∞ λ -2(N+2) a(λ) = (-1) N+1 κ + κ -α -β -α + N n=1 b n c n q 3+N β + (ζ + -1/ζ + ) (ζ - -1/ζ -) , (5.5) a 0 = lim λ→0 λ 2(N+2) a(λ) = (-1) N+1 q 3+N κ + κ -β + N n=1 a n d n α -β -α + (ζ + -1/ζ + ) (ζ - -1/ζ -) , (5.6)
and

F (λ) = 2N b=1   λ p ζ (0) b p - ζ (0) b p λ p   , (5.7
det p D τ (λ) = F (λ) λ 2p - 1 λ 2p 2 p-1 k=0 (τ ∞ -(q k a ∞ + q -k a 0 )).
(

τ |τ ′ = 0 (4.24)
The computation of such scalar products is the very first step towards the dynamics, several further steps being required to reach this characterization for the models associated to cyclic representations of the 6-vertex reflection algebra: the reconstruction of the local operators in separate variables, the identification of the ground state, the homogeneous and the thermodynamic limit.

For example a rewriting of the determinant representations for the form factors obtained from separation of variable will be necessary to overcome the standard problems related to the homogeneous limit. This problem has been addressed and solved for the XXX spin 1/2 chain, linking the separation of variable type determinants with Izergin's, Slavnov's and Gaudin's type determinants [118,119].

Functional equation characterizing the T -spectrum

The purpose of this section is to characterize the spectrum by functional relations analogous to Baxter's TQ-equation. To begin with, we first need the following property. Lemma 5.1.

Let τ (λ) be a function of λ invariant under the transformation λ → 1/λ and λ → -λ

then det p D τ (λ) is a function of Z = λ 2p + 1 λ 2p , (5.1)
i.e. it is a function of λ p invariant under the transformations λ p → 1/λ p and λ → -λ.

Moreover, if τ (λ) is a Laurent polynomial of degree N + 2 in Λ then det p D τ (λ) is a Laurent polynomial of degree N + 2 in Z.
Proof. The first part of this lemma about the dependence w.r.t. Z of det p D τ (λ) has been already proven in Lemma 5.2 of our previous paper [1] while the second part of this lemma can be proven following the proof given in Proposition 6.1 of the same paper. To adapt this proof here, let us observe that the matrix D τ (i a q h+1/2 ) for a ∈ {0, 1} and h ∈ {0, ..., p -1} contains one row with two divergent elements, i.e. -a(±1) and -a(±i), respectively for a = 0 and a = 1. Nevertheless the determinants det p D τ (i a q h+1/2 ) are all finites for any a ∈ {0, 1} and h ∈ {0, ..., p -1} if τ (i b q k+1/2 ) are finite for any b ∈ {0, 1} and k ∈ {0, ..., p-1}. Indeed, by the symmetries λ p → 1/λ p and λ → -λ all the determinants det p D τ (q h+1/2 ) coincide as well as all the determinants det p D τ (iq h+1/2

). So that we have to prove our statement for one value of q h+1/2 and one value of iq h+1/2

. Now, we can use the expansion of the determinant w.r.t. the central row:

det p D τ (λq 1/2 ) = τ (λ)det p-1 D τ,(p+1)/2,(p+1)/2 (λq 1/2 ) + x(λ)det p-1 D τ,(p+1)/2,(p+1)/2-1 (λq 1/2 ) λ 2 -1/λ 2 - x(1/λ)det p-1 D τ,(p+1)/2,(p+1)/2+1 (λq 1/2 ) λ 2 -1/λ 2 , (5.2)
The discrete characterization of the spectrum given in Theorem 4.1 can be reformulated in terms of Baxter's type TQ-functional equations and the eigenstates admit an algebraic Bethe ansatz like reformulation, as we show in the next theorem. These type of reformulations of the spectrum holds for several models once they admit SoV description, see for example [69,[108][109][110][115][116][117].

In the following we denote with Q(λ) a polynomial in Λ of degree N Q of the form:

Q(λ) = NQ b=1 (Λ -Λ b ) .
(5.24)

Theorem 5.1. For almost all the values of the boundary-bulk parameters such that:

τ ∞ = q -k a ∞ + q k a 0 ∀k ∈ {0, ..., p -1}, (5.25) τ (λ)
∈ Σ T if and only if τ (λ) is an entire function and there exists and is unique a polynomial Q(λ) of the form (5.24) with N Q = (p -1) N, satisfying the following functional equation: (5.26) and the conditions:

τ (λ)Q(λ) = a(λ)Q(λ/q) + a(1/λ)Q(λq) + τ ∞ -(q -NQ a ∞ + q NQ a 0 ) Λ 2 -X 2 F (λ),
(Q(ζ (0) a ), ..., Q(ζ (p-1) a
)) = (0, ..., 0) ∀a ∈ {1, ..., N}.

( ) are finites in the limits λ → ±1, ±i, this is a consequence of the identities:

a(λ)Q(λ/q) + a(1/λ)Q(λq) = x(λ)Q(q/λ) -x(1/λ)Q(λq) λ 2 -1/λ 2 (
D τ (λ)         X 0 (λ) X 1 (λ) . . . . . . X p-1 (λ)         p×1 = τ ∞ -(q -NQ a ∞ + q NQ a 0 ) F (λ)         Λ 2 0 -X 2 Λ 2 1 -X 2 . . . . . . Λ 2 p-1 -X 2         p×1 , (5.30
τ (±i a q 1/2 ) = a(±i a q 1/2
), a(±i a q -1/2 ) = 0, ∀a ∈ {0, 1}.

(5.12)

Explicitly, we have that the nonzero elements of the rows 1, (p + 1) /2 and p are:

Dτ (±i a q 1/2 ) 1,1 = r a,± , Dτ (±i a q 1/2 ) 1,2 = s a,± , (5.13)
Dτ (±i a q 1/2 ) 1,p-1 = - s a,± , Dτ (±i a q 1/2 ) 1,p = - r a,± , (5.14)
Dτ (±i a q 1/2 ) (p+1)/2,(p+1)/2-1 = -x(±i a ), (5.15)
Dτ (±i a q 1/2

) (p+1)/2,(p+1)/2+1 = x(±i a ), (5.16
)

Dτ (±i a q 1/2 ) p,1 = - τ (±i a q 1/2 ), Dτ (±i a q 1/2 ) p,p = τ (±i a q 1/2
),

(5.17)

where we have defined:

r a,± = (-1) a lim λ→±1 τ (i a q 1/2 λ) -a(i a q 1/2 /λ) λ 2 -1/λ 2 , s a,± = (-1) a lim λ→±1 a(i -a q -1/2 /λ) λ 2 -1/λ 2 .
(5.18)

The remaining rows of Dτ (±i a q 1/2 ) produce the tridiagonal part of this matrix. Then, it is possible to prove that this matrix has linear dependent rows; so that det p Dτ (±i a q 1/2 ) = 0. Finally, we can compute the following asymptotic formulae:

∆ ∞ ≡ lim λ→∞ λ -2p(N+2) det p D τ (λ) = det p lim λ→∞ λ -2(N+2) D τ (λ) (5.19) = lim λ→0 λ 2p(N+2) det p D τ (λ) = det p lim λ→0 λ 2(N+2) D τ (λ) t (5.20) = det p           τ ∞ -a 0 0 • • • 0 -a ∞ -xa ∞ xτ ∞ -xa 0 0 • • • 0 0 -x 2 a ∞ x 2 τ ∞ -x 2
-x 2l-1 a ∞ x 2l-1 τ ∞ -x 2l-1 a 0 -x 2l a 0 0 . . . 0 -x 2l a ∞ x 2l τ ∞           , (5.21)
where we have denoted with t the transpose of the matrix and x = q 2(N+2) . We have that ∆

∞ is a degree p polynomial in τ ∞ whose zeros are known from the identities:

∆ ∞ | τ∞=q k a∞+q -k a0 = 0 ∀k ∈ {0, ..., p -1}, (5.22)
so that we get:

∆ ∞ = p-1 k=0 (τ ∞ -(q k a ∞ + q -k a 0 )).
(5. 23) This means that we have determined det we get the symmetry:

X 0 (λ) = X 0 (1/λ), (5.40)
which together with the symmetry X 0 (λ) = X 0 (-λ) implies that X 0 (λ) is a function of Λ.

By using this last result we can rewrite the first equation of the system (5.30) as it follows

∀λ ∈ C : τ (λ)X 0 (λ) -a(λ)X 0 (λ/q) -a(1/λ)X 0 (λq) = τ ∞ -(q -NQ a ∞ + q NQ a 0 ) Λ 2 -X 2 F (λ). ( 5
τ (±i a q 1/2-h λ) at least one of the three non-zero elements is diverging under the limit λ → ±1, ±i. We can proceed as done in the previous theorem, we define the matrix D(1) τ,h (λ) as the matrix with all the rows coinciding with those of D

τ (λ) except the row h + 1, which is obtained by summing the row h and h + 1 of D

τ (λ) and dividing them by ((i

a q h-1/2 λ) 2 -1/(i a q h-1/2 λ) 2
), and the row h, obtained multiplying the row h of D

τ (λ) by

((i a q h-1/2 λ) 2 -1/(i a q h-1/2 λ) 2
). Clearly we have:

det p D(1) τ,h (λ) = det p D (1) τ (λ), (5.42)
and the interesting point is that now all the rows of the matrix D

τ,h (±i a q 1/2-h λ) are finite in the limits λ → ±1, ±i. We have that the nonzero elements of the rows h, h+1 and h of D(1) τ,h (±i a q 1/2-h ) reads:

D(1) τ,h (±i a q 1/2-h ) h+1,h-1 = -s a,± (1 -δ h-1,1 ) + δ h-1,1 ω a , (5.43)
D(1) τ,h (±i a q 1/2-h ) h+1,h = -r a,± (1 -δ h,1 ) + δ h,1 ω a , (5.44)
D(1) τ,h (±i a q 1/2-h ) h+1,h+1 = r a,± (1 -δ h+1,1 ) + δ h+1,1 ω a , (5.45)
D(1) τ,h (±i a q 1/2-h ) h+1,h+2 = s a,± (1 -δ h+2,1 ) + δ h+2,1 ω a , (5.46)
D(1) τ,h (±i a q 1/2-h ) h, h-1 = -x(±i a )(-1) a (1 -δh -1,1 ), (5.47)
D(1) τ,h (±i a q 1/2-h ) h, h+1 = x(±i a )(-1) a (1 -δh +1,1 ), (5.48)
D(1) τ,h (±i a q 1/2-h ) h,h-1 = τ (±i a q 1/2 )(1 -δ h+1,1 ), (5.49)
D(1) τ,h (±i a q 1/2-h ) h,h+1 = -τ (±i a q 1/2 )(1 -δ h+1,1 ), (5.50)
where we have defined:

ω a = (-1) a q 2 -1/q 2 . ( 5
τ (±i a q 1/2-h ) = 0 and the following factorization holds:

det p D (1) τ (λ) = λ 2p - 1 λ 2p P τ (λ) .
(5.52)

18

where we have used the notations:

Λ i = q 2i λ 2 + 1 q 2i λ 2 .
(5.31)

From the condition τ (λ) ∈ Σ T and the assumption of general values of the boundary-bulk parameters (5.25), we know that det p D τ (λ) is a non-zero polynomial, so defining:

Z detpDτ = ±i a q h+1/2 , ±ζ (h) n ∀a ∈ {0, 1} , n ∈ {1, ..., 2N}, h ∈ {0, ..., p - 1} , (5.32)
we can solve the previous system of equations for any value of λ ∈ C\Z detpDτ by the Cramer's rule:

X i (λ) = τ ∞ - (q -NQ a ∞ + q NQ a 0 ) (Z 2 - 4) p-1 k=0 [τ ∞ - (q k a ∞ + q -k a 0 )] det p D (i+1) τ (λ), (5.33)
where D (i) τ (λ) is the p × p matrix obtained replacing the column i by the column at the r.h.s. of (5.30). Let us now rewrite the system of equation (5.30) bringing the first element in the last one for the two column vectors:

Dτ (λ)        X 1 (λ) X 2 (λ) . . . X p-1 (λ) X 0 (λ)        p×1 = τ ∞ - (q -NQ a ∞ + q NQ a 0 ) F (λ)        Λ 2 1 - X 2 Λ 2 2 - X 2 . . . Λ 2 p-1 - X 2 Λ 2 0 - X 2        p×1 , (5.34)
where it is easy to see that Dτ (λ) = D τ (λq). Rescaling now the argument of the functions, we can rewrite it as it follows:

D τ (λ)        X 1 (λ/q) X 2 (λ/q) . . . X p-1 (λ/q) X 0 (λ/q)        p×1 = τ ∞ - (q -NQ a ∞ + q NQ a 0 ) F (λ)        Λ 2 0 - X 2 Λ 2 1 - X 2 . . . Λ 2 p-2 - X 2 Λ 2 p-1 - X 2        p×1 , (5.35)
so that it must hold:

X i+1 (λ/q) = X i (λ) ∀λ ∈ C\Z detpDτ , i ∈ {0, ..., p - 1} (5.36)
where we have used the notation X p (λ) ≡ X 0 (λ), or equivalently:

X a (λ) = X 0 (λq a ) ∀λ ∈ C\Z detpDτ , a ∈ {1, ..., p - 1} .
(5.37)

Let us observe now that, from their definition, X a (λ) are continuous functions of λ so the above equation must be indeed satisfied for any value of λ ∈ C. Moreover, from the identity:

det p D (1) τ (λ) = det p D (1) τ (1/λ), (5.38)
which we can prove by some simple exchange of rows and columns, and from the fact that:

∀i ∈ {0...p - 1}, λ → 1/λ ⇒ Λ i → Λ p-i
(5.39)

Taking the limit λ → ±i a with a ∈ {0, 1}, we obtain:

τ (±i a )P τ (±i a ) = 1 ±2i a dx dλ (±i a ) (P τ (±i a /q) -P τ (±i a q)) + x(±i a ) lim λ→±i a P τ (λ/q) λ 2 -1/λ 2 - P τ (λq) λ 2 -1/λ 2 , (5.63)
so that using the previous result (5.61) and the identity: for a ∈ {0, 1}, ǫ ∈ {-1, 1}, we obtain: τ (±i a q)P τ (±i a q) = a(±i a q)P τ (±i a ) + a(±i a /q)P τ (±i a q 2

lim λ→±i a P τ (λ/q) λ 2 -1/λ 2 = lim
),

(5.66) τ (±i a /q)P τ (±i a /q) = a(±i a /q)P τ (±i a /q 2 ) + a(±i a q)P τ (±i a ), (5.67) implying:

P τ (±i a /q 2 ) = -P τ (±i a q 2 ) = 0, (5.68)
being a(±i a q ǫ ) = 0 for a, ǫ ∈ {0, 1}. We can iterate these computations for λ = ±i a q bǫ for any a ∈ {0, 1}, ǫ ∈ {-1, 1} and b ∈ {2, ..., (p -3) /2} obtaining that: (5.69)

P τ (±i a /q 2b ) = -P τ (±i a q 2b ) = 0,
In the cases λ = ±i a q ±1/2 as a(±i a /q 1/2 ) = 0 the functional equation for P τ (λ) give us:

τ (±i a q ±1/2 )P τ (±i a q ±1/2 ) = a(±i a q 1/2
)P τ (±i a q ∓1/2 ), (5.70) which being P τ (±i a q 1/2 ) = -P τ (±i a q -1/2 ) and τ (±i a q ±1/2 ) =a(±i a q 1/2 ) = 0 implies the identity:

P τ (±i a q 1/2 ) = -P τ (±i a q -1/2 ) = 0,
(5.71) so that the factorization (5.54) is proven and we get that:

X 0 (λ) = τ ∞ -(q -NQ a ∞ + q NQ a 0 ) p-1 k=0 [τ ∞ -(q k a ∞ + q -k a 0 )] Qτ (λ), (5.72) is a polynomial of degree N Q = (p -1)
N in Λ which has the form (5.24). This follows by taking the asymptotic of its functional equation so that we can fix: Q(λ) ≡ X 0 (λ), (5.73) hence giving a constructive proof of the existence of the polynomial Q-function solution of the equation (5.26). The fact that it is unique is shown observing that if Q(λ) is another polynomial solution then:

D τ (λ)         Q(λ) -Q(λ) Q(λq) -Q(λq) . . . . . . Q(λq p-1 ) -Q(λq p-1 )         p×1 =         0 0 . . . . . . 0         p×1 , (5.74) 20
Here P τ (λ) is a Laurent polynomial of degree 2(p -1)N + 2p in λ, with the following odd parity:

P τ (1/λ) = -P τ (λ) , (5.53)
being det p D (1) τ (λ) a polynomial of degree (p -1)N + 2p in Λ. Here, we want to prove that in fact:

det p D (1) τ (λ) = λ 2p - 1 λ 2p 2 Qτ (λ) , (5.54)
where Qτ (λ) is a polynomial of degree (p -1)N in Λ. In order to do so we write down the equation:

τ (λ)R τ (λ) = a(λ)R τ (λ/q) + a(1/λ)R τ (λq) + Z 2 -4 Λ 2 -X 2 p-1 k=0 τ ∞ -(q k a ∞ + q -k a 0 ) F (λ), (5.55)
where for convenience we have denoted R

τ (±i a )R τ (±i a ) = 1 ±2i a dx dλ (±i a ) (R τ (±i a /q) -R τ (±i a q)) + x(±i a ) lim λ→±i a R τ (λ/q) λ 2 -1/λ 2 -R τ (λq) λ 2 -1/λ 2 , (5.56)
now by using the known identities:

R τ (±i a ) = R τ (±i a /q) = R τ (±i a q) = 0, (5.57) R τ (λ/q) λ 2 -1/λ 2 = R τ (q/λ) λ 2 -1/λ 2 , (5.58)
we get:

lim λ→±i a R τ (λ/q) λ 2 -1/λ 2 = -lim λ→±i a R τ (λq) λ 2 -1/λ 2 , (5.59)
and so being x(±i a

) = 0 lim λ→±i a R τ (λ/q) λ 2 -1/λ 2 = 0.
(5.60)

These results imply the identities:

P τ (±i a /q) = -P τ (±i a q) = 0.
(5.61)

We can now write the functional equation for P τ (λ):

τ (λ)P τ (λ) = a(λ)P τ (λ/q) + a(1/λ)P τ (λq) + λ 2p - 1 λ 2p Λ 2 -X 2 p-1 k=0 τ ∞ -(q k a ∞ + q -k a 0 ) F (λ).
(5.62)

Now replacing the first row R 1 with the following linear combination of rows:

R1 = R 1 + p-2-h i=0 i j=0 a(1/ζ ( h+j) a ) τ (ζ ( h+j+1) a ) R 2+i , (5.83)
we get R1 = Wa, h 0

• • • 0 0 1×p (5.84)
where:

Wa, h = W a, h + p-2-h i=0 i j=0 a(1/ζ ( h+j) a ) τ (ζ ( h+j+1) a ) W a, h+1+i (5.85)
and so:

Q(ζ ( h) a ) = Wa, h p-1 k = h,k=0 τ (ζ (k) a ) = 0, (5.86
τ ∞ = q -k a ∞ + q k a 0 , (5.87
(λ)Q(λ) = a(λ)Q(λ/q) + a(1/λ)Q(λq), (5.88)
and the conditions:

(Q(ζ (0) a ), ..., Q(ζ (p-1) a
)) = (0, ..., 0) ∀a ∈ {1, ..., N}.

(5.89)

Proof. First let us assume that τ (λ) and Q(λ) satisfies the homogeneous Baxter equation with τ (λ) entire function and Q(λ) polynomial of the form (5.24) with N

Q ≤ (p -1) (N+1) and N Q = k mod p,
then from this same equation it follows that τ (λ) is a polynomial of the form (4.1). Moreover, for any fixed λ ∈ C we can construct the following homogeneous system of equations:

D τ (λ)         Q(λ) Q(λq) . . . . . . Q(λq p-1 )         p×1 =         0 0 . . . . . . 0         p×1 , (5.90
T (ζ (k) a ) = p-1 k=0 a(1/ζ (k) a ), (5.78)
from which in particular follows:

p-1 k=0 τ (ζ (k) a ) = p-1 k=0 a(1/ζ (k) a ).
(5.79)

Let us remark now that the r.h.s and the l.h.s of the above equation are continuous w.r.t. the boundary-bulk parameters so that the above identity holds also if we take the special limit µ By definition of the function Q(λ) under these conditions and limit on the bulk parameters we get:

Q(ζ ( h) a ) ∝ det p                   W a, h -a(1/ζ ( h) a ) 0 • • • 0 0 W a, h+1 τ (ζ ( h+1) a ) -a(1/ζ ( h+1) a ) 0 • • • 0 W a, h+2 0 τ (ζ ( h+2) a ) -a(1/ζ ( h+1) a ) . . . . . . • • • . . . W a,p-1 0 • • • 0 τ (ζ (p-1) a ) 0 • • • . . . . . . . . . 0 W a, h . . . 0 0 τ (ζ ( h-2) a ) -a(1/ζ ( h-2) a ) W a, h
Once we recall that C

C 1,p (λ) = y 1,1 C 1,1 (λq -1 ), C 1,2 (λ) = y -1 1,1 C 1,1 (λq)
(5.102) and defined x

1,1 ≡c 1,1 /c 1,2 =c 1,p /c 1,1
, we obtain the following Baxter equation in the polynomial

C 1,1 (λ): t(λ)C 1,1 (λ) = (x 1,1 y 1,1 ) a(λ)C 1,1 (λq -1 ) + (1/(x 1,1 y 1,1 )) a(1/λ)C 1,1 (λq), (5.103)
and computing the above equation in λ = q 1/2 we get:

t(q 1/2 )C 1,1 (q 1/2 ) = (x 1,1 y 1,1 ) a(q 1/2 )C 1,1 (q -1/2 ), (5.104)
from which it follows x

1,1 y 1,1 = 1 once we recall that C

1,1 (q 1/2 ) = 0 and that C 1,1 (λ) is even under λ → 1/λ. So, we can define: ), (B.12) ) - Moreover, these gauge transformed operators satisfy the following parity properties: (q -1/q)(λ 2 β/q -q/λ 2 β) Proof. Both the commutation relations and the parity properties here presented coincide with those derived in [68] for the case of the XXZ spin 1/2 quantum chain with general integrable boundaries. This is the case as they are clearly representation independent. Here we are just writing them in a Laurent polynomial form instead of a trigonometric form.

Q(λ) ≡ C 1,1 (λ), (5.105)
A -(λ 2 |βq 2 )B -(λ 1 |β) = (λ 1 q/λ 2 -λ 2 /qλ 1 )(λ 1 λ 2 /q -q/λ 1 λ 2 ) (λ 1 /λ 2 -λ 2 /λ 1 )(λ 1 λ 2 - 1/λ 1 λ 2 ) B -(λ 1 |β)A -(λ 2 |β) + (λ 1 λ 2 /q -q/λ 1 λ 2 )(λ 1 β/qλ 2 -λ 2 q/βλ 1 )(q - 1/q) (λ 1 /λ 2 -λ 2 /λ 1 )(λ 1 λ 2 - 1/λ 1 λ 2 )(β/q -q/β) B -(λ 2 |β)A -(λ 1 |β) + (λ 1 λ 2 /β -β/λ 1 λ 2 )(q - 1/q) (λ 1 λ 2 - 1/λ 1 λ 2 )(β/q -q/β) B -(λ 2
1 λ 2 /q -q/λ 1 λ 2 ) (λ 1 /λ 2 -λ 2 /λ 1 )(λ 1 λ 2 - 1/λ 1 λ 2 ) D -(λ 2 |βq 2 )B -(λ 1 |β) - (λ 1 λ 2 /q -q/λ 1 λ 2 )(λ 2 βq/λ 1 -λ 1 /λ 2 βq)(q - 1/q) (λ 1 /λ 2 -λ 2 /λ 1 )(λ 1 λ 2 - 1/λ 1 λ 2 )(βq - 1/βq) D -(λ 1 |βq 2 )B -(λ 2 |β) - (λ 1 λ 2 β - 1/λ 1 λ 2 β)(q - 1/q) (λ 1 λ 2 - 1/λ 1 λ 2 )(
(λ 1 λ 2 /β - 1/λ 1 λ 2 )(q - 1/q) (λ 1 λ 2 - 1/λ 1 λ 2 )(β/q -q/β) B -(λ 1 |β)C -(λ 2 |βq 2 ) = A -(λ 2 |βq 2 )A -(λ 1 |βq 2 ) - (λ 1 λ 2 /β - 1/λ 1 λ 2 )(q - 1/q) (λ 1 λ 2 - 1/λ 1 λ 2 )(β/q -q/β) B -(λ 2
A -(λ|β) = - (q - 1/q)(λ 2 q/β -β/λ 2 q) (β/q 2 -q 2 /β)(λ 2 - 1/λ 2 ) D -(λ|β) + (β/q -q/β)(λ 2 /q -q/λ 2 ) (β/q 2 -q 2 /β)(λ 2 - 1/λ 2 ) D - (
(β/q 2 -q 2 /β)(λ 2 - 1/λ 2 ) A -(λ|β) + (β/q -q/β)(λ 2 /q -q/λ 2 ) (β -

B.2 Representation of the gauge transformed Reflection algebra

In the bulk of the paper we have anticipated that for almost all the values of the boundary, bulk and gauge parameters the operator family B -(λ|β) is pseudo-diagonalizable. We will show this 2λ det q M (1)( The identities: From it follows the fact that the matrix M is diagonal:

λ q 1/2 + q 1/2 λ ) N b=1 Λ -X (hb) b X -X (hb) b + |β, h (-1) N iq 1/2 2λ ζ -+ 1/ζ - ζ - -1/ζ - det q M (i)( λ q 1/2 - q 1/2 λ ) N b=1 Λ -X (hb) b X + X (hb) b + q λ 2 /q -q/λ 2 N b=1 (Λ -X (
A ∞,0 - = q ∓1 D 0,∞ -, B 0 -= -q 2 B ∞ -, C 0 -= -q 2 C ∞ -, (
β/q 2 , h|β, k = β,

C Properties of cofactor

In this appendix we prove a lemma giving the main properties of the cofactors of the matrix D τ (λ). Proof. Let us remark that independently from the explicit form of τ (λ) the following identities hold:

C 1,p (q 1/2-p /µ a,+ ) = p-1 j=1
a(µ a,+ q j ) = 0 ∀a ∈ {1, ..., N}, (C.7) so that C 1,p (λ) is a non-zero polynomial in λ which implies the statement on the rank of D τ (λ).

The proof of the above symmetry properties is standard we just need to make some exchange of rows and columns to bring the matrix in the determinant defining the cofactor in the l.h.s into the matrix defining the cofactor in the r.h.s..

M (h)
1,1 (±i a q h λ) are finite in the limits λ → 1 and so the same is true for their determinants. In fact, it is possible to show that these lines are linear dependents in each one of the matrices

M (h)
1,1 (±i a q h ), so that: ) so that we cannot remove here the divergence as we have done before. However, we can proceed differently, let us explain it in the case h = (p + 1)/2 as in the other case we can proceed similarly. In the last row of M 1,1 (±i a q (p+1)/2 λ) under the limit λ → 1 the last element tend to τ (i a q 1/2 ), finite nonzero value, and the next to last tend to a(±i a q -1/2 ) = 0, all the others on this row are zero. So that C 1,1 (±i a q (p-1)/2

det p-1 M (h) 1,1 (±i a
) is finite iff det p-2 D (1,p),(1,p) (±i a q (p+1)/2 ) is finite. This is shown using the following expansion of the determinant: det p-2 D (1,p),(1,p) (±i a q (p+1)/2 λ) = τ (λ)det 1

As for h = 0 the matrix M1,1(±i a ) does not contain any singular elements.

Résumé de thèse

Les principaux outils pour la compréhension du comportement macroscopique de systèmes quantiques à partir de leur description microscopique sont la détermination du spectre du Hamiltonien associé et le calcul des fonctions de corrélation. Cette thèse se place dans le cadre du développement d'un tel programme de recherche afin d'étudier des systèmes intégrables quantiques avec des conditions aux bords intégrables générales, le but à long terme étant la description exacte d'une physique quantique hors équilibre.

Chapitre un : modèles intégrables quantiques

Pour un nombre fini de degrés de liberté, il existe une définition rigoureuse de l'intégrabilité en mécanique classique due à Liouville. Cependant, déjà pour les modèles classiques continus, il n'y a pas d'analogue universel du théorème de Liouville, car la notion "d'un nombre infini de charges conservées indépendantes" doit être minutieusement examinée au cas par cas. La situation pour les systèmes quantiques n'est pas meilleure, et ce même pour un nombre fini de degrés de liberté, car il n'existe pas non plus un analogue du théorème de Liouville. Cependant, depuis les travaux pionniers de Bethe, de grands efforts ont été menés pour inventer des méthodes puissantes afin de résoudre le spectre et même la dynamique de nombreux systèmes quantiques d'intérêt. Le cadre de travail le plus fructueux dans cette logique est fourni par la méthode de diffusion inverse quantique (QISM) introduite par Fadeev, Sklyanin et Taktadjan. Celle-ci peut-être considérée comme une version quantifiée de la méthode de diffusion inverse classique, qui permet de résoudre des théories continues avec des solutions de type solitons, tout en fournissant également une version algébrique de la méthode de l'ansatz de Bethe. Il hérite également de la stratégie développée pour les modèles sur réseau à 2 dimensions en utilisant la notion de matrices de transfert, qui commutent. En fait, ce cadre de travail contient également les ingrédients nécessaires à la définition d'un analogue quantique à la méthode de séparation des variables (SoV), qui, dans un certain sens, pourrait être considéré comme une définition pratique de l'intégrabilité quantique. Comme pour le cas classique, la première étape pour être capable de diagonaliser un Hamiltonien quantique est de trouver un ensemble complet de charges conservées qui commutent, ayant un spectre commun simple, menant ainsi à une caractérisation complète des états propres en utilisant les nombres quantiques correspondants. Bien sûr ceci n'est a priori pas suffisant car il faut encore construire les vecteurs propres correspondants et calculer leurs niveaux d'énergie.

En fait, nous voulons également déterminer les fonctions de corrélations dynamiques. La méthode QISM fournit le cadre algébrique pour aborder un tel problème pour une large classe de systèmes, principalement définis sur des réseaux 1D. Le premier chapitre présente cette méthode pour l'exemple particulier de l'algèbre de Yang-Baxter à 6-vertex, qui peut décrire une large classe de modèles d'intérêts physiques, tels la chaine de spins de Heisenberg, le modèle de sine-Gordon sur réseau ou bien le modèle de chiral Potts. Ensuite, nous rappelons l'ansatz de Bethe algébrique en nous focalisant sur l'exemple de la chaine de spins XXZ-1/2. Une petite partie du chapitre est dédiée au calcul des facteurs de forme des opérateurs locaux et aux fonctions de corrélations de ce modèle. Ensuite, nous définissons rigoureusement la classe de modèle à l'étude dans cette thèse, c'est-à-dire ceux associés à des représentations cycliques de l'algèbre de Yang-Baxter à 6-vertex.

Chapitre deux : modèles intégrables quantiques avec bords

L'étude des modèles avec bords a soulevé un grand enthousiasme de recherche, car ils peuvent décrire à la fois une physique à l'équilibre et hors équilibre. Quelques applications intéressantes concernent la description de processus de relaxation stochastique classique, comme le modèle ASEP et les propriétés de transport dans les chaines de spin.

Dans le premier chapitre, nous avons rappelé que le Hamiltonien d'un système intégrable quantique peut-être obtenu à partir de la matrice de transfert. Cependant, avec la définition précédente comme la trace de la matrice de monodromie, il se trouve que les modèles que l'on peut reconstruire possèdent des conditions aux bords périodiques. Dans le but de décrire des conditions aux bords intégrables plus générales, le chapitre deux rappelle une première généralisation, en considérant une matrice de transfert twistée. En utilisant une représentation scalaire de l'algèbre de Yang-Baxter, on peut en effet déformer la matrice de monodromie, ce qui in fine mène à la description de conditions aux bords quasi-périodiques.

Cependant, le principal but de ce chapitre est de souligner le rôle majeur joué par l'algèbre de réflexion introduite par Sklyanin. En particulier, celle-ci permet de définir une matrice de transfert avec bords (définie comme la trace d'une matrice de monodromie avec bords), qui peut être utilisée pour générer des Hamiltoniens quantiques intégrables avec des bords intégrables généraux.

Le dernier paragraphe est dédié au rappel de la procédure de fusion, qui permet de définir des Hamiltonians de chaines de spins s quelconque. Dans tout ce chapitre, la chaine de spin XXZ est utilisée comme l'exemple principal pour illustrer et expliquer les différents outils et concepts.

Chapitre trois : séparation quantique des variables

Nous introduisons ici l'importante notion de separation des variables pour les systèmes intégrables. En commençant par le point de vue classique, nous rappelons la définition usuelle de séparation des variables. Ensuite, nous montrons l'implémentation de ce concept aux modèles quantiques (Sklyanin), en insistant sur l'analogie entre les descriptions classiques et quantiques. La séparation quantique des variables permet la caractérisation du spectre de la matrice de transfert dans le cadre de la méthode de diffusion inverse quantique (QISM). De façon remarquable, elle permet la description de modèles pour lesquels un état de référence |0y, utilisé dans la méthode de l'ansatz de Bethe algébrique, ne peut pas être obtenu directement. Une fois encore, afin d'illustrer nos propos nous utilisons la chaine de spin XXZ-1/2 (antiperiodique) et construisons de façon explicite pour ce modèle la base séparée (le principal objet associé à la méthode de séparation quantique des variables). Nous profitons également de cet exemple pour souligner les principales motivations à travailler avec cette méthode.

Chapitre quatre : le spectre de la matrice de transfert avec bords par la séparation quantique des variables

Dans ce chapitre nous étudions le problème spectral de la matrice de transfert avec bords pour des représentations cycliques de l'algèbre de réflexion à 6-vertex associée aux opérateurs de Lax de Bazhanov-Stroganov. Comme déjà mentionné, des cas particuliers importants sont contenus dans ce modèle, tels les chaines de spin s avec anisotropie à la racine de l'unité, le modèle de sine-Gordon sur réseau à la racine de l'unité ou encore le modèle de Potts chiral.

En implémentant la séparation quantique des variables pour ces représentations avec les bords intégrables les plus généraux, nous caractérisons de deux façons différentes le spectre de la matrice de transfert. La première caractérisation fait intervenir les solutions d'un système discret d'équations polynomiales dans une certaine classe de fonctions. La seconde caractérisation, équivalente, fait quant à elle intervenir les solutions d'une équation de Baxter fonctionnelle de type TQ. Cela permet de faire le lien dans certains cas particuliers avec la méthode de l'anstaz de Bethe algébrique, qui ne permet pas d'étudier ces modèles en toute généralité.

Les conditions aux bords sont décrites par une matrice K ´pλq à gauche et K `pλq à droite. Afin d'être plus compréhensible, nous présentons notre méthode en deux étapes. Dans un premier temps, nous considérons une matrice K ´pλq générale et une matrice K `pλq triangulaire. Dans ce cas, la base séparée est obtenue comme la base de diagonalisation de l'opérateur Bpλq (une des entrées de la matrice de monodromie avec bords). Ensuite, afin de décrire des bords intégrables complètement généraux, nous avons introduit une transformation de jauge, principalement en adaptant pour notre situation, une méthode introduite par Baxter. En choisissant astucieusement les paramètres de jauge, nous pouvons ramener ce problème général à l'étude d'une matrice de bords (jaugée) triangulaire, nous ramenant ainsi au cas traité précédemment. Dans ce cas, la base séparée est obtenue comme la base de (pseudo-)diagonalisation de l'opérateur Bpλq jaugé, qui est une combinaison linéaire de toutes les entrées de la matrice de monodromie avec bords.

Les résultats décrit dans ce chapitre ont donnés lieu aux deux publications I et II.

Chapitre cinq : Hamiltoniens locaux associés aux représentations cycliques de l'algèbre de réflexion Nous résolvons le problème de définir des Hamiltoniens locaux avec des bords intégrables qui commutent avec la matrice de transfert avec bords associée à des représentations cycliques de l'algèbre de réflexion à 6vertex. Ce résultat permet de modéliser les interactions à un niveau microscopique, et d'ainsi comprendre la physique du système étudié.

Nous suivons les étapes suivantes afin de déterminer une expression pour les Hamiltoniens. Tout d'abord, nous considérons la matrice R fondamentale, solution de l'équation de Yang-Baxter qui entremêle deux opérateurs de Lax de Bazhanov-Stroganov agissant sur deux espaces quantiques différents. Nous définissons ensuite une équation de réflexion mélangée, entremêlant ces opérateurs de Lax et deux matrices scalaires de bords, définies respectivement sur l'espace auxiliaire (de dimension 2) et l'espace quantique (de dimension p). En résolvant explicitement cette équation mélangée, nous trouvons des solutions scalaires diagonales pour les matrices de bords sur l'espace quantique, associées aux matrices scalaires diagonales connues sur l'espace auxiliaire (les matrices K ´pλq et K `pλq du chapitre quatre, mais choisies ici diagonales).

Nous pouvons alors définir une famille à plusieurs paramètres (spectraux) de matrices de transfert fondamentales, qui commutent entre elles et avec la matrice de transfert avec bords associée à l'algèbre de réflexion cyclique originelle. De plus, nous prouvons que ces matrices de transfert fondamentales génèrent des familles à plusieurs paramètres de Hamiltoniens locaux avec bords intégrables. Il est utile de mentionner que ces Hamiltoniens locaux sont obtenus comme des dérivées secondes, par rapport aux paramètres spectraux, des matrices de transfert fondamentales.

Le chapitre finit en explicitant les Hamiltoniens locaux pour quelques modèles d'intérêt, pour des espaces quantiques locaux de dimension 3 : la chaine XXZ de spin 1 à la racine de l'unité, le modèle de Potts chiral super intégrable, le modèle de sine-Gordon à la racine de l'unité et une généralisation de celui-ci. Ils sont tous obtenus à partir de l'opérateur de Lax de Bazhanov-Stroganov, avec un choix approprié des paramètres.

Les résultats décrit dans ce chapitre ont donnés lieu à la publication III.

La conclusion de la thèse énonce des directions dans lesquelles ces travaux peuvent être poursuivis et utilisés. Par exemple, le rôle de la matrice R fondamentale dans les équations de réflexions pour des représentations cycliques doit être clarifié. Une autre direction concerne l'étude des équations de Baxter de type TQ dans le contexte de bords intégrables généraux, et en particulier la compréhension du lien entre la matrice de transfert fondamentale et l'opérateur Q. Une autre façon encore de continuer ces travaux est de s'attaquer au problème de la dynamique en utilisant la séparation quantique des variables.

Même s'il reste encore beaucoup à faire, par exemple en ce qui concerne la caractérisation de l'état fondamental, ces techniques devraient permettre de calculer le paramètre d'ordre du modèle de Potts chiral d'une façon assez directe.

Thesis abstract

The main theoretical tools to understand the macroscopic behaviour of quantum systems from their microscopic description are the determination of their Hamiltonian spectrum and the computation of their correlation functions. This thesis takes place in the development of such a research program to study quantum integrable models with general integrable boundary conditions, the long-range goal being to be able to exactly describe out of equilibrium physics.

More specifically, we have analysed the class of integrable quantum models on the lattice associated to cyclic representations of the 6-vertex reflection algebra, including as particular cases the lattice sine-Gordon model at root of unity and the chiral Potts model with general integrable boundaries.

A large part of the work has been devoted to the development of the quantum separation of variables method to solve the spectral problem for these models with general integrable boundary conditions, by generalising the Baxter's gauge transformations to these cyclic reflection algebras.

We have completely characterised the transfer matrix spectrum (both eigenvalues and eigenstates) in terms of the set of solutions to a discrete system of polynomial equations and equivalently as the set of solutions, in a given class of functions, to a Baxter like functional equation. This last point allows in particular cases to make a link with the Algebraic Bethe Ansatz approach, which in general, cannot be used for the study of these models.

We have then constructed families of new local Hamiltonians with integrable boundaries commuting with the above transfer matrix. To that end, we have defined a hierarchy of new mixed reflection equations, involving different representations of the 6-vertex algebra and using, among others, the fundamental R-matrix. keywords: Yang-Baxter algebra, 6-vertex reflection algebra, cyclic representations, quantum separation of variables, local Hamiltonians

Résumé de thèse

Les principaux outils pour la compréhension du comportement macroscopique de systèmes quantiques à partir de leur description microscopique sont la détermination du spectre du Hamiltonien associé et le calcul des fonctions de corrélation. Cette thèse se place dans le cadre du développement d'un tel programme de recherche afin d'étudier des systèmes intégrables quantiques avec des conditions aux bords intégrables générales, le but à long terme étant la description exacte d'une physique quantique hors équilibre.

Plus spécifiquement, nous avons analysé la classe des systèmes intégrables quantiques sur réseau associés aux représentations cycliques de l'algèbre de réflexion à 6-vertex, avec comme exemples les modèles de sine Gordon et de Potts chiral avec conditions aux bords intégrables.

Une large partie du travail a été consacrée au développement de la méthode de séparation quantique des variables pour résoudre le problème spectral de la matrice de transfert de ces modèles avec conditions de bords intégrables les plus générales, en étendant l'idée des transformations de jauge de Baxter à ces algèbres de réflexion.

Nous avons caractérisé complètement le spectre de la matrice de transfert (valeurs propres et vecteurs propres) en termes des solutions d'un système discret d'équations polynomiales et d'une façon équivalente en termes des solutions, dans une certaine classe de fonctions, d'une équation de type Baxter fonctionnelle. Cela permet de faire le lien dans certains cas particuliers avec la méthode de l'anstaz de Bethe algébrique qui ne permet pas d'étudier ces modèles en toute généralité.

Nous avons ensuite construit des familles de nouveaux Hamiltoniens locaux avec conditions aux bords intégrables qui commutent avec la matrice de transfert. Pour ce faire nous avons défini une hiérarchie de nouvelles équations de réflexion mélangeant différentes représentations de l'algèbre quantique à 6-vertex et utilisant, entre autres, la matrice R fondamentale cyclique. mots-clés : algèbre de Yang-Baxter, algèbre de réflexion à 6-vertex, représentations cycliques, séparation quantique des variables, Hamiltoniens locaux

  (4.26) and using the identities p4.24q-p4.25q, one easily obtains that xΩ| and ˇˇΩ D are Bpλq-eigenstates: xΩ| Bpλq " b 0 pλq xΩ| and Bpλq ˇˇΩ D " ˇˇΩ D b p-1 pλq (4.27) with non-zero eigenvalues: b 0 pλq " b ´pλqapλ{q 1{2 qap1{pλq 1{2 qq and b p-1 pλq " b ´pλqapq 1{2 λqapq 1{2 {λq (4.28)

(4. 32 ) 4 . 2 . 1 .

 32421 Proposition For generic boundary and bulk parameters satisfying (4.23) and:

ånd H respectively: xh|Bpλq " b h pλqxh| ( 4 . 48 )

 448 Bpλq|hy " |hyb h pλq (4.49) The eigenvalues b h pλq have the same expressions (4.30)-(4.31), but now with the more general roots ξ phq n .

(4. 63 )

 63 Separate basisLet |τ y be a right eigenvector of the boundary transfer matrix associated to the eigenvalue τ pλq: T pλq |τ y " τ pλq |τ y (4.64) By acting with T pξ phnq n q on the left and on the right, the computation of the element xh| T pξ phnq n q |τ y leads to: τ pξ phnq n qΨ τ phq " apξ phnq n qΨ τ pT ń phqq `ap1{ξ phnq n qΨ τ pT ǹ phqq (4.65)

Proposition 4 . 3 . 1 .

 431 The gauge transformed boundary operators satisfy, among others, the following com-VARIABLES mutation relations:

5 .

 5 The boundary transfer matrix admits the following expression in terms of the gauged boundary matrix K`p λ|α, βq and of the gauged generators:

Proposition 4 . 3 . 9 .

 439 For almost any values of the bulk and boundary parameters, the set t|τ yu of all the eigenvectors of the boundary transfer matrix forms a basis of the Hilbert space H. VARIABLES Proof. Let us consider the following special representation, where the bulk parameters satisfy: c n " ´bn ; d n " ´an and α nβ n " a nb n(4.177)

Proposition 4 . 3 . 10 .

 4310 Let tpλq be a function of λ invariant under the transformation λ Ñ 1{λ and λ Ñ ´λ then det D t pλq is a function of Z " λ 2p `1 λ 2p (4.181)

4. 3 .Proposition 4 . 3 . 11 .

 34311 THE MOST GENERAL INTEGRABLE BOUNDARIES polynomial of degree N `2 in Λ, even if the elements ap˘1q and ap˘iq diverge. Let us thus introduce the following asymptotic behaviours: a 8 " lim λÑ`8 λ ´2pN`2q apλq and a 0 " lim λÑ0 λ 2pN`2q apλq (4.182) as well as the function: Then we have a functional expression for the determinant of the matrix D τ pλq: For almost all the values of the boundary and bulk parameters, T pλq has simple spectrum and τ pλq of the form p4.68q is an element of Σ τ if and only if det D τ pλq is a Laurent polynomial of degree N `2 in the variable Z which satisfies the following functional equation:

Corollary 4 . 3 . 14 ." 1

 43141 The left and right transfer matrix eigenstates associated to τ pλq P Σ τ admit the following Bethe ansatz like representations: xτ | " xβ, ω| N Q ź bBpλ b |α, βq and |τ y " Bpλ b |α, βq|β, ωy (4.195) where the λ b (fixed up to the symmetries λ b Ñ ´λb and λ b Ñ 1{λ b ), for b P 1, N Q , are the zeros of Qpλq and we have imposed the condition p4.154q on the gauge parameters.

4. 4 .

 4 A FIRST STEP TOWARD THE DYNAMIC4.4.1 Separate states and their scalar productsLet us introduce a class of left and right states, the so-called separate states, characterised by the following type of decompositions in the left and right separate basis:

Proposition 5 . 1 . 6 (

 516 Unitarity of the fundamental R-matrix). When the fundamental R-matrix exits, it satisfies:S ab pP a |P b |P C qS ba pP b |P a |P C q " ρpP a |P b |P C q 1 (5.47) with the scalar function ρpP a |P b |P C q " Σ pQq ab Σ pRq ab (5.48) 5.1. FUNDAMENTAL R-MATRIX Proof. By the mixed Yang-Baxter equation (5.1), it follows that S ab pP a |P b |P C qS ba pP b |P a |P C q 9 1. Using the next relation on the Boltzmann weights, clear from the definition of proposition 5.1.1: @h P 0, p ´1 , W P Q phqW QP phq " 1 for any points pP, Qq P `C4 ˘2 (5.49) the expression (5.17) for the matrix elements, and the lemma 5.1.5 we can in fact compute the value of the proportionality coefficient. The symmetry Σ pXq ab " Σ pXq ba concludes the proof.

Proposition 5 . 1 . 7 .

 517 The Υ operation leaves invariant the different sets P C of proposition 5.1.4. It means that one can apply the Υ operation on S ba , the existence of S Υa ba , S Υ b ba and S Υ b Υa ba is guaranteed without any other requirements. Proposition 5.1.8 (Crossing-unitarity of the fundamental R-matrix). When the fundamental R-matrix S ba exists, it holds: S ta ba pP b |P a |P C qS ta ab pP a |P Υ b Υ b b |P C q " ρpP a |P b |P C q1 (5.52) with the scalar function ρpP a |P b |P C q " ρpP Υ ´1 a a |P b |P C q (5.53)

Proposition 5 . 2 . 1 .

 521 Let V a and V b be two vector spaces. If one knows a representation T ´,a in the space V a of an (ABCD)-type quantum quadratic algebra A, then one can dress the solution thanks to the operators A ab and B ab or thanks to the operators C ab and D ab to get a representation in the space V a b V b . For example, A ab T ´,a B ab and `Ct b ab ˘´1 T ´,a `D´1 ab ˘tb are representations of the (ABCD)-quantum quadratic algebra A in the vector space V a b V b .

5 . 82 )

 582 Let us recall that in chapter 4 we have shown that the boundary transfer matrices associated to the Bazhanov-Stroganov Lax operators (1.72) have simple spectrum. Let us see what happens when considering two non isomorphic spaces.

T 2 "

 2 1 for parameters of point C) of proposition 5.1.4(5.96)

177 )

 177

4 . 2 .

 42 Indeed, in the expression (5.154) the bulk interactions `1 are eliminated by the zero factor tr a tK `apξ|P ´|P C qu (proposition 5.4.4), and the remaining terms are scalars (proposition 5.4.5).

183 )Proposition 5 . 4 . 6 .

 183546 This form(5.182) for the second order derivative is very interesting as it is similar to the one obtained for the first order derivative: we can see the interactionsř N ´1 k"1 H pxq k,k`1 and two terms only acting on site 1 and site N, the boundaries. What is more, contrary to what happens with the first order derivative, 98 5.4. EXPRESSION OF LOCAL HAMILTONIANS the next proposition makes usable the second order derivative both to compute local Hamiltonians and to show their commutation with the boundary transfer matrix: For generic parameters, and for any direction pxq of derivation, it holds: α pxq ‰ 0 (5.184)

100 5. 5 .

 1005 SOME EXPLICIT (NEW) MODELS FOR 3-DIMENSIONAL LOCAL QUANTUM SPACES • The Fateev-Zamolodchikov model. With the set of parameters:

2 14 1

 214 Representation of the reflection algebra in -(λ)-eigenstates basis 9 4.3 Change of basis and SoV spectral decomposition of the identity 13 4.4 Separate states and their scalar products SciPost Phys. 2, 009 (2017)

3 , 4 SciPost

 34 we define the associated representations of the cyclic 6-vertex reflection algebra. In section 4, we prove the diagonalizability of the generator -(λ) of the reflection algebra generated by -(λ) for the most general K -(λ) boundary matrix while we impose one constraint on the parameters of the Bazhanov-Stroganov Lax-operator for any quantum site to make easier the explicit construction of the --eigenstates basis. Moreover, we compute the scalar product for the so-called separate states in the --eigenstates basis. In section 5, we show that the --eigenstates basis is the SoV-basis for the transfer matrix spectral problem associated to the most general K --boundary matrix and upper triangular K + -boundary matrix and we solve in this SoV basis this spectral problem. In section 6, we show that the SoV characterization of the transfer matrix spectrum is equivalent to inhomogeneous Baxter's like TQ-functional equation with polynomial Q-functions. We present four appendices, in the first one we extend the proof of section 4 for the diagonalizability of the operator -(λ) to the case of general values of the boundary and bulk parameters. The remaining three appendices deal with the reduction of our representations to those associated to the chiral-Potts, the sine-Gordon and the XXZ spin s-chains at the 2s+1 roots of unit. 2 Cyclic representations of the 6-vertex Yang-Baxter algebra In this section we recall the basics of the cyclic representations of the 6-vertex Yang-Baxter algebra associated to the Bazhanov-Stroganov Lax-operator. We consider the representations defined by the tensor product of N local representations of the 6-vertex Yang-Baxter algebra on the local Hilbert spaces n . Each local representation is defined as the representation of a to a root of unit q, where u n and v n are the Weyl algebra generators on the Hilbert spaces n . Here, we assume that u n and v n are unitary operators and that it holds: p even and p = 2l + 1 odd. This type of representation can be defined on a p-dimensional linear space n , imposing that the v n spectrum coincides with the p-roots of the unit: n is defined a p-dimensional representation of the Weyl algebra by setting: u n |k, n〉 = |k + 1, n〉 ∀k ∈ {-l, ..., l}, Phys. 2, 009 (2017) Sklyanin has used the 6-vertex case and the associated XXZ spin 1/2 quantum chains [80, 106-108] to develop an explicit example of this construction. However as pointed out in [3] similar constructions applies also to other 6-vertex cases like the non-linear Schrödinger and the Toda chains as well as for models associated to the 8-vertex case like the XYZ spin 1/2 quantum chains. Further integrable quantum models with open boundary conditions have been presented following the Sklyanin's construction. Interesting examples are the higher spin open quantum chains [17, 18], the higher rank open quantum spin chains [40-42] and the Hubbard model [109-112] with integrable open boundaries [43-48]. In fact, this mainly results in the possibility to associate to any closed integrable quantum model (characterized by a solution of the Yang-Baxter equation) new open integrable quantum models (characterized by the associated Sklyanin's solutions of the reflection equation).

[ 3 ]

 3 which us allows to associate to any solution M a (λ) ∈ End( 2 ⊗ ) of the 6-vertex Yang-Baxter equation a solution a

) 6 SciPost

 6 Phys. 2, 009 (2017) n is also called the right local quantum space at the site n of the chain. Let n be the dual space of n then we can define the following scalar products: 〈k, n|k , n〉 = ((〈k, n|) † , |k , n〉) ≡ δ k,k , (2.6) for any k, k ∈ {-l, ..., l}.

. 10 ) 2 ⊗ 2

 1022 where a and b denote two bidimensional spaces V a , V b ≡ 2 and R a b (λ) is an endomorphism on their tensor product, i.e. R a b (λ) ∈End(

) 4

 4 SoV representation of cyclic 6-vertex reflection algebra In this section we construct the left and right basis which diagonalize the one-parameter family of commuting operators -(λ) associated to the most general K -(λ) matrix. Here we impose one constraint on the parameters of the representation at any quantum site: b p n + a p n = 0, ∀n ∈ {1, ..., N}.

) 8 SciPost

 8 Phys. 2, 009 (2017) with a,- (λ) and a,+

  both solution of the reflection equation (3.1), where:

. 13 ) 3 . 1 .

 1331 PropositionThe following boundary quantum determinant:

(4. 22 )

 22 Theorem 4.1 (Left -(λ) SOV-representations). If b -(λ) = 0 and it holds:

  ..., p -1} and N is a free normalization, define a -(λ)-eigenstates basis of * : 〈h| -(λ) = B h (λ)〈h|.

(4. 26 )

 26 Here we have denoted 〈h| ≡ 〈h 1 , ..., h N |. The quantum determinant and the following left action on the generic state 〈h|:

  completely determine the representation of the other generators of the reflection algebra in the -(λ)-eigenstates basis. Indeed, the representation of -(λ) follows from the identity(3.18) while -(λ) by the quantum determinant.

( 4 .

 4 14) it follows that 〈Ω| is a -(λ)-eigenstate with non-zero eigenvalue:

  -(λ)-eigenstates basis The left and right SoV-representations of the cyclic 6-vertex reflection algebra are now defined by constructing the left and right -(λ)-eigenstates basis and by determining in this new basis the representation of the other generators of the algebra. In order to present our results

( 4 . 40 )

 440 Indeed, the representation of -(λ) follows from the identity (3.18) while -(λ) is given by the quantum determinant.

( 4 . 42 )

 442 By using them, the definition of the states |h〉 and the quantum determinant, we get our interpolation formula for the right action of -(λ) on |h〉. Let us remark that in fact, the chosen gauge for the coefficients of -(λ) is consistent with the quantum determinant condition as we have:

12 SciPost

 12 h ∈ {0, ..., p -1} and a ∈ {1, ..., N}. Phys. 2, 009 (2017)

[ 99 ]

 99 to prove the validity of (4.26). The action of -(ζ (h b ) b ) for b ∈ {1, ..., 2N} follows from the definition of the states 〈h|, the reflection algebra commutation relations (4.31) and the quantum determinant relations. Let us show now that the conditions (4.23) and (4.24) imply that the set of p N states 〈h| is a -(λ)-eigenstates basis of * . As by condition (4.23) each such state is associated to a different eigenvalue of -

N

  + x N | with x j ∈ {0, 1} for any j ∈ {1, ..., N} are nonzero, which just prove the validity of the induction.

Theorem 4 . 2

 42 (Right -(λ) SOV-representations). If b - (λ) = 0 and (4.23)-(4.24) are satisfied, the states:

  )with N the same coefficient as in (4.25), define a - (λ)-eigenstates basis of :

  the following choice of the normalization: N

  numbers, meaning that the coefficients of these separate states have a factorized form in this basis. These separate states are interesting at least for two reasons : they admit simple determinant scalar products, as it will be shown in the next proposition, and the eigenstates of the transfer matrix are special separate states, as we will show in the next section. Proposition 4.2. Let us take an arbitrary separate left state 〈α| (separate covector) and an arbitrary separate right state |β〉 (separate vector) then it holds:

  p N × p N matrices U (L) and U (R) defining respectively the change of basis from the original left and right basis, formed by the v n -eigenstates basis: 〈h| ≡ ⊗ N n=1 〈h n , n| and |h〉 ≡ ⊗ N n=1 |h n , n〉, (4.46) to the left and right --eigenstates basis:

. 51 )

 51 Proof.Note that the action of a left --eigenstate on a right --eigenstate is zero, for two different --eigenvalues. This implies that the matrix M is diagonal; then to compute its diagonal elements we compute the matrix elements θ a ≡ 〈h 1 , ..., h a , ..., h

( 4 . 52 )

 452 Using the left action of the operator-(ξ (h a +1) a

. 7 )

 7 Lemma 5.2. Let τ(λ) be a function of λ invariant under the transformation λ → 1/λ then det p D τ (λ) is a function of λ p invariant under the transformation λ p → 1/λ p . Proof. Let us observe that for the invariance of the function τ(λ) under λ → 1/λ, we have that: we have denoted by O R the operation on a p × p matrix which exchanges the couple of rows p -i with i + 2 for any i ∈ {0, ..., (p -3) /2}, similarly O C is the operation on a p × p matrix which exchanges the couple of columns p -i with i + 2 for any i ∈ {0, ..., It is then trivial to see that:

  ) for any n ∈ {1, ..., N} and h∈ {0, ..., p -1} N , i.e. a system of p N Baxter-like equations in the wave-functions:

  ) for any n ∈ {1, ..., N} and for any h m =n in {0, ..., p -1}. Then the condition τ(λ) ∈ Σ implies the compatibility equations for these linear systems, i.e. it must hold: detD τ (ξ (0) a ) = 0, ∀a ∈ {1, ..., N}

( 4 .

 4 23), (4.24) and (5.15). Then (up to an overall normalization) the solution is unique and independent from the h m =n ∈ {0, ..., p -1} for any n ∈ {1, ..., N}. So fixing τ(λ) ∈ Σ there exists (up to normalization) one and only one corresponding -eigenstate |τ〉 with coefficients of the factorized form given in (5.17)-(5.18); i.e. the -spectrum is simple. Vice versa, if τ(λ) is in the set of functions (5.1) and satisfies (5.16), then the state |τ〉 defined by (5.17)-(5.18) satisfies:

.34) 18 SciPost

 18 Phys. 2, 009 (2017) Note that the condition b + (λ) = 0, keeping instead if desired a c + (λ) = 0, can be simply realized by the following renormalization of the boundary parameters κ + = e -γ κ + and e τ + = e τ + -γ by sending γ → +∞. Under this limit the asymptotic of the transfer matrix reads:

  , p -1}, n ∈ {1, ..., N} (5.15) and (4.23)-(4.24) are satisfied, then (λ) has simple spectrum and Σ coincides with the set of polynomials τ(λ) of the form (5.1) with (5.14) which satisfy the following discrete system of equations: detD τ (ζ (0) a ) = 0, ∀a ∈ {1, ..., N}. (5.16) I) The right -eigenstate corresponding to τ(λ) ∈ Σ is defined by the following decomposition in the right SoV-basis:

  and Dτ (λ) is the family of p × p matrices defined substituting in D τ (λ) the coefficient A(λ)

. 23 ) 1 ≤ p - 1 .

 2311 which in particular implies that the action of 〈τ| on |τ 〉 is zero. SciPost Phys. 2, 009 (2017) are satisfied for any h ∈ {1, ..., x} and let us prove it for h = x + We have that by the SoV equations it holds:

  these conditions we can explicitly construct the left and right basis which diagonalize the one-parameter family of commuting operators -(λ) and -(λ) associated to the most general K -(λ) matrix. In the previous section we have done this construction for the -(λ) eigenstates basis, clearly we can do a similar construction also for -(λ). In the previous sections we have explained how the spectral problem of the transfer matrix (λ) can be characterized by SoV in the -(λ) eigenstates basis when b + (λ) = 0 keeping instead if desired a c + (λ) = 0, by the same approach we can characterize the (λ)-spectrum by SoV in the -(λ) eigenstates basis when c + (λ) = 0 keeping instead if desired a b

= - 1

 1 where ε = +1 for b + (λ) = 0 (keeping instead a c + (λ) = 0) and ε for c + (λ) = 0 (keeping instead a b + (λ) = 0). Let us moreover introduce the following notations:

. 42 )= - 2

 422 by the SoV Baxter's like equation and the condition: This means that the second formula in (5.40) and (5.41) are both satisfied for h = 1 once the initial conditions t (-1) τ,a = 0, t (0) τ,a = 1 are imposed. So let us assume that these two identities λ → ±i. This implies that in both the limits λ are the only points for which det p Dτ (λ) may have divergencies, then, from τ(λ) and X(λ) Laurent's polynomial in λ of degree 2N+4, it follows that det p Dτ (λ) is a polynomial of degree N + 2 in the variable Z. Let us now remark that by the SoV characterization of the spectrum we have that τ(λ) ∈ Σ if and only if it has the form we are assuming that (6.7) is satisfied. Finally, it is simple to verify that the following asymptotic hold

2 in

 2 1)N which admits the following interpolation formula: then (λ) has simple spectrum and τ(λ) of the form (5.1) with (5.14) is an element of Σ if and only if det p Dτ (λ) is a Laurent polynomial of degree N + which satisfies the following functional equation: det p Dτ (λ) -F (λ) 1 a=0

. 9 ) 2 ( 1 1 (λq 1/ 2 ) λ 2 - 1 /λ 2 - 1 (q 1/ 2 /

 9211221212 Here Dτ (λ) is obtained from D τ (λ) by substituting in it A(λ) with A(λ). Proof. Let us observe that det p Dτ (λ) is an even function of λ as a consequence of the parity of τ(λ) and A(λ) moreover following the same steps of Lemma 5.2 we can prove that det p Dτ (λ) is invariant under the transformations λ → 1/λ and λ → qλ, so that det p Dτ (λ) is indeed a function of Z = λ 2p + 1/λ 2p . Let us now observe that in the points row of the matrix Dτ (λ) has two elements which are divergent as proportional to A(±q p/2 ) and A(±iq p/2 ), respectively. In the following we prove that: we ask that the function τ(λ) has the functional form (5.1). Let us first introduce the notausing it let us expand the determinant: det p Dτ (λq 1/2 ) =τ(λ)det p-1 Dτ,(p+1)/2,(p+1)/Dτ,(p+1)/2,(p+1)/2regards to the central row. Here, we have denoted with Dτ,i,j (λ) the (p -1) × (p -1) matrix defined by removing the row i and the column j to the matrix Dτ (λ). The following identity holds: det p-1 Dτ,(p+1)/2,(p+1)/2+1 (λq 1/2 ) = det p-1 Dτ,(p+1)/2,(p+1)/2it follows that just exchanging the row j with the row p -j, for any j ∈ {1, .., (p + 1) /2}, and then the column j with the column p -j, for any j ∈ {1, .., (p + 1) /2}, in the matrix Dτ,(p+1)/2,(p+1)/2+1 (λq 1/2 ). Note that the determinants det p-1 Dτ,(p+1)/2,(p+1)/2+1 (λq 1/2 ) and det p-1 Dτ,(p+1)/2,(p+1)/2-1 (λq 1/2 ) are Laurent's rational functions both finite for λ → ±1 and SciPost Phys. 2, 009 (2017) in the form (6.27).

1 )

 1 ∈ Σ . So we are left with (6.33)-(6.35) a linear system of N}. Let us define by induction the following coefficients: a,h+1 = x a,0,h a,h + x a,-,h a,h-1 , ∀h ∈ {1, ..., p + 1} and where we have defined: ξ s(n,h n ) ≡ ξ (h n -1) n , ξ (p-1)N+1 is arbitrary. Moreover, we use the notation:

  1)N of the power expansion of the polynomial Q(λ). Once this notation are introduced, the previous characterization of the spectrum can be reformulated in terms of Baxter's type TQ-functional equations and the eigenstates admit an algebraic Bethe ansatz like reformulation, as we show in the next theorem. Theorem 6.1. Let the conditions (4.23), (4.24), (5.15)and (6.7) be satisfied and let τ(λ) be an entire function for which there exists a polynomial Q(λ) of the form(6.22)

( 6 . 26 )

 626 with (p -1)N -1 ≤ N Q , then τ(λ) ∈ Σ and (up to normalization) the left and right transfer matrix eigenstates associated to it admit the following Bethe ansatz like representations: where the λ b (fixed up the symmetry λ b → -λ b , λ b → 1/λ b ) for b ∈ {1, ..., N Q } are the zeros of Q(λ). Vice versa, if τ(λ) ∈ Σ then there exists a polynomial Q(λ) of the form (6.22) with (p -1)N -1 ≤ N Q satisfying with τ(λ) the Baxter's equation (6.26).

26 ) 2 )( 6 . 43 )

 262643 that τ(λ) is a polynomial in Λ of the form (5.1) with (5.14). This together with the equations: det p Dτ (ξ (0) a ) = 0, ∀a ∈ {1, ..., N}, (6.29) which are trivial consequences of (6.26) and of (6.25), imply by the SoV characterization that τ(λ) ∈ Σ . Let us show now that the eigenstates associated to this τ(λ) ∈ Σ can be written Proof. The condition ζ + = iz + with z + ∈ {-1, +1} implies: τ(iq 1/2 ) = A(iq 1/implies that G(λ|q ∞ , q 1

  So finally we are left with a system of N inhomogeneous equations in theN unknown Q(ξ (0) a )for all a ∈ {1, ..., N}, which always admits a nontrivial solution, i.e.{Q(ξ(0) 1 ), ..., Q(ξ (0) N )} = {0, ..., 0}.(6.41) Finally, let us point out that the degree of the polynomial Q(λ) is constrained by the Baxter's equation (6.26). Indeed, it is trivial to remark that if N Q ≤ (p -1)N -2, then the equation (6.26) admits only the trivial solution Q(λ) = 0 which is not compatible with equation (6.41). Instead, the equation (6.26) may still be satisfied with a nontrivial Q(λ) only if the following condition: lim Λ→∞ Λ (p-1)N-1

  .42) is satisfied. It is clear that this represents one supplementary condition and one can expect that up to some exceptional case, related to the values of the parameters of the representation and to some special choice of the τ(λ) ∈ Σ , it is not satisfied and so that N Q = (p -1)N. It is then simple to prove the following corollary which provides under some further constraints a complete characterization of the transfer matrix eigenvalues and eigenstates in terms of solution of ordinary Bethe equations. Corollary 6.1. If the conditions (4.23), (4.24), (5.15) and (6.7) are satisfied and if we fix the boundary parameter ζ + = iz + with z + ∈ {-1, +1} and the following global condition: κ + e ε(τ --τ + ) 0, then (λ) has simple spectrum and τ(λ) ∈ Σ if and only if τ(λ) is entire and there exists a polynomial Q(λ) of the form (6.22), with N Q ≤ (p -1)N, which satisfies the following homogeneous Baxter equation:

(6. 44 )

 44 The (λ 1 , ..., λ N Q ) entering in the Bethe ansatz like representations of the eigenstates (6.27) are solutions of the associated ordinary Bethe equations. SciPost Phys. 2, 009 (2017) or they are identically zero or they can be zero only over subspaces of nonzero codimension in the space of the bulk parameters. So here we have just to prove that the bn,m are not identically zero to derive that (A.8) holds for almost all the values of the parameters. To do so we can just recall that from the results derived in the previous section it holds: bp -,m = q p/2 ..., N }. From which the condition (A.8) holds as soon as we impose: Once we have proven this statement, then we have just to use the reflection algebra to construct an eigenbasis of -(λ) and this is done just by a repeated action of the generators -(λ) computed in the zeros of -(λ) on the eigenstates |Ω R 〉 and 〈Ω L |. That is we repeat the construction of the eigenbasis presented in Section 4.2 by substituting to the reference states defined in (4.1) with the -(λ) on the eigenstates |Ω R 〉 the state 〈Ω L |. Note that such an action can generate a null vector only if some of the zeros of -(λ) coincides with the zeros of the quantum determinant anyhow as discussed in Section 4.2 under the condition (4.24) we are always able to chose an appropriate set of p N zeros of -(λ) which do not coincides with those of the quantum determinant so that we generate exactly p N independent states.

. 12 )

 12 each local representation being defined as the representation of a local Weyl algebra to a root of unit q, where U n and V n are the Weyl algebra generators on the Hilbert space n . Let us introduce the monodromy matrices for the Yang-Baxter and reflection equations of the lattice sine-Gordon model, they read:

  also the boundary transfer matrix of the sine-Gordon model, which reads: Let us remark that we have used the upper index sG in the above two monodromy matrices and transfer matrix to point out that they are related to the sine-Gordon model while we will continue to denote with M a (λ), a,- (λ) and (λ) those associated to the original τ 2 -model. The following lemma establishes the connection between the monodromy and transfer matrices of the sine-Gordon model and the original τ 2 -model. 28 SciPost Phys. 2, 009 (2017) then it trivially follows that -(λ)

2 )

 2 the operator family is not nilpotent, in particular does not exist any state annihilated by -(λ) for any λ ∈ . That is does not exist a reference state and we cannot use ABA to analyze the spectral problem of the transfer matrix. Here, we show that under the condition (A.2) for general values of the bulk parameters the -(λ) is indeed diagonalizable and with simple spectrum. Let us first prove the following lemma: Lemma A.1. There exists at least one simultaneous eigenstate: |Ω R 〉, 〈Ω L | (A.4) of the one parameter family of commuting operators -(λ):

-

  We can always put -,1 in a Jordan normal form, let us denote with B 1 the right eigenspace associated to a given eigenvalue b-,1 = 0 of -,1 , as -,1 is invertible. If this eigenspace is one dimensional we have found our simultaneous eigenstate of -(λ). If this is not the case then by the commutativity we have that B 1 is an invariant space with regards to -,n for any n ∈ {1, ..., N }. So we can always put -,2 in a Jordan normal form in B 1 , let us denote with B 1,2 the eigenspace associated to a given eigenvalue b-,2 = 0 of -,2 . If this eigenspace is one dimensional we have found our simultaneous eigenstate of -(λ) otherwise we can reiterate this procedure. We can have two possibilities both at the step n ≤ N we find that the eigenspace B 1,...,n is one-dimensional or we arrive up to the eigenspace B 1,...,N in both the cases we have (at least) one simultaneous eigenstate of the -,n for any n ∈ {1, ..., N } and so one eigenstate of -(λ). Similarly, we can prove the existence of 〈Ω L |. From the previous Lemma and the reflection algebra equation we can prove the following proposition. Proposition A.1. Under the condition (A.2) for almost all the values of the bulk parameters, the operator family -(λ) is diagonalizable and it has simple spectrum and its average value is central and it holds: -(λ) = p a=1 Let us observe that by definition the operator family -(λ)is a polynomial in the bulk parameters so the same must be true for its spectrum. This implies in particular that defined bn

2 -

 2 once we define the parameters of the τ model in terms of those of the lattice sine-Gordon model as it follows: a 2n+y = κ 2 2n+y

2 -

 2 model and the lattice sine-Gordon model, we get:

-

  (p + 1) ∈ {-2s, -2(s -1), ..., 2s} mod 2p (A.53) Proof. Let us denote |a, n〉 = 0

2 =

 2 Let us now impose that in our representation the v n -eigenstates coincide with the elements of the canonical basis: |p -a, n〉 = |a + 1, n〉 ∀a ∈ {0, ..., p -1}, (A.58) we can verify now the formulae (A.52)-(A.53). The formula (A.53) is equivalent to: holds for the following identities: q (S z n +p+1)/2 |a + 1, n〉 = |a + 1, n〉q (2(s-a)+p+1)/|p -a, n〉 q p-a

  n〉 ∀a ∈ {1, ..., p}.Under these constraints the class of the inhomogeneous chiral Potts representations is charac-terized by the following identity:

  ∀n ∈ {1, ..., Here we show that imposing a set of conditions on the parameters of the τ 2 -Lax operator we can reduce it to the one of the spin s = (p -1) /2 XXZ case at the p roots of unit. This has the interesting consequence that the analysis done of the open τ 2 -chain reduces for these special representations to that of an open spin chain under the same boundary conditions.

  is the two sites local Hamiltonian of the spin s XXZ chain, K scalar solutions of the reflection algebra obtained by doing the fusion p-1 times starting from the original 2×2 scalar solutions K 0,± (λ), respectively. Lemma A.3. Let us fix the parameters of the τ 2 -representations as follows:

. 8 )

 8 Theu n ∈End(R n ) and v m ∈End(R m )are unitary Weyl algebra generators:

  Hilbert spaces and the full representation space of the cyclic Yang-Baxter and reflection algebra is defined by the tensor product of the local quantum spaces, i.e.H = ⊗ N n=1 R n .Moreover, we consider here the most general boundary matrices defined as:

)

  and the operator families A -(λ|β) and D -(λ|β) in the zeros of B -(λ|β) act as simple shift operators:

. 23 ) 8 3

 238 Let us comment that the existence of the states Ω β | and |Ω β can be proven by a general argument which we present in Appendix B. For general representations, the pseudo-spectrum of B -(λ|β), i.e. the values of b -,n (β) and b -(β), must be computed by recursion on the number of sites. However, in Appendix B we present the explicit expression for b -,n (β) and b -(β) in some particular representations.The interest in these gauge transformed boundary generators is due to the possibility to use them to rewrite the transfer matrix associated to the most general cyclic 6-vertex reflection algebra representations in a simple form, as presented in the following proposition: Gauged cyclic reflection algebra and SoV representationsIn our previous paper we solved the spectral problem associated to the transfer matrix of the cyclic representations under the requirement that one of the boundary matrices is triangular, i.e.b + (λ) ≡ 0.

Theorem 3 . 1 .

 31 For almost all the values of the boundary-bulk-gauge parameters there exit a left Ω β | and a right |Ω β pseudo-eigenstate of B -

  for h= (h 1 , ..., h N ) ∈ {0, ..., p

Theorem 4 . 1 .

 41 For almost all the values of the boundary-bulk parameters T (λ) is diagonalizable and it has simple spectrum and Σ T coincides with the set of polynomials τ (λ) of the form (4.1) which satisfy the following discrete system of equations: det D τ (ζ (0) a ) = 0, ∀a ∈ {1, ..., N}.

(4. 7 )

 7 I) The right T -eigenstate corresponding to τ (λ) ∈ Σ T is defined by the following decomposition in the right SoV-basis:

( 3 .

 3 25) ǫ = ±1. Moreover, if we set the gauge parameter α to:

Lemma 4 . 1 .

 41 Denote by Σ T the transfer matrix spectrum, then any τ (λ) ∈ Σ T is an even function of λ symmetrical under the transformation λ → 1/λ which admits the following interpolation formula:

(4. 12 )

 12 Proof. The Theorem 3.1 implies that for almost all the values of the gauge-boundary-bulk parameters the conditions (3.7)-(3.8) hold. Here, we need to prove also that for almost all the values of the boundary-bulk parameters we have, b 2 -,n

  with those in (B.64) and (B.65) and impose the N conditions (B.63). Under these conditions, Theorem 3.1 implies the pseudo-diagonalizability of B -(λ|β) and fixes the spectrum of its zeros b -,n (β) by (B.66); so that the inequality (4.13) is satisfied. As we have proven that for almost all the values of the boundary-bulk parameters the inequalities (4.13), (B.64) and (B.65) hold, to prove this theorem we have just to follow the same proof given in the non-gauged case, i.e. the proof of Theorem 5.1 of our previous paper.Let us comment that with respect to this last theorem here we are stating also the diagonalizability of the transfer matrix for almost any value of the parameters of the representation. This last statement can be proven as it follows. Let us consider the following special representation, where the bulk parameters satisfy:

  0

p

  D τ (λ) in N + 2 different values of Z together with the asymptotic for Z → ∞. From which the characterization (5.8) trivially follows.

. 41 )

 41 Let us now prove that det p D (1) τ (λ) is indeed a polynomial of degree (p -1)N + 2p in Λ. Note that in the following when we refer to a row k ∈ Z what we mean is the row k ′ ∈ {1, ..., p} with k ′ = k mod p. In the row h = (p + 1)/2 + h of D

  τ (±i a ) = 0. Let us now compute the functional equation for P τ (λ) in the points λ = ±i a q ǫ

  for any b ∈ {1, ..., (p -3) /2} .

τ

  (λ) =det p D (1) τ (λ). The above equation is a direct consequence of the equation satisfied by X 0 (λ) and of the definition of this last function in terms of det p D (1) τ (λ). Now let us consider the following limit on the above equation λ → ±i a with a ∈ {0, 1}:

) 22 from

 22 which it follows Q(λ) ≡ Q(λ) as D τ (λ) is invertible for any λ ∈ C\Z detpDτ . Finally, let us show that Q(λ) satisfies the condition (5.27). By the definition (5.33), Q(λ) is a continuous function of the boundary-bulk parameters, then it is enough to prove this statement for some value of these parameters to show that it holds for almost all the values of these parameters. Let us impose the condition (B.63), where the ratio β/α is fixed by (3.26), then the following identities are satisfied: a(ζ (0) a ) = 0 ∀a ∈ {1, ..., characterization of the transfer matrix spectrum holds for any value of the boundarybulk parameters satisfying the inequalities (B.64)-(B.65). So in particular if we impose: µ nk,-= 1/(q 1+k µ a,+ ) ∀k ∈ {1, ..., p -1}, (5.76) for some n k ∈ {1, ..., N}\{a} once we have chosen any a ∈ {1, ..., N}. Under these conditions it holds: a(ζ (k) a ) = 0, ∀k ∈ {1, ..., p -1}, (5.77) and the SoV representation implies the following centrality condition:

a,-→ q 1

 1 -p /µ a,+ for which it holds a(1/ζ (p-1) a ) = 0 and so we get: ∃! h ∈ {0, ..., p -1} : τ (ζ ( h)

a

  polynomial in Λ of maximal degree (p -1) (N + 1), which satisfies the homogeneous Baxter equation as required. Let us introduce now the following states: β, ω| = is a degree N polynomial in Λ. As first remarked in the papers [61,104], from the polynomial characterization of the Q-function and the SoV characterization it follows the Bethe-like rewriting of the transfer matrix eigenstates stated in the following: Corollary 5.1. The left and right transfer matrix eigenstates associated to τ (λ) where the λ b (fixed up the symmetry λ b → -λ b , λ b → 1/λ b ) for b ∈ {1, ..., N Q } are the zeros of Q(λ) and we have imposed the condition (3.26) on the gauge parameters. 24 which is satisfied as a consequence of the Baxter equation. Finally, being (Q(λ), ..., Q(λq p-1 )) non-zero for any λ ∈ C, up to at most a finite number of values, we get: detD τ (λ) = 0 ∀λ ∈ C (5.91) so that Proposition 5.1 implies τ (λ) ∈ Σ T .To prove the reverse statement we use the results of the Lemma C.1 on the matrix D τ (λ) and on its cofactors:

  so by Lemma C.1 it follows that rankD τ (λ) = p -1 for any λ ∈ C\K, where K is a finite set of complex numbers if not empty. Then the matrix composed of the cofactors of the matrix D τ (λ) has rank 1 for any λ ∈ C\K. This just means the proportionality:V i (λ) = a i,j (λ)V j (λ) ∀λ ∈ C\K, ∀i, j ∈ {1, ..., p} (5.94)where we have defined:V i (λ) ≡ (C i,1 (λ), C i,2 (λ), ..., C i,p (λ)) ∀λ ∈ C\K, ∀i ∈ {1, ..., p}(5.95)and a i,j (λ) are some functions such that: a i,j (λ) = 0 and finite for any λ ∈ is the set of the p-roots of unit andK a ≡ {x ∈ C : V a (x) ≡ (0, ..., 0)} ∀a ∈ {1,..., p}(5.97) such sets are finite if not empty, being the elements of the vectors (Λ p -X p )V i (λ) Laurent polynomials. The above identities in particular imply:

(5. 99 )

 99 Hence it holds for any λ ∈ C using continutiy properties of the cofactors, being {K ∪ set of values. Similarly, the fact that the vectorial condition D(λ)V 1 (λ) = 0 ¯holds true for any λ ∈ C\K implies that it is indeed satisfied for any λ ∈ C. Here, we write explicitly the first element of this vectorial condition:τ (λ)C 1,1 (λ) = a(λ)C 1,p (λ) + a(1/λ)C 1,2(λ), (5.100)together with the rewriting of (5.99) by using the identity (C.1):C 1,1 (λ)C 1,1 (λq) = C 1,2 (λ)C1,p (qλ).

(5. 101 ) 34 Proposition B. 1 .

 101341 statement in the last subsection of this appendix, but for now we want to write explicitly the representation of the other gauge transformed boundary operator families in the left and right basis formed out of the pseudo-eigenstates of B -(λ|β). Theorem B.1. The action of the reflection algebra generator A -(λ|βq 2 ) on the generic state β, h| is given by the following expression: β, h|A once the parameter α has been fixed by (3.26). Proof. The following interpolation formula: β, h|A -The gauge transformed boundary operators satisfy the following commutation relations:

15 )-(λ|q 2 /

 152 Similar commutation relations involving C-(λ|β) can be written by using the following β-symmetries:B -(λ|β) = C

+

  Similarly, the following theorem characterizes the right SoV representation of the gauged cyclic reflection algebra: Theorem B.2. The action of the reflection algebra generators D -(λ|β) on the generic state |β, h , can be written as it follows: D -(λ|β)|β, h = |β, h (-1) N q 1/2

-

  The following interpolation formula is derived as in the previous theorem using the polynomiality of the operator family D -(λ|β): D -(λ|β) = follows by the definition of the operator family D -(λ|β), from which we get: qD ∞ -(β)/β + βD 0 consequence of the functional dependence with respect to λ: are representation independent. Instead the asymptotic operators A ∞,0 -(βq 2 ) depend on the representation and we can compute them observing that using the definition (3.1) of A limits of the ungauged elements of U -(λ).

31 )U (R,βq 2 )

 312 This identity allows to compute these asymptotic operators once we use the interpolation formula to writeA 0 -(βq 2 ) in terms of A ∞ -(βq 2 ) as it follows: β, h|A 0 -(βq 2in the left and right pseudo-eigenstates, then thep N ×p N matrix M ≡ U (L,β)is the following invertible diagonal matrix:M κ(h)κ(k) = β, h|βq 2The following identity holds:b h (λ|β) β/q 2 , h|β, k = β, h|B -(λ|β)|β, k = b k (λ|β) β, h|βq 2 , k . (B.50)

h|βq 2 , 2 ) 2 , 2 ) 43 )U (R,βq 2 ) 2 )(B. 45 )

 2222432245 there exists at least a n ∈ {1, ..., N} such that h from the choice of the nonzero normalization factor n β , the Theorem 3.1 implies that the matrices U (L,β) and U (R,βq are invertible for almost all the values of the boundary-bulk-gauge parameters so that the same must be true for the diagonal matrix M , i.e. it must holds:M κ(h)κ(h) = h,β|βq we can define the normalization factor according to (B.47). The computation now of the remaining diagonal matrix elements M κ(h)κ(h) for h = p -1 can be done in a standard way by computing the matrix elements:θ a,ha (β) ≡ β, h 1 , ..., h a , ..., h N |A -(ζ (ha+1) which the statement of the theorem follows easily . B.3 SoV spectral decomposition of the identity The Theorem 3.1 states the pseudo-diagonalizability of B -(λ|β) for almost all the values of the boundary-bulk-gauge parameters, so that for almost all the values of these parameters the left and right states β, h| and |β, k are well defined nonzero left and right states describing a left and right basis in the space of the representation. We can now defines the following p N × p N matrices U (L,β) and U (R,βq defining the change of basis from the original left and right basis: h| ≡ ⊗ N n=1 h n , n| and |h ≡ ⊗ N n=1 |h n , n , (B.41) composed by v n -eigenstates, to the left and right pseudo-eigenbasis of B -(λ|β): β, h| = h|U (L,β) κ is an isomorphism between the sets {0, ..., p -1} N and {1, ..., p N } defined by: κ : h ∈ {0, ..., p It follows from the pseudo-diagonalizability of B -(λ|β) that the p N × p N square matrices U (L,β) and are invertible matrices for which it holds:U (L,β) B -(λ|β) = ∆ B-(λ|β)U (L,β) , B -(λ|β)U (R,βq -(λ|β) is the p N × p N diagonal matrix defined by: ∆ B-(λ|β) i,j ≡ δ i,j b κ -1 (i) (λ|β) ∀i, j ∈ {1, ..., p N }.We can prove that it holds, with the same notation as in Theorem 3.1: Proposition B.2. For almost all the values of the boundary-bulk-gauge parameters it holds: and right states are well defined and nonzero states which are pseudo-eigenstates of B -(λ|β) associated to different pseudo-eigenvalues as a consequence of the gauge transformed commutation relations. The proof of the fact that the states (3.9) and (3.10) are all nonzero is done reproducing the argument presented in the proof of Theorem B.3. The statements about the spectral decomposition of the identity of the theorem have been already given in Proposition B.2.

Lemma C. 1 . 3 )- 1

 131 The matrix D τ (λ) has at least rank p -1 for any λ ∈ C, up to at most a finite number of values. The following symmetries:C i+h,j+h (λ) = C i,j (λq h ) ∀i, j, h ∈ {1, ...,p}, hold. Moreover, the cofactors C 1,1 (λ), C 1,2 (λ) and C 1,p (λ) are polynomials in λ of maximal degree (p -1) (2N + 4) which admit the following decomposition: (λ) is a polynomial in Λ of degree (p (λ) and C 1,p (λ) are polynomials of maximal degree 2 (p -1) (N + 1) in λ.

44

  Note that in the bulk of the paper we have chosen to present the construction of the SoVbasis starting from a state |Ω β associated to the pseudo-eigenvalue b 0 (λ|β) just to simplify the simultaneous presentation of the left and right basis; in fact, we can construct the right basis also starting from the state | Ωβ associated to b 1 (λ|β), which is the state constructed directly here for the considered special representations. B.4.2 Pseudo diagonalizability and simplicity of B -(λ|β): general representations In this section we prove the Theorem 3.1 stating the pseudo diagonalizability and simplicity of the operator family B -(λ|β) for almost all the values of the boundary-bulk-gauge parameters. Let us first prove the following lemma: Lemma B.1. There exists at least one left and one right pseudo-eigenstate |Ω β and Ω β | of the one parameter family of pseudo-commuting operators B -(λ|β) satisfying the condition (3.5) with pseudo-eigenvalue b 0 (λ|β) satisfying the conditions (3.7) and (3.8). Proof. The operator family B -(λ|β) admits the following representation: B -= 0 ∀a, b ∈ {1, ..., N}, (B.94) as a consequence of the commutation relations (B.12). The result of the previous section implies that for some special choice of the boundary-bulk-gauge parameters all the operators B-,a,β are invertible as B -(λ|β) is pseudo-diagonalizable and it admits the following representation: B -(λ|β) = b -(β)( λ 2 B -,a (β) are commuting and invertible operators. Then the fact that this operators depend continuously on these parameters implies that this statement is true for almost any values of these parameters. This also implies that for almost all the value of the boundary-bulk-gauge parameters we can use the above representation for B -(λ|β). We can now recall that, thanks to the result of the Lemma A.1 of our previous paper, we can always find a nonzero simultaneous eigenstate of commuting operators such as the B -,a (β) for any a ∈ {1, ..., N}. This is a pseudo-eigenstate of the operator family B -(λ|β). Now, for the same set of representations considered in the previous section we know that the pseudo-eigenvalues of B -(λ|β) satisfy the conditions (3.7) and (3.8). Then, we can use once again the continuity argument to argue that the eigenvalues on the common eigenstate still satisfy (3.7) and (3.8). We can now prove the Theorem 3.1, by using the results of the previous sections. Proof of Theorem 3.1. The proof of the pseudo-diagonalizability of B -(λ|β) is a direct consequence of the previous lemma. Indeed, under the conditions (3.7) and (3.8) we can prove that all the Let us show our statement on the form of C 1,1 (λ). In order to do so we have to prove that C 1,1 (λ) is finite in the points 1 ..., p -1}. More precisely, in the line p -h there is at least one element of the matrix M 1,1 (λ) associated to C 1,1 (λ) which is diverging in the limit λ → ±i a q h . Here, we have to distinguish three cases. For the case h = (p ± 1)/2, we can proceed as done in the bulk of the paper. We can define the matrix M (h) 1,1 (λ) as the matrix with all the rows coinciding with those of M 1,1 (λ) except the row (p + 1)/2 -h, which is obtained by summing the row (p-1)/2-h and (p+1)/2-h of M 1,1 (λ) and dividing them by ((λ/q h (λ) = (-1) i+j C 1,1 (λ) and all the rows of the matrix

8 )

 8 In the remaining cases, ifh = (p ± 1)/2 then the row (p ± 1)/2p -h = p mod(p) is not contained in M 1,1 (±i a q h

p- 3 2 λ) λ 2 - 1 /λ 2 , 2 - 1 /λ 2 )

 32212212 D τ,(1,(p+1)/2,p),(1,(p+1)/2,p) (±i a q (p+1)/2 τ,(1,(p+1)/2,p),(1,(p+1)/2-1,p) (±i a q (p+1)/2 τ,(1,(p+1)/2,p),(1,(p+1)/2+1,p) (±i a q (p+1)/(C.11)and the identity:det p-3 D τ,(1,(p+1)/2,p),(1,(p+1)/2-1,p) (±i a q (p+1)/2 ) = det p-3 D τ,(1,(p+1)/2,p),(1,(p+1)/2-1,p) (±i a q (p+1)/2).(C.12)Finally, let us remark that in the case h = 0 the lines (p -1)/2 and (p + 1)/2 ofM 1,1 (±i a) are one the opposite of the other so that det p-1 M 1,1 (±i a ) = 0. We can so define the matrix M with all the rows coinciding with those of M 1,1 (λ) except the row (p + 1)/2, which is obtained by summing the row (p -1)/2 and (p + 1)/2 of M 1,1 (λ) and dividing them by (λ , this matrix has finite elements on the row (p + 1)/2 also in the limit λ → ±i a . Similarly to the previous cases one can show that the rows of M which our statement on the form of C 1,1 (λ) follows. Similarly, we can prove our statement on C 1,p (λ).

  QUANTUM INVERSE SCATTERING METHODfrom which the involution of the conserved quantities follows. If there exists sufficiently many independent conserved quantities, then there exists a canonical transformation mapping the original variables to a new set of coordinates, the so-called action-angle variables. The latter are defined such that their evolution in time is linear, so easily computable. Thereafter, thanks to the inverse transformation (this is the Inverse Scattering Problem part), one can deduce the time evolved original variables (see e.g. the books[18,[START_REF] Faddeev | Hamiltonian methods in the theory of solitons[END_REF]), hence solving the problem. Let us finally remark that the above structures admit an interesting discretisation preserving the integrability properties. The discrete analogue of the Lax matrix L, namely Lpn, t; λq depends now on the dynamical variables at site n of a lattice having N sites. In fact it is useful to see such an Lpn, t; λq as the monodromy matrix of the continuum theory T px n , x n ´δ; t; λq, with δ the lattice spacing. As a consequence, the proper Poisson brackets becomes quadratic also for the L's at a given site n while two L's on different sites have vanishing Poisson brackets. We get : tL 1 pn, t; λq, L 2 pn, t; µqu " rr 12 pλ, µq, L 1 pn, t; λqL 2 pn, t; µqs(1.8) The full monodromy matrix is then just the ordered product all along the chain of such discrete Lax

, t; λq, L 2 py, t; µqu " δpx ´yqrr 12 pλ, µq, L 1 px, t; λq `L2 py, t; µqs (1.6) leading to the quadratic Poisson bracket for T [225]: tT 1 pb, a; t; λq, T 2 pb, a; t; µqu " rr 12 pλ, µq, T 1 pb, a; t; λqT 2 pb, a; t; µqs (1.7) 1.2. matrices : T pN, 1; t; λq " LpN, t; λq . . . Lp1, t; λq (1.9) and satisfies the Poisson brackets : tT 1 pN, 1; t; λq, T 2 pN, 1, t; µqu " rr 12 pλ, µq, T 1 pN, 1; t; λqT 2 pN, 1; t; µqs (1.10)

  12 pλ{µqU 1 pλqR 12 pλµ{qqU 2 pµq " U 2 pµqR 12 pλµ{qqU 1 pλqR 12 pλ{µq

	2.2. THE 6-VERTEX REFLECTION ALGEBRA AND INTEGRABILITY
	(2.8)

  Baxter algebra, and Kpλq is a scalar solution of the reflection equation (2.8), i.e.: R 12 pλ{µqK 1 pλqR 12 pλµ{qqK 2 pµq " K 2 pµqR 12 pλµ{qqK 1 pλqR 12 pλ{µq (2.11)

  ă12ą3 pλq " P 12 R 13 pλqR 23 pqλqP 12 R ă12ą3 pλ{µqR ă12ą4 pλqR 34 pµq " R 34 pµqR ă12ą4 pλqR ă12ą3 pλ{µq (2.51) R 12 pλ{µqR 1ă34ą pλqR 2ă34ą pµq " R 2ă34ą pµqR 1ă34ą pλqR 12 pλ{µq (2.52)

		(2.49)
	R 3ă12ą pλq " P 12 R 32 pλ{qqR 31 pλqP 12	(2.50)
	satisfy the Yang-Baxter equations:

R 0ă12ą pλ{µqK 0 pλqR ă12ą0 pλµ{qqK ă12ą pµq " K ă12ą pµqR ă12ą0 pλµ{qqK 0 pλqR 0ă12ą pλ{µq

  , they show that: K ă12ą pλq " P 12 K 1 pλqR 12 pλ 2 qK 2 pλqqP 12 R ă12ąă34ą pλq " P 12 R 1ă34ą pλqR 2ă34ą pqλqP 12 " P 34 R ă12ą4 pλ{qqR ă12ą3 pλqP 34

	2.3. LOCAL HAMILTONIANS ASSOCIATED TO REPRESENTATIONS OF THE REFLECTION
		ALGEBRA
		(2.53)
	satisfies the following reflection equation:
		(2.54)
	This equation is a mixed reflection equation, as it involves different representations. This kind of equation
	will be the starting point to our study of more general reflection equations in chapter 5. Lastly, by
	considering the fusion on both the auxiliary and quantum spaces [238, 239], one can construct:
		(2.55)
	which satisfies the Yang-Baxter equation:
	R ă12ąă34ą pλ{µqR ă12ąă56ą pλqR ă34ąă56ą pµq " R ă34ąă56ą pµqR ă12ąă56ą pλqR ă12ąă34ą pλ{µq (2.56)
	and which leads to the reflection equation:
	R ă12ąă34ą pλ{µqK ă12ą pλqR ă34ąă12ą pλµ{qqK ă34ą pµq "	(2.57)
	K ă34ą pµqR ă34ąă12ą pλµ{qqK ă12ą pλqR ă12ąă34ą pλ{µq	(2.58)
	Thus, from this last equation one can apply Sklyanin's procedure to compute local Hamiltonians.
	We emphasise that from the 6-vertex R-matrix, which has an auxiliary space of dimension 2, we can get

  H 2 " L 2 and H 3 " H where L prq and Y l,m pθ, φq " F l,m pθqe imφ are respectively the radial part, solution of (3.22) (expressed via Laguerre polynomials), and the spherical harmonics. F l,m pθq and e imφ are the respective solutions of (3.21) and

	where R k,l
												z " ´i	B Bφ	(3.18)
	as these operators satisfy:
					rH, L 2 s " rH, L z s " rL 2 , L z s " 0	(3.19)
	Similarly to the classical case, the operators position X i and impulsion P i satisfy, in these coordinates,
	separate relations of the form (3.15) with:
							F 1 ˆφ, ´i	B Bφ	, h 1 pmq ˙" ´i	B Bφ	´h1 pmq	(3.20)
	F 2 ˆθ, ´i	B Bθ	, h 1 pmq, h 2 plq	˙"	1 tanpθq	B Bθ	`B2 B 2 θ	´h2 1 pmq 2	1 sin 2 pθq	´h2 plq	(3.21)
	F 3 ˆr, ´i	B Br	, h 2 pkq, h 3 pk, lq ˙" ´	2 2m	B 2 B 2 r `1 2mr 2 h 2 plq	´e2 r	´h3 pk, lq	(3.22)
	One can solve these three equations, and the wave functions are separate and expressed as the product
	of the solutions:
					Ψpr, θ, φq " xr, θ, φ|Ψy " R k,l prqF l,m pθqe imφ	(3.23)

  .., η N | are simply shifts on the zero η i solely:

	xη 1 , ..., η i , ..., η N | Apη i q 9 xη 1 , ..., q η i , ..., η N |	(3.26)
	xη 1 , ..., η i , ..., η N | Dpη i q 9 xη 1 , ..., η i {q, ..., η N |	(3.27)

  Thus, by acting on the state x0| with Āpµq, for µ a root of dpλq we have that x0| Āpµq is another eigenstate of Bpλq, with the eigenvalue f pλ, µqdpλq. Repeating this procedure for the different roots ξ n of dpλq, we can consider xh 1 , ..., h N | the following states:

							3.2. QUANTUM SEPARATION OF VARIABLES
	with the scalar functions:
	f pλ, µq "	qλ{µ ´µ{pλqq λ{µ ´µ{λ	and gpλ, µq "	q ´1{q λ{µ ´µ{λ	(3.38)
	xh 1 , ..., h N | " xh| "	1 N	x0|	N ź n"1	ˆĀpξ n q āpξ n q	˙hn	with @n P 1, N , h n P t0, 1u	(3.39)
	where N is a global normalisation (it is moreover convenient to make explicit the term āpξ n q of the quantum determinant ). Then one easily gets that the states xh| are eigenvectors of Bpλq:
	xh| Bpλq " b h1,...,h N pλq xh| with b h1,...,h N pλq "	N ź a"1	ˆλq ha ξ a	´ξa λq ha ˙(3.40)
									Āpλq	(3.37)

pλ, µq P C 2 , Āpµq Bpλq " f pλ, µq Bpλq Āpµq `gpλ, µq Bpµq

  Following the general procedure introduced in 3.2.2, let us now compute the action of the operator Āpη i q on the eigenbasis elements containing η i , namely xη 1 , ..., η i , ..., η N |. As we explain CHAPTER 3. QUANTUM SEPARATION OF VARIABLES here, this allows to show that the eigenbasis of Bpλq is indeed a separate basis.From the definition of the basis elements xh|, it follows:xh| Āpξ i {q hi q " āpξ i {q hi q xh 1 , ..., h i `1, ..., h N | Āpξ i {q hi q and Dpξ i {q hi`1 q simply shift the eigenvalue η i when they act on the states xh 1 , ..., h i , ..., h N | and xh 1 , ..., h i `1, ..., h N | respectively. With these known actions, a simple Lagrange interpolation formula gives the action of Āpλq and Dpλq. tpξ n {q hn qΨtphq " āpξ n {q hn qΨtph 1 , ..., h n `1, ...h N q `dpξ n {q hn qΨtph 1 , ..., h n ´1, ...h N q (3.49)

						(3.45)
	Using then the quantum determinant (3.31) and (3.34), it holds:
			xh 1 , ..., h i `1, ..., h N | Dpξ i {q hi`1 q " dpξ i {q hi`1 q xh|	(3.46)
	As we highlighted in 3.2.2, the operators We can now consider the spectral problem of the transfer matrix T pλq " Āpλq `Dpλq. So let	tD	be
	an eigenvector of the transfer matrix, associated to the eigenvalue tpλq:
			T pλq	tD	" tpλq	tD	(3.47)
	and let us denote by Ψtphq the wave functions in this basis:
			Ψtphq "	@ h ˇˇt D	(3.48)
	Computing xh| T pλq	tD	thanks to the action of T pλq on the right and on the left, one easily obtains the
	so called Baxter equations:
	@n P 1, N ,
	These are the separate relations (3.15) in the eigenbasis. Indeed, each of these N equations only involves
	the parameter η n (or h n equivalently). Thus, let Qtph n q be a solution of the Baxter equation (3.49). It
	holds the following separate form of the wave functions:
					N
			Ψtphq "	ź
					n"1

  D ´pλq " kpλqA ´pq{λq where kpλq " `λ2 ´1{λ 2 ˘{pλ 2 {q 2 ´q2 {λ 2 q (4.16)

2 {q 2 ´q2 {λ 2 qA ´pλq 1{2 qA ´pq 1{2 {λq (4.14) with the function: A ´pλq " g ´pλqapλq ´1{2 qdp1{pq 1{2 λqq (4.15) 4.2. THE CASE OF A TRIANGULAR RIGHT BOUNDARY MATRIX Similarly, we introduce (for further use) the function:

  -(1.31)) for some relevant couples of algebra generators:

	Apλ|α, β, γqApµ|α, β{q, γ{qq " Apµ|α, β, γqApλ|α, β{q, γ{qq	(4.94)
	Bpλ|α, βqBpµ|α, β{qq " Bpµ|α, βqBpλ|α, β{qq	(4.95)
	Cpλ|α, β, γqCpµ|α, βq, γ{qq " Cpµ|α, β, γqCpλ|α, βq, γ{qq	(4.96)
	Dpλ|α, βqDpµ|α, β{qq " Dpµ|α, βqDpλ|α, β{qq	(4.97)
	Apλ|α, β, γqBpµ|α, β{qq "	qpλ{µ ´µ{λq λq{µ ´µ{qλ	Bpµ|α, βqApλ|α, β{q, γqq
	`pq ´1{qqµ{λ λq{µ ´µ{qλ	Apµ|α, β, γqBpλ|α, β{qq	(4.98)

Bpλ|α, βqApµ|α, β{q, γqq " λ{µ ´µ{λ qpλq{µ ´µ{qλq Apµ|α, β, γqBpλ|α, β{qq `pq ´1{qqλ{µ λq{µ ´µ{qλ Bpµ|α, βqApλ|α, β{q, γqq (

  βq " " ´λq 3{2 {β Apλq ´αq Bpλq `1{pαqq C ´pλq `β{ ´λq 3{2 ¯Dpλq Moreover, one can expand this gauged boundary monodromy matrix in terms of the gauged generators of the Yang-Baxter algebra. In particular, it holds:

					ı	{pβ{q 2 ´q2 {βq	(4.105)
	Bpλ|α, βq "	" ´λβ{q 1{2 Apλq ´αq Bpλq `β2 { pαqq Cpλq `βq 1{2 {λDpλq ı	{pβ ´1{βq	(4.106)
	Cpλ|α, βq "	" λq 3{2 {β Apλq `αq Bpλq ´q3 { `αβ 2 ˘Cpλq ´q5{2 { pλβq Dpλq ı	{pβ{q 2 ´q2 {βq	(4.107)
	Dpλ|α, βq "	" λβ{q 1{2 Apλq `αq Bpλq ´1{pαqqCpλq ´q1{2 { pλβq Dpλq ı	{pβ ´1{βq	(4.108)

ˆApλ|α, βq 2 q Cpλ|α, βq 2 q ˙" M pλ|α, β, γq Kpλ|γq ˆĀpλ|α, βq, γqq Cpλ|α, βq, γqq ˙(4.109) ˆBpλ|α, βq Dpλ|α, βq ˙" M pλ|α, β, γq Kpλ|γq ˆBpλ|α, β{q, γqq Dpλ|α, β{q, γqq ˙(4.110)

  βqA ´pλ 2 |α, βq `pλ 1 λ 2 {q ´q{λ 1 λ 2 qpλ 1 β{qλ 2 ´λ2 q{βλ 1 qpq ´1{qq pλ 1 {λ 2 ´λ2 {λ 1 qpλ 1 λ 2 ´1{λ 1 λ 2 qpβ{q ´q{βq Bpλ 2 |α, βqApλ 1 |α, βq `pλ 1 λ 2 {β ´β{λ 1 λ 2 qpq ´1{qq pλ 1 λ 2 ´1{λ 1 λ 2 qpβ{q ´q{βq Bpλ 2 |α, βqDpλ 1 |α, βq (4.115) Bpλ 1 |α, βqDpλ 2 |α, βq " pλ 1 q{λ 2 ´λ2 {qλ 1 qpλ 1 λ 2 {q ´q{λ 1 λ 2 q pλ 1 {λ 2 ´λ2 {λ 1 qpλ 1 λ 2 ´1{λ 1 λ 2 q Dpλ 2 |α, βq 2 qB ´pλ 1 |α, βq ´pλ 1 λ 2 {q ´q{λ 1 λ 2 qpλ 2 βq{λ 1 ´λ1 {λ 2 βqqpq ´1{qq pλ 1 {λ 2 ´λ2 {λ 1 qpλ 1 λ 2 ´1{λ 1 λ 2 qpβq ´1{βqq Dpλ 1 |α, βq 2 qBpλ 2 |α, βq ´pλ 1 λ 2 β ´1{λ 1 λ 2 βqpq ´1{qq pλ 1 λ 2 ´1{λ 1 λ 2 qpβq ´1{qβq Apλ 1 |α, βq 2 qB ´pλ 2 |α, βq (4.116)

and Apλ 1 |α, βq 2 qApλ 2 |α, βq 2 q ´pλ 1 λ 2 {β ´1{λ 1 λ 2 qpq ´1{qq pλ 1 λ 2 ´1{λ 1 λ 2 qpβ{q ´q{βq Bpλ 1 |α, βqC ´pλ 2 |α, βq 2 q " Apλ 2 |α, βq 2 qApλ 1 |α, βq 2 q ´pλ 1 λ 2 {β ´1{λ 1 λ 2 qpq ´1{qq pλ 1 λ 2 ´1{λ 1 λ 2 qpβ{q ´q{βq Bpλ 2 |α, βqC ´pλ 1 |α, βq 2 q (4.117)

  N , b ´,n pβq " µ n, n q 1{2 , (i.e. independent of βq (4.140) and b ´pβq " F rα, β, t n , k n u, κ ´, τ ´, ξ ´s (4.141) Proof. The proof of this proposition follows quite straightforwardly the steps of the proof of diagonalisability of the operator Bpλq given in proposition 4.2.1. More explicitly, for Bpλ|α, βq it holds the next expression from the definition (4.110) of the gauged generators:

	The explicit value of the function F is not of prime interest in this presentation, we just emphasise
	that it is a rational fraction in all the bulk, boundary and gauge parameters. The exact expression
	can be found in (II, B.67).
	The left pseudo-eigenbasis characterised by the formulas p4.127q is obtained by fixing:
	xΩ β | " xΩ, α, β, γ|	(4.142)
	Bpλ|α, βq
	f pα, β{q, γqq

h 1 , ..., h N | (4.139) 4.3. THE MOST GENERAL INTEGRABLE BOUNDARIES and has simple pseudo-spectrum characterised by fixing: @n P 1, " K´p λ|γq 12 Apλ|α, β, γqAp1{λ|α, β{q, γqq ´K ´pλ|γq 11 Apλ|α, β, γqBp1{λ|α, β{qq `K ´pλ|γq 21 Bpλ|α, βqBp1{λ|α, β{qq ´K ´pλ|γq 22 Bpλ|α, βqAp1{λ|α, β{q, γqq (4.143) This way we easily find a first pseudo-eigenvector of Bpλ|α, βq: xΩ, α, β, γ|Bpλ|α, βq " b 0 pλ|βq xΩ, α, β{q 2 , γ| (4.144) with b 0 pλ|βq " f pα, β{q, γqq K´p λ|γq 21 bpλ|α, βqbp1{λ|α, β{qq (4.145)

This pseudo-eigenvalue is not identically zero as soon as γ ‰ γ ˘. This fixes the values of the b ´pβq and b ´,a pβq to those stated in this proposition, recalling that an explicit expression for the coefficients K´p λ|γq ij can be found in paper II. Thereafter, one can repeat the standard procedure already developed using now the modified commutation relations (4.116)-(4.117) to show that the relations (4.139) of the proposition is satisfied.

3.4 Discrete characterisation of the spectrum of the boundary transfer ma- trix In

  .159) VARIABLES Then, thanks to the boundary quantum determinant (4.121)-(4.122) and the simplicity of pseudo-spectrum of Bpλ|α, βq, one can check via the same procedure already seen for the ungauged case that the operator families Apλ|α, βq and Dpλ|α, βq in the zeros of Bpλ|α, βq act as simple shift operators: , ..., h a , ..., h N |T ȃ " xβ, h 1 , ..., h a ˘1, ..., h N | (4.162) and T ȃ |β, h 1 , ..., h a , ..., h N y " |β, h 1 , ..., h a ˘1, ..., h N y (4.163) and ϕ a is simply defined in (4.59). The simple representations p4.155q-p4.156q of the transfer matrix in terms of the gauge transformed boundary generators and the known actions p4.160q-p4.161q of these operators imply that the transfer matrix spectral problem is separate in the pseudo-eigenbasis of Bpλ|α, βq. Explicitly, it follows the Baxter equations:

	xβ, h|Apζ phaq a	|α, βq 2 q " A ´pζ phaq a	qxβ, h|T ´ϕa a	(4.160)
	and Dpζ phaq a	|α, βq|β, hy " T ´ϕa a	|β, hyD ´pζ phaq a	q	(4.161)
	where
	xβ, h 1 τ pξ phnq
					phaq a	are given by (4.159), and the
	separate basis are different.
	As before, we can state a proposition on the scalar products of the elements of the left and right
	separate basis:
	Proposition 4.3.7. Using an appropriate choice for the norm n β in the basis elements (4.127) and
	(4.128), it holds:

n qΨ τ pβ{q 2 , hq " apξ phnq n qΨ τ pβ{q 2 , T ń phqq `ap1{ξ phnq n qΨ τ pβ{q 2 , T ǹ phqq (4.164) where we introduced: apλq " a `pλqA ´pλq and Ψ τ pβ{q 2 , hq " xβ{q 2 , h|τ y (4.165) for the characterisation of the eigenvalue τ pλq and |τ y the associated eigenvector. While for the eigencovector xτ | it holds: @n P 1, N , τ pξ phnq n q Ψτ pβ, hq " dpξ phnq n q Ψτ pβ, T ń phqq `dp1{ξ phnq n q Ψτ pβ, T ǹ phqq (4.166) with: dpλq " d `pλqD ´pλq and Ψτ pβ, hq " xτ |β, h 1 , ..., h N y (4.167) 4.this paragraph we characterise the spectrum of the boundary transfer matrix associated to general integrable boundaries. In view of what has been done previously, the results have a very similar form as for the ungauged case, proposition 4.2.7. Simply here the coefficients ζ

  These separate states are interesting at least for two reasons: the eigenstates of the boundary transfer matrix are special separate states; they admit a simple determinant scalar product, as it is stated in the next proposition: Let us take an arbitrary separate left state xα| and an arbitrary separate right state |βy. Then it holds: xα|βy " det M pα,βq (4.198)

										2 , h 1 , ..., h N |	(4.196)
		p´1		N
	|βy "	ÿ		ź	β phaq a	ź	pX phaq a	´Xph b q b	q|β, h 1 , ..., h N y	(4.197)
	h1,...,h N "0	a"1		1ďbăaďN
	where the coefficients α a phaq	and β	phaq a	are arbitrary complex numbers.
	Proposition 4.4.1.

  If for each site n P 1, N we have pQ n , R n q P C k chP ˆCk chP , then the set of the Lax operators L 0n pλ|Q n , R n q describes the chiral Potts model.

	Definition 5.1.1. The algebraic chiral Potts curve C k chP of modulus k is defined by 1 :
		"			*
	C k chP "	S P C 4 | x p S	`yp S " kp1 `xp S y p S q ; kx p S " 1	´k1 s p S	and ky p S " 1 ´k1 s p S , with k 2 `k12 " 1 (5.8)

  2 , xh 1 , h 2 | " xh 1 , a| û b xh 2 , b| û and |h 1 , h 2 y " |h 1 , ay û b |h 2 , by û (5.14)

  In our study, we do not consider degenerate cases for which one of the parameters x Qn , x Rn , y Qn , y Rn or σ n is zero. Moreover, we take parameters for which the different scalars a i are well defined, i.e. we suppose:

2 1 and h 2 2 . From now on, we can emphasise the fact that: a 1 a 2 a 3 a 4 " 1 (5.19) 5.1. FUNDAMENTAL R-MATRIX Remark.

  13 A 23 " A 23 A 13 A 12 (5.59) 5.2. REFLECTION EQUATIONS and similar conditions involving the other operators, for instance A 12 B 31 B 32 " B 32 B 31 A 12 .The point is that if the matrix R, describing the reflection, satisfies the Yang-Baxter equation: R 12 pP 1 , P 2 qR 13 pP 1 , P 3 qR 23 pP 2 , P 3 q " R 23 pP 2 , P 3 qR 13 pP 1 , P 3 qR 12 pP 1 , P 2 q (5.60)

  In view of the application of this structure to the obtention of commuting families, it is defined the following algebra, that can be considered as the dual of (5.65):`,11 1 R 11 1 ,221 (5.68) Indeed, one directly can check that for K `a scalar representation of this dual algebra, it holds (by simply evaluating (5.66) on the bi-covector K

	CHAPTER 5. LOCAL HAMILTONIANS ASSOCIATED TO CYCLIC REPRESENTATIONS OF
	THE REFLECTION ALGEBRA
	T `,11 1 T t 11 1 `,22 1 " T t 22 1 `,22 1 T t 22 1	t 11 1
	t 22 1 `,22 1 K `,11 1 ): t 11 1
	K `,11 1 T ´,11 1 K t 11 1 `,22 1 T ´,22 1 " K t 22 1	t 22 1 `,22 1 T ´,22 1 K	t 11 1 `,11 1 T ´,11 1	(5.69)
				1 1 1 2	(5.67)
			81

  71) This operation leaves unchanged P a and P C and it is consistent with the Yang-Baxter equations. That is we can apply σ 0 to the Yang-Baxter equation (1.12),(1.13) and (5.1), leaving all them satisfied. The reflection equation (5.57) is thus: R σ2σ1 21 pµ, λqK 1 pλqR σ2 12 pλ, µqK 2 pµq " K 2 pµqR σ1 21 pµ, λqK 1 pλqR 12 pλ, µq (5.72) MIXED REFLECTION EQUATION the 6-vertex reflection equation is then written in the usual form [177]: R 12 pλ{µqK 1 pλqR 12 p λµ q qK 2 pµq " K 2 pµqR 12 p λµ q qK 1 pλqR 12 pλ{µq (5.75)This R-matrix satisfying the Yang-Baxter equation (1.13), the associated algebra:

	R 12 pλ{µqT ´,1 pλqR 12 p	λµ q	qT ´,2 pµq " T ´,2 pµqR 12 p	λµ q	qT ´,1 pλqR 12 pλ{µq	(5.76)

Using the following properties of the 6-vertex R matrix:

R

12 pλ, µq " R 12 pλ{µq (5.73) R 21 pλ{µq " R 12 pλ{µq (5.74) 82 5.3.

  , let us consider:

	`R´1 12 pλ{µq	˘t1t2 T `,1 pλq	" `Rt1 12 pλµ{qq	˘´1 ı t2	T `,2 pµq
		" T `,2 pµq	" `Rt1 12 pλµ{qq	˘´1 ı t2	T `,1 pλq `R´1 12 pλ{µq ˘t1t2	(5.77)
	By using the crossing unitarity of the 6-vertex R-matrix:
	R t1 12 pλqR t1 12 p1{λq 2 q " ˆq	´1 q	˙2 ´ˆλq	´1 λq	˙2	(5.78)
	as well as the unitarity property:
		R 12 pλqR 12 p1{λq "	ˆq λ ´λ q	˙ˆλq	´1 λq	˙(5.79)
	and the fact that for this R-matrix, it also holds:
				R t1 12 pλq " R t2 12 pλq	(5.80)
	this dual reflection algebra reads:

R 12 pµ{λq T `,1 pλq R 12 p1{λµqq T `,2 pµq " T `,2 pµq R 12 p1{λµqq T `,1 pλq R 12 pµ{λq

(5.81)

  a1 pP a |P 1 |P C qu (5.205) 5.5. SOME EXPLICIT (NEW) MODELS FOR 3-DIMENSIONAL LOCAL QUANTUM SPACES Then, for each of the four models previously described, we give the explicit bulk Hamiltonians h

	pxq
	k,k`1

  1 (5.247) THE REFLECTION ALGEBRA (they are defined up to additive constants cst and cst1 ) In that case, the boundary matrices (in the v-basis) are given by:K a pξ ´|y Qa , y Ra |P C q "We can equivalently write them in terms of the generators. It holds:K a pξ ´|y Qa , y Ra |P C q " 1{N py Qa , y Ra , ξ ´q " α 0 py Qa , y Ra , ξ ´q1 `α1 py Qa , y Ra , ξ ´qv a `α2 py Qa , y Ra , ξ ´qv 2 The factor N py Qa , y Ra , ξ ´q is simply a normalisation, and the three coefficients α i py Qa , y Ra , ξ ´q are polynomials in the variables y Qa y Ra and ξ ´. Explicitly, for the choice of normalisation (5.248) , it holds: N py Qa , y Ra , ξ ´q " 3p1 `yQa y Ra ξ 2´`y Qa y Ra ξ 4

	¨yQa y Ra ´q2 ξ 2 1´q 2 y Qa y Ra ξ 2 ´0 0 pq 2 y Qa y Ra ´ξ2 ´qpy Qa y Ra pq 2 ´yQa y Ra ξ 2 ´qp1´q 2 y Qa y Ra ξ 2 ´q2 ξ 2 ´q ´q	0 0	' ‹ ‹	(5.248)
		0	0	1
	and K `apξ `|y Qa , y Ra |P C q "	¨yQa y Ra ´ξ2 q2 ´qy Qa y Ra ξ 2 `0 0 py Qa y Ra ´ξ2 `qpy Qa y Ra ´qξ 2 `q p1´y Qa y Ra ξ 2 `qpq´y Qa y Ra ξ 2 `q	0 0 ' ‹ ‹	(5.249)
		0	0	1
					‰
					a
					(5.250)

´q

(5.251)

and the coefficients α i py Qa , y Ra , ξ ´q read: α i py Qa , y Ra , ξ ´q " α i0 pξ ´q `αi1 pξ ´qy Qa y Ra `αi2 pξ ´qpy Qa y Ra q 2

  For the general model, let us consider the parameters (5.197) with k " 1 and i " 2. The boundary matrices have the following form:K a pξ ´|y Qa , y Ra |P C q " Ra ξ 2´qpq´y Qa y Ra ξ 2

	5.5. SOME EXPLICIT (NEW) MODELS FOR 3-DIMENSIONAL LOCAL QUANTUM SPACES
	The general model
		¨1 0 0	0 y Qa y Ra Qa y Ra ξ 2 ´ξ2 1´y ´0 0 0 py Qa y Ra ´ξ2 ´qpqy Qa y Ra	´ξ2 ´q ‹ ‹ '	(5.283)
		¨1	0	0
	and K `apξ `|y Qa , y Ra |P C q "	0 0	y Qa y Ra ´yQa y Ra ξ 2 ´q2 ξ 2 q2 `0 0 py Qa y Ra ´ξ2 `qpy Qa y Ra	´q2 ξ 2 `q
	110

´q p1´y Qa y

  py Qa , y Ra , ξ ´q " α 0 py Qa , y Ra , ξ ´q1 `α1 py Qa , y Ra , ξ ´qv a `α2 py Qa , y Ra , ξ ´qv 2 The factor N py Qa , y Ra , ξ ´q is simply a normalisation, and the three coefficients α i py Qa , y Ra , ξ ´q are polynomials in the variables y Qa y Ra and ξ ´. Explicitly, for the choice of normalisation (5.248), it holds: N py Qa , y Ra , ξ ´q " 3p1 ´yQa y Ra ξ 2 ´qp1 ´q2 y Qa y Ra ξ 2

	‰
	a
	(5.285)

´q

(5.286)

and the coefficients α i py Qa , y Ra , ξ ´q read: α i py Qa , y Ra , ξ ´q " α i0 pξ ´q `αi1 pξ ´qy Qa y Ra `αi2 pξ ´qpy Qa y Ra q 2

  The simple representations (3.27)-(3.28) of the transfer matrix in terms of the gauge transformed boundary generators and the known actions (3.18)-(3.19) of these operators imply that the transfer matrix spectral problem is separated in the pseudo-eigenbasis of B

-(λ|β).

  .18) Thus for this special representation the boundary transfer matrix is normal. Then it follows that the determinant of the p N Noticing that this determinant is a fractional function of the bulk and boundary parameters, non zero for the special choice of the parameters above defined, it follows that it is non zero for almost every choice of the parameters. Which concludes the proof.It is also interesting to remark that we can obtain the coefficients of a left transfer matrix eigenstates in terms of those of the right one. The following lemma defines this characterization and can be proven as in the standard case[1]: Lemma 4.2. Let τ (λ)

	given basis of covectors and | τ × p N matrix of elements e i |τ j , where e i | is the generic element of a j is the generic transfer matrix eigenvector, is non zero.

  These separate states are interesting at least for two reasons: the eigenstates of the boundary transfer matrix are special separate states, and they admit a simple determinant scalar product, as it is stated in the next proposition: Proposition 4.1. Let us take an arbitrary separate left state α| and an arbitrary separate right state |β . Then it holds:

	hb) )|β, h b 1 , ..., h N (4.21)	where the coefficients α (ha) a and β (ha) a are arbitrary complex numbers, meaning that the coefficients	of these separate states have a factorised form in these basis.

  satisfied for almost all the values of the boundary-bulk parameters it is enough to prove that we can find some values of these parameters for which (4.13) is satisfied.Indeed, we can chose arbitrary boundary-bulk parameters satisfying the following inequalities:

	p -1}, n ∈ {1, ..., N}, (4.13)	once we set the ratio α/β as in (3.26). Let us first observe that B -(λ|β) is a Laurent polynomial in the inner boundary parameters and the bulk parameters. So that by (3.26), the one parameter α, β, family B	µ p	+,n

-(λ|β) becomes Laurent polynomial in the outer boundary parameters too. Consequently, to prove that (4.13) is

  For the symmetry it is enough to consider the above limit in the case h = 0. Let us denote with Dτ (λq 1/2 ) the matrix whose first row is the sum of the first and the last row of DThe proof is quite straightforward, it is based on the fact that one can see a Vandermonde determinant when computing the scalar product. One of the main corollary is the orthogonality of two eigenstates τ | and |τ ′ of the boundary transfer matrix associated to two different eigenvalues

	.8)	Proof. The SoV characterization of the spectrum implies that τ (λ) ∈ Σ T if and only if it holds:	det p D τ (ζ (0) a ) = 0, ∀a ∈ {1, ..., N}, (5.9)	and τ (λ) has the form (4.1). In the previous lemma we have shown that det p D τ (λ) is a Laurent	polynomial of degree N + 2 in Z, here we show that from τ (λ) of form (4.1) it follows the identities:	lim λ→±1,±i det p D τ (λq 1/2+h ) = 0 ∀h ∈ {0, ..., p -1}. (5.10)	τ (λq 1/2 -1/λ 2 ) ) divided for (λ 2 -1/λ 2 while all the others rows of Dτ ) and whose row (p + 1) /2 is the row (p + 1) /2 of D τ (λq 1/2 ) multiplied for (λ 2 (λq 1/2 ) and D τ (λq 1/2 ) coincide. Clearly it holds:	(5.11) det p Dτ (λq 1/2 ), τ (λq 1/2 p D ) = det	14

τ (λ) and τ ′ (λ):

  .27) Proof. Let us prove first that if it exists a Q(λ) of the form (5.24) with N

	Q = (p -1) N satisfying T . The r.h.s of the equation (5.26) is
	∈ Σ
	(5.27) and (5.26) with τ (λ) an entire function, then τ (λ)	a Laurent polynomial in λ as we have:

  ) for generic values of the boundary-bulk parameters. Indeed, as the W

	a, h+1+i are functions only ( h+j) of the bulk parameter µ a,+ while the ratios a(1/ζ a )/τ (ζ ( h+j+1) a ) are functions of both the boundary and the bulk parameters then we can prove that Wa, h = 0. Explicitly we can compute the asymptotic of Wa, h in the limit µ a,+ → ∞, by using the know asymptotic of the transfer matrix, therefore showing that it is non-zero for general values of boundary-bulk parameters.	In the previous theorem we have excluded the boundary-bulk one-constraint cases leading to	an identically zero detD τ (λ) for any τ (λ) ∈ Σ theorem. T , these specific cases are considered in the next	Theorem 5.2. Let us assume that there exists a k ∈ {0, ..., p -1} such that it holds:

)

  then, for almost all the values of the boundary-bulk parameters, τ (λ) is an entire function and there exists and is unique a polynomial Q(λ) of the form (5.24) with N

	∈ Σ

T if and only if τ (λ) Q ≤ (p -1) (N + 1) and N Q = k mod p, satisfying the following homogeneous Baxter equation: τ

There is some intricate history of the inverse square law for gravity, involving several scientists as more particularly Bullialdus, Hooke, Borelli, Wren and Halley, but it was Newton that finally found the proper formulation, still acknowledging their contributions in Principia[5].

For large time, asymptotically, the N-body problem is generically chaotic and perturbation theory does not apply. However, sophisticated perturbation and numerical techniques have been developed to give high accuracy in the determination of trajectories even for rather large scales of time[11]. Nevertheless, for scales of the order of 100 millions years or above, the behaviour of the solar system is unpredictable for uncertainties of the order of a few meters on the initial positions of the planets[12].

This model can describe the melting and freezing of atomic monolayers on crystalline surfaces. This liquid layer orders itself into domains that either line up with the substrate structure or follow the layer own ordering, leading to the so-called commensurate and incommensurate phases, describing for instance the adsorbtion phenomenon.

In principle one can add a left hand side term to this equation that commutes with L 1 `L2 `L3 , giving rise to the modified Yang-Baxter equation.

Notice that in general these parameters could be sets of complex parameters and not only complex parameters. See for instance the chiral Potts Lax operator (1.72).

Indeed, in that case the Lax operators have an auxiliary space of dimension 2 and quantum spaces of dimension 2s `1. We can fuse the auxiliary space up to the obtention of an auxiliary space of dimension 2s `1.

The states xjn ´1, n| v and |jn, ny v, where jn is such that (4.23) is satisfied by imposing bn " ´q2jn´1 an.

Let us comment that the three relations in this definition are not independent, as for example the first one can be obtained from the others.

We recall that the mean value of an operator Opλq, belonging to a commutative family of operators, is by definitionă Opλq ą" p´1 ś k"0 Opq k λq " Opλ p q

constants of the model (L and N for example, cf point A. of proposition 5.1.4) and on another free parameter z.

SciPost Phys.2, 009 (2017)

Remerciements

Acknowledgements J. M. M. and G. N. are supported by CNRS and ENS Lyon; B. P. is supported by ENS Lyon and ENS Cachan.

du jury, et Olivier Babelon

The study of quantum models with integrable open boundary conditions has attracted a large research interest, e.g. see and references therein. These models are of physical interest as they can describe both equilibrium and out of equilibrium physics; e.g. some interesting applications concern the description of classical stochastic relaxation processes, like ASEP [53,54], [48][49][50][51], and quantum transport properties in spin systems [55,56].

In this paper we start the analysis of the class of open integrable quantum models associated to cyclic representations [57][58][59][60][61] of the 6-vertex reflection algebra. The literature of these models is so far rather sparse with the exception of some rather special representations and boundary conditions, like the open XXZ chains at the roots of unity, that can be traced back to these representations under some special constraints, and for which some results are known in the framework of algebraic Bethe ansatz (ABA) [52,62,63], fusion of transfer matrices and truncation identities [6,7,[64][65][66][67].

In order to study the general representations and boundary conditions we have to go beyond traditional methods which do not apply for these general settings. This is done by developing the Sklyanin's SoV method [84][85][86][87] for this class of models, a method that has the advantage to lead (mainly by construction) to the complete characterization of the spectrum (eigenvalues and eigenvectors) and has proven to be applicable for a large variety of integrable quantum models [28][29][30][31][32][33][34][35][36][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102], where traditional methods fail. Moreover, the SoV approach has the advantage to allow also for the study of the dynamics of the models as it leads to universal determinant formulae for matrix elements of local operators on transfer matrix eigenstates as shown for different classes of models, first in [97], and then in many other cases in [33,88,89,99,100,103]. Moreover, the analysis developed in [104,105] makes it possible to compute homogeneous and then thermodynamic limit of these matrix elements opening the way to the computation of the corresponding correlation functions.

Let us recall that in [3], Sklyanin has shown how to construct classes of quantum models with integrable boundaries in the framework of the so-called Quantum Inverse Scattering Method [62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78][79], constructing in particular associated families of commuting transfer matrices The generator -(λ) admits the following boundary bulk decomposition:

if the inner boundary matrix is non-diagonal and the bulk parameters are generals, i.e. if it holds:

while for

x = 1, we can write:

which together with:

A.3 Reduction to inhomogeneous chiral Potts representations

In this appendix we want to point out that a nontrivial class of representations of the inhomogeneous chiral Potts model can be described on the closed chain in the space of the parameters of the τ 2 -model considered in this paper. In order to do so let us recall that the transfer matrix T chP λ of the inhomogeneous chiral Potts model [113] is characterized by the following kernel: and where:

The algebraic curve k of modulus k is by definition the locus of the points in the four-

which satisfy the equations: 37) where: Let us here directly characterize the class of the inhomogeneous chiral Potts representations once we restrict the space of the parameters to that used in the section 6; in particular, we assume that it holds:

(A.39)

The parameters of the τ 2 -Lax operators are written in terms of the coordinate of the points p, r n , q n by using the equations (5.3) of the paper [96]. Then we have that the points r n , q n are elements of k if and only if beyond (A.39) the parameters of the Abstract. This article is a direct continuation of [1] where we begun the study of the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax operator. There we addressed this problem for the case where one of the K-matrices describing the boundary conditions is triangular. In the present article we consider the most general integrable boundary conditions, namely the most general boundary K-matrices satisfying the reflection equation. The spectral analysis is developed by implementing the method of Separation of Variables (SoV). We first design a suitable gauge transformation that enable us to put into correspondence the spectral problem for the most general boundary conditions with another one having one boundary K-matrix in a triangular form. In these settings the SoV resolution can be obtained along an extension of the method described in [1]. The transfer matrix spectrum is then completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions and equivalently as the set of solutions to an analogue of Baxter's T-Q functional equation. We further describe scalar product properties of the separate states including eigenstates of the transfer matrix. where:

Moreover, the identity:

and the corollaries:

imply the following two equivalent expressions of the quantum determinant by the gauge transformed generators:

plus other two equivalent rewriting. The gauge transformed Yang-Baxter generators are of special interest as they define a closed set of commutation relations: A(λ|α, β, γ)A(µ|α, β/q, γ/q) = A(µ|α, β, γ)A(λ|α, β/q, γ/q) (A.13)

We can prove these commutation relations by direct computations using the properties of the gauge transformations and their action on the Yang-Baxter equation.

A.2 Pseudo-reference state for the gauge transformed Yang-Baxter algebra

In the following, we want to study the conditions for which a nonzero state identically annihilated by the action of the operator family A(λ|α, β, γ) exists:

6

Conclusions

In this second article we have shown how to implement the SoV method to characterize the transfer matrix spectrum for integrable models associated to the Bazhanov-Stroganov quantum Lax operator and to the most general integrable boundary conditions. For that purpose it was necessary to perform a gauge transformation so as to recast the problem in a form similar to the one studied in our first article, i.e., such that one of the boundary K-matrices becomes triangular after the gauge transformation. Let us stress that the separate basis was designed again as the (pseudo)-eigenvector basis of some gauged operator of the reflection algebra having simple spectrum. What remains to be done is the construction of integrable local cyclic Hamiltonian having appropriate boundary conditions and commuting with the boundary transfer matrices considered here. This amounts to use trace identities involving the fundamental R-matrix acting in the tensor product of two cyclic representations [121,124,125] and to construct the associated K-matrices, hence also acting in these cyclic representations. The reflection equations will have to be written for arbitrary choices (and mixing) of the spin-1/2 and cyclic representations. Correspondingly, there will be compatibility conditions between the different K-matrices acting in these two different representations. We will address this question in a forthcoming article [138].

Appendices A Gauge transformed Yang-Baxter algebra

A.1 Gauge transformed Yang-Baxter generators For arbitrary complex parameters α and β let us introduce the following two matrices:

and their inverses:

Now we can construct the gauge transformed bulk monodromy matrix: M (λ|α,

and, in a similar way, we can define:

The definition here chosen of these gauge transformations differ w.r.t. that used previously in the literature on one hand for the particular choice of the right transformation in M (λ|α, β, γ) and, on the other hand, as the parameters on the left and the right transformation are a priori independent.

It is simple to prove by direct computations that:

for some fixed ǫ n = ±1 and k n ∈ {0, ..., p -1}, moreover this state is uniquely defined by:

These are pseudo-eigenstates of the operator B n (λ|α, β):

where:

Proof. The lemma is proven by direct construction. Let us introduce a state:

and look for the conditions to be imposed on c h (n, α, β, γ) in order to satisfy the equation:

By the definition of A n (λ|α, β, γ) it is easy to verify that we have:

(A. 37) where:

),

(A.38)

and we omit to write explicitly the dependence on n, α, β, γ in c h when it is not misleading. So that we get the following system of equations:

together with the cyclicity condition:

It is an easy consequence of the gauge transformed Yang-Baxter commutation relations that under the condition that this state exists and is unique then it is a pseudo-reference state for the gauge transformed Yang-Baxter algebra, i.e. it holds: Ω, α, β, γ|A(λ|α, β, γ) = 0, Ω, α, β, γ|B(λ|α, β) = b(λ|α, β) Ω, α, β/q, γq|,

Ω, α, β/q, γq|C(λ|α, βq, γq) = c(λ|α, βq) Ω, α, β, γ|, Ω, α, β, γ|D(λ|α, β) = 0, (A. 21) with:

Here, we show that we can construct such a pseudo-reference state if and only if we impose at least N + 1 constraints on the bulk and gauge parameters. Let us start our analysis looking to the local conditions to be imposed, in order to do so let us define the local gauge transformed bulk operators: for some fixed ǫ n = ±1 and k n ∈ {0, ..., p -1}, moreover this state is uniquely defined by: c n q rn-1/2 (A.55)

Proof. The operator family A(λ|α, β, γ) is a degree N Laurent polynomial of the form:

where the A n (α, β, γ) are operators, for example we write explicitly:

For general values of the parameters these are invertible operators so that we have to impose at least N + 1 constraints to have that their common kernel is at least one dimensional. We can find the set of constraints by using induction and decomposing A(λ|α, β, γ) in terms of gauged operators on two subchains one of N -1 sites and one of 1 site. The most general decomposition reads:

where we have defined:

|x, y) = A(λ|α, β, x, y) B(λ|α, β, x, y) C(λ|α, β, x, y) D(λ|α, β, x, y) , (A.60)

Then it is easy to show that the only solution of this system of equation is obtained fixing the two gauge parameters by (A.28) which correspondingly fixes the form of the state (A.29).

Let us compute now the action of the operator B n (λ|α, β) on this state; by definition it holds:

so that:

where to get the third line we used the identity C + hn = 0. Now remarking that:

as the effect of q hn is to bring k n to k n + 1 in the state Ω n,α,β,γ |, this, for the gauge choice (A.28), being equivalent to the above redefinitions of the gauge parameters. So that we get:

and so:

Similarly, one can prove our statements for the right state and the action on it of B n (λ|α, β).

Let us remark that if the condition (A.27) are not satisfied we can still derive the left and right local reference states imposing some case dependent condition on the gauge parameters; here for simplicity we have chosen to omit the description of these cases. q hn(kn+1) hn rn=1

B Gauge transformed Reflection algebra B.1 Gauge transformed boundary operators

The gauged two-row monodromy matrix can be defined as it follows:

Note that one can expand this last gauged monodromy matrix in terms of the gauged bulk ones. Moreover, U -(λ|α, β) does not depend on the internal gauge parameter γ, so we are free to chose it at will. The following decompositions hold:

B(λ|α, β/q, γq) D(λ|α, β/q, γq) ,

/λ|γq).

(B.4)

Explicitly, for B -(λ|α, β), it holds:

K-

K-

K- from which it follows:

(A.65)

Then A N,...,1 (λ|α, β, γ) admits a non-zero state annihilated by its action once we impose that it is true for A N,...,2 (λ|α, β, x y ) and A (λ|1, 1/x y , γ) or for A (λ|1, y /x , γ) and A N,...,

), and this state is given by the tensor product of the ones on the two subchains. As the parameters x and y are arbitrary in fact these two conditions are equivalents and so we can chose just one of them.

So let us say we ask the second one and we repeat the same argument for A N,..., So on by induction we get that the existence condition is equivalent to the existence conditions for the following N local operators:

where we have denoted

while the y n /x n for any n ∈ {1, ..., N -1} are free parameters to be used to satisfy the existence condition for the local operators

). From the previous lemma for

, the existence condition is equivalent to:

for any ǫ n = ±1 and the right state annihilated by

From this it is clear that the existence conditions of such a state for A N,...,1 (λ|α, β, γ) coincides with the simultaneous existence for the N local operators (A.66) and that the state is just the tensor product of the states (A.69) so that our proposition is proven. Similarly, we can prove the statement for the right state and using the previous lemma we can prove our statement on the action of the operator B(λ|α, β) on these states.

)

and the left pseudo-eigenbasis characterized by the formulae (3.9) by fixing:

We prove the pseudo-digonalizability and pseudo-simplicity of B -(λ|β) in two steps. We first consider some special representation for which such statement is proven by direct computation then we use this result to prove our statement for general representations. so that β, h

| is nonzero. Using this we can prove that all the states β, h and let us act on it with the following product of operator:

where the generic monomial in it: