Sarah

Théo Sruthi

-J'ai

Éric, Éric, Evgeny, François, Gabriel, Julie, Julien, Lidice, Maxime, Maxime, Mickaël, Olivier Adrien Bastien

Vincent Qi

Affleck Kennedy Lieb

Tasaki Cft

K T Kosterlitz

Thouless Lsm

Lieb Schultz

Mattis Mps

Rokhsar Rk

Kivelson

Wess Zumino Witten

CTMRG Corner Transfer Matrix Renormalization Group DMRG Density Matrix Renormalization Group irrep irreducible representation

Remerciements

Au terme de ces trois années de travail, il convient de remercier celles et ceux grâce à qui j'ai pu mener cette thèse à bien. En premier lieu, je me dois de remercier Didier et Sylvain. Je fus un doctorant souvent plus préoccupé par réécrire un code plus beau et plus performant que par obtenir les résultats que vous vouliez, merci de m'avoir laissé la liberté de faire les choses à ma façon. Je suis heureux que nous ayons toujours réussi à converger à la fin. Merci pour votre disponibilité, entre les cours, les enfants et les travaux de maison à gérer, vous avez toujours su trouver le temps pour discuter. J'ai appris beaucoup de choses au cours de cette thèse, en sens physique comme en méthodes numériques, et cela grâce à vous. Enfin Didier, merci pour le ski à Bénasque.

Je remercie Andreas Läuchli et Norbert Schuch qui ont accepté d'être rapporteurs de ce manuscrit, et par la même occasion Philippe Corboz, Laura Messio et Nathalie Guihéry qui ont participé à mon jury. Je remercie Nathalie Guihéry d'avoir également réalisé le suivi de ma thèse pour l'école doctorale.

Je suis rentré dans le monde des électrons fortement corrélés grâce à Michel Ferrero, qui m'a enseigné le B-A-BA de la physique numérique lors de mon stage au CPhT de l'École polytechnique. C'est lui qui m'avait conseillé de chercher une thèse à Toulouse et je lui en sais gré.

Au sein du groupe fermions fortement corrélés, je suis redevable à Fabien pour ses explications sur la physique RVB, à Matthieu et Ji-Yao pour nos nombreuses conversations sur l'implémentation des symétries dans les réseaux de tenseurs et à Pierre pour ses excellentes recommandations de lecture. Je n'oublie pas Aleksandra, Nicolas, Revaz et Zoran, dont j'ai apprécié les journal clubs. Plus généralement, je remercie tous les membres du Laboratoire de Physique Théorique avec qui j'ai pu interagir au cours de ces trois années. Je dois beaucoup à Malika, notre secrétaire, capable de rendre triviale n'importe quelle démarche administrative ainsi qu'à Sandrine, notre informaticienne, grâce à qui j'ai pu calculer en local comme à distance. Je remercie également l'équipe CALMIP, quand le cluster du LPT ne suffisait plus.

Je rends hommage aux doctorants et postdocs du LPT qui ont toléré mes laïus répétés sur les mérites respectifs du C, du C++, du FORTRAN et du python pour la programmation scientifique. J'écris ces quelques lignes pour vous témoigner ma reconnaissance : avec vous j'ai mangé, bu, discuté, ri, polémiqué, fait du sport, joué, en un mot vécu. Je suis arrivé à Toulouse sans connaître personne, avec vous je repars d'une ville où j'ai des amis.

Je ne suis jamais plus heureux que quand on m'apporte la contradiction : avec Bertrand j'étais donc aux anges, vu qu'il s'avère difficile de nous trouver un seul sujet d'accord -sans doute un manque de pensée complexe de ma part.

Merci Giuseppe de m'avoir sorti alors que je débarquais tout juste à Toulouse, je te pardonne d'avoir encadré ce groupe de L1 qui avait classé la supraconductivité au-dessus de l'astrophysique.

Je remercie Maxime (le blond) de m'avoir initié à iTensor et à l'optimisation de courbes sous python.

Je salue Benjamin (Senior), Grand Maître de l'ordre de Wolfram. Avec toi j'ai rencontré un vrai passionné de physique fondamentale, tu m'as aidé à donner du sens à mes cours de théorie des champs.

Wei-Lin, sache que je suis très admiratif de ta capacité à apprendre d'autres langues. Tu fus un excellent colocataire à Bénasque.

Je ne pouvais côtoyer Benjamin (Junior) que la moitié du temps, parce qu'il suffit que tu dises au LPT que tu es à l'IRAP et à l'IRAP que tu es au LPT et tu es iv tranquille n'est-ce pas. Expert en dinosaures et en interprétation de la mécanique quantique, tu restes à ce jour la seule personne à m'avoir spontanément demandé ce qu'est un spineur.

Julie un immense merci pour ta bonne humeur, ton entrain, ta gentillesse et ta bonne volonté pour organiser des sorties en tout genre -il suffit juste de décaler l'heure de rendez-vous initiale. Avec toi mes horaires au laboratoire paraissaient tout de suite plus raisonnables et j'avais de la compagnie pour rentrer le soir (On rentre bientôt ?) On retourne courir dès que je repasse à Toulouse ! Ce fut inattendu mais ô combien heureux de recroiser Nicolas des années après la promotion Majorana. Merci pour ta gentillesse et ta bonne humeur. Avec toi on pouvait désespérer à deux de l'avenir de notre planète et avoir une conversation basée sur des faits sourcés.

Le niveau en échecs du LPT a fortement progressé grâce au Camarade Hugo, secrétaire général du comité de planification du machine learning et LPT héraut de la France insoumise.

Merci à Jordan de nous avoir emmenés au théâtre, je me garderais de faire ici une critique des différentes pièces vues. Promis je t'envoie des nouvelles de Ponyta dès que j'en ai.

Je remercie Francesca de m'avoir invité chez elle et de m'avoir fait découvrir la réalité virtuelle -quelques vidéos du plus haut intérêt en gardent la trace.

Enfin c'est avec grand plaisir que j'ai partagé mon bureau avec un autre sceptique en la personne de Maxime (le brun). Tu m'as fait découvrir Feyerabend, et on a toujours pu rigoler sans se prendre la tête.

Parmi les stagiaires du laboratoire -Antoine, Célestin, Étienne, Gabriel, Jordy,

Introduction

This thesis presents the results of three years of work under the direction of Sylvain Capponi and Didier Poilblanc inside the Laboratoire de Physique Théorique of the Université Paul Sabatier, Toulouse. It is mostly based on the results published in two articles [START_REF] Gauthé | Entanglement properties of the twodimensional SU(3) Affleck-Kennedy-Lieb-Tasaki state[END_REF] and [START_REF] Gauthé | SU(4) topological resonating valence bond spin liquid on the square lattice[END_REF], with a few additions. The source code used to run the simulations can be found at the url https://framagit.org/ogauthe/PEPS-python.

In this work, we aim to construct and determine the physical properties of paradigmatic wavefunctions of SU(N ) systems. The new possibilities offered by recent advances in cold atoms motivated us to explore the physics of solid states systems made of SU(N )-symmetric components for N > 2. To this extend, we use the numerical methods of tensor networks, which underwent a rapid expansion in the last decade. We follow here an uncommon approach in the spirit of the Bardeen-Cooper-Schrieffer and Affleck-Kenney-Lieb-Tasaki states, focusing not on a given Hamiltonian but on wavefunctions. We construct fully symmetric wavefunctions of SU(N ) systems on a lattice, explore their properties and only afterwards search for an associated Hamiltonian. The exciting features of topological physics inspired us first to construct an SU(3) symmetry protected topological phase phase and second to explore a family of SU(4) topologically ordered states.

Introduction the splitting of a beam of silver atoms through an inhomogeneous magnetic field and understood it as a proof of the existence of space quantization: for an unknown reason, angular momentum had to be quantized in the real world, this was not just a theoretical trick. The orbital momentum of the silver atom is actually zero: the result of Stern and Gerlach is a manifestation of spin, but the concept was not born yet and their interpretation was wrong. Only in 1927 was the Stern and Gerlach experiment correctly interpreted, not until spin had been thought up from other considerations. Another of its manifestation, the anomalous Zeeman effect, was known since 1896 and kept defying theorists.

The concept of spin was first introduced in 1924 by Wolfgang Pauli while he worked on electronic shell [START_REF] Pauli | Nobel Lecture: Exclusion Principle and Quantum Mechanics[END_REF]. To explain the doublet structure of the alkali spectra, he introduced a new electronic degree of freedom that had to be two-valued and had no classical description. This led him to the formulation of his exclusion principle the following year. In 1926, Uhlenbeck and Goudsmit interpreted this degree of freedom as an intrinsic angular momentum due to the self-rotation of the electron and explained the anomalous Zeeman effect with it [START_REF] Jackson | Classical electrodynamics[END_REF]. Hence, at that time, the concept of spin as a quantized intrinsic angular momentum was accepted and the orbital angular momentum was experimentally proven to be quantized. Both of them were relevant and well-defined quantities that could be measured, but no theory explained their quantization.

In 1927, Pauli proposed the first quantum theory of spin in the new formalism of quantum mechanics. He used Pauli matrices to describe the three components of the spin operator and introduced the concept of spinor. The following year, Dirac published his relativistic equation of the electron where spin naturally appears: the modern theoretical description of the spin was born [START_REF] Basdevant | Mécanique quantique[END_REF]. He was also among the first to realize the importance of group in quantum physics.

From there, Weyl and Wigner applied the mathematical framework of group theory to gain a deeper understanding of the mathematical foundations of quantum theory and the role of symmetries in this theory. In 1931, Wigner proved the theorem that bears his name, which stipulates that the bijective transformations of a wavefunction have to be linear or antilinear unitary maps of the Hilbert space. With these tools, spin and orbital angular momentum can be unified by the study of the rotational group SO(3) and their quantization explained. This group is closely tied to the complex unitary group SU(2), their appearance in quantum physics is the consequence of the rotation invariance of the law of physics, which is a subpart of the larger Lorentz invariance.

In high energy physics, the concept of continuous symmetry group turned out to be extremely fruitful with the development of gauge theories. Gauge invariance was already known from classical electrodynamics, but quantum theory brought a new perspective on it. The first gauge theory, quantum electrodynamics, describes electromagnetism by the action of the gauge group U(1) on the wavefunction. The next step was taken by Yang and Mills in 1954, when they extended the concept of gauge theory to non-abelian groups. This fruitful work allowed to treat general SU(N ) gauge group in quantum field theory. In 1961, Gell-Mann proposed his Eightfold Way based on SU(3) representation theory. He applied it to classify observed subatomic particles and accurately predict new ones. This was the first experimentally relevant use of SU(N ) for N > 2, which laid the basis of quantum chromodynamics.

Nowadays, the Standard Model describes the strong, weak and electromagnetic interactions through the gauge group SU(3) × SU(2) × U(1). Beyond Standard Model theories try to pursue unification of interactions with symmetries from larger SU(N ) groups or exceptional Lie groups, however these theories have so far failed to provide new insight experimentally relevant.

The successes of SU(N ) gauge theories in high-energy physics encouraged theorists to look for SU(N ) symmetry in other domains of physics, starting with solid state. There, group theory already plays a major role with Landau theory of phase transition, which is based on spontaneous symmetry breaking. However, SU(N ) symmetry is absent from the fundamental description of condensed matter systems for N > 2. Indeed, the only relevant interaction is electromagnetism, with gauge group U(1), and general space-time invariance is ruled by the Lorentz group SO(1,3), associated with SU(2) only.

However, SU(N ) can emerge as an effective model and be relevant to describe certain systems with additional symmetries. Furthermore, recent advances in the domain of cold atoms systems bring new possibilities to simulate systems with atoms trapped in optical lattices. With these techniques, experimental realization of systems exhibiting SU(N ) physics is no more the exclusivity of the domain of high energies and can be achieved within a laboratory the size of a classroom.

Organization of this manuscript

This manuscript consists in five chapters. The first three ones deal with the general physical context, the mathematical theory and the algorithms. The two last ones present the results obtained using these elements.

The first chapter details how physicists turned to SU(N ) in the domain of condensed matter physics. It starts with an overview of quantum magnetism and the theoretical description of spins, which are ruled by the group SU(2), as well as the conditions for this symmetry to enhance to SU(N ). It then explains how the recent domain of cold atom physics allows to design new types of matter, including quantum simulators for SU(N ) systems. It concludes with an introduction of topological physics, which is an important motivation for our latter work.

In the second chapter we introduce the conceptual tools of representation theory, which is the mathematical framework needed to address SU(N ) physics. We start with general definitions and then focus on the representation of matrix Lie groups. We dive into the structure of the finite-dimensional representations of SU(N ) and describe the formalism of Young tableaux used to label them. Finally, we get back to physics and review how this mathematical structure can be used to obtain major results on physical systems.

The third chapter is dedicated to tensor network methods. We summarize the core ideas of these algorithms and the key role of entanglement. We then detail the type of tensor network we use, the PEPS, and explain how to implement SU(N ) symmetry directly inside the elementary tensor, which dramatically reduces the number of degrees of freedom. We conclude with the description of the corner transfer matrix algorithm, which is the main algorithm we used in our work.

In the fourth chapter, we discuss the Affleck-Kenney-Lieb-Tasaki state, introduced in the binlinear-biquadratic spin-1 chain. We extend its construction to two dimensional lattices and replace SU(2) spins by representations of SU(3). We explore the entanglement properties of this new state and prove it belongs to the class of symmetry protected topological states.

The fifth chapter covers resonant valence bond-like states. We generalize the concept to any representation of SU(N ) on a bipartite lattice and apply it to the staggered fundamental-conjugate representations on the square lattice. We then Introduction consider the special case of the two-fermion representation 6 of SU(4), which is selfconjugate and allows a translation-invariant wavefunction. We construct a family of resonant valence bond-like states that can be either gapless or gapped spin liquids and exhibits its topological order. Lastly, we search for a reasonable, local Hamiltonian that could host this phase.

We tried to base our dissertation on published sources whenever possible, however we acknowledge inspiration from a few unpublished ones. Those documents include A. Zheludev's Advanced Solid State Physics course for the first chapter. For the second chapter, we refereed to D. Bernard and D. Renard's lecture notes Éléments de théorie des groupes et symétries quantiques, partially published in [START_REF] Renard | Groupes et représentations[END_REF], to online course notes on group theory by J.-B. Zuber and to an online document on Young tableaux by G. Ferrera. The third chapter includes many hints from course notes and slides by P. Corboz. Chapter 1

Physics of SU(N ) systems

Before entering the technical aspects of our work, we propose in this chapter an overview of spin physics. We first derive spin Hamiltonians from ab initio principles and list a bunch of relevant magnetic phases. We then recall the basics of the quantum description of spins and show how additional degeneracies enlarge the symmetry group. After that we give a brief description of the domain of cold atoms and how they can be used as a simulator for specific quantum Hamiltonians. We finish with an introduction to topological physics.

Quantum spin systems

1.1.1 From ab initio to spin systems SU(N ) physics is a generalization of the SU(2) theory that describes quantum magnetism, therefore before going to SU(N ), we will first recall the basics of magnetism in solid state physics, focusing on insulators. Solid state physics aims to describe the low-energy properties of large numbers of particles, when nuclei arrange around a stable, fixed configuration. While the first principles that rule the physics of one or two particles at those energies are very well understood, macroscopic systems exhibit brand new properties that cannot be easily derived from those first principles [START_REF] Anderson | More Is Different[END_REF]. The first notion to emerge from considering a large amount of particles in the concept of phase, which is irrelevant to describe a small amount of particles but naturally appears for large systems with a number of particle of the order of the Avogadro number. Since a solid can exist in more than one phase, solid state theory needs to address phases transitions and in order to do this heavily relies on the concept of symmetry breaking. Thus a deep understanding of symmetries is of first importance in this domain of study.

The wide variety of phenomena to be observed in materials, such as magnetism, conductivity or quantum Hall effect all emerge from the very same basic bricks that are nuclei, electrons and Coulomb interaction between them. Because of the large number of degrees of freedom involved, solving the general problem of interacting solid states systems is totally out of reach and new approaches based on effective models are needed. An effective model is a model that exhibits the same physical behavior as the original system in a certain limit while being much simpler. Depending on the problem one wants to understand, different effective models can be derived from the same initial Hamiltonian, the objective being to keep only the relevant terms that account for the dedicated phenomenon and remove all the others for simplicity.

To explain how the magnetism of insulators emerges as a branch of solid state physics, two approaches can be followed. We could start from the simplest model and gradually add elements to describe more complex systems. We would begin with the free electron gas, then go to Bloch waves, mean-field theory, Fermi liquid theory Chapter 1. Physics of SU(N ) systems and finally arrive at strongly correlated fermions. We will rather take the opposite path, starting from the general many-body Hamiltonian of interacting particles and gradually make approximations and truncations to simplify it to interacting spins on a lattice. This derivation and the mechanisms we explain are commonly found in textbooks, e.g. [START_REF] Alloul | Physique des électrons dans les solides. Tome I : Structure de bandes, supraconductivité et magnétisme[END_REF], therefore we will not refer to original articles.

The first limit we will consider is low-energy physics. Solid state physics is relevant at room temperature or below 1 , the energy scale to consider is at most a few electronvolts. We can truncate any state of the Hilbert space above this energy scale and still be accurate, thus we will not consider internal degrees of freedom of the nuclei, which are outside the scope of solid state physics.

As we stated, the general solid state system is composed of electrons and nuclei that interact with each other via Coulomb interaction. The electrons are described by a Hamiltonian containing the kinetic and the Coulomb terms:

H e = i p 2 i 2m e + 1 2 i =i e 2 |r i -r i | . (1.1)
The nuclei follow the same pattern, but their masses and their charges may differ:

H n = j P 2 j 2M j + 1 2 j =j Z j Z j e 2 R j -R j . (1.2)
and the electron-nuclei interaction writes:

H n-e = - i,j Z j e 2 |r i -R j | . (1.3)
The most general Hamiltonian of solid state physics is the sum of those terms:

H 0 = H n + H e + H n-e .
(1.4)

First, we will assume that the nuclei form a perfect crystal lattice, without any disorder. We will not take into account deformations of the lattice either. In this limit, the nuclei are totally decoupled from the electrons and only form a background potential in which electrons move without generating any retroaction on it. This system naturally decomposes into positive ions and outer shell electrons. Indeed, core electrons stay tightly bound to the nuclei and their role can be reduced to a static screening that renormalizes the potential of the nuclei. Away from the nuclei, outer shell electrons have more freedom to hop between sites or interact with other particles at a distance. Summing only on those electrons and renormalizing the ion potential, we get:

H 1 = i p 2 i 2m e + 1 2 i =i e 2 |r i -r i | + i V (r i ).
(1.5)

Those approximations already removed some possibilities of ordering, such as for instance conventional, BCS-superconductivity which rely on electron-phonon interaction. However, the many different configurations of outer shell electrons still allow for a very rich variety of phases to appear.

1.1. Quantum spin systems 7 This Hamiltonian is still tremendously complicated, with long-distance interactions and continuous variables. The next step is therefore to assume the Hilbert space of the electrons can be decomposed on a basis of atomic orbitals -or more rigorously, Wannier functions. This is the so-called tight-binding approximation, since electrons can now only exist together with the lattice sites. We consider only one orbital per site, therefore according to Pauli principle only two electrons of opposite spins can be on the same site. We will consider only half-filling, with an average occupation number of one electron per site. Since they carry kinetic energy, electrons can hop from site to another. In our model, the kinetic term becomes an amplitude t that couples two different sites orbitals, we restrict this hopping to first neighbor only. Under the effect of screening, the effective Coulomb interaction decreases faster and can be neglected at large distances: we keep only repulsion between two electrons on the same site. We arrive at the Hubbard model:

H 2 = -t i,j ,σ c † i,σ c j,σ + U i n i↑ n i↓ . (1.6)
This model is totally contained into the previous ones, but thanks to our simplifications the relevant parameters now appear more clearly. It has two well-defined limits: for t/U 1, the effective interaction between electrons is small and electrons are weakly correlated. Since kinetic term dominates, they delocalize on the whole lattice to form a conduction band and at half filling the material is a metal. For t/U 1, electrons are strongly correlated, the energy cost for two electrons to be on the same site is high and they localize on the lattice sites, with exactly one electron per site at half-filling. Without conduction, the material is an insulator, called Mott insulator. The transition from metal to insulator under the effect of interactions is the Mott transition.

On the localized side of the Mott transition, electrons are attached to a site but they can still make virtual hopping from one site to another to minimize their kinetic energy. Because of Pauli principle, hopping from one site to another is only possible if the electrons from two neighboring sites have opposite spins. No minimization by virtual hopping can occur for parallel spins and antiferromagnetic ordering is favored. This process can be formalized using second order perturbation theory in t/U and leads to an effective spin Hamiltonian as a strongly interacting limit of the Hubbard model at low temperature:

H 3 = J i,j S i • S j .
(1.7)

with an antiferromagnetic coupling constant J = 4t 2 /U . This Hamiltonian is the final step of our simplification: ions and electrons moves are frozen, the only degrees of freedom are spins on a lattice. Because of electron localization, there is no conductivity.

The system is a magnetic insulator and it is described by a spin-1/2 wavefunction

|Ψ = c i 1 ...i N |i 1 ...i N .
(1.8)

Looking backward, we have been extremely crude in our unrolling, putting aside some elements that are not negligible compared to the ones we kept and making many assumptions on the form of those ones, especially on geometry and interaction range. The key point is that in magnetic insulators, low-energies properties can be deduced by considering only spin degrees of freedom in simple effective Hamiltonians. Proceeding in a more subtle and cautious way does not change this hand-waving picture.

Chapter 1. Physics of SU(N ) systems

We note that the microscopic origin of the spin-spin coupling J is not the magnetic dipolar interaction but the very strong Coulomb repulsion between electrons. This dipolar interaction is orders of magnitude smaller than the ones resulting from the interplay of Coulomb's law and kinetic energy and cannot account for magnetic ordering at room temperature. However, other interactions can emerge from the combination of Coulomb repulsion and Pauli principle. The first one to consider is the exchange interaction. A toy-model to understand it considers two sites with an electron localized on each of them, on the same orbital. The total electronic wavefunction is a product of the orbital wavefunction and the spin wavefunction. Since electrons are fermions, it has to be antisymmetric by exchange of particles. It can either be orbital-symmetric and spin-antisymmetric, where the two electrons form a spin singlet, or orbital anti-symmetric, spin-symmetric, where the two electrons form a triplet. Without electron-electron interaction, those four states are degenerate. Now, if we turn it on, it lifts the degeneracy between the singlet and the triplet, since electrons are not at the same distance from each other in the two configuration. The effective interaction is again a spin-spin S i • S j term. The exact value of the coupling constant is very hard to obtain theoretically and is usually fit from experiments, its sign itself depends on the overlap of the two orbital wavefunctions. Due to its origin in orbital wavefunction geometry, it is very short-ranged. To account for interaction at longer distances, one has to consider superexchange. There, hybridization between the orbitals of an anion and two cations generates a strong coupling of the cations that can be either ferromagnetic or antiferromagnetic depending on the geometry.

Thus, depending on the microscopic mechanism causing it, the sign, the value and the range of the effective interaction constant may change. Considering general, long-range spin-spin interactions, we get the more general Hamiltonian:

H 4 = i,j J ij S i • S j .
(1.9) This is the Heisenberg Hamiltonian, which lays the foundation of quantum magnetism. We note that it does not privilege any direction: for any state, rotating every spins of the system by the same angle does not change the energy. This rotation invariance is the basis of SU(2) physics.

The eigenvectors of the Heisenberg Hamiltonians depends on the values of the coupling constants J ij , but also on the dimension and the lattice we consider. Even in the simplest case of uniform, first neighbor interaction, its resolution is difficult. It is exactly solvable in one dimension and for spins 1/2 only, using Bethe ansatz. In higher dimensions, one has to make approximations or rely on numerics.

Other terms, not necessarily Heisenberg-like, can appear in magnetic systems. Since we are interested in symmetries, we will not consider models that explicitly break it, such as the Ising or XXZ models, or any model including an external field. We will also restrict ourselves to two-sites terms only, although SU(2) symmetric three-spins or more terms can be constructed, such as S i • (S j × S k ).

Magnetic order and disorder

In section 1.1.1, we detailed how spins Hamiltonians emerge as an effective model in the context of solid state physics. In this section, we will review a couple of magnetic phases that can also be found in SU(N ) systems. The subject is very rich and we will not try to be exhaustive, instead we concentrate on phases we will meet in subsequent chapters and keep our focus on the Heisenberg Hamiltonian. At high energy, when the temperature is way larger than coupling constants, spins are free and independent. They do not point in any preferential direction and the full SU(2) symmetry is preserved. The state is thermal disordered and paramagnetic. At lower temperatures, this symmetry may spontaneously breaks down, giving birth to a multitude of magnetic phases. Different phases are separated by phase transitions, which are driven by thermal fluctuations.

At zero temperature, there are no thermal fluctuations: the system is in its ground state, but depending on the Hamiltonian, this ground state can exhibit tremendously different properties. Indeed, different states belonging to different phases can have very close energies. If the Hamiltonian depends on a continuous parameter λ, there is a competition between those states and at a critical value λ c , the system encounters a quantum phase transition and its phase changes. Reaching zero-temperature is not experimentally achievable. More, zero temperature phase can differ from any thermal state since at any finite temperature there can be no ordering in one dimension and in two dimensions, the Mermin-Wagner theorem states that no continuous symmetry can be broken. Yet describing quantum phases at zero temperature is still very much relevant. The first reason is that low-temperature physics of materials are highly dependent on the nature of the ground state. More, the presence of a quantum critical point generates peculiar properties even at high temperature, a phenomenon refereed as quantum criticality [START_REF] Sachdev | Quantum Phase Transitions[END_REF]. As an example, the high-temperature conductivity in cuprates is closely tied to the vicinity of a quantum critical point between magnetic and Fermi liquid phases [START_REF] Proust | The Remarkable Underlying Ground States of Cuprate Superconductors[END_REF]. Hence we will now consider zero-temperature phases of the Heisenberg Hamiltonian.

When all the coupling constants J ij are negative, the system chooses a given orientation and all spins point towards the same direction: the system is a ferromagnet, Chapter 1. Physics of SU(N ) systems as seen in figure 1.1 (a). The magnetization acquires a finite mean value m and SU(2) symmetry is spontaneously broken. Although this behavior is well-known for metals, it is very rare in magnetic insulators, yet still exists for instance in EuS or EuO [START_REF] Wachter | Chapter 19 Europium chalcogenides: EuO, EuS, EuSe and EuTe[END_REF].

On a bipartite lattice with only positive coupling constants between first neighbors, the system also chooses a given orientation and the two sublattices point towards opposite directions, as shown in figure 1.1 (b). The mean magnetization is zero but the staggered magnetization is non-zero and breaks SU(2): this is the antiferromagnetic phase. Nonetheless, the purely staggered state |↑↓↑↓ . . . , or Néel state, cannot be the ground state because it is not an eigenvector of the first-neighbor Heisenberg Hamiltonian. Indeed, the term S • S exchanges spins and therefore modifies the Néel state. The exact ground state is more complicated, with quantum fluctuations lowering its energy by lowering the amplitude of the order parameter, but the classical Néel state captures its key features, including the doubling of the elementary cell size. This phase is the most common for Mott insulators on a bipartite lattice, in particular it is the low-doping limit of cuprate superconductors [START_REF] Proust | The Remarkable Underlying Ground States of Cuprate Superconductors[END_REF].

All the states we surveyed are low-entanglement states which can be conceived with a classical picture, even though the true quantum state is more subtle. This is no more true for some other, purely quantum phases that have no classical equivalent, where entanglement plays a key role. They usually appear in the context of frustrated magnetism, when the different constraints appearing in the Hamiltonian cannot be all fulfilled at the same time [START_REF] Mila | Frustrated Spin Systems[END_REF]. Frustration prevents magnetic ordering and at low temperature such systems are often spin singlets, with S tot = 0 [START_REF] Lhuillier | Introduction to Quantum Spin Liquids[END_REF].

An accurate description of the total singlet subspace of the Hilbert space is given by valence bond theory. A valence bond, or dimer, consists of two spins pairing into a singlet: this is a purely quantum object since the two spins are maximally entangled and cannot be described separately. Many different states can then be constructed as tensor products of valence bonds that do not break SU(2) symmetry and are therefore spin singlets. The set of states which are products of long-distance valence bond indeed form an overcomplete basis of the total singlet subspace. Even when the pairing is restrained to first neighbor only, there is a macroscopic large number of those dimer covering states.

Valence bonds can assemble themselves into larger structures, or plaquettes, that are independent from each other, with more than two spins forming a singlet. When the dimers or plaquettes form a periodic structure that covers the lattice, the state is called a valence bond crystal (VBC). An example is presented in figure 1.1 (c) with a regular dimer covering, another with a larger unit cell would be 2×2 plaquettes on the square lattice. Such a crystal has a long-range order that breaks lattice symmetries and a local order parameter can be defined although it only has short-range spin-spin correlations.

As soon as the lattice contains an odd number of spins per unit cell, as in the square lattice, any dimer covering state breaks lattice symmetries. This is not surprising because a dimer covering can be seen as a solid made of valence bonds. However, when the dimers are not static but allowed to fluctuate lattice symmetries can be recovered. Systems without any long-range order can occur even at zero temperature, favored by quantum fluctuations induced by frustration. These systems are strongly correlated yet have no obvious ordering 2 and exhibit only short-range spin-spins correlations: in this they are similar to liquids and therefore are called quantum spin liquids (QSL). The QSL is a very exotic phase with surprising properties. It does not break any symmetry, neither spatial or SU(2), and therefore yields no local order parameter: it is completely featureless (see figure 1.1 (d)). In ferromagnets and antiferromagnets, the breaking of SU(2) symmetry generates massless Goldstone modes, known as spin waves. In VBC states, no continuous symmetry is broken and a gap protects the state from excitations. On the other hand, quantum spin liquids can be either gapped or gapless and have fractional excitations, the spinons, which occur only by pair but are deconfined and decouple spin and charge degrees of freedom.

The first example of such a spin liquid is the resonating valence bond (RVB) state and was proposed by Anderson in 1973 [And73], who later developed a theory of high-temperature superconductivity based on it [START_REF] Anderson | The Resonating Valence Bond State in La 2 CuO 4 and Superconductivity[END_REF]. It is an equal-weight superposition of all possible first-neighbor dimer coverings. While it as been ruled out in its initial proposal as a ground state for the spin-1/2 Heisenberg model on the triangular lattice [START_REF] Bernu | Signature of Néel order in exact spectra of quantum antiferromagnets on finite lattices[END_REF] , it is still a plausible ground-state candidate on the kagome spin-1/2. Short-range parents Hamiltonian designed to stabilize it as their ground state have been constructed, but they are rather complicated with many spins involved in the elementary terms [START_REF] Cano | Spin Hamiltonians with Resonating-Valence-Bond Ground States[END_REF].

The experimental search for a quantum spin liquid is very challenging because of the absence of any local order parameter, and more generally of any exclusive feature experimentally accessible [START_REF] Balents | Spin liquids in frustrated magnets[END_REF]. Furthermore, QSL seems to be a very rare phase in real materials, antiferromagnets being much more common in Mott insulators. Searches concentrate on the Kagome lattice with spin 1/2 and convincing arguments account for the Herbertsmithite to be a QSL [START_REF] Shaginyan | Identification of strongly correlated spin liquid in herbertsmithite[END_REF][START_REF] Norman | Colloquium: Herbertsmithite and the search for the quantum spin liquid[END_REF].

Finding the ground state phase of a given Hamiltonian is a hard problem. Frustrated systems usually carry a sign problem and cannot be solved with Monte Carlo simulations (see section 2.4.3), other analytical or numerical approaches are needed but do not always concur. In spite of a general solution, some knowledge on the solution can be obtained from exact theorems. Specifically, the Lieb-Schultz-Mattis (LSM) theorem [START_REF] Lieb | Two soluble models of an antiferromagnetic chain[END_REF], later extended by Oshikawa [START_REF] Oshikawa | Commensurability, Excitation Gap, and Topology in Quantum Many-Particle Systems on a Periodic Lattice[END_REF] and to 2D by Hastings [START_REF] Hastings | Lieb-Schultz-Mattis in higher dimensions[END_REF], states that in the thermodynamic limit, any half-integer spin Hamiltonian with an odd number of sites per unit cell either has a degenerate ground state, indicating symmetry breaking or topological ordering, or has gapless excitations. When no symmetry breaking is observed, the ground state must therefore have long-range correlations.

SU(2) physics

Our presentation of quantum magnetism can very well be summed up in whether and how the SU(2) symmetry is broken in magnetic phases. As we stated in our historical section, the group SU(2) already appeared in atomic physics to understand the electronic shell of the atom. In fact, the same mathematical framework unifies orbital angular momentum and spins and gives a solid basis to their quantization. In this section, we recall this framework and its consequences for spin systems and show how it can give rise to SU(N ).

Quantization of angular momentum

A full mathematical description of the angular momentum using group theory will be developed next chapter, for now we will only derive basic properties of SU(2) operators as can be found in elementary quantum mechanics textbooks, as for instance [START_REF] Basdevant | Mécanique quantique[END_REF]. In this subsection only, we will put the labels of the operators as indices instead of exponents to improve readability.

Chapter 1. Physics of SU(N ) systems

We define the orbital angular momentum observable by replacing the variables x and p by their quantum equivalent in its classical definition:

S = x × p (1.10)
S is a vector operator with three components S x , S y and S z . Imposing the commutators of the canonical variables [x, p] = i implies the following commutation relations for S:

[S α , S β ] = i αβγ S γ , (1.11)
where is set to 1 and αβγ is the Levi-Civita symbol. Equation (1.10) has little meaning for spins, since quantum particles are punctual and any tentative to see them as rotating spheres leads to faster-than-light speeds. Hopefully, we do not actually need it: every necessary information is contained inside the commutators (1.11) only.

Thus we impose any angular momentum observable, including spin, to obey these commutation rules and build our derivation from them.

Let us define the norm of the angular momentum S 2 = S 2 x + S 2 y + S 2 z . By virtue of the commutators, it commutes with the three components of S. Our goal here is to understand the structure of the Hilbert space H on which the operators S α acts. In order to do it, we will decompose H into blocks of S 2 eigenspaces. These subspaces are stable under the action of S α since S 2 commutes with each of them. We will then look at the action of these operators inside these blocks.

We consider a given characteristic subspace H s of S 2 . The square of an hermitian operator is an hermitian positive operator, as a sum of squares the observable S 2 has to be hermitian positive. The eigenvalue associated with this eigenspace is therefore real and positive and we without loss of generality can write it s(s + 1) with s real and positive.

Since S 2 and for instance S z are commuting hermitian observables, we can codiagonalize S z inside H s in an orthonormal basis. Let m be an eigenvalue of S z , m has to be real. By definition of S 2 , S 2 -S 2 z is also a positive operator so that

s(s + 1) -m 2 ≥ 0 (1.12)
and the eigenvalues of S z are bounded. We define the operators S + and S -as

S + = S x + iS y , S -= S x -iS y .
(1.13) These operators are not hermitian, instead S † ± = S ∓ . They do not represent any physical observable but they are useful mathematical tools. As linear combinations of S x and S y , they also commute with S 2 and they can be restrained to H s . They do not commute with the operator S z and yield the commutators

[S z , S ± ] = ±S ± .
(1.14)

Let |s, m be an element of H s that is also an eigenvector of S z associated with the eigenvalue m. Then

S z S ± |s, m = (S ± S z ± S ± ) |s, m = (m ± 1)S ± |s, m (1.15)
by definition of |s, m . Thus S ± |s, m is an eigenvector of S z associated with eigenvalue m ± 1 or it is the null vector. This means that the operators S + and S -act as ladder operators on the Hilbert space H s , in a similar way to the creation and annihilation operators of bosons and fermions. Starting from vector |s, m , they 1.2. SU(2) physics 13 generate orthogonal vectors associated with eigenvalues m ± 1. Since m is bounded, the set of vectors generated from the same starting point has to be finite because applying iteratively S + or S -shifts its value by 1 until it gets out of bounds. Note that we did not impose the dimension of the eigenspace H s to be finite, but it naturally splits into independent, finite dimensional subspaces. Any state from such a subspace spans all the other by applying the ladder operators.

To quantize s and m, we have to look at the operator S -S + . We have

S -S + = (S x -iS y )(S x + iS y ) = S 2 x + S 2 y + i[S x , S y ] = S 2 -S 2 z -S z = S 2 -S z (S z -1)
and therefore

S 2 = S -S + + S z (S z + 1) = S + S -+ S z (S z -1) (1.16)
Let us consider the highest-m vector from this set. By virtue of equation (1.15), this vector belongs to the kernel of S + , else it would not be maximal. Applying the first part of equation (1.16) on this vector gives

S 2 |s, m = S z (S z + 1) |s, m = m(m + 1) |s, m ,
(1.17) but |s, m is also an eigenvalue of S 2 with eigenvalue s(s+1), which imposes m(m+1) = s(s + 1). The negative solution has to be discarded because it violates equation (1.12), therefore m = s. Going in the other direction with S -also reaches a last non-zero vector. Applying the second part of equation (1.16), we find m(m -1) = s(s + 1). This time we keep the negative solution and we conclude m = -s.

Applying the operator S -a finite number of time q shifts the eigenvalue of S z by -q. If we apply it on the full range of values from s to -s, we find s -q = -s and therefore 2s = q: s is either integer or half integer. The eigenvalue m of S z takes all the values -s, -s + 1, . . . , s -1, s. Since S x and S y play exactly the same role as S z , the same result stands for their eigenvalues.

We have found the well-known results on spins. Just by imposing commutators, the angular momentum is quantized to integer or half-integer values. The Hilbert space splits into blocks called spins, indexed by a positive integer or a half-integer s. This subspace has dimension 2s + 1 and is an eigenspace for the observable S 2 associated with eigenvalue s(s + 1). One operator, here S z , can be diagonalized inside this subspace and takes integer or half-integers values. Reciprocally, for each integer of half-integer s there exist such a Hilbert space of dimension 2s + 1.

From a mathematical point of view, what we just did is the determination of every finite-dimension representation of the Lie algebra su(2). We will generalize theses results in section 2.2.

Addition of angular momentum

For a system of more than one particle, the eigenvalues of the observables of each particles are not good quantum numbers. One has to consider the sum of all the local operators. For simplicity, we will restrict ourselves to the case of two particles of spin s 1 and s 2 , the generalization is straightforward.

Chapter 1. Physics of SU(N ) systems

The Hilbert space to consider is the tensorial product of each particle Hilbert space:

H = H 1 ⊗ H 2 , (1.18)
where H 1 is the space of spin s 1 , with dimension 2s 1 + 1 and H 2 is the space of spin s 2 , with dimension 2s 2 + 1. The dimension of H is the product of the dimensions of H 1 and H 2 and every observable on one site is obtained by taking the tensorial product with the identity on the other site:

O 1 H = O 1 H 1 ⊗ I H 2 . (1.19)
Consequently any observable on one subsystem commutes with any observable on the other one. Using this definition we can define the total angular momentum as

S = S 1 + S 2 . (1.20)
This observable obeys the commutation relations (1.11) and all the previous results apply on the Hilbert space H. We can diagonalize the operator S 2 and split H as a direct sum of different spins s. Of course, the spins that appear in this decomposition are linked to the spins s 1 and s 2 . This leads us to the concept of fusion rules: the tensor product of two spins decomposes in a direct sum of spins that is totally determined by s 1 and s 2 . The tensor product and direct sum of spins pictures are totally equivalent, therefore their total dimension have to agree. By convention, we denote the space of spin s by its dimension in bold figures. In the case of two spins 1/2, we find the well known fusion rule

2 ⊗ 2 = 1 ⊕ 3, (1.21)
that is the Hilbert space of two spins 1/2 decomposes as a direct sum of a singlet and a triplet. We check that the total dimension is the same on the two sides of the decomposition.

The common eigenvectors of S 2 and S z , obtained by diagonalization, are commonly expressed in the tensor product basis of H. In our case, we find

|0, 0 = 1 √ 2 (|↑↓ -|↓↑ ) |1, 1 = |↑↑ |1, 0 = 1 √ 2 (|↑↓ + |↓↑ ) |1, -1 = |↓↓
We note that the singlet is antisymmetric by exchange of the two particles while the triplet is symmetric. The coefficients that appear in this basis are the so-called Clebsh-Gordon coefficients. Their general expression is quite complicate, for smalldimensional Hilbert spaces it is easier to obtain them by codiagonalization. However, it is not necessary to compute them to obtain the fusion rules: they can be deduced from s 1 and s 2 only, without any linear algebra. The general formula for spins is that the product of s 1 and s 2 decomposes into the sum of the spins s 1 + s 2 , s 1 + s 2 -1, . . . , |s 1 -s 2 |. In term of fusion rules, this can be written

(2s 1 +1) ⊗ (2s 2 +1) = 2(s 1 +s 2 )+1 ⊕ 2(s 1 +s 2 )-1 ⊕ . . . ⊕ 2|s 1 -s 2 |+1.
(1.22)

SU(2) invariant Hamiltonian

We define that a Hamiltonian is SU(2) invariant if it commutes with the spin observables S x , S y and S z of the system. This definition is equivalent to the rotation invariance from section 1.1.1, the demonstration will be given in the next chapter. For now we simply admit that as a consequence, the spectrum of such a Hamiltonian has a multiplet structure. This means it can be split into different spins s, with every 2s + 1 basis states having the same energy. Note that a given spin value s can appear more than once in this decomposition, each of the occurrences having a different energy.

The simplest example is the Heisenberg Hamiltonian for two spins-1/2 H = S 1 • S 2 . Diagonalizing it yields two eigenvalues: -3/4, of degeneracy 1, corresponding to the singlet state and +1/4, of degeneracy 3, corresponding to the triplet, or spin 1. Introducing the projectors on the singlet and the triplet P 1 and P 3 , we can rewrite this Hamiltonian as

H = - 3 4 P 1 + 1 4 P 3 (1.23)
This decomposition generalizes to any number of spins, for any spin s. Any SU(2) invariant Hamiltonian can be rewritten as a sum of projectors on the different spins of the Hilbert space. If the Hilbert space is a direct sum of n different spins, the Hamiltonian is fully determined by the n different coefficients in front of the projector, independently of the dimension of the space, which depends on the different spins involved. Therefore SU(2) symmetry dramatically reduces the number of parameters of a Hamiltonian. It is also a powerful tool in computations: for large systems, Hamiltonians are huge matrices that are not easy to deal with. However, knowing the decomposition in term of spins of the space is enough to diagonalize them. Such a decomposition may not be easy to reach, but at least the decomposition in blocks of fixed S z value is very simple coming from the tensor product basis, and the Hamiltonian can be block-diagonalized inside them.

SU(N ) points of a spin Hamiltonian

SU(2) physics appears from considering rotation-invariant systems and enforcing this symmetry inside the Hamiltonian by making it commute with the three operators S x , S y and S z . These operators are tensor products of local observables S x i , S y i and S z i , which are traceless, linearly independent matrices acting on spin i. SU(N ) physics is a generalization of SU(2) physics where we consider not only 3 but N 2 -1 linearly independent operators S α that commute with the Hamiltonian. Again, the spectrum decomposes into degenerate multiplets which are necessarily block diagonal for S α and the Hamiltonian can be decomposed as a sum of projectors on those multiplets.

The first appearance of SU(N ) symmetry in condensed matter physics is through finely-tuned spins Hamiltonians. For special ratios of the coupling constants, some spins multiplets become degenerate and SU(2) symmetry is enhanced to larger groups, leading to SU(N ) physics.

The simplest example is a system of spins-1 with two-sites interactions. The general fusion rule (1.22) writes here

3 ⊗ 3 = 1 ⊕ 3 ⊕ 5, (1.24)
Chapter 1. Physics of SU(N ) systems meaning the Hilbert space obtained by tensor products of two spins 1 can be decomposed as a spin 0, a spin 1 and a spin 2. Therefore, introducing the projectors on the spins subspaces, the most general SU(2) invariant Hamiltonian acting on two sites is:

H = αP 1 + βP 3 + γP 5 . (1.25)
Now, at the precise point β = γ, the triplet and the quintet are degenerate. The Hamiltonian makes no difference between spin 1 and spin 2 and this degeneracy extends to the tensor products of those spins when more than two sites are considered. The fusion rule becomes 3 ⊗ 3 = 1 ⊕ 8 (see next chapter for the explanation of the overline), it is more restrictive and implies a larger symmetry group. Indeed at this point the Hamiltonian commutes not only with S x , S y and S z but with a total of eight linearly independent and traceless matrices acting on a given site. This is an SU(3) point of the Hamiltonian. Another SU(3) point arises when α = γ, with an associated fusion rule 3 ⊗ 3 = 3 ⊕ 6.

The form of equation (1.25) is unusual, but it is just another way of writing the spin-1 bilinear-biquadratic Hamiltonian. Up to an irrelevant energy-shift and setting the coupling constant to 1, the bilinear-biquadratic spin-1 chain is parameterized by a single parameter θ and its Hamiltonian reads

H 1BB = i cos θ S i • S i+1 + sin θ (S i • S i+1 ) 2 .
(1.26)

The rule 1 ⊕ 8 corresponds to the points θ = ±π/2, the rule 3 ⊕ 6 to the points θ = π/4 (integrable Uimin-Lai-Sutherland point) and -3π/4. At these points, the whole chain is described by an SU(3) theory: in the first case the chain is equivalent to a staggered quark-antiquark chain, in the second to a quark chain. These cases are only discrete points in the phase diagram, but their presence gives precious information in whole regions of the phase diagram, which has been extensively studied [IK97; LST06; Man+11]. Indeed, for π/4 < θ < π/2, the system is gapless and even though it does not exhibit the full SU(3) symmetry, its long-range properties are ruled by an SU(3) 1 Wess-Zumino-Witten (WZW) theory [START_REF] Witten | Global aspects of current algebra[END_REF] with some perturbations added. Moreover, in the presence of strong disorder their characteristic properties extends to the whole phase diagram, which shows an emergent SU(3) symmetry [START_REF] Quito | Emergent SU(3) Symmetry in Random Spin-1 Chains[END_REF]. Note that we considered a 1D chain for simplicity but the argument stands for any lattice with a nearest-neighbor interaction. The points θ = π/4 and -3π/4 always have an additional symmetry and play a key role in the phase diagram, while for the points θ = ±π/2 the lattice has to be bipartite. Another possibility for SU(2) physics to enhance to SU(N ) arises when different atomic orbitals are taken into account. These materials are described by the Kugel-Khomskii Hamiltonian [START_REF] Kugel | The Jahn-Teller effect and magnetism: transition metal compounds[END_REF], which defines a pseudo-spin to account for the different orbitals and couples the spin and orbital degrees of freedom. The crystal field usually lifts the degeneracy of the orbitals and generate anisotropic effective interactions that depend on the relative orientation of the orbitals with respect to the lattice orientation. However in some cases, more than one orbital is compatible with the crystal field, leading to orbital degeneracy [START_REF] Van Den Brink | Frustration in Systems with Orbital Degrees of Freedom[END_REF].

The pseudo-spin is still highly anisotropic due to Hund's coupling. In the isotropic limit, the Hamiltonian becomes

H = i,j (4S i • S j + 1)(4τ i • τ j + 1) (1.27) 1.3. Cold atom systems
where the τ are Pauli matrices acting on the orbital pseudo-spin. Since rotation invariance is assumed both in spin space and in the two orbital space, the Hamiltonian has an SU(2) × SU(2) symmetry. In fact, due to the symmetric role played by the spin and the pseudo-spin, its symmetry is enhanced to SU(4) [START_REF] Li | SU(4) Theory for Spin Systems with Orbital Degeneracy[END_REF]. This can be checked by looking at the eigenvalues, which follow the fusion rule is 2 ⊗4 = 6 ⊕ 10. In real materials, Hund's coupling is not small and this model has to be understood as a simplified limit of the Kugel-Khomskii Hamiltonian. Yet this unrealistic assumption facilitates its study and can bring new insight on the physics of orbital degeneracy.

For instance, the symmetric version of the model has been shown to host an SU(4) quantum spin liquid phase on the honeycomb lattice [START_REF] Corboz | Spin-Orbital Quantum Liquid on the Honeycomb Lattice[END_REF]. The compound α-ZrCl 3 has been proposed as a material realizing this phase thanks to strong spinorbit coupling, still neglecting Hund's coupling [START_REF] Yamada | Emergent SU(4) Symmetry in α-ZrCl 3 and Crystalline Spin-Orbital Liquids[END_REF].

Cold atom systems

In condensed matter physics, we saw that SU(N ) physics appears as an enhancement of SU(2) in a given limit of a model or at discrete points of a family of Hamiltonians. While stimulating for a theoretician, its physics is unfortunately rarely relevant for experimentalists who cannot access those singular conditions. Although many theoretical tools have been developed to investigate SU(N ) systems in solid state physics, they remain mostly theoretical objects with little link to existing materials. However things are different in the domain of cold atoms, where dramatic advances in the last decades offer a new playground to probe condensed matter results. In this section, we will review the core principles of this field and how cold atoms can be used as a quantum simulator for SU(N ) physics.

Optical lattices

The first experimental realization of a Bose-Einstein condensate with cold atoms was achieved in 1995 by Cornell, Wieman and co-workers [START_REF] Anderson | Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor[END_REF]. It brought many advances in the ability of experimentalists to cool down and control atoms at very low temperatures. This control allows a new paradigm for physicists, not only to to probe matter but to design it along their wishes. Cold atoms can thus be used as a quantum simulator for many different models, with an excellent control on the parameters of the model.

The key point of control is cooling: atoms are cooled to very low temperatures, of the order of 1 µK or below, where they can be dealt with extraordinary precision. Because of the very low temperature, most of the atom degrees of freedom are frozen, including electronic excitations to higher orbitals or other atoms. The atoms can be seen as neutral objects that interact through induced electric dipole moments.

Cooling is done by several steps: first atoms are sent in a high-quality vacuum where their velocity is reduced by Doppler cooling inside a magnetic field. When their energy is low enough, they cannot escape the magneto-optical trap. The next step is evaporative cooling: atoms that have a too high velocity are removed from the trap and only the slowest ones are kept. The result is a net loss in the variance of velocity distribution and therefore of the temperature of the system.

Lasers are assembled in order to create a periodic electric potential. Cold atoms interact with it and when the temperature is low enough they localize at the minima of the potential. Atoms as a whole are trapped in the optical potential, where they can hop from one minimum to another by tunneling. The result is an optical lattice of atoms, completely similar to the crystal potential binding electrons in a solid. Here, the lattice properties are totally tunable: the lattice type and spacing are controlled by the laser disposition and the depth of the optical potential controls the hopping amplitude. Depending on their number of nucleons, the atoms can have fermionic or bosonic statistics, which dramatically changes their behavior at such low temperatures. In the case of fermions, nuclear spins degrees of freedom still allow for more than one atom per site. Atoms have a Van der Waals-like, dipolar interaction with each other, which decay as 1/r 6 and can be well approximated as on-site. This interaction is the analog of the local Coulomb repulsion from strongly correlated electrons. It is therefore possible to simulate the Hubbard model (see figure 1.2) while tuning all its parameter, which cannot be done in solid state materials. Well known phenomena from solid state have been observed in cold atoms, including Mott transition as well as Anderson localization.

While it should be possible to simulate every aspects of this model, it is still very hard to reach temperature low enough for quantum magnetism to be the effective theory because the strongly interacting limit t U is only seen for T ∼ t 2 /U ∼ 1 nK [START_REF] Wall | Quantum Magnetism with Ultracold Molecules[END_REF]. Reaching this temperature for a relatively large number of atoms is still an experimental challenge.

SU(N ) quantum simulators

Cold atoms offer many possibilities to design new types of matter, one of the most exciting being SU(N ) systems. The key ingredient is nuclear spin: in some cases, a subset or all of the N = 2I + 1 nuclear spin states are degenerate for the effective Hamiltonian that rules the system and the atoms acquire an emergent SU(N ) symmetry. The Bose-Einstein condensate (left) is characterized by large particle number fluctuations, whereas in the Mott insulating state (right) -dominated by strong repulsive interactions between the particlesthese are largely suppressed. Individual quantum and thermal fluctuations of the strongly interacting many-body system can be directly probed in the system. Figure and caption taken from [START_REF] Bloch | Quantum simulations come of age[END_REF].

The original successes of cold atoms experiments were obtained using alkali atoms, which have a simple electronic structure with only a single valence electron. Under a strong magnetic field, nuclear-spin degrees of freedom become decoupled and 6 Li acquires an approximate SU(3) symmetry, which has been observed experimentally [START_REF] Ottenstein | Collisional Stability of a Three-Component Degenerate Fermi Gas[END_REF][START_REF] Huckans | Three-Body Recombination in a Three-State Fermi Gas with Widely Tunable Interactions[END_REF]. However this simplicity comes to a price since this lonely electron interacts with other atoms with an effective spin-spin interaction that is not easy to tackle. More, unless a strong magnetic field is applied, nuclear-spin interacts with electronic degrees of freedom and this coupling breaks down SU(3) symmetry.

Subsequent researches were made using more complex alkaline-earth atoms and some other with similar electronic structure, such as Ytterbium. The advantage is that in the ground state, the electronic spin and the orbital momentum are zero, therefore they cannot interact with the nuclear spin and I is a good quantum number [START_REF] Cazalilla | Ultracold Fermi gases with emergent SU(N ) symmetry[END_REF]. This property even extends to some excited electronic states. These atoms have been used to realize the SU(N ) Hubbard model:

H = -t i,j N -1 α=1 c † iα c jα + U 2 i n i (n i -1) (1.28)
which was realized for SU(6) with the fermionic 173 Yb [START_REF] Taie | Realization of a SU(2)×SU(6) System of Fermions in a Cold Atomic Gas[END_REF]. Other experiments have been done with 87 Sr and report probing SU(N ) magnetism with a nuclear degeneracy up to N = 10 [Zha+14].

Topological phases

One of the most exciting advances in recent years in solid state physics is the discovery of topological systems. These systems do not break any symmetry yet they exhibit non-trivial properties: they lie beyond Landau paradigm of spontaneously broken symmetries and new conceptual tools are needed to address them. Two categories can be defined: symmetry protected topological (SPT) phases and topological ordered systems.

Chapter 1. Physics of SU(N ) systems [START_REF] Haldane | Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly[END_REF] with an external field vanishing on average but locally inhomogeneous. (b) In the quantum spin Hall effect [START_REF] Kane | Quantum Spin Hall Effect in Graphene[END_REF], two distinct modes with opposite spins (red and blue) and opposite directions appear at the edges of the system. The edge modes are protected by time-reversal symmetry.

SPT phases

Topological insulators are non-interacting systems that have non-trivial topological properties, with a gapped bulk and conducting edge modes. Examples include the integer quantum Hall effect (see figure 1.4 (a)) characterized by a non-zero Chern number, which is a topological invariant. The spin-Hall effect shown in figure 1.4 (b) constitute another example, with gapless edge modes insensitive to disorder. They have been classified in the tenfold way [START_REF] Ryu | Topological insulators and superconductors: tenfold way and dimensional hierarchy[END_REF].

SPT phases are the generalization of topological insulators to interacting systems. They are short-range entangled and characterized by their gapless edge modes, which distinguish the phase from a totally trivial phase that does not exhibit them. The trivial phase is adiabatically connected to a product state without any entanglement while the SPT phase is not [START_REF] Senthil | Symmetry-Protected Topological Phases of Quantum Matter[END_REF]. They do not spontaneously break any symmetry of the Hamiltonian and are actually symmetry protected, that is any local perturbation that preserves the symmetry cannot localize the edge modes. It is not possible to go from the topological to the trivial phase without closing the bulk gap as long as the Hamiltonian does not break the symmetry. The best example is the Haldane phase for spin-1 chains, in 1D, that has fractional spin-1/2 edge modes. We will discuss it in details in chapter 4, where we use this phase as a starting point to propose a 2D SU(3) SPT phase with fractional edge modes. These systems have been classified using group cohomology theory for bosons [START_REF] Chen | Symmetry protected topological orders and the group cohomology of their symmetry group[END_REF], later extended to supercohomology theory for fermions [START_REF] Gu | Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory[END_REF].

Topological order

On the other hand, topological ordered systems have global properties that cannot be deduced from local configurations but a global order parameter can be defined. They are long-range entangled. There have at least two degenerate ground states with different topological parameters, the number depending on the associated group. For instance, this topological invariant can be a winding number around a hole in a surface with non-zero genus. In Kitaev's toric code (TC) [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF], it is defined by the configuration of non-contractible loops in the system (see figure 1.5). A local perturbation would change the shape of the loop but cannot make it disappear.

It is not possible to connect the degenerate states by any local Hamiltonian the relevant symmetry, a global action involving a Wilson loop is needed. Such a system exhibits fractional excitations with neither fermionic nor bosonic statistics. By 1.4. Topological phases 21 Figure 1.5 -On a surface with genus g, Kitaev's toric code has a ground state degeneracy of 4 g . Indeed, each hole can host two distinct non-contractible loops whose presence do not change the energy. Here g = 2.

defining projective symmetry group to characterize quantum orders, Wen was able to classify quantum spins liquids into four different classes, depending on their statistics, their excitations and their long-range properties [START_REF] Wen | Quantum orders and symmetric spin liquids[END_REF]. In chapter 5, we propose a Z 2 topological state with SU(4) symmetry based on an RVB state.

Chapter 2

Representation theory of SU(N )

Mathematically, a symmetry is nothing but the group of transformations that leaves an object invariant, therefore group theory is a very powerful tool to treat systems with symmetries. In quantum mechanics, a category called Lie groups play a key role and the application of the representation theory of Lie groups and algebras has led to key predictions. In this chapter, we introduce the most important elements of this theory and apply it to the specific case of SU(N ). As far as possible, we will derive the consequences of these results on the group SU(2) to recover and shed a new light on the spin physics we detailed in section 1.2. We will not provide the demonstrations of the properties we enunciate, they are to be found in the relevant mathematics literature, for instance [START_REF] Hall | Lie groups, Lie algebras, and representations: an elementary introduction[END_REF][START_REF] Renard | Groupes et représentations[END_REF] or the chapter 13 of the Big Yellow Book [START_REF] Di Francesco | Conformal Field Theory[END_REF]. With this new framework, we will be able to rigorously define SU(N )-symmetric Hamiltonian and demonstrate their key properties. We will then review the different analytic and numerical methods used to probe them.

Definitions and formalism

Heuristic

We will try to introduce the keys ideas of representation theory in a simple way before going to the technical details. Our goal here is to give a straight answer to the question what is a spin?

Consider an example of classical electrodynamics. We consider the 3-dimensional space with a system of coordinates (O, e x , e y , e z ). A point M of space with coordinate r = ae x has an electric potential V (r) = V 0 and an electric field E(r) = Ee y .

The laws of physics are rotation invariant, which means no direction in space is privileged and any experiment will yield exactly the same results if it is rotated, although the description an observer gives may change. As a consequence for our experiment, another observer with a different system of coordinates (O, e x , e y , e z ) will see the same physics but with different conventions. Let us assume for simplicity a π/2 rotation along the z-axis: e x = e y , e y = -e x and e z = e z . Going from one description to another is like acting with a rotation R on every quantities of the system. The rotation maps r to r , V to V and E to E . Let us examine in details the effects of this rotation.

First, for the potential, V (r ) = V 0 = V (r): nothing changed, the potential is the same before and after rotation. For the electric field however, we have E (r ) = Ee y = -Ee y = E(r). We observe that the rotation acts differently on the potential and on the electric field: while V (r) = V (r ), E (r ) = RE(r). We can consider a more complicate object, the polarization tensor χ, that will transform yet another way. Therefore we conclude that the rotation R acts differently on different mathematical objects. Furthermore, R, is a 3 × 3 matrix, yet it can operate on spaces that are Chapter 2. Representation theory of SU(N ) not 3-dimensional. We can spot that the way it operates is directly linked to the number of components of the field: V is 1-dimensional, E is 3-dimensional and χ is 5-dimensional if we require it to be symmetric and traceless.

The next step is to enumerate all the ways a rotation can act on different mathematical objects. This enumeration is actually what we did in section 1.2.1 although this was not apparent: the result is there is an infinite number of ways a rotation can acts, each of them labeled by an integer or half-integer. Reciprocally, for each integer or half integer there is one and only one way the rotation act: this is the spin. Mathematically, the spin is the integer or half-integer that labels how a physical system behaves under the action of a rotation.

We can classify our findings in electromagnetic in this language:

-the electric potential belongs to the class of 1-dimensional objects that are unaffected by rotations: they are spin 0, called scalars ; -the electric field is a 3-dimensional object that transforms the same way as regular space vectors: these object are spin-1, or vectors ; -a 3 × 3 matrix decomposes as a scalar trace, a vector antisymmetric part and finally a 5-dimensional symmetric part, such as the polarization tensor: these are spin-2, or tensors. In the classification also appeared half-integer spin that we did not find in electromagnetism, where only integer spin appear. The simplest is a 2-dimensional field called spinor. Their description cannot be simply explained in terms of real space rotation and rely on complex numbers: as such, they are purely quantum objects.

Group representation

Our heuristic showed a matrix with a given size can act on spaces that have different dimensions. In this section we will formalize the concepts we sketches there using the language of group theory.

We recall that a group is a set G together with an intern operation G × G → G, (g, h) → gh that is associative and admits an identity element e such that every element has an inverse. If this operation is commutative, G is called abelian.

We say that a group G acts on a set X if there is a map

G × X → X, (g, x) → g • x such that ∀x ∈ X, e • x = x and ∀g, h ∈ G, (hg) • x = h • (g • x).
As an example, the permutation group S 3 acts on the set {1, 2, 3}. More generally, for any set X the permutation group of X naturally acts on X and the action of the group can be seen as a group morphism between G and the group of all bijections of X. In our heuristic, the rotation group of the three-dimensional space SO(3) acts on the different spaces of the system. This notion of group action is a very general concept with infinities of applications. Representation theory is the branch of mathematics that studies the special case where the set X has a structure of vector space and the action preserves this structure. A representation of a group G is the linear action of G on a vector space E, or in other terms a group morphism ρ : G → GL(E), g → ρ(g).

(2.1)

2.1. Definitions and formalism 25 Such a map exists for any group G since we can always define the trivial representation E = C, ρ(g) = Id. Setting aside this one, being a representation of a group is a strong statement that gives a lot of information on the structure of E. Representation theory in its most general form is very rich but bears many difficulties if the group G does not have nice topological properties. Hopefully we will not need to consider those cases and in all our discussion we will require G to be compact (possibly finite), which is true for SU(N ), and we will focus on finite-dimensional representations.

If E is a Hilbert space with a dot product, we say the representation ρ is unitary if the action preserves this product:

∀g ∈ G, ∀u, v ∈ E ρ(g) • u | ρ(g) • v = u|v , ( 2.2)
i.e. the values of ρ belong to the unitary group of E. The first consequence of our restriction on the group G is the theorem:

Theorem 1 Any finite-dimensional representation of a compact group can be made unitary.

This means up to a redefinition of the dot product, acting with a group does not change the norm of the vector, which is expected if the group operation consists in a change of axes or a lattice symmetry operation; this is also the case in any gauge theory. Since we deal with compact groups only, we will always consider this unitary condition to be fulfilled. In physics, Wigner's theorem guarantees that the bijective transformations of a wavefunction (more precisely, of a ray of the projective Hilbert space), which represent the action of a group, are either linear or antilinear unitary maps.

Irreducible representations and equivalent representations

If we consider the trivial representation of G on a space E of dimension higher than 2, we will of course not get any information on its structure: we can always "split" the representation in separate subspaces that are not linked by G. This is not always the case: if we consider the 3-dimensional real space, a given vector of norm 1 spans the whole unit sphere under the action of the group SO(3) and the associated representation is necessary 3-dimensional without any splitting possible. This brings us to the notions of reducible and irreducible representations. We say that a subspace

F of E is ρ-invariant if for all g ∈ G, ρ(g) • F ⊂ F . We say a representation E is irreducible if it is
non-zero and its only ρ-invariant subspaces are 0 and E. We will denote these objects as irrep for irreducible representation. These are the fundamental objects of representation theory. We say that a representation is completely reducible if it can be decomposed as a direct sum of irreps. There stands the second important theorem:

Theorem 2 Any finite-dimensional representation of a compact group is completely reducible.

In particular, the tensor product of two representations of G is a representation of G since we can act with G on the different components of the product. Therefore this theorem imposes that a tensor product can be always decomposed as a direct sum of irreps, which is what we did in section 1.2.2 when we decomposed the Hilbert space of two spins into different spin subspaces. In this language, spins are the irreps of SU(2) and the fusion rules are the dictionary that decomposes a tensor product into a direct sum of irreps. Now, we guess that two spins 1/2 will be totally similar for group representation while a spin 1 will be different. To put it formally, if ρ and τ are two representations of G acting on spaces E and F respectively, we define an intertwining map of representations as a linear map T :

E → F such that ∀g ∈ G ∀v ∈ E T (ρ(g) • v) = τ (g) • T (v).
(2.3)

We say that two representations are equivalent, or isomorphic, if there exists an invertible intertwining map between them. We will consider two equivalent representations to be basically the same, as if it was just a change in basis. Our goal now will be to characterize the different irreducible representations of a given group. We will need the Theorem 3 (Schur's lemma) Let (ρ, E) and (τ, F ) be complex irreducible representations of a group. Then 1. if ρ and τ are not equivalent then the only intertwining map is zero. 2. if E = F then any intertwining map between ρ and τ is a complex scalar times identity.

3. if T 1 and T 2 are two non-zero intertwining maps between (ρ, E) and (τ,

F ) then ∃λ ∈ C, T 1 = λT 2 .
This is a very powerful theorem that states the identity is basically the only way to go from one irrep to another. A basic consequence is that a complex irreducible representation of an abelian group is one-dimensional. The last notion to introduce is the concept of projective representation. A protective representation of G is a map π together with a complex vector space E respecting the following rules:

π : G → GL(E) ∀g, h ∈ G ∃ c(g, h) ∈ C * such that π(g)π(h) = c(g, h)π(gh), (2.4)
or in other words it is a representation up to a scalar. When c is 1, we recover the definition of a representation. A projective representation is in fact the representation of a larger group G that contains G.

Finite groups

In the case of a finite group, the number of different irreps is finite and is equal to the number of conjugacy classes. An irrep is totally defined by its character which is the set of traces of the matrices representing the group elements. A character table lists each irrep of a group together with its characters: all the information of the group representation theory is contained in the table. As an example that we will use later to classify tensors, we consider the finite group C 4v , which is the symmetry group naturally associated to the square lattice. This group has eight elements, as shown in figure 2.1:

irrep dim 2 C 4 C 2 2 σ v 2 σ d A 1 1 1 1 1 1 A 2 1 1 1 -1 -1 B 1 1 -1 1 1 -1 B 2 1 -1 1 -1 1 E 2 0 -2 0 0
-the identity; -two rotation of π/4 labeled C 4 ; -one rotation of π/2 labeled C 2 ; -two symmetries exchanging two opposite axes, labeled σ v ; -two symmetries along the diagonals, labeled σ d . It has five irreducible representations, traditionally labeled A 1 , A 2 , B 1 , B 2 and E. We give its character table in the table 2.1.

Representation of Lie groups

In the case of finite groups, there is only a finite number of non-equivalent irreducible representation and their character table is enough to fully describe them. This is no more the case for infinite groups, which can have an infinite number of irreps. We need other tools to determine the irreducible representations of such groups. Again, the full theory bears numerous difficulties but we will assume another strong property on our groups of interest: to be differentiable. Such groups are called Lie groups and their representation theory is well understood.

Lie groups and Lie algebras

A Lie group has the structure of a smooth manifold, i.e. it is locally isomorphic to a vector space at each point. The strategy is to note that the group is a complicate object and characterizing its representation is hard, but its tangent space has the simple structure of a vector space. Therefore we will forget the representations of the group for a moment and only work on the tangent space: first we will explore its properties and its relation with the group. We will then extend the concept of representation to it and show how they are linked to those of the group. Finally, we will characterize the finite-dimensional representations of the algebra and find their structure, which will give the needed results for the group.

From now, we suppose G is a compact matrix Lie group that is a subgroup of GL N (C). We will denote an irrep of G by its dimension in bold letters as we did for spins in section 1.2.2. Besides the trivial representation 1, another irrep can automatically be defined, the fundamental, or definition representation, where the group naturally acts on C N with the matrix product operation. This irrep is of course N-dimensional. Furthermore, for any complex representation ρ we can always define a conjugate representation by τ : g → -ρ(g) * , the minus sign being a mere convention. Physically, this operation corresponds to charge conjugation. This new representation has the same dimension as the original one, we will refer to it by adding a line over its dimension. A representation that is equivalent to its conjugate is called self-conjugate. We will refer to the conjugate of the fundamental irrep as just the conjugate, or N (it is also sometimes called anti-fundamental).

We recall that SU(N ) is the group of special unitary matrices of size N , or in other terms

SU(N ) = {U ∈ M N (C) | U † U = I and det U = 1}.
(2.5)

As any matrix group, it is naturally embedded into the vector space M N (C). It is compact as closed and bounded in finite dimension and it is differentiable since the properties of the definition (2.5) are.

The first difference between the general SU(N ) group and SU(2) is that a given representation is not necessarily self-conjugate. Indeed, the charge-conjugation of SU(2) is actually the famous rotation over a π angle that sends |↑ to |↓ and |↓ to -|↑ , and since this operation belongs to the group SU(2) it can act on any representation, which is therefore equivalent to its conjugate. Consequently it is always possible to make a singlet out of two same-size spins. This is no more true for SU(3) for instance, where the fundamental irrep, or quark, differs from its antiquark conjugate. A tensor product of two given irreps of SU(N ) yields a singlet if and only if the two irreps are the conjugate of each other.

Let us now take a closer look at the group manifold structure. If we consider elements close to the identity, we can define the tangent space of G, labeled with Fraktur letters g (see figure 2.2). With our requirements for G, it is a subspace of M N (C) and therefore of finite dimension. Hence we can find a set of linearly independent matrices T α that spansit and all elements g of the group close to the 2.2. Representation of Lie groups 29 identity can be written

g = I + α θ α T α + o( θ ) (2.6)
A more formal definition for the tangent space relies on the exponential map: the (complex) tangent space g of a matrix Lie group G is the subspace of matrices

X ∈ M N (C) such that ∀X ∈ g ∀t ∈ R exp(tX) ∈ G, (2.7)
the differentiation being well defined thanks to the real variable t. We can now go back and forth from the group to the tangent space, using the exponential to map the tangent space to the group and differentiating it to go back to the tangent space.

The main consequence of this definition is that the tangent space has an algebra structure defined by its commutators, or Lie brackets

(X, Y ) → [X, Y ] = XY -Y X.
(2.8)

This operation is bilinear and anti-symmetric, it fulfills the Jacobi identity :

[X, [Y, Z]] + [Z, [X, Y ]] + [Y, [Z, X]] = 0.
(2.9)

These commutators reflect the properties of the group product inside the algebra. They encode all the local properties of the group, which is locally isomorphic to the algebra in any point. The limitation is topology: since the exponential is continuous and exp 0 = Id, it can only reach the connected component of the identity in the group. The tangent space only captures the local structure of the group, if the group has non-trivial global properties the algebra cannot share them. Note that we defined the Lie algebra as the tangent space of the group and deduced its properties from those of the group. Another possibility would be to start from the Lie algebra defined as a vector space together with a Lie bracket that fulfills the Jacobi identity and show the exponential maps the Lie algebra to a matrix Lie group. To define a Lie algebra, we only need a basis of matrices T α of the algebra with commutations relations obtained from the Lie brackets :

[T α , T β ] = f αβγ T γ .
(2.10)

The operators T α are called the generators of the algebra. Their choice is of course not unique and is equivalent to a redefinition of the tensor f αβγ . The elements f αβγ are called the structure constants, they contain all the information needed to define the algebra unequivocally. The structure constants are real as long as the generators are hermitian.

In the case of SU(N ), we can differentiate (2.5) and define its Lie algebra as :

su(N ) = {M ∈ M N (C) | M † + M = 0 and Tr M = 0}, (2.11)
that is the subspace of traceless skew-hermitian matrices, of dimension N 2 -1. The usual convention is to put a factor i in front of the matrices of the algebra and to consider hermitian matrices instead of skew-hermitian. Hence, the structure constants are always real.

Representation of a Lie algebra

A representation of a Lie algebra is a linear map that preserves the Lie brackets. In other words, it is an algebra isomorphism. The concepts of unitarity, reducibility Chapter 2. Representation theory of SU(N ) and Schur's lemma extends straightforwardly to the representations of a Lie algebra. To represent a given Lie algebra of dimension k, we need to find k matrices that fulfills the commutations relations (2.10). Then thanks to linearity any element A of the algebra can be decomposed in the basis as X = θ α T α and is represented by

ρ(X) = α θ α ρ(T α ).
The action of the algebra on itself through the Lie brackets X : Y → [X, Y ] is linear and automatically fulfills the commutators, therefore it defines an irrep of the algebra called adjoint representation. Its dimension is the dimension of the Lie algebra, N -1 in the case of su(N ). Since it encodes the action of the algebra, it allows to use the algebra to navigate through different states of any irrep of the algebra.

Any representation of the group gives rise to a representation of the algebra since we can always differentiate t → ρ(exp(tX)) and define the action of the matrices close to the identity, which belong to the algebra. However, the reciprocal is not true: the Lie algebra is the vector space tangent the group, therefore it only encodes the local properties of the group through the Lie brackets. As we stated before, if the group has non-trivial global (topological) properties, the algebra cannot know them since its topology is trivial. The necessary condition is given by the theorem:

Theorem 4 If G is a simply connected matrix Lie group then any representation of its algebra gives rise to a representation of G.

The representation of the group is obtained by taking the exponential of the matrices representing the algebra: if

G g = exp(-i θ α T α ) then ρ(g) = exp(-i θ α ρ(T α )).
By convention, we note S α = ρ(T α ) the representatives of the generators in a given representation. We already saw that any representation of the group gives rises to a representation of the algebra, therefore when G is simply connected the representations of the group and those of the algebra are in one-to-one correspondence. When G is not simply connected, some representations of the algebra only gives rise to a projective representation, that is a representation up to an extra phase. This is the case of the group SO(3), as we will see.

A convenient basis of the Lie algebra su(2) is the set of three Pauli matrices {σ x , σ y , σ z }, with commutation relations

[σ α , σ β ] = 2i αβγ σ γ (2.12)
which are the commutation relations of angular momenta defined in equation (1.11) up to a factor 2. Therefore the whole section 1.2.1 is to be understood as the determination of all finite-dimensional irreps of su(2). The matrices S x , S y and S z are the representatives of the generators of the Lie algebra in a given representation and we know their form in any irrep. Since SU(2) is simply connected, its irreps are obtained by applying the exponential to the irreps of the algebra. This is also true for the general case SU(N ), which is always simply connected. Now we can also understand the nature of half-integer spins and their surprising -1 sign when acting with a 2π rotation. The starting point to define spins was the rotation invariance, which is associated with the group SO(3), whose Lie algebra is isomorphic to su(2), with the same commutators. The action of the rotations of SO(3) on a quantum system is generated by the angular momentum observables, who live in the algebra su(2). However, because SO(3) is not simply connected, the representations of its algebra do not all give rise to representations of the group: some of them, the half-integer spins, only yield a projective representation. Hence we demonstrated our initial statement, that is spins are irreducible representations of SU(2). The generators of the group are the angular momentum operators, and they generate the rotation of the system through exponentiation.

For su(3), a common choice is the set of eight Gell-Mann matrices:

λ 1 =    0 1 0 1 0 0 0 0 0    , λ 2 =    0 -i 0 i 0 0 0 0 0    , λ 3 =    1 0 0 0 -1 0 0 0 0    , λ 4 =    0 0 1 0 0 0 1 0 0    , λ 5 =    0 0 -i 0 0 0 i 0 0    , λ 6 =    0 0 0 0 0 1 0 1 0    , λ 7 =    0 0 0 0 0 -i 0 i 0    , λ 8 = 1 √ 3    1 0 0 0 1 0 0 0 -2    .
with a structure factor f αβγ defined by the commutation relations

[λ α , λ β ] = 2if αβγ λ γ . (2.13)
This construction can be generalized to any N , keeping the generators of su(N ) as the N 2 -1 first generators of su(N + 1). There are of course other possibilities, another common one being to introduce Schwinger boson operators that explicitly exchange two colors. In a fermionic language they write

S α β = c † α c β , with commutations relations [S α β , S γ δ ] = δ γ β S δ α -δ α δ S γ β . (2.14)
This choice results in a very simple structure factor, but one has to deal with N 2 non linearly independent generators.

Structure of SU(N ) representations

In this subsection, we aim to characterize every finite-dimensional representations of SU(N ) for any N . Most of our discussion here will actually remain valid for a more general class of algebras called semi-simple Lie algebras but we will not make use of any other case than su(N ) in the following chapters.

First of all, we recall that SU(N ) is compact and any finite-dimensional representation is totally reducible as a direct sum of unitary irreps. Second, any irrep of a matrix Lie group yields an irrep of the algebra through differentiation and since SU(N ) is simply connected, the exponential maps any irrep of the algebra to an irrep of the group, so that the representations of su(N ) and SU(N ) are in one-to-one correspondence. Therefore we reduced our problem to determining the irreps of su(N ). This is a simpler task thanks to the simple structure of a Lie algebra: we just need to find N 2 -1 matrices S α with the appropriate commutators that represent the generators and then thanks to linearity we can represent the whole algebra. We already did the work for su(2) in section 1.2, with a rigorous construction of all finite-dimensional irreps.

Among the generators of su(N ), some of them commute. The Cartan subalgebra is defined as the maximal algebra of commuting elements, the generators that spans it are the Cartan operators. This algebra is not unique, but two different Cartan algebras are always linked by an automorphism. In the case of su(N ), it is of dimension N -1, also refereed as the rank of the Lie algebra. Since all the elements of the subalgebra commute, they can be diagonalized in the same basis in any representation. They can Chapter 2. Representation theory of SU(N ) be chosen to have integer eigenvalues in all irreps, in our case if E ij is the canonical basis M n (C), we will use the operators that in the fundamental irrep write

H i = E ii -E i+1,i+1
(2.15)

for 1 ≤ i ≤ N -1. For N = 2 we retrieve the operator S z up to a factor 2.

For each vector of the Cartan-diagonal basis of a given irrep, we define its weight as the set of N -1 integers eigenvalues of the Cartan operators. Note that a given weight may not be unique in the irrep. Conjugation opposes the signs of all the weights, so that an irrep can only be self-conjugate if each of its weights comes with its opposite.

By using with the commutators, that is acting with the adjoint irrep, we can navigate through the different states of this basis. Each of these operations is associated with a change in the weights, called root; these roots are exactly the non-zero weights of the adjoint irrep. A root is said to be positive if its first nonzero coefficient is positive, a simple root is a root that cannot be written as the sum of two positive roots. In su(N ) there are N -1 simple roots, which are associated with N -1 decreasing ladder operators that reduce the weight of a state by the value of the simple root. They allow us to define a partial order between weights: a weight is higher than another if the difference between them is a linear combination of simple roots with positive coefficients.

In the case of su(2), any irrep has a unique highest weight state corresponding to the maximal eigenvalue 2m of 2S z . All the other states are obtained by applying the ladder operator S -iteratively to this highest weight state, with an associated root 2. This structure generalizes to su(N ) with the highest weight theorem:

Theorem 5 (highest weight) Any finite-dimensional irreducible representation of su(N ) has a unique highest weight state, which only include positive integers.

This highest weight can then be used to label irreps. Since there are N -1 Cartan operators for SU(N ) with integer eigenvalues, this means an irrep of SU(N ) is labeled by N -1 positive integers 1 . Reciprocally, for any set µ of N -1 positive integers there is one and only one irrep of SU(N ) with highest weight µ. Again for SU(2), we retrieve the result of one integer (twice the spin) to label an irrep that is the maximal value of 2S z .

The structure of a given irrep is given by the arrangement of the weights, linked by simple roots. Again, starting from the highest weight state, ladder operators are successively applied to generate all the vectors of the irrep. This structure can be visualized in weight diagrams, we show two examples in figure 2.3.

Lastly, we remember that in section 1.2, we used the operator S 2 to identify a spin s with its eigenvalue s(s + 1). This operator is called Casimir operator, it commutes with all the elements of the algebra. Any operator invariant under the action of the algebra is actually an intertwining map of the irrep. Hence, according to the theorem 3, it has to be diagonal in any irrep. The construction of all the Casimir operators is a difficult problem, we will only use the quadratic Casimir operator defined as S • S = α S α S α for any irrep of SU(N ). This operator is independent of the choice of the generators S α and commutes with every elements of the algebra.

Young tableau formalism

33

(2) 2), with highest weight (2). The unique Cartan operator is 2S z and the unique decreasing ladder operator S -reduces the weight by 2. (right) Two-fermion irrep of SU(4) (0, 1, 0) = 6. There are three Cartan operators and three ladder operators S 1-, S 2-and S 3-associated with the simple roots (2, -1, 0), (1, -2, 1) and (0, -1, 2) respectively.

(0) (-2) S - S - (0,1,0) (1,-1,1) (-1,0,1) (1,0,-1) (-1,1,-1) (0,-1,0) S 2- S 1- S 3- S 3- S 1- S 2- Figure 2.3 -Weight diagrams of two irreps of SU(N ). (left) Spin-1 irrep of SU(

Young tableau formalism

There exists a simple, graphical way to denote all the irreps of SU(N ), the Young tableaux. The Young tableau formalism is a powerful tool that allows to visualize every irreps, but also compute their dimension as well as the decomposition of any tensor product with very simple algorithms involving only combinatorics. We will make extensive use of this language throughout the following chapters.

A Young tableau is an arrangement of left-aligned boxes such that the rows are sorted by decreasing lengths from top to bottom. The convention is to label a Young tableau by r integers between brackets, each integer corresponding to the length of each of the r rows. The fundamental irrep of SU(N ) is a simple box:

= [1] = N.
(2.16)

Every irreps can then be generated by considering tensor products of a certain number of fundamental irrep, or in physical terms by adding particles. In terms of tableaux, adding a box in a row means symmetrizing the states in a bosonic way, while adding a box in a column means anti-symmetrizing the states in a fermionic way. There can be at most N anti-symmetrized particles with SU(N ) symmetry, which define a one-dimensional space: this is the trivial representation, or singlet. Thus a column of height N is a singlet and can be removed, we write it down with a bullet • and no column can be higher than N -1 boxes. On the other hand there is no limit in the number of symmetrized particles and the rows can be of any length.

A hole or anti-particle can be seen a "missing" fermion in a column to make a singlet, and more generally any irrep acts as a hole of the conjugate irrep corresponding to missing boxes. Therefore to conjugate a given Young tableau, we replace each column of height h by a column of height N -h. In particular the conjugate irrep is represented by a column of N -1 boxes, written [1, 1, .., 1] with N -1 ones. For SU(2), there can be only one row and we retrieve that any irrep is self-conjugate, the length of the row being twice the spin. The adjoint irrep is the (self-conjugate) symmetric combination of the fundamental and the conjugate irreps, and therefore its Young tableau is two columns of height N -1 and 1 respectively. We give here the example of SU(3), with the Young tableaux corresponding to the fundamental and the conjugate, the trivial and the adjoint and finally the two-boson irrep and its conjugate:

= [1] = 3 = [1, 1] = 3 = • = 1 = [2, 1] = 8 = [2] = 6 = [2, 2] = 6,
(2.17) the two-fermion irrep being the irrep of one hole, that is to say the conjugate, or antiquark.

Any Young tableau fulfilling these constraints corresponds to one and only one irrep of SU(N ). Reciprocally, any irrep of SU(N ) corresponds to a Young tableau. The correspondence is given by the highest weight: we saw that an irrep of SU(N ) is labeled by the N -1 integers of its unique highest weight. The weights of an irrep can be accessed by codiagonalizing the Cartan operators, then by applying ladder operators corresponding to simple roots one finds the highest weight. With the conventions (2.15), the highest weight (n 1 , n 2 , . . . , n N -1 ) corresponds to the Young tableau with n 1 columns of height 1, n 2 columns of height 2, and so on, or more formally the tableau with entries N -1 j=1 n j . Let us consider SU(4) this time, with the fundamental and the conjugate, the trivial and the adjoint and finally the two-boson and two-fermion irreps. For each irrep we give its Young tableau, the label of the tableau between brackets, its highest weight between parentheses and its dimension in bold figures:

= [1] = (1, 0, 0) = 4 = [1, 1, 1] = (0, 0, 1) = 4 = • = (0, 0, 0) = 1 = [2, 1, 1] = (1, 0, 1) = 15 = [2] = (2, 0, 0) = 10 = [1, 1] = (0, 1, 0) = 6.
(2.18)

The dimension of any irrep can be obtained from its Young tableau as a simple ratio of two integers corresponding to two different fillings of the tableau. To get the numerator, we write the number N in the top left box of the Young tableau, then for each box in the row we increment the number and write it in the box. We repeat the operation for all rows while decreasing by one the starting number in the row at each floor. The numerator is the product of the entries in each box. The denominator, or hook length, is obtained by writing in each box the number of boxes being to its right plus the number of boxes being below it plus one and taking the product of all the 2.4. SU(N ) Hamiltonians 35 entries. In the case of the adjoint irreps of SU(3) and SU(4), we get

SU(3) : num = 3 4 2 den = 3 1 1 4 × 4 × 2 /3 = 8 SU(4) : num = 4 5 3 2 den = 4 1 2 1 4 × 5 × 3 × 2 /(4 × 2) = 15 (2.19)
The last thing to consider is how to obtain fusions rules. There is a simple algorithm to compute the tensor product of two Young tableaux in four steps:

1. fill the second tableau with indices corresponding to the row, one different index per row.

2. Attach the boxes from the second to the first tableau one by one following the order of the rows in all the possible way. The resulting tableaux must fulfill the Young tableaux rules and they do not have more than one occurrence of a given index in a column.

3. Remove any copy of a tableau with exactly the same index filling.

4. Going through each tableau row by row from right to left, discard the tableau if at some point there are more indices from a given row then from one of the rows above.

It can be checked that applied to SU(2), this algorithm gives the accurate rules of angular momentum addition we saw in section 1.2.2. Now we can compute the decomposition of any tensor product of SU(N ) representations by first decomposing them into a sum of irreps and then applying the algorithm to the corresponding Young tableaux. We have every mathematical tools we need to address SU(N ) systems.

SU(N ) Hamiltonians

Hilbert space and representations

Let us first take a fresh look at our previous chapter with the framework of representation theory. SU(N ) physics arises when the Hilbert space realizes a representation of the group SU(N ). Since the laws of physics do not favor any direction, rotations naturally act on the wavefunctions with the angular momentum observables as associated generators. These generators obey su(2) commutations rules, the Hilbert space has the structure of its representations and therefore the angular momentum of a quantum system is quantized. As we have seen in the previous chapter, this SU(2) symmetry may enhance to a larger symmetry group. In high-energy physics, gauge theories rely on another symmetry group acting on the Hilbert space. The wavefunction is SU(N )-symmetric if it is an SU(N ) singlet, or in other words if it belongs to a trivial representation, both of the group and of the algebra.

The representation ρ of a group G on a vector space naturally induces a representation on its matrix space through the action M → ρ(g)M ρ(g) -1 . In particular, if SU(N ) acts on a wavefunction then it also acts on the Hamiltonian of the system. The Hamiltonian is SU(N )-symmetric if it is invariant under this action for any element of the group. By differentiating the representation, we see the Hamiltonian is also invariant under the action of the Lie algebra, hence it commutes with all its elements. Reciprocally if an operator commutes with the generators of su(N ) it is invariant under the action of the whole algebra, and by applying the exponential we see it Chapter 2. Representation theory of SU(N ) is also invariant under the action of the group SU(N ). This explains why rotation invariance is equivalent to commuting with S x , S y and S z .

By the same argument we used for Casimir operators, on any irrep an SU(N )invariant Hamiltonian is a scalar times identity, and since the Hamiltonian is hermitian this scalar is real. Hence we demonstrated the result we used in section 1.2.4: if a Hamiltonian is SU(N ) is invariant then its spectrum has a multiplet structure corresponding to the decomposition of the Hilbert space as a direct sum of irreps.

For the same number of particles in the fundamental irrep, the representations of SU(N ) with N > 2 are higher-dimensional than those of SU(2) (any non-trivial irrep is at least N -dimensional), as can be checked with the Young tableaux. Hence systems ruled by SU(N ) symmetry have many more degeneracies than spin systems. Consequently, these systems have more room for quantum fluctuations to destroy a local order parameter, favoring QSL phases which are harder to characterize. At the same time, SU(N ) symmetry allows for new, exotic phases without any equivalent in SU(2), such as for instance mixing dimerization and Néel order [START_REF] Corboz | Simultaneous Dimerization and SU(4) Symmetry Breaking of 4-Color Fermions on the Square Lattice[END_REF]. We will briefly review analytic and numeric methods used to investigate those systems. Since SU(N ) is nothing but a generalization of SU(2), the methods developed for SU(2) can usually be generalized with some tailoring.

Analytic methods

Let us first take a look at exact results. The LSM theorem from section 1.1.2 can be reformulated as any SU(2)-symmetric 1D system with an odd number of boxes in the Young tableaux of the local irrep is either gapless or has a degenerate ground state in the thermodynamic limit. It had many extensions to the more general SU(N ) case. Affleck and Lieb proved that in the thermodynamic limit, the ground state of a 1D chain made of irreps of SU(2N ) with an odd number of boxes in the Young tableau is either degenerate or gapless [START_REF] Affleck | A proof of part of Haldane's conjecture on spin chains[END_REF]. Many reasonable conjectures exist, depending on the congruence modulo N of the number of boxes and the shape of the Young tableau of the local irreps [START_REF] Lajkó | Generalization of the Haldane conjecture to SU(3) chains[END_REF][START_REF] Wamer | Self-conjugate representation SU(3) chains[END_REF]. The results in 2D are rarer and lattice-dependent. For instance, on the square lattice with staggered fundamentalconjugate irreps states that a featureless gapped ground state is permitted for odd integers N [START_REF] Jian | Lieb-Schultz-Mattis theorem and its generalizations from the perspective of the symmetry-protected topological phase[END_REF].

Beyond generic no-go theorems, many analytical tools can be used to tackle SU(N ) systems. The specificity of SU(N ) compared to SU(2) is to allow a new parameter for approximations and series expansions, the number of colors N itself [START_REF] Affleck | Large-n Limit of SU(n) Quantum "Spin" Chains[END_REF]. Many exact results can be obtained in the limit N → ∞, then perturbation in 1/N are added. This approach was originally conceived as a way to get back to spins in the limit N → 2. However the results with large N depend largely on the shape of the Young tableaux, especially its row number, which has no equivalent in SU(2) [START_REF] Read | Spin-Peierls, valence-bond solid, and Néel ground states of low-dimensional quantum antiferromagnets[END_REF]. Note that this approach can be problematic since some phase transitions appear between two different values of N .

A common way to tackle a quantum problem is to find a classical state with similar features then add quantum perturbations. This is what the classical ferromagnetic and Néel states consist in, then spin waves are added to the classical picture. In the case of SU(N ), this can be done as long as a classical equivalent can be found. In the case of ferromagnetic ordering, the solution is straightforward: imposing the quantum state to be fully symmetric, under any site permutation gives an exact, classical ferromagnet, as for SU(2). This state saturates the value of one of the color on each site. Acting with SU(N ) rotations to this state spans the whole magnetically saturated eigenspace. The classical "large-spin" limit is the same as for SU(2), approximating every local irreps by a Young tableau that has infinitely long rows. Spin waves are generalized to SU(N ) into flavor waves (referring to the flavor theory of quarks in particle physics) that decrease the value of the order parameter. They can also be used in the antiferromagnetic case [START_REF] Kim | Linear flavor-wave theory for fully antisymmetric SU(N ) irreducible representations[END_REF], starting from a "classical" Néel order of staggered colors on the lattice. Note that contrary to SU(2), the choice of colors to alternate is not unique and more than one classical Néel state can be defined. These states are no more eigenvectors of the Heisenberg Hamiltonian since it still exchanges colors on two sites. Again, flavor-waves can be defined, they perturb the classical order and those quantum fluctuations lower the energy. Some common approaches from SU(2) can be used for the general SU(N ) without much change. Critical systems in dimension 1 + 1 or 2 + 0 are fully described by a conformal field theory (CFT) [START_REF] Di Francesco | Conformal Field Theory[END_REF]. SU(N ) symmetry -actually any Lie algebra symmetry -naturally takes its place in this framework. These theories have proven extremely successful to describe critical phases, such as the WZW theory for the spin-1 chain we mentioned in section 1.2.4. Some discrete points of the phase diagram of a quantum systems are integrable. In the case of one-dimensional systems, they can be exactly solved using the Bethe ansatz, without any difference between SU(2) and SU(N ) [START_REF] Cao | Nested off-diagonal Bethe ansatz and exact solutions of the su(n) spin chain with generic integrable boundaries[END_REF].

Numerical methods

From a conceptual point of view, the simplest way to treat a Hamiltonian is through exact diagonalization. The problem is that the matrix dimension is exponential in the size of the system and this method is limited to very small systems. Symmetries are required to break the matrix into smaller submatrices with fixed quantum numbers, corresponding as well to lattice symmetries as to a group acting on the local variables. This can be very powerful for SU(N ) Hamiltonians because they are very symmetric, with the multiplet structure of their spectra and the N conserved quantities. The difficulty is that SU(N ) is non-abelian: the different invariant quantities cannot be all defined at the same time. The simplest solution is to limit the symmetry use to the commuting Cartan operators, considering blocks of fixed Cartan eigenvalues, effectively restraining SU(N ) to U(1) N -1 . This can easily be done but is not fully efficient. A recent dedicated method allows to directly split the Hamiltonian into the different irreps blocks, which are way smaller than the Cartan blocks, allowing for larger system sizes [START_REF] Wan | Exact diagonalization of SU(N ) Heisenberg and Affleck-Kennedy-Lieb-Tasaki chains using the full SU(N ) symmetry[END_REF]. When diagonalizing the whole Hamiltonian is not achievable, it is still possible to access the ground state and a few excited ones using Lanczos algorithm. Shift-invert method generalizes it to any part of the spectrum.

Monte Carlo methods are another widely used category of methods that rely on stochastic processes. Quantum Monte Carlo realizes an importance sampling of the partition function, generating a large amount of different states at finite temperature and assigning them a weight proportional to their Boltzmann weight. Some methods imply simulations running in the many-body Hilbert space with no constraint, then the Gutzwiller projector is used to project the wavefunction into a subspace with a fixed number of particles per site. Some other, such as the stochastic series expansion (SSE), work directly with variables statically attached to the lattice. These methods are however limited by the famous sign problem. In the presence of a fermionic system or magnetic frustration, the weight of a given configuration is negative and it is not possible to interpret it as a probability. This is a fundamental limitation of the method, which has been linked to NP-hard problems in computer science [START_REF] Troyer | Computational Complexity and Fundamental Limitations to Fermionic Quantum Monte Carlo Simulations[END_REF]. However, when the sign problem is absent or can be circumvented, these techniques usually offer the best numerical results. The second limit is the size of the system, which can Chapter 2. Representation theory of SU(N ) be way bigger than the limitations of exact diagonalization but is still rather small and finite-size scaling is required, which is a source of uncertainties. These methods have been successfully applied to SU(N ) systems [Ass05; Oku+15], notably when the number of colors is even, determinant Monte Carlo allows to tackle SU(2N ) systems without any sign problem [START_REF] Wang | Competing Orders in the 2D Half-Filled SU(2N ) Hubbard Model through the Pinning-Field Quantum Monte Carlo Simulations[END_REF][START_REF] Zhou | Quantum Monte Carlo simulations of thermodynamic properties of SU(2N ) ultracold fermions in optical lattices[END_REF].

Variational Monte Carlo is another Monte Carlo method where the wavefunction follows an ansatz with a few free parameters [START_REF] Becca | Quantum Monte Carlo Approaches for Correlated Systems[END_REF]. Each wavefunction is given a weight obtained by computing a Slatter determinant. The Hilbert space is then explored following a stochastic processes, aiming to minimize the energy after Gutzwiller projection. Again, the size of the system is limited and finite-size scaling is needed. Furthermore, the ansatz limits the variety of states that can be generated with this method. It still yields many interesting results, including in our domain of interest [START_REF] Dufour | Variational Monte Carlo investigation of SU(N ) Heisenberg chains[END_REF].

All of these methods have unpleasant limitations: exact diagonalization is only possible for very small systems and heavily relies on finite-size scaling. Quantum Monte Carlo methods still require scaling and cannot address a large amount of signfrustrated problems. Variational Monte needs a decisive ansatz as an input and also requires finite-size scaling. This leads us to tensor network methods, which can deal with infinite-size systems and are not impacted by the sign problem. They are the subject of the following chapter.

Chapter 3

Tensor network algorithms

In this chapter we detail the principles of tensor network (TN) algorithms and how they can be used to construct wavefunctions of quantum systems. While tensor networks can also be used to compute the partition function of classical systems as well as density matrices of thermal states, in this thesis we will only consider their use to describe pure quantum states. We will first introduce the diagrammatic formalism for tensors and the general ideas of TN. We will then describe which family of TN algorithm we use, the so-called PEPS, and explain how to implement symmetries in this framework. Finally, we will detail an efficient algorithm to contract the TN, the corner transfer matrix renormalization group.

Tensor description of a quantum state

Tensor formalism

In mathematics, a tensor is a generalization of matrices to higher dimensions: a rank-p tensor is an array of scalar with p axes, each of them labeled by a different index. Scalars identifies to rank-0 tensors, vectors to rank-1 and matrices to rank-2. Each axis has its own dimension and the total size of the tensor is the product of all its dimensions. The matrix product is generalized to tensor contraction: two tensors are contracted over one or more indices with same dimensions by taking the product of the elements of the two tensors and summing over the contracted indices. Typically if A i 1 ,...,ip and B ip,j 2 ,...,jq are two tensors with ranks p and q, the rank p + q -2 tensor C is obtained by contracting the common index i p :

C[i 1 , . . . , i p-1 , j 2 , . . . , j q ] = ip A[i 1 , . . . , i p ] * B[i p , j 2 , . . . , j q ]
To avoid writing down complicate formulas with many indices, we use diagrammatic notations for tensors, described in figure 3.1: a tensor is represented by a box with one leg per axis. Contracting or tracing over an index naturally transcribes as merging legs, as described in figure 3.2. This notation allows to write complicate contractions in a TN in a clear and readable way, however the implementation is still not trivial. Depending on the order in which the legs are contracted, the number of operations to execute and the amount of memory needed varies dramatically, unfortunately determining the optimal way to contract a tensor network is a NP-hard problem [START_REF] Pfeifer | Faster identification of optimal contraction sequences for tensor networks[END_REF].

Virtual variables and tensor product state

We consider a quantum system with p discrete degrees of freedom, typically spins on a lattice, each degree of freedom labeled by an integer index i k . The wavefunction of the system is expressed in the product basis as

|Ψ = i k c i 1 i 2 ...ip |i 1 i 2 . . . i p . (3.1)
This wavefunction can naturally be seen as a rank-p tensor, each axis corresponding to a different degree of freedom. This tensor is huge: if all degrees of freedom have d possible values, the tensor has a size d p exponential in the number of variables. For a small system of 8 × 8 spin 1/2 on a square, just storing this tensor would require more than 10 8 hard disk drives of 1 TiB: it is unreasonable to store or even compute all these coefficients. We must find other ways to compute the relevant properties of the system. The core idea of TN algorithms is to decompose the tensor as a product of smaller, local tensors that are connected by virtual variables. Each coefficient of the wavefunction can then be accessed by selecting the relevant physical indices and contracting over all the virtual indices. The one dimensional case is shown in figure 3.5: the wavefunction of a system of four variables i 1 , i 2 , i 3 , i 4 is decomposed as a product of matrices and the coefficients of the wavefunction are computed by taking the trace

c i 1 i 2 i 2 i 4 = Tr A i 1 B i 2 C i 3 D i 4 . (3.2)
More generally for any geometry, a tensor network is a set of tensors linked by virtual legs and whose free legs stand for physical variables. This tensor networks 3.1. Tensor description of a quantum state Figure 3.3 -The wavefunction |Ψ of a quantum system with discrete variables i k can be seen as a tensor. The rank of the tensor is the number of variables and its dimension is the dimension of the tensor product space k d k . The tensor can be reshaped as a product of local tensor involving only one physical variable i k at the cost of introducing virtual variables α k . The decomposition into a matrix product state can be exact if the virtual dimension is large enough, else it is an approximation.

represents a tensor product state whose coefficients in the product basis are obtained by contracting the whole tensor network along all its virtual legs.

It is easy to see that if the dimension of the virtual variables is large enough, the decomposition into several tensors is just a reshape and no information is lost, but there is no compression gain. The very objective of tensor network methods is to approximate the wavefunction with a virtual dimension χ small enough for computations to be possible and still be accurate. This allows a huge gain in memory: the number of coefficients is reduced from an exponential d p to a polynomial number in d and χ. Hence the virtual dimension χ is the decisive parameter in TN: small χ only allow to construct a small number of states with limited possibilities, while large χ allow more features but are memory intensive. Finite-entanglement scaling can then be used to extrapolate results to infinite virtual dimension.

The simplest case is to have a virtual dimension χ = 1: all degrees of freedom are restrained to a single scalar so that the total state is a product state of those scalars. The well-known limit is that product states cannot carry entanglement: for instance, it is not possible to give a product state ansatz for a singlet made of two spins-1/2. However, it becomes possible if we consider matrix product states (MPS) with a virtual dimension χ = 2. Indeed if we consider the matrices

A ↑ = 0 1 0 0 , A ↓ = 0 0 1 0 , B ↑ = 0 -1 0 0 , B ↓ = 0 0 1 0 (3.3)
then the state |Ψ αβ = Tr A α B β is exactly the (unnormalized) singlet. We see in this example that virtual variables carry entanglement.

Entanglement entropy and area law

To quantize this assertion, we need to consider the entanglement entropy. This entropy measures how much a real-space subregion A and its complementary B are entangled. It is defined as the von Neumann entropy of the reduced density matrix ρ A of region A, obtained by tracing out the states of the region B in the density matrix ρ: ρ A = Tr B ρ and

S A = -Tr ρ A ln ρ A . (3.4)
It has been shown that for any tensor network with a finite virtual dimension, the entanglement entropy of a given subregion A is bounded. Indeed, if the product of the dimensions of all virtual legs crossing the boundary of A is χ then the entanglement entropy of A is dominated by: S A ≤ ln χ.

(3.5)

Chapter 3. Tensor network algorithms

Thus tensor network methods give good approximations of weakly-entangled states only, including classical ones [START_REF] Orús | A practical introduction to tensor networks: Matrix product states and projected entangled pair states[END_REF].

The entanglement entropy of a random state of the Hilbert state is high, it grows linearly with the volume of the region A and is out of reach with these methods. Fortunately, we are not interested in any state but in a very special category of states, those that are the low-energy states of a physical Hamiltonian. Here "physical" implies the Hamiltonian to be a sum of local terms involving only a small number of sites, with short-range interactions that decay exponentially or faster with distance. From a mathematical point of view, this is a strong requirement on the Hamiltonian with many consequences [START_REF] Hastings | Solving gapped Hamiltonians locally[END_REF], but this is of course the case for any real-world Hamiltonian. The low-energy states of theses Hamiltonian have a precious feature: their entanglement is small. Indeed, they follow an area law: the entanglement entropy of a region A do not grow as the volume of A but only as its area. This theorem has been rigorously proven for the ground sates of one-dimensional gapped quantum systems [START_REF] Hastings | An area law for one-dimensional quantum systems[END_REF], it is expected to hold in higher dimensions and for the first excited states. The large amount of non-trivial results obtained with TN supports this conjecture. Critical states in 1D have an additional logarithmic correction to the area law. This is also the case for some critical ground states in 2D, but not all of them.

In conclusion, being a low-energy states of a local Hamiltonian imposes strong constraints on a quantum state. Such states are highly uncommon in the Hilbert space: the vast majority of this space is totally irrelevant with states that can never be explored during the lifetime of the universe. The low-energy states are the ones we are interested in solid state physics. They have a very low entanglement entropy and can be well approximated with TN. TN methods are therefore the natural framework to deal with them, automatically truncating the Hilbert space to its small relevant part. They do not suffer fundamental limitations such as the sign problem of quantum Monte Carlo and can be applied to any lattice problem, the unique constraint being the virtual dimension which cannot be too large. TN methods are therefore particularly suited for frustrated magnetism problems, which suffer the sign problem, and their generalizations to SU(N ), with great successes in 1D [START_REF] Nataf | Density matrix renormalization group simulations of SU(N ) Heisenberg chains using standard Young tableaus: Fundamental representation and comparison with a finite-size Bethe ansatz[END_REF] and various 2D lattices [START_REF] Corboz | Simultaneous Dimerization and SU(4) Symmetry Breaking of 4-Color Fermions on the Square Lattice[END_REF][START_REF] Corboz | Simplex solids in SU(N ) Heisenberg models on the kagome and checkerboard lattices[END_REF]. Some implementations propose to encode generic non-abelian symmetries in the TN [START_REF] Weichselbaum | Non-abelian symmetries in tensor networks: A quantum symmetry space approach[END_REF].

Projected entangled pair states

The simplest tensor network is a one-dimensional chain, which gives a MPS. The density matrix renormalization group (DMRG) [START_REF] White | Density matrix formulation for quantum renormalization groups[END_REF] is the variational technique that optimizes MPS to approximate the quantum state of a one-dimensional chain. Critical systems and their long-range properties are hard to study with this method and another type of TN was designed to study them, the so-called multi-scale entanglement renormalization ansatz (MERA) [START_REF] Vidal | Class of Quantum Many-Body States That Can Be Efficiently Simulated[END_REF]. It can deal with different length scales and has been extended to two dimensions. DMRG can still be used in higher dimensions using a "snake" to map the lattice to a chain. However, this brings difficulties because two nearest neighbor sites, which are typically strongly entangled, can be far away from each other in the snake and the virtual dimension has to be very large to account for this effective long-distance entanglement. Thus a new family of TN algorithm has been defined to deal with two-dimensional systems, the projected entangled pair states (PEPS) [START_REF] Verstraete | Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems[END_REF], which are better suited for gapped systems 3.2. Projected entangled pair states Figure 3.4 -PEPS-representation of a quantum wavefunction on the square lattice. The tensor has four virtual legs and one physical leg. The area law is naturally fulfilled since the number of virtual legs grows linearly with the border size.

with short-range correlations. In all our work, we will only use PEPS, this section details their key features.

The area law has been rigorously demonstrated in 1D and explains the successes of DMRG since the entanglement entropy then scales as a constant. In 2D, the result has been postulated but no exact demonstration is known. Yet 2D tensor networks have proven extremely efficient to simulate various quantum SU(2) spin models [PSA16; PM17; Poi17; NC17]. The absence of demonstration is not a limit in itself and empirical studies show the manifold of the ground states of gapped, local Hamiltonians is very well described by PEPS.

PEPS are a natural extension of MPS to 2D, where each site is described by a single tensor A. On a lattice with coordination number z, the tensor has one physical leg of dimension d corresponding to the physical variable of the system and z virtual variables of dimension D, as we can see in figure 3.4. Contracting all the tensors of the TN over the virtual indices yields the coefficients of the wavefunctions. Unit cells of any sizes can be considered, which allow to break translation invariance, however for simplicity we will only consider translation-invariant TN. Bond operators, which are matrices with all singular values equal, are inserted in the bonds between two tensors to project the pair on the most entangled states so that the TN is maximally entangled, hence the name. PEPS automatically produce states that fulfill the area law in 2D since each virtual leg carry an entanglement (at most) ln D and the border of a given region crosses a number of virtual legs that grows linearly with its length.

The tensor A is of course not unique for a given wavefunction. The wavefunction lives in the Hilbert space and is characterized by its coefficients only: any transformation of the tensors that affect the virtual space only yields the same wavefunction. For any invertible matrix M , a pair M and M -1 can be inserted between two connected virtual legs and absorbed into the corresponding tensors with no effect on the contracted TN, i.e. the wavefunction. With N b bonds in the TN, the gauge group associated to this degree of freedom is the very large [GL D (C)] N b [START_REF] Jiang | Symmetric tensor networks and practical simulation algorithms to sharply identify classes of quantum phases distinguishable by short-range physics[END_REF]. On the square lattice, imposing a normalized tensor for e.g. the Frobenius norm and a translation-invariant TN still yields a gauge group U(D) 2 . In 1D, the fundamental theorem of MPS states that this is an equivalence: two MPS give the same wavefunction if and only if they are linked by such a gauge transformation on the two Chapter 3. Tensor network algorithms virtual legs. In 2D, the larger number of legs allows for different geometries and a fundamental theorem of PEPS is more subtle.

The wavefunction still has a number of coefficients that grows exponentially with the size of the system but we do not actually need to compute them. Indeed, what we really want is to compute observables of the system Ψ|O|Ψ / Ψ|Ψ . We can achieve this by considering a two-layer TN obtained by contracting a ket tensor A and its conjugate bra A * over the physical variable, which results in an elementary tensor E (see figure 3.5 (a)). All the computation are done with this tensor in the network. A local observable acts on the physical variable of A, it is sandwiched between the bra and the ket tensors and can then be inserted inside the TN (see figure 3.5 (b)). The same procedure applies to multi-sites observables, with more than one bra and one ket legs to contract with physical variables of several tensors A.

Until now, we only talked about wavefunctions and not about Hamiltonians, let us consider some exact results on them. First, it has been rigorously proven that for all short-range PEPS, there exists a gapped, short-range parent Hamiltonian that admits it as its ground state. This Hamiltonian is unique up to topological degeneracy and can be explicitly constructed, however it may not be simple and involve interactions within large minimal clusters. The reciprocal is also true: for any gapped, short-range Hamiltonian, there exist a 2D PEPS with finite virtual dimension that approximate the ground state with an arbitrary precision [START_REF] Hastings | Solving gapped Hamiltonians locally[END_REF].

The case of gapless Hamiltonians and wavefunctions with long-distance correlations is more complicate and there are no proven theorems to guide us. Contrary to the 1D case of DMRG, PEPS allow to construct critical wavefunctions with long-range correlations with a finite virtual dimension D. For instance, the RVB wavefunction we introduced in section 1.1.2 has an exact PEPS formulation with D = 3, we will investigate it in details in chapter 5. However, computational limitations makes the effective correlation length always finite in any algorithm based on PEPS.

Last, as MPS in 1D, PEPS are particularly useful to study entanglement properties and edge physics. Actually, on a finite system with open boundary conditions, there is a correspondence between the virtual degrees of freedom and the physical edge modes of the system. This is particularly useful to study the edge modes of SPT phases, which naturally appear in this framework. This correspondence gave a deep insight into the origin of the fractional spin-1/2 at the edge of the Haldane chain thanks to the Affleck-Kennedy-Lieb-Tasaki (AKLT) state, which naturally writes as an MPS.

To be more precise, if we consider a subregion A of a quantum system, the spectrum of its reduced density matrix is very similar to the one expected for a 1D critical theory [START_REF] Li | Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States[END_REF]. This spectrum gives keys insight on the entanglement of A with the rest of the system, and has been conjectured to be linked to the physical edge modes of a finite system. In the PEPS framework, the 1D system described by this density matrix can be interpreted as the boundary state living at the virtual edge of A, written in terms of the virtual variables located on the virtual legs along the cut [START_REF] Cirac | Entanglement spectrum and boundary theories with projected entangled-pair states[END_REF]. From the PEPS itself, one can construct an isometry U that maps virtual variables from the edge to the physical variable of the bulk, defining a bulk-edge correspondence.

Typically, on an infinite cylinder cut along the vertical axis, one can define the boundary density matrix σ L (resp. σ R ) of the left (resp. right) boundary state. Using the isometry U , the reduced density matrix ρ of the left subregion can then be written as:

ρ = U σ T L σ R σ T L U † (3.6)
We will construct these edge density matrices in chapter 4 and use the virtual variables of a PEPS to study the edge modes of an SU(3) SPT phase.

Symmetries implementation

The PEPS framework can be used for any weakly entangled state, but when the Hamiltonian has symmetries, either global or lattice, it is a powerful tool to implement them in the algorithm. Indeed, it is possible to encode global SU(N ) invariance and lattice symmetries directly at the level of a unique site tensor A which completely characterizes the quantum state [START_REF] Mambrini | Systematic construction of spin liquids on the square lattice from tensor networks with SU(2) symmetry[END_REF]. For simplicity, we will consider tensors on the square lattice, but this discussion can be generalized straightforwardly to other lattices.

Note that if symmetries are encoded inside A, it will never be possible to break them inside the TN. Thus such tensors can only describe QSLs phases and cannot reach phases that break either SU(N ) or lattice symmetries. However, implementing symmetries results in a dramatic reduction in the number of different coefficients to consider since symmetries impose many constraints on the coefficients of the tensor. More, the symmetries are known to be exact in the whole TN algorithm and they can be used inside it to get a computational boost in a similar way to exact diagonalization, although the use of the full SU(N ) symmetry is difficult.

We consider a PEPS whose physical variable lives in a Hilbert space H that is an irrep d of SU(N ) and we aim to define SU(N )-symmetric tensors. The starting point is to impose not only the Hilbert space but also the virtual space V to be a representation of SU(N ) and the tensor A to map the representation of the virtual space to the irrep d. d has to appear in the decomposition of the tensor product of the four virtual variables into a direct sum of irreps:

V ⊗4 = g • d ⊕ . . . , (3.7)
where g is the number of occurrences of the irrep d in the tensor product. The coefficient of the tensor A are the (generalized) Clebsh-Gordan coefficients associated with this fusion rule. Indeed, the tensor is exactly the projector from the product Hilbert space of dimension D 4 to the desired irrep reshaped into a (d, D, D, D, D) tensor, which can easily be obtained by exact diagonalization of the quadratic Casimir. Every irrep in the tensor product defines a tensor and g linearly independent tensors can be defined with physical variable d. When g = 1, additional symmetries can be used to select one irrep in the projector, including occupation numbers of the different irreps forming the virtual space representation as well as lattice symmetries1 . One must be very careful with the phase conventions of the different basis states of d: to be able to add two tensors, they have to share the same convention. A different one would break SU(N ) symmetry in the sum. The safest way to ensure this is to compute the projector on the unique highest weight state only, as defined in section 2.2.3. Then all the other basis states of the irrep can be generated by applying lowering operators to this state, following the structure of the weight diagram such as those in figure 2.32 . In order not to break the symmetry, special attention is needed for the bond operator: indeed, it must not only maximally entangle two neighboring tensors but also preserve the SU(N ) symmetry. Since after contraction there is only one degree of freedom in the bond, the bond operator has to project the virtual space on the SU(N ) singlet. This imposes strong restrictions on the virtual space V : it must be possible to make at least one singlet out of V ⊗2 and the bond operator is the projector on this singlet, reshaped from a (D 2 , 1) projector on a 1-dimensional space to a (D, D) square matrix (which is no more a projector in the mathematical meaning). To ensure maximal entanglement, all the singular values of the bond operator must have the same value regardless of the dimension of the irreps involved in the singlets of V ⊗2 . In other words, the projector on the singlet of two irreps a and a but me multiplied by √ a when more than one singlet appears in the decomposition of V ⊗2 . The operation the bond operator realizes is actually the charge conjugation C. Indeed, the singlet made from a and a can always be written as the trace over the diagonal of the tensor product, that is

|0 = 1 √ a 11 + 22 + • • • + |aa , (3.8)
which is exactly the tensor contraction. In the case of SU(2), this is not a problem since V is automatically self-conjugate. The projector on the singlet is the rotation of π, which is an operation of the group SU(2). On a bipartite lattice, it can be absorbed in the tensors of the sublattice B and is equivalent to a rotation of π of the physical variable that disappears on a bilayer tensor without physical variable. This is no more possible in the case of SU(N ) because the charge conjugation is not an operation of the group, which admits non self-conjugate irreps. The simplest solution is to require V to be staggered, as we will do in chapter 4 and section 5.1, which automatically grants a charge conjugation on the bonds. However, the price to pay is to impose staggered tensors in the TN and translation invariance is lost. When V consists in only one staggered irrep, the bond operator is just identity and it is not necessary to add it explicitly. Another solution is to include the square root of the projector on the local tensor: the projector on the singlet can always be chosen to be symmetric, its square root is a complex symmetric matrix that can be contracted on every virtual leg of the tensor A.

Lattice symmetries are implemented in a similar way: the point group C 4v naturally acts on the tensor A and therefore this tensor can be decomposed into irreps of C 4v , we listed them in the character table 2.1. The irreps of SU(N ) already impose some constraints on those of C 4v : for instance, the most symmetric irrep made of four virtual variables, as the one we will use in chapter 4, has to be invariant under any leg permutation and therefore belongs to the irrep A 1 . A subset of commuting matrices representing the operations of the group can be codiagonalized with the Cartan operators of SU(N ) to obtain irreps of both groups. Mixing tensors that belongs to different irreps of C 4v breaks the lattice symmetry. However, we can mix tensors belonging to the irreps A 1 and A 2 by adding an i in front of the A 2 tensors to get a symmetry A 1 + iA 2 . Such a tensor breaks time-reversal symmetry T because of the imaginary part as well as parity inversion P but preserves the product PT . It is rotation-invariant and undergoes complex conjugation under a reflection, yielding a chiral tensor that describes a chiral QSL. Let us consider the example of SU(2) with a spin-1/2 physical variable and a virtual variable of dimension D = 3 that decomposes as a spin-1/2 and a singlet: V = 2 ⊕ 1. The tensor A has shape (2, 3, 3, 3, 3) and a total size of 162 coefficients. We consider the tensor product V ⊗4 :

(2 ⊕ 1) ⊗4 = 12 • 2 ⊕ . . .

(3.9) therefore twelve SU(2) invariant tensors can be constructed with this choice of physical and virtual variables. But we can get more information from this fusion rule: to get a half-integer spin, the number of half-integer spins in the product has to be odd, therefore there are two categories of tensors with different occupation numbers: first one spin-1/2 and three singlets, or occupation numbers {1, 3}, and second three spins-1/2 and one singlet {3, 1}. We then look at the group C 4v : in the first case, the tensors dispatches into one irrep A 1 (which is the local tensor for the RVB wavefunction, as we will see in chapter 5), one irrep B 1 and one (2-dimensional) irrep E. In the second case, there are one A 1 , one A 2 , one B 1 , one B 2 and two E. The results are summarized in figure 3.6. We check we recover the multiplicity twelve from the fusion rule (3.9): from an initial number of 162 coefficients, imposing SU(2) symmetry reduces this number by a factor 13! We can even go further and impose rotation invariance, leaving only 3 coefficients. This family of tensors has been studied by Chen and Poilblanc [START_REF] Chen | Topological Z 2 resonating-valence-bond spin liquid on the square lattice[END_REF].

In table 3.1, we follow this method to classify three families of tensors of interest. The first row corresponds to the SU(2) case detailed above. The second is the SU(3) tensor that projects four quarks 3 = [1] on the most symmetric irrep 15 = [4], which we will use in chapter 4. The last row lists all the SU(4)-symmetric tensors for a physical variable 6 = [1, 1] with virtual variable 6 ⊕ 1, we will need them in chapter 5. We indicate occupation numbers in the second column, note that only a few of them are allowed by the fusion rules. We check in the last column that we recover the number of occurrences g of the irrep in the tensor product V ⊗4 . This recipe allows to construct and classify SU(N ) tensors. For small D, there is most of the time at most one tensor in each symmetry class (the irrep E can easily be split into two complex E 1 and E 2 tensors that are eigenvectors of the rotations Chapter 3. Tensor network algorithms

n occ A 1 A 2 B 1 B 2 E g N = 2, H = 2 {1, 3} 1 0 1 0 2 12 V = 2 ⊕ 1 {3, 1} 1 1 1 1 4 N = 3, H = 15 {4} 1 0 0 0 0 1 V = 3 N = 4, H = 6 {1, 3} 1 0 1 0 2 16 V = 6 ⊕ 1 {3, 1} 2 1 2 1 6
Table 3.1 -Classification of three family of SU(N )-symmetric tensors in terms of occupation numbers and C 4v irreps. The first column indicates the symmetry group SU(N ), the physical Hilbert space H and the virtual space V . The values in column E are doubled to account for its dimension and recover the multiplicity g.

Figure 3.7 -An approximate environment is constructed around any rectangular core that simulate an infinite number of tensors E around it. Since we consider only rotation and translation invariant tensors E, there is only one corner C and one edge T . This environment can be used to compute any observable.

C 4 with eigenvalues ±i) and we obtain an exact expression of the tensor coefficients (they are typically rationals or square roots of rationals). Linear combination of these tensors can then be considered, with the certainty not to break any symmetry unless different classes are mixed. Instead of an extensive number of tensor coefficients, only the limited number of weights of the different tensors need to be considered: in a similar way of TN describing only the tiny relevant part of the Hilbert space, this classification allows to restrain the large vector space of tensors to the few relevant elements with the desired symmetries.

Corner transfer matrix algorithm

The contraction of a tensor network is a hard problem that requires an extensive amount of memory. In the thermodynamic limit, an infinite number of sites are considered and the memory needed is infinite, for finite systems the memory grows exponentially with the size. Thus approximate algorithms that use less memory and computation power are needed to realize an efficient contraction. Many methods have been developed, including some based on MPS-matrix product operator (MPO) [START_REF] Murg | Variational study of hard-core bosons in a two-dimensional optical lattice using projected entangled pair states[END_REF] or the tensor renormalization group (TRG) [LN07] that we will use in chapter 5, which has numerous extensions. In this section, we introduce the corner transfer matrix renormalization group (CTMRG) algorithm developed by Orús and Vidal [START_REF] Orús | Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction[END_REF], from an initial idea by Nishino and Okunishi [START_REF] Nishino | Corner Transfer Matrix Renormalization Group Method[END_REF].

We work directly in the thermodynamic limit, i.e. on an infinite 2D lattice, with the double-layer TN representing the norm Ψ PEPS |Ψ PEPS (infinite PEPS, or iPEPS). The principle is to construct border tensors that simulate an infinite environment for the tensor E as shown in figure 3.7. We will only consider rotation invariant tensors E. The procedure involves controlled approximations, using a real space renormalization group technique, depicted in figure 3.8. The infinite-volume environment of any (rectangular) region of space is approximated using an edge tensor T of dimension χ × D 2 × χ and a corner χ × χ matrix C, where χ is the environment dimension taken as large as possible. We restrict ourselves to tensors E that are in the representation A 1 + iA 2 of the point group C 4v . This symmetry imposes the corner matrix C to be hermitian and we can diagonalize it easily. More, rotation invariance implies the four corners and the four edges to be identical and only one of them needs to be renormalized. C is obtained iteratively by adding tensors E to the corner, diagonalizing the resulting hermitian matrix and keeping only the χ largest eigenvalues in magnitude. A new tensor T is then computed using the diagonalization basis U . The algorithm can also be used when the tensor network is not translation invariant, the renormalization process only requires to add the unit cell as a whole. The initial values for C and T are obtained by contracting respectively one and two pairs of bra and ket legs of E. The process is iterated until the spectrum s of C has converged in magnitude: the sign of the eigenvalues is not well defined, but their absolute value converge. Since we only use SU(N ) symmetric tensors, every tensors involved will also obey this symmetry. In particular, s has a multiplet structure. In order to preserve the symmetry, the exact value of χ has to fit this decomposition and cannot be fixed arbitrarily. At every steps, the cut is made between two multiplets around a fixed value χ 0 , and the exact value of χ fluctuates before convergence. Note that the tensor T itself does not converge because of its gauge freedom: any unitary matrix can appear in the degeneracy blocks of s, yet any observable computed from this tensor does converge. The result must be normalized by the wavefunction norm and the connex part S1 • S2 2 must be subtracted.

When the environment is converged, any observable can be evaluated, padding it with tensors E to get a rectangle that can be surrounded by corner and edge tensors. The result is normalized by the norm of the wavefunctions, obtained by replacing all observables tensors by E. We give the example of the dimer-dimer expectation value in figure 3.9. Long-distance dimers-dimer correlations can be computed by adding tensors E between the two dimers.

We used the C 4v symmetry extensively in this algorithm when we considered only one corner and one edge. The implementation of SU(N ) symmetry is more tricky. The bottleneck of the algorithm is the diagonalization part, we would like to speed it up by using some exact diagonalization techniques since the matrix C is SU(N )-symmetric. While not impossible, implementing non-abelian symmetries is very hard because the different invariant values cannot be defined all at the same time.

Here, even the eigenvalues of the Cartan operators, or colors, are not well defined. Indeed, the tensor E is bilayer, with one ket with fixed colors and one bra with fixed anti-colors that are conjugate from each other. Unfortunately the tensors C and T do not have this structure and the diagonalization matrix mixes every ket vector with its conjugate bra. As a result, colors can only be defined up to a global sign, that is no distinction between color and anti-color can be made. Starting from the second iteration of the algorithm, it is no more possible to trace the colors of the corner eigenvectors: when a color ±1 is combined with color 1 from a vector of E, the result can be either 0 or 2: the value is not well-defined. However, the parity of this color stays well defined. Thus, it is possible to define a boolean pseudo-color for every state of E and every corner eigenvector which is conserved during the CTMRG process. Since there are N -1 Cartan colors in SU(N ), we can define N -1 pseudo-colors, which correspond to a symmetry group Z N -1 2 . Extensive empirical tests confirm the existence and the relevance of this symmetry. At each iteration, the pseudo-colors of E are combined with those of the previous corner in a new matrix C that conserves them. These pseudo-colors can then be used as invariants to block-diagonalize this large matrix with a huge gain in performance, allowing to reach very large χ and transferring the bottleneck of the algorithm to the contraction part. When cutting the spectrum, the pseudo-colors of the kept vectors are stored to be used in the next iteration. It is also possible to use them in tensor contractions but it imposes to contract non-contiguous elements of the tensors. This is particularly problematic because this operation cannot be performed by BLAS. In any cases, cache issues limit the possible gains independently from the implementation.

Chapter 4 SU(3) AKLT state

The AKLT state [START_REF] Affleck | Rigorous results on valence-bond ground states in antiferromagnets[END_REF] has been a breakthrough in the comprehension of integer spin chains and the so-called Haldane phase. In this chapter, we will recall the key points of AKLT physics and extend it to 2D and to SU(3). We show that it can be represented as a simple tensor network, allowing extensive studies. We explore its bulk properties on an infinite cylinder using transfer matrix methods. The edge physics is investigated by computing the entanglement spectrum and the related entanglement Hamiltonian. We show that the latter can be very well approximated by a simple SU(3) Heisenberg Hamiltonian with exponentially decaying interactions. This Hamiltonian acts on virtual variables that are quarks and anti-quarks, therefore the edge modes are fractional and attest an SPT phase. This chapter is adapted from the article [START_REF] Gauthé | Entanglement properties of the twodimensional SU(3) Affleck-Kennedy-Lieb-Tasaki state[END_REF].

AKLT physics

As we stated in section 1.1.2, the LSM theorem imposes the ground state of a 1D half-integer spin Hamiltonian to be critical or degenerate. In 1983, Haldane showed that unexpectedly integer spins behave differently [START_REF] Haldane | Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State[END_REF]. While the bilinear, halfinteger spin chain is critical with massless spinons excitations, he showed that the bilinear spin-1 chain with periodic boundary conditions has a unique, gapped ground state. Since the Hamiltonian is SU(2) symmetric and the ground state is unique, it has to be a singlet, otherwise each state of the multiplet would be a ground state. The underlying phase is called the Haldane phase, characterized by a gapped, symmetrypreserving ground state and fractional, spin-1/2 edge modes. This phase can only exist for integer spins since it derogates the LSM theorem. It is experimentally relevant for several systems described by an effective spin-1 chain.

To understand this phase and especially its edge modes, Affleck, Kennedy, Lieb and Tasaki proposed in 1987 a paradigmatic, exact state for the spin-1 chain that does belong to the Haldane phase. The principle is to decompose every spin-1 of the chain into two spins 1/2, entangle all pairs of spins on the bonds into singlets and project pair of spins on every site onto physical spin-1. A pictorial representation of the 1D spin-1 AKLT state is shown in figure 4.1 (a). The advantage of this construction is that its parent Hamiltonian is easy to find and connected to the purely bilinear Haldane chain. Indeed one can rewrite the most general SU(2)-invariant nearest-neighbor Hamiltonian presented in section 1.2.4 with convenient constants and obtain

H 1BB = 1 2 i [S i • S i+1 + β(S i • S i+1 ) 2 + 2/3], (4.1)
which is the Haldane chain Hamiltonian for β = 0. For β = 1/3, the Hamiltonian can be expressed as a sum of non-commuting projectors on the spin-2 subspace of the nearest neighbor bonds, i.e.

H 1D SU(2) = H 1BB (β = 1/3) = i P S=2 i,i+1 , (4.2)
where we have used the decomposition in terms of projectors previously detailed. Since H 1D SU(2) is a sum of projectors, it is positive. It also annihilates the AKLT state of figure 4.1 (a), since two sites consists in four spins 1/2, and after entangling two of them in a singlet it is no more possible to make a spin 2 out of them. Hence the AKLT state is a ground state of the Hamiltonian (4.1) for β = 1/3. It has been shown that on an infinite chain or with periodic boundary conditions, this ground state is unique. While it is not the ground state of the Haldane chain, numerical studies with varying β from 1/3 to 0 have shown that the two states are adiabatically connected and belong to the same phase. With open boundary conditions, the two edge spins-1/2 are totally free and the ground state has a degeneracy of four. These are fractional excitations that also exist in the Haldane chain and are protected by a symmetry [START_REF] Pollmann | Symmetry protection of topological phases in one-dimensional quantum spin systems[END_REF], defining a particular SPT class [Che+12; GW14; Sen15].

The AKLT construction can be straightforwardly extended to 2D lattices. On the square lattice (or any 2D lattice of coordination z = 4), one attaches four virtual spin-1/2 on each site, and then projects them onto the most symmetric (i.e. spin-2) irrep, as shown in figure 4.1(b). Again, the parent Hamiltonian takes the simple form of a sum of projectors over all nearest neighbor bonds i, j , H 2D SU(2) = i,j P S=4 i,j . In 2D, the family of AKLT states are protected by SU(2) spin-rotations and one-site translation symmetries [START_REF] Takayoshi | Field theory of symmetryprotected valence bond solid states in (2+1) dimensions[END_REF], a direct consequence of the LSM-Hastings theorem.

The 1D or 2D SU(2) AKLT states have extremely simple representations in terms of MPS [START_REF] Pollmann | Symmetry protection of topological phases in one-dimensional quantum spin systems[END_REF] and PEPS [START_REF] Cirac | Entanglement spectrum and boundary theories with projected entangled-pair states[END_REF], respectively, which make the analysis of their bulk and edge properties accurately computable. Indeed, it is easy to see that the 1D SU(2) AKLT state of figure 4.1(a) is in fact a MPS defined from a set of three 2 × 2 matrices labeled by the physical spin-1 with virtual spin-1/2 variables (i.e. d = 3 and D = 2). This construction can easily be generalized in 2D by replacing the d matrices by d rank-z tensors, where z is the lattice coordination number. In our case, we consider the square lattice with z = 4. We take the most symmetric arrangement of the four spins 1/2 and the elementary tensor has the coefficients

A 2 [↑, ↑, ↑, ↑] = 1 A 1 [↑, ↑, ↑, ↓] = 1/2 A 0 [↑, ↑, ↓, ↓] = 1/ √ 6 A -1 [↑, ↓, ↓, ↓] = 1/2 A -2 [↓, ↓, ↓, ↓] = 1
where all permutations of the four virtual variable have the same coefficient and therefore the five subtensors with fixed physical variable are normalized.

Although AKLT parent Hamiltonians are fine-tuned, the AKLT states provide in fact simple paradigms for the simplest (non-topological) gapped spin liquid phases, which can occupy a rather extended region in the parameter space of realistic Hamiltonians. Since localized SU(N ) spin systems can now be realized on optical 1D and 2D lattices, SU(N ) AKLT states are expected to describe generic spin liquid phases in such systems and are therefore of high interest. In the case of a SPT phase, the edge modes of the AKLT wave function will also be generic of the whole phase, being protected by symmetry.

SU(3) AKLT wavefunction

We now extend the recipe for the construction of SU(2) AKLT states to SU(3), in a straightforward way: the principle is to replace spins by irreducible representations of SU(3). First, in order to realize SU(3) singlets on all nearest neighbor bonds of the square lattice, four "quarks" in the fundamental [1] = 3 irrep ("antiquarks" in the anti-fundamental [1, 1] = 3 irrep) are attached on each even (odd) site. This way, neighboring virtual spins on every nearest neighbor bond belong to 3 and 3 irreps and can then be projected onto SU(3) • = 1 singlets. Then, in order to entangle this simple product of singlets, one projects the group of four quarks on each even (odd) site onto the most symmetric [4] = 15 ([4, 4] = 15) irrep corresponding to the actual physical degrees of freedom, as seen in figure 4.2(a). Note that the assignment as fundamental or anti-fundamental is arbitrary, the same tensor being placed on every site. As for SU(2), a simple parent Hamiltonian can be build from bond projectors on the largest, most-symmetric [8, 4] (self-conjugate) irrep obtainable from the tensor-product 15 ⊗ 15,

H 2D SU (3) = i,j P [8,4] i,j , (4.3)
where the sum runs over all nearest neighbor bonds. We already mentioned the tensor corresponding to this wavefunction in table 3.1. It can be constructed by diagonalizing the appropriate matrix and applying the two lowering operators S 1-and S 2-to the known highest weight (4, 0) to obtain the generalized Clebsch-Gordon coefficients as we explained in section 3.3. Here, thanks to the simple form of the tensor it is not even necessary and the tensor can be obtained by pure combinatorics. We want a fully symmetric combination of four indistinguishable particles that have three different states, or in other words four bosons three times degenerate. The dimension 15 corresponds to the number of 4-combination with three repetitions. The possible configurations are: -4 quarks in the same color: 3 1 = 3 distinguishable color states corresponding to the selected color, each of them with only 1 state of the Hilbert space with the appropriate color so a coefficient 1.

-3 quarks in one color, 1 quark in another one: 3 1 × 2 1 = 6 color states, 4 1 occurrences in the Hilbert space each so the coefficient is 1/2 -2 quarks in one color, 2 in another one: 3 2 = 3 color states, 4 2 occurrences, coefficient 1/ √ 6. -2 quarks in one color, 1 quark in a second, 1 quark in the third:

3 1 = 3 color states, 4 2 × 2 1 = 12 occurences, coefficient 1/2 √ 3.
We check this makes the 15 required states and the 81 color configurations of the Hilbert space are involved. The same argument gives the same tensor on odd site with conjugate "anti-colors".

We now focus on the tensor network algorithm. For simplicity, let us first start with a periodic (L-site) 1D chain with d on-site physical degrees of freedom labeled by α (e.g. the components of the physical spin). As we detailed in chapter 3, the amplitudes

c α 1 α 2 •••α L of a (translation-invariant) MPS of virtual dimension D are given solely in terms of d D × D matrices A α as c α 1 α 2 •••α L = Tr{A α 1 A α 2 • • • A α L }.
Here in 2D, the amplitudes of the PEPS are obtained from the tensor network defined by attaching a tensor on each lattice site and by contracting the site tensors over the virtual indices [CV09; Cir12; Sch13; Orú14b]. The S = 2 AKLT state of figure 4.1(b) can then be viewed as a simple PEPS with D = 2 virtual degrees of freedom (corresponding to the attached virtual spin-1/2) and d = 5 physical spin components [START_REF] Cirac | Entanglement spectrum and boundary theories with projected entangled-pair states[END_REF]. Similarly, the SU(3) AKLT state can be interpreted as a PEPS of virtual dimension D = 3 (for the three colors of the quarks) and d = 15 physical dimension, as depicted in figure 4.2 (b). The virtual space being one staggered irrep, there is no bond operator to insert as we discussed in section 3.3.

To compute the PEPS wave function norm Ψ|Ψ and expectation values Ψ|O|Ψ of local operators O, we defines the two-layer tensor network as discussed in section 3.2, each layer representing the ket and bra wave functions. By contracting two identical tensors on their physical indices one gets a new tensor E of dimension D 2 = 9, as shown in figure 4.2 (c). This way, the physical index disappears and its large dimension (15) is irrelevant for computations. We form an infinite cylinder by imposing periodic boundary conditions in one direction with circumference N v . Each row of the cylinder can then be seen as a transfer matrix, propagating states from the left to the right. This matrix acts on boundary states expressed in terms of virtual variables of the tensor network as shown in figure 4.3. To construct the fixed point boundary state of size (D 2 ) Nv , one uses iterated powers / Lanczos algorithm to converge to the leading eigenvector / leading eigenvalues of the transfer matrix. Note that since the latter is The gap ∆ in the bulk can easily be computed from the two largest eigenvalues of the transfer matrix, ∆ = ln (E 1 /E 2 ), with E 1 > E 2 , the correlation length ξ being defined as the inverse of the gap. We have computed ∆ for cylinders of perimeter N v = 2, 4, 6, 8 and extrapolated the result in the limit N v → ∞, as shown in figure 4.4 (a). We find that the extrapolation of ξ for the SU(3) case is very short (ξ 3 1.2), even shorter than the SU(2) value (ξ 2 2.1). Note that the extrapolation is very accurate, the scaling being exponential and the system size being large compared to ξ.

Entanglement properties

In order to construct the entanglement Hamiltonian, the fixed-point state is reshaped as a D Nv × D Nv boundary density matrix Σ b , acting on virtual variables. As we discussed in section 3.2, it has previously been shown [START_REF] Cirac | Entanglement spectrum and boundary theories with projected entangled-pair states[END_REF] that this matrix can be mapped onto the reduced density matrix of the half cylinder ρ via an isometry U . Here, thanks to time-reversal and reflection symmetry, the left and right boundary states are real and identical and we can just rewrite Eq. (3.6) as ρ = U † (Σ b ) 2 U . The entanglement Hamiltonian H acting on virtual boundary configurations is defined via (Σ b ) 2 = exp(-H), with an irrelevant temperature of 1. Since U is an isometry, the spectrum of H -the entanglement spectrum -is exactly the logarithm of the spectrum of the reduced density matrix ρ up to a factor 2. Such a spectrum has been conjectured by Li and Haldane [START_REF] Li | Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States[END_REF] to be in one-toone correspondence with the physical edge modes of the system. We compare the entanglement spectra of SU(2) and SU(3) AKLT wavefunctions in figure 4.5 (a) and (b). We observe they are very much similar at low energy: (i) the ground state is a singlet with momentum k = 0 (when N v = 4n), (ii) low-energy excitations follow a sinusoidal dispersion typical of the lower edge of a 2-spinon continuum, shown in figure 4.5 (c). This can be explained from the simple (approximate) analytical form of the entanglement Hamiltonian derived next.

S irrep 0 • 1 [2, 1] 2 [3] 3 [4, 2] 4 [5, 1] 5 [6, 3] 6 [6] -[7, 2] -[8, 4]
To understand its nature we decompose the entanglement Hamiltonian on the canonical basis of SU(3) operators acting on the virtual degrees of freedom at the boundary. The latter are being defined in a fermionic representation as

S α β (i) = c † α,i c β,i -δ α,β /3 if i is even -c α,i c † β,i + δ α,β /3 if i is odd (4.4)

Entanglement properties

57

where α, β label the three SU(3) colors. Note that the definition takes into account the anti-fundamental representation on odd sites [START_REF] Affleck | Large-n Limit of SU(n) Quantum "Spin" Chains[END_REF], which in the fermion language is obtained via a particle-hole transformation 1 Since the Hamiltonian is SU(3) invariant, there is a limited number of combination of operators that can appear, in particular no linear term can appear. The only second order SU(3) invariant terms are Heisenberglike terms, S i • S j = α,β S α β (i)S α β (j). Hence,

H = E 0 + i =j J(|i -j|) S i • S j + H rest , (4.5)
where E 0 = Tr(H). The higher order terms H rest are corrections of much lower weights -only 5% (6%) of the euclidean norm of H -E 0 for N v = 8 (N v = 6) -and are expected to be irrelevant. We show in figure 4.4(b) that the weights J(r) follow an exponential decay with distance, from with we can extract a typical decay length λ.

The sign of those weights is staggered, meaning the interaction is antiferromagnetic between quarks and anti-quarks and ferromagnetic between two quarks. By comparing SU(3) and SU(2), we see that λ 3 < λ 2 , fulfilling the same inequality than the bulk correlation length ξ 3 < ξ 2 . This is in agreement with a general argument based on PEPS that the range λ of the entanglement Hamiltonian tracks the bulk correlation length ξ [START_REF] Cirac | Entanglement spectrum and boundary theories with projected entangled-pair states[END_REF]. Interestingly, the entanglement Hamiltonian of the SU(3) AKLT state is adiabatically connected to the nearest neighbor 3 -3 Heisenberg chain [START_REF] Affleck | Large-n Limit of SU(n) Quantum "Spin" Chains[END_REF]. The latter can be mapped to a spin-1 chain with a purely negative biquadratic coupling and was shown to exhibit a small spontaneous dimerization [AH87; Aff89; BB89; SY90]. It is however plausible that the extra J(2) ∼ 0.3 J(1) coupling will close the gap and lead to a gapless spectrum. Indeed, the numerical entanglement spectra shown in figures 4.5 (b,c) do not show any hint of spontaneous translation symmetry breaking (implying ground state two-fold degeneracy in the N v → ∞ limit), but such a feature is hard to see at high temperature. The conformal field theory (CFT) description of our entanglement Hamiltonian is an open problem which would require the numerical treatment of very long chains. Interestingly, the parent Hamiltonian [TNS14; BQ14] for a CFT wave function constructed from the SU(3) 1 WZW models is, once truncated, quite similar to our quasi-local entanglement Hamiltonian, although with a larger ratio J(2)/J(1) 0.56 and a three-body term of significant amplitude. Hence a description of the entanglement Hamiltonian in terms of a SU(3) 1 WZW theory seems natural and, at least, agrees with our low-energy entanglement spectrum shown in figure 4.5 (c). Tu et al. [START_REF] Tu | Quantum spin models for the SU(n) 1 Wess-Zumino-Witten model[END_REF] report critical properties deviating from the expected behaviors of the SU(3) 1 WZW model. We note however that the two (local) models may sit in different critical phases.

Another interesting question is the possible correspondence between the entanglement spectrum and the edge physics [START_REF] Li | Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States[END_REF]. As for the SU(2) case, one can construct a local SU(3)-invariant parent Hamiltonian or "PEPS model" [START_REF] Pérez-García | PEPS as unique ground states of local Hamiltonians[END_REF][START_REF] Yang | Edge Theories in Projected Entangled Pair State Models[END_REF] for which any region with an open 1D boundary ∂R will have a degenerate manifold of (at most) D |∂R| ground state. As for any PEPS models in a trivial (i.e. short-ranged entangled) phase, any Hamiltonian can be realized on the edge [START_REF] Yang | Edge Theories in Projected Entangled Pair State Models[END_REF] by slightly perturbing the (fine-tuned) SU(3) PEPS model. However, it is still possible to protect edge properties by symmetries in the bulk [START_REF] Chen | Symmetry-Protected Topological Orders in Interacting Bosonic Systems[END_REF]. For example, similarly to the SU(2) AKLT model, SU(3) symmetry and translation invariance rule out a gapped edge which does not break any symmetry [START_REF] Lieb | Two soluble models of an antiferromagnetic chain[END_REF]. This is in direct correspondence with the properties of the (infinite size) entanglement spectrum discussed above.

Lastly, we comment on the relevance of this work to cold atom physics. We detailed in section 1.3 how to simulate SU(N ) systems in cold atomic gases. The difficulty here is that this the irreps are staggered conjugate, which is realizable experimentally although very challenging [START_REF] Laflamme | CP(N -1) quantum field theories with alkaline-earth atoms in optical lattices[END_REF]. It requires SU(3) fermions with a staggered optical potential such that the occupation numbers are exactly 1 and 2 on the two sublattices. To enforce the same irrep on every site, a different AKLT construction is needed, involving virtual states belonging to the smallest self-conjugate irrep. For SU(3) it corresponds to the adjoint [2, 1] = 8 irrep.

Chapter 5

SU(N ) RVB states

We introduced the concept of RVB in section 1.1.2, we recall that the RVB wavefunction is defined as an equal-weight sum of all nearest-neighbor dimer covering states. It is expected to play an important role in low-temperature magnetic systems, including superconductors. RVB-like states are also relevant in the domain of cold atoms and have already been experimentally realized with 87 Rb [START_REF] Nascimbène | Experimental Realization of Plaquette Resonating Valence-Bond States with Ultracold Atoms in Optical Superlattices[END_REF]. In this chapter, we generalize the RVB construction to any representation of SU(N ) with staggered conjugation on the square lattice and we use the CTMRG algorithm detailed in section 3.4 to approximate these wavefunctions. We first look at the case of staggered fundamental-conjugate irreps N -N and explore their properties for several N . We then consider self-conjugate representations, which allow to construct translation invariant RVB wavefunctions. We apply this construction to the irrep = 6 of SU(4) and add three other tensors to the PEPS ansatz. We argue the wavefunction obtained by linear combination realizes a Z 2 topological QSL. We conclude by proposing a reasonable Hamiltonian which could host it.

RVB N -N wavefunctions

In this section, we generalize the RVB wavefunction to SU(N ) on the square lattice. In the case of SU(2), the conjugate irrep is equivalent to the fundamental, it corresponds to the rotation of π on one sublattice. After this rotation has been done, the same tensor can be used on every sites and the wavefunction is translation invariant. For the general SU(N ) case, it is not generally not possible to make a valence bond out of two irreps. However there is only one (and only one) singlet appearing in the decomposition of the tensor product of an irrep and its conjugate. We can therefore generalize the concept of RVB wavefunction by considering staggered representations on a bipartite lattice that form bonds with one of their nearest neighbor and consider the equal-weight sum of all the possible configurations.

RVB tensor

The nearest-neighbor RVB state has a very simple tensor representation on a bipartite lattice [START_REF] Schuch | Resonating valence bond states in the PEPS formalism[END_REF]: for a given physical irrep d1 , the virtual space is V = d ⊕ 1 of dimensions D = d + 1, with conjugation on the sublattice B. In order to fully cover the lattice with all possible dimer configurations, the tensor dispatches the physical representation between one virtual and z -1 singlets, or holes, summing over the z different geometries with an equal weight, with z the coordination number of the lattice. Such a tensor is invariant under any leg permutation with a symmetry group S z , more crucially on the square lattice it belongs to the irrep A 1 of the point group C 4v (refer to the character table 2.1). The non-zero coefficients of the tensor on this lattice are given by:

∀ i, 1 ≤ i ≤ d, A i [i, D, D, D] = 1 A i [D, i, D, D] = 1 A i [D, D, i, D] = 1 A i [D, D, D, i] = 1
Again, there is no projector to add explicitly in the tensor network thanks to the staggered representations: first the singlet dd is 1/ √ d Î as we stated in Eq. 3.8 Second, the singlet made of two holes is of course diagonal and maximal entanglement imposes the coefficient to be 1/ √ d too. Hence the resulting bond operator is an irrelevant scaling constant times identity matrix. The elementary tensor is the same for the fundamental representation and its conjugate, therefore as for the AKLT SU(3) wavefunction the tensor network is translation invariant but the wavefunction itself is not unless d is self-conjugate and the group operators acting on the wavefunction need to be staggered. The number of holes is a good quantum number of the tensor on the two sublattices, thus in addition to the global SU(N ) symmetry the wavefunction has an additional U(1) local gauge symmetry.

Correlation length and central charge

We first use this construction in the simplest case, the staggered fundamentalconjugate irrep of SU(N ) on the square lattice for N ≤ 6. The resulting PEPS has a physical dimension of d = N and a virtual dimension of D = N + 1. This construction generalizes the RVB wavefunction we described in section 1.1.2: the wavefunction is an equal weight superposition of all the nearest neighbor dimer covering states, the dimer being the SU(N ) singlet made of the tensor product N-N. This model can be mapped on a classical model of interacting dimers at finite temperature [START_REF] Damle | Resonating Valence Bond Wave Functions and Classical Interacting Dimer Models[END_REF], where different N correspond to different temperatures. As a consequence, the wavefunction is critical with long-range correlations for every N since starting from N = 2 the temperature corresponds to the disordered Kosterlitz-Thouless (KT) phase, which is critical, and the temperature only rises for larger N . The N → ∞ limit corresponds to the infinite temperature of the classical system, where the partition function is an equal weight sum of all micro-states regardless of interactions. This state is the ground state of the Rokhsar-Kivelson (RK) point of the quantum dimer model [START_REF] Rokhsar | Superconductivity and the Quantum Hard-Core Dimer Gas[END_REF]. This model can be solved by a mapping to a scalar height field theory which is known to be critical at the RK point [START_REF] Fradkin | Bipartite Rokhsar-Kivelson points and Cantor deconfinement[END_REF].

We now define the bilayer elementary tensor E by contracting the physical index and apply the CTMRG algorithm to construct an environment for it. We implement the Z N -1 2 symmetry explained in section 3.4, we can even enhance it to Z N 2 by considering the parity of the hole number as another pseudo-color. In the case of SU(2), this additional pseudo-color is identical to the first one and the symmetry group is just Z 2 . After convergence, we obtain the corner matrix C and the edge tensor T . As we show in figure 5.2 (a), the N -N RVB wavefunctions are very similar for all N . The spectrum s of the corner matrix decays exactly in the same way, with the same algebraic exponent that does not depend on the value of χ.

By (approximately) contracting the E tensor network from infinitely far away on the left and right of the 2D plane, one ends up with an infinitely long (vertical) two-leg ladder (T ⊗ T ) L , L → ∞ (where the D 2L virtual indices between the two chains of T tensors are contracted). Using the Lanczos algorithm, one can extract the two leading eigenvalues λ 1 and λ 2 of the corresponding ladder transfer matrix T (see figure 5.1(a)) and compute the maximal correlation length ξ of the system from the transfer matrix gap ∆ as

∆ = 1 ξ = ln λ 1 λ 2 . (5.1)
Note that the finiteness of the corner dimension χ automatically implies a finite correlation length even if the RVB wavefunctions are critical, so that a finite-χ scaling analysis is necessary. In figure 5.2 (b), we see the correlation length ξ increases linearly with χ, diverging in the relevant χ → ∞ limit. In this case, the maximum correlation (left) "Spin" correlation functions S • S. We observe an exponential decay with a "spin" correlation length ξS. (right) Dimer-dimer correlation functions. We observe an algebraic decay with a critical exponent α which is cut at large distances by an exponential fall-off.

length corresponds in fact to the diverging dimer correlation length (see section 5.1.3).

We can compute the central charge of the corresponding CFT by looking at the entanglement entropy of the one-dimensional quantum state |Ψ Env ∈ [C D 2 ] ⊗L made of an infinite (L → ∞) chain T ⊗L of χ-contracted T tensors (see figure 5.1 (b)), which can be seen as a MPS. We compute the leading eigenvector Σ of T (see figure 5.1 (c)) and reshape it as a χ × χ matrix. The so-called "environment" entanglement entropy is defined as S Env = -Tr Σ 2 ln Σ 2 . For a critical wavefunction, the relation [START_REF] Calabrese | Entanglement entropy and quantum field theory[END_REF] S Env = c 6 ln ξ + S 0 (5.2)

allows to compute the central charge c. We show the results in figure 5.2(c) and find c = 1 for every N , consistent with the physics of the KT phase.

Correlation functions

From the converged C and T tensors one can construct the environment of any rectangular subsystem. Using an (infinitely long) strip delimited by two chains of T tensors, we have computed first the expectation value of the observable S i • S j for all distance |i -j| in the strip direction, similarly to the dimer case showed in figure 3.9. The resulting correlation functions are plotted in figure 5.3 (a), we observe a clear exponential decay with a very short "spin" correlation length ξ S . We have also computed the dimer-dimer correlation function in 5.3 (b) (this time exactly as shown in figure 3.9). This observable has long range correlations with algebraic decay, below the maximum correlation length ξ that finite-χ induces. As a consequence, only for large enough χ can one fit the algebraic behavior on a sufficiently large range of distances to obtain accurate values of the critical exponent. Here luckily those 5.2. SU(4) topological RVB spin liquid [Alet] χ are easily reachable and the data can be fitted fairly well, with only a trend to underestimate the value of the exponent.

63 N ξ S [PEPS] ξ S [Stéphan] α[PEPS] α
As we stated, the N-N RVB wavefunctions can be related to a classical interacting dimer model, thus allowing to estimate their critical dimer exponent obtained from this classical model using transfer matrix and classical Monte Carlo methods [START_REF] Alet | Classical dimers with aligning interactions on the square lattice[END_REF] 2 . We compare our results with these ones and some other obtained using Monte Carlo methods directly on the SU(N ) RVB wavefunction [START_REF] Stéphan | Entanglement in gapless resonating-valence-bond states[END_REF] in table 5.1; we find good agreement within error margins. The mapping on the quantum dimer model predicts a limit α → 2 for N → ∞, which is compatible with our findings. Those results were obtained using totally different algorithms that are subject to different kinds of systematic errors, yet the results are very similar: this convergence make us reasonably confident in our TN algorithms.

SU(4) topological RVB spin liquid

While not impossible, charging optical lattices with a staggered number of particles is an experimental challenge and it is more reasonable to consider translation invariant systems. This is compatible with our RVB construction as soon as the physical irrep is self-conjugate. We develop here the case of the two-fermions irrep of SU(4) = 6 (d = 6), aiming to construct a topologically ordered phase as seen in section 1.4.2. Indeed, the local gauge symmetry acting on the tensor can encode topological order [START_REF] Schuch | PEPS as ground states: Degeneracy and topology[END_REF], which makes the PEPS formalism particularly suited to construct simple ansätze of topological QSLs. As for the case of SU(2) treated by Chen and Poilblanc [START_REF] Chen | Topological Z 2 resonating-valence-bond spin liquid on the square lattice[END_REF], we consider a family of extended SU(4) RVB states on the square lattice. We show that, when longer-range SU(4)-singlet bonds are included, the local gauge symmetry is broken down from U(1) to Z 2 , leading to the emergence of a short-range spin liquid. Evidence for the topological nature of this QSL is provided by the investigation of the Renyi entanglement entropy of infinitely-long cylinders and of the modular matrices. The results presented in this section are the subject of the article [START_REF] Gauthé | SU(4) topological resonating valence bond spin liquid on the square lattice[END_REF].

Elementary tensors and bond operator

We follow the recipe given in section 3.3 to construct the elementary tensors of our state. We consider a virtual space V = ⊕ •, with D = 7. The relevant fusion rule writes T1, T2 and T3 tensors with three virtual 6 states and one hole. (c) The square root of the projector on the singlet P is applied to every virtual legs of the ket-tensor.

⊕ • ⊗4 = 16 ⊕ . . . ( 5
Hence there are 16 possible tensors, we already gave their classification in table 3.1. We only consider rotation invariant tensors in the irreps A 1 and A 2 , which limits this number to 4. The first tensor to consider, T 0 has occupation numbers {1, 3}. This tensor is exactly the same as the SU(6) 6 -6 nearest neighbor RVB tensor defined in the previous section (see figure 5.4 (a)) and the PEPS wavefunctions are identical. More precisely, the physical wavefunctions are not the same because they belong to different Hilbert spaces and the operators S α acting on a site are not the same, they are the 15 generators of SU(4) 6 instead of the staggered 35 generators of SU(6) 6 -6, however these wavefunctions share the same expression of their coefficients in the product basis. Hence this wavefunction has a symmetry enhanced to SU(6) and its physical properties, including correlation length, entanglement entropy, "spin" correlation length and dimer critical exponent are the same as the SU(6) case formerly discussed.

The three other tensors, T 1 , T 2 and T 3 have occupation numbers {3, 1} (see figure 5.4 (b)). The tensors T 1 and T 2 have the same occupation numbers and C 4v symmetry, however their internal structure is different. If we do not consider the hole, we can look at the irreps of C 3v formed with the three remaining : T 1 belong to the class A 1 and is more symmetric than T 2 and T 3 , which form an irrep E. Another way to realize this higher symmetry is to look at the permutation group of the legs of the square tensor, which is the symmetric group S 4 . As well as T 0 , T 1 is invariant under this group, while T 2 and T 3 are not. The expressions of these four elementary tensor coefficients can be found in the supplemental materials of [START_REF] Gauthé | SU(4) topological resonating valence bond spin liquid on the square lattice[END_REF].

The most general rotation-invariant tensor describing a short-range RVB state can be written as

A = a 0 T 0 + a 1 T 1 + a 2 T 2 + ia 3 T 3 , (5.4)
with a symmetry A 1 + iA 2 . Note that our generalized RVB state not only contains nearest neighbor singlet bonds but also longer range singlet bonds, including some connecting the same sublattice (see figure 5.5). The state fully covers the lattice with all possible configurations of 6 -6 nearest neighbor singlet bonds or 1 -1 nearest neighbor "empty" bonds. This time the virtual space is not staggered and we had to explicitly insert a bond operator. Since two linearly independent singlets can be made on a bond, either 6 ⊗ 6 → 1 or 1 ⊗ 1 → 1 there exist a one-dimensional family of valid bond projector parameterized by an angle κ, P (κ) = cos κ(P 6⊗6 ⊗ Î) + sin κ( Î ⊗ P 1⊗1 ). We chose κ = π/4 to treat the two singlets in a similar way. The states of the 6-representation, labeled as |1 , |2 , |3 , |4 , |5 , |6 , are defined by their weights (1, 0, -1), (1, -1, 1), (0, -1, 0), (0, 1, 0), (-1, 1, -1) and (-1, 0, 1), respectively (see the weight diagram of the irrep in figure 2.3). The last state |7 corresponds to the 1-singlet. With this convention of ordering the vectors of the 6 ⊕ 1 representation, the projector we apply reads:

P = 1/ √ 12            
0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

√ 6             . ( 5.5)
This choice does not give a maximal entanglement on the bonds and does not rigorously follow the requirement of a bond operator we detailed in section 3.2, which would be an angle κ = arctan 1/ √ 6 with a matrix of ones. However it is easy to see that changing the angle κ is equivalent to a renormalization of the coefficients a i in equation (5.4): going from the maximally entangled matrix to any κ is done by contracting all virtual legs with a diagonal matrix M = diag(1, 1, 1, 1, 1, 1, tan κ/ √ 6), up to an irrelevant scaling constant. Since the tensors T i have a well defined hole number h i = 3 for i = 0 and h i = 1 otherwise, it is equivalent to a scalar multiplication by (tan κ/ √ 6) h i . In our case, κ = π/4 implies a weight 6 times higher for tensor T 0 . This is an example of the GL D (C) gauge freedom mentioned in section 3.2. To avoid dealing explicitly with the projector in our tensor network algorithm, we absorb P in the definition of the tensor A. More precisely, we consider the square root of Pwhich is a complex symmetric matrix -and contract it on every physical leg of the initial tensor A as shown in figure 5.4 (c). The double layer tensor E is then computed after this operation is done and it turns out that it also exhibits the same A 1 + iA 2 symmetry as the initial mono-layer tensor A. Note that while √ P is complex, E happens to be real when a 3 = 0, we expect this is due to the fact 6 is self-conjugate.

Critical spin liquids

We first analyze the properties of these four elementary tensors alone, which define critical states with long-distance correlations. We roughly follow the same plan as in the previous section, with the same methods. As we see in figure 5.6, these Chapter 5. SU(N ) RVB states to the T0, T1, T2, and T3 tensors. We observe an exponential decay with a "spin" correlation length ξS.

10 0 10 1 10 2 10 3 i 10 -2 10 -1 10 0 |s i | T 0 χ = 450 T 1 χ = 2004 T 2 χ = 593 T 3 χ = 553
wavefunctions have dramatically different computational behavior for the CTMRG algorithm, unlike the different nearest neighbor SU(N ) RVB wavefunctions. Very large plateau corresponding to high-dimensional irreps appear in the corner spectrum and the corner eigenvalues decay at different speeds. In particular they decay very slowly for tensor T 1 , which requires large χ in order to make significant finite-entanglement scaling. Indeed, the correlation lengths ξ(χ) plotted in figure 5.7 is always linear in χ, which attests critical wavefunctions, but they do not grow as fast as the nearest neighbor RVB case from figure 5.2 (b) and in the case of T 1 the estimation of the central charge and the critical exponent are difficult. Hopefully, the implementation of symmetries authorizes very large values of χ in the CTMRG algorithm. First, T 0 inherits the Z 6 2 symmetry of the SU(6) nearest neighbor RVB tensor. While we only expected Z 3 2 in the other cases, empirical tests showed that the Z 6 2 symmetry extends to every tensors A of the form (5.4), even though we do not understand its origin. This symmetry allows us to break the corner matrix C into (at most) 64 submatrices that can easily be diagonalized and χ as big as 2000 can be reached.

We then use the results from the CTMRG algorithm to compute observables. The four "spin" correlation functions are plotted in figure 5.8, they follow the expected exponential decay with a correlation length ξ S . The dimer case plotted in figure 5.9 is more interesting. Indeed, these four wavefunctions can all be viewed as RK ground states of quantum dimer models, this time with the "dimers" corresponding either to 6 -6 (for T 0 ) or to 1 -1 (for T i , i = 1, 2, 3) singlet bonds. The same mapping onto a (coarse-grained) height field theory implies algebraically-decaying dimer-dimer correlations C d (r) ∼ (1/r) α versus distance r. Note that the height representation is linked directly to a local U(1) gauge symmetry of the A tensor. The critical behavior can only be observed for distances smaller than the maximal correlation length of the system. Since this maximal correlation length is rather small compared to the previous section, we can clearly see the extension of the critical domain when χ rises. This correlation length, which we computed from the transfer matrix, can be recovered by fitting the exponential queue of the dimer correlation functions introduced by the finite χ.

In order to characterize the underlying conformal field theory by its central charge c, we have investigated the entanglement entropy of the one-dimensional quantum state |Ψ Env the same way we did for N -N. We found c = 1 in all cases, consistently with the prediction of the height field theory. For T 1 we had to include a higher order correction to the fit to account for the very small correlation length.

10 0 10 1 10 2 r 10 -4 10 -3 10 -2 10 -1 10 0 (-1) r ( (S 0 • S 1 )(S r • S r+1 ) -S 0 • S 1

Phase diagram

We now consider linear combination of the tensors T i . For simplicity, we shall not consider the T 1 tensor any further and restrict to the two-dimensional PEPS family parameterized by two angles θ and φ, defining the coefficients of the A ≡ A(θ, φ) tensor in equation (5.4) as

A(θ, φ) = cos θ cos φ T 2 + sin θ cos φ T 0 + i sin φ T 3 (5.6)
For each elementary tensor, a fix number of dimers -either 1 for T 0 or 3 for T 1 , T 2 and T 3 -connect to each site. Hence the number of (fluctuating) dimers cutting a given closed loop is conserved. This means the wavefunction defined by one tensor alone has a U(1) gauge symmetry, as for the general RVB construction. For any linear combination of the fundamental tensors with a 0 = 0, long-range dimers appear and this number of fluctuating dimers is no longer constant, but its parity is. Thus U(1) symmetry breaks into Z 2 and we expect the above critical QSLs to become unstable.

Chapter 5. SU(N ) RVB states Figure 5.12 -We consider the (restricted) two-dimensional tensor family given by A = cos θ cos φ T2+ sin θ cos φ T0 + i sin φ T3 and we plot the maximal value of ξ we obtained (the value of χ is not exactly the same for every circle). The red points label the three elementary tensors, the green stars label the points where the boundary entanglement entropy and the entanglement spectrum were computed (see figures (5.14) and (5.15)) Their exact positions are: (a) A = cos(π/8)T2 + sin(π/8)T0 with χ = 538, (b) A = cos(3π/8)T2 + sin(3π/8)T0 with χ = 320, (c) A = cos(π/4)T0 + i sin(π/4)T3 with χ = 364. As shown in figure 5.11 (a), along the line interpolating between the T 2 and T 0 tensors, one can extract the correlation length from a fit of ξ(χ) vs χ. In contrast to the critical case, for a short-ranged wavefunction, ξ(χ) converges exponentially fast towards a finite length ξ ∞ with an exponential law ξ(χ) = ξ ∞ + a exp(-χ/l). As can be seen in figure 5.11 (b), we found an extended critical region around the nearest neighbor RVB state. However, the complete scan of the whole two-dimensional phase diagram in figure 5.12 reveals no sign of other extended critical regions in the vicinity of the critical states defined by the T 2 and T 3 tensors (the different weight associated with the tensor T 0 may explain this difference). In this figure, we show the values of the (maximum) correlation length in this space in figure for several values of θ and φ. The cut parameter χ was taken around 350 but may be slightly different for the different circles. We note that ξ gets larger when A is approaching one of the T 0 , T 2 or T 3 corners of the triangular parameter space.

PEPS bulk-edge correspondence and topological entanglement entropy

We now turn to the investigation of the topological properties of the short-range SU(4) QSL of our phase diagram. To do so, we compute the topological entropy [START_REF] Levin | Detecting Topological Order in a Ground State Wave Function[END_REF] of the system.

For this, we use the bulk-edge correspondence theorem of PEPS detailed in section 3.2 applied to an infinitely long (horizontal) cylinder (of finite perimeter N v ) partitioned in two halves by a vertical plane. The reduced density matrix

ρ C = Tr C |Ψ PEPS Ψ PEPS | of the half-cylinder C can be expressed as ρ C = U ρ b U † ,
where ρ b is a boundary density matrix acting on the virtual space [C D ] ⊗Nv at the boundary of C and U is an isometry. Naturally, due to the tensor Z 2 gauge symmetry, ρ b splits into two Z 2 -even and Z 2 -odd invariant blocks. These blocks correspond to an even / odd number of holes in the N v virtual indices of the boundary virtual space; they are normalized separately with a trace equal to 1/2. Using the CTMRG algorithm, ρ b can be well approximated by ρ b = σ 2 b , where σ b is the product of (reshaped) edge T tensors shown in figure 5.13 (note that the left and right boundary states are conjugate from each other, hence σ T L = σ † R = σ R ). This matrix can be used to compute Renyi entanglement entropy defined as S q = 1 1-q ln Tr(ρ b ) q . Although finite-χ effects are large for q > 1, we have obtained converged results for q = 1/2 and q = 1/3 [START_REF] Jiang | Accuracy of Topological Entanglement Entropy on Finite Cylinders[END_REF]. As shown in figure 5.14 for three different PEPS, S 1/3 versus N v can be well fitted by a straight line which intersects the vertical axis at a finite value consistent withln 2, the topological entropy expected for a Z 2 QSL of the same class as Kitaev's TC [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF]. Note the fit in (b) and (c) may be less reliable since N v remains smaller than ξ. However, in (a) ξ N v . the three wavefunctions defined in figure 5.12, with respect to the cylinder circumference Nv for q = 1/3, 1/2 and 1. Squares (circles) label the odd (even) sector. The stars stand for the average of the two sectors and the dashed lines are linear fits. The arrows point to the valueln 2. for the location in the parameter space). The two topological even and odd sectors are normalized separately.

The entanglement spectra are shown in figure 5.15 for the three points (a), (b) and (c) defined in figure 5.12. The ground state of the entanglement Hamiltonian is always an even singlet and the first excitation is an odd 6 irrep. We believe the excitation energy remains finite for N v → ∞, i.e., the entanglement spectra remain gapped. The points (a) and (b) correspond to real wavefunctions and we indeed observe a k ←→ -k symmetry in the spectrum associated to time-reversal and parity symmetries. The point (c) labels a complex wavefunction with A 1 + iA 2 symmetry that breaks T and P. We observe that the even sector of the entanglement spectrum still exhibits the k ←→ -k symmetry, but not the odd sector, which follows a k ←→ π -k symmetry. We believe this is due to the fact that the product T P is still preserved. Note also that we have not observed the emergence of gapless chiral edge modes since the entanglement spectrum remains gapped.

When a 0 = 0 and the U(1) symmetry is not broken, we also expect different sectors that follow this U(1) symmetry, and since Z 2 is a subgroup of U(1) we could always group the sectors just according from Z 2 and observe a split spectum. Our results are therefore consistent with a Z 2 topological order but a more subtle analysis is required to assert it.

TRG and modular matrices

To obtain further evidence for the topological nature of our family of short-ranged QSLs, we have used the TRG algorithm detailed in figure 5.16 to compute the S and T modular matrices [START_REF] He | Modular matrices as topological order parameter by a gauge-symmetry-preserved tensor renormalization approach[END_REF]. At the PEPS tensor level, the Z 2 gauge transformation simply amounts to multiplying by -1 (respectively +1) the virtual states in the 6 (respectively 1) irrep, and this gauge symmetry has to be preserved at each TRG iteration [START_REF] He | Modular matrices as topological order parameter by a gauge-symmetry-preserved tensor renormalization approach[END_REF][START_REF] Mei | Gapped spin liquid with Z 2 topological order for the kagome Heisenberg model[END_REF]. The accuracy of the TRG is also controlled by the maximum number of singular values χ that are kept at each step, always being careful not to cut multiplets. After convergence, we can easily act with elements of the Z 2 group in order to compute the modular matrices [START_REF] Chen | Topological Z 2 resonating-valence-bond spin liquid on the square lattice[END_REF].

The two modular matrices S and T are actually linked by permutation of the lines. A critical or a trivial gapped phase (a topological phase with Z 2 topological order) is characterized by the modular matrix S shown in figure 5.17 (a) (figure 5.17 (b)). The trace of the converged modular matrix S plotted in Fig. 5.17 (c) vs θ at fixed φ = 0 shows two sharp transitions revealing the existence of an extended topological QSL phase. Note that, when increasing the control parameter χ, the extent of the topological phase gets bigger, in agreement with our previous finding that criticality exists only at θ = 0 and in the close vicinity of θ = π/2 (T 0 PEPS).

By generalizing Oshikawa's argument [START_REF] Oshikawa | Commensurability, Excitation Gap, and Topology in Quantum Many-Particle Systems on a Periodic Lattice[END_REF], one can extend LSM-Hastings theorem to conjecture that it is not possible to obtain a unique featureless gapped state for SU(4) model with physical 6-representation on each site3 . We believe our topological QSL could be observed both in theoretical models and in experiment. We develop the search for a Hamiltonian stabilizing this phase in the following section.

Host Hamiltonian

We now search for a reasonable Hamiltonian that could stabilize the Z 2 topological phase constructed in the previous section. We slightly change our conventions compared to the previous section by considering a bond operator that maximizes the entanglement entropy. This imposes the bond operator to have all singular values equal to one, corresponding to an angle κ = arctan 1/ √ 6 between the projectors P 6⊗6 and P 1⊗1 and replacing √ 6 by 1 in equation (5.5). This change is equivalent to dividing the coefficient a 0 from previous section by 6 and we can retrieve our results. This work is not published yet. The value 2 corresponds to a topological phase while 4 implies a trivial phase. We show two sets of data obtained for two ranges of the cut parameter χ and we note that larger χ are needed when the correlation length grows.

S =     1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1     S =     1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1     0 π/8 π/4 3π/8 π/2

Hamiltonian and conventions

We consider a lattice where we attach an SU(4) irreducible representation 6 on each site and we consider nearest-neighbor coupling only. We start from the fusion rule on two sites

⊗ = • ⊕ ⊕ .
(5.7)

Thus three SU(4) symmetric projectors can be defined on two sites: P 1 , P 15 and P 20 . Using them as a natural basis, the operator S • S writes S • S = -5 P 1 -P 15 + P 20 (5.8) and exactly as the spin-1 chain we considered in section 1.2.4, the most general twosites SU(4) symmetric Hamiltonian can be parameterized by a single parameter θ.

Following the conventions of [START_REF] Affleck | SU(2n) quantum antiferromagnets with exact C-breaking ground states[END_REF], the Hamiltonian becomes

H(θ) = cos θ S • S + sin θ 4 (S • S) 2 .
(5.9) On a bipartite lattice there are four SU(6) points, when the coefficients in front of two projectors are identical:

-at θ = π/4 and θ = 5π/4, the fusion rule is enhanced to 6 ⊗ 6 = 15 ⊕ 21 -at θ = ±π/2, the fusion rule becomes 6 ⊗ 6 = 1 ⊕ 35. On a given lattice with coordination number z and nearest neighbor coupling, the energy per site is one half of the average value of all the H(θ) taken on the z bonds.

Chapter 5. SU(N ) RVB states

The energy of a given wavefunction for all θ is a sinusoid parameterized by its value in two points only: H(θ) = cos(θ) H(0) + sin(θ) H(π/2).

A pure bilinear model θ = 0 is expected to stabilize an ordered phase that breaks SU(4) [START_REF] Kim | Linear flavor-wave theory for fully antisymmetric SU(N ) irreducible representations[END_REF]. We build our work starting from early calculations based on projected wavefunctions [START_REF] Paramekanti | SU(N ) quantum spin models: a variational wavefunction study[END_REF]. They suggest the existence of a narrow QSL region around θ = 0.19π (thus for a sign of the biquadratic interaction appropriate to a half-filled fermionic Hubbard model [START_REF] Wang | Competing Orders in the 2D Half-Filled SU(2N ) Hubbard Model through the Pinning-Field Quantum Monte Carlo Simulations[END_REF]), whose exact nature remains unclear. Our topological QSL is therefore a good candidate for this phase, although other alternatives like a chiral QSL -as for the SU(3) triangular lattice [Boo+20] -may exist.

Exact results

We now introduce a bunch of exact states on the square lattice to compare to our PEPS. These states correspond to magnetic phases described in section 1.1.2 generalized to SU(4).

Ferromagnetic

Hamiltonian minimization

Our CTMRG algorithm allows us to compute the energy of any linear combination of the form (5.4). We first consider the elementary tensors T i . As we previously stated, we only need to compute the expectation value of the Hamiltonian in θ = 0 and θ = π to generate the full curve for any θ. To account for the finite-χ, we scale the infinite entanglement limit as (χ) = ∞ + a/χ. Again, T 1 has the most difficult scaling.

We then use the CTMRG to optimize the four coefficients a i to minimize the Hamiltonian (5.9) for a given θ. We start from an initial guess for the a i , then at each point we converge the environment tensors C and T . We evaluate the gradient numerically and give it to a standard minimization algorithm such as the conjugate gradient. To take into account the error induced by the finite corner dimension χ, we restart the process with increasing χ, starting from the previous converged point. When a maximal value of χ is reached, we take the last converged point and use finite-entanglement scaling to extrapolate the χ → ∞ limit of the energy. Since the coefficients are a function of θ, the minima are only valid for a given θ and may not follow a sinus wave.

In figure 5.18, we plot the energy as a function of θ for the critical wavefunctions defined by the elementary tensors T i along with the different exact wavefunctions from the previous section. The main limit is we do not have an exact expression for the energy of the quantum antiferromagnetic state, which play an important role around θ = 0. We observe two main regions of crossing, around θ = 0.19π and θ = -0.7π.

The first one corresponds to the region where Paramekanti and Marston propose two possibilities, either a direct transition from a Néel phase to a charge-conjugation breaking one or a thin QSL phase. In our case, the A 1 plaquette state has the best energy around, better than our minimizations, and the antiferromagnetic state is probably even better. Our phase is not be found there.

However, things are different in the region around θ = -0.7π, we zoom on it in figure 5.19. In this region, Paramekanti and Marston proposed a phase transition from the ferromagnetic phase to a dimerized one, with a dimerization parameter close to 1 at the transition. This fully-dimerized state is the D state we considered and we see our minima have an energy lower than both the dimerized and the ferromagnetic states. Hence even though we cannot access the partially dimerized state they consider, we know that our state has a lower energy near the level crossing FM -D. In these minima, the coefficient a 3 is 0 up to minimization precision therefore this phase does not break time-reversal symmetry. We conclude our Z 2 topological phase is a good candidate to appear in the phase diagram of the bilinear biquadratic nearest-neighbor Hamiltonian around θ = -0.7π. Note that on the square lattice SU(4) differs from SU(2) where no QSL survives in the case of nearest neighbor interactions only. This leaves open the possibility of experimental realizations using any ultracold alkaline-earth atoms realizing SU(N ) symmetry by simply tuning the number of species [START_REF] Pagano | A onedimensional liquid of fermions with tunable spin[END_REF] to e.g. N = 4. Fixing a filling of two particles per site should avoid three-body losses and thus allow controlled experiments.

Conclusion

To summarize, we have used tensor networks to construct two different kinds of SU(N ) topological states. In chapter 4, we constructed an SU(3) SPT phase with fractional, quark-antiquark edge modes. We investigated its EH and proved it to be close to the Heisenberg Hamiltonian with exponentially decaying coupling.

In chapter 5 we have extended the construction RVB state on the square lattice first to SU(N ) then to a new family of SU(4)-symmetric QSLs with physical 6representation on each site. Physically, such states involve resonance between shortrange 6 -6 singlets, in close analogy with their SU(2) analogs. Using an exact PEPS representation of these QSLs we showed that (i) when singlet dimers are restricted to NN bonds, the QSLs are critical, (ii) away from these fine-tuned states (or small region) the QSLs have short-range correlations and (iii) exhibit Z 2 topological order.

To conclude, tensor network methods are powerful numerical tools to investigate solid states systems, especially when sign problems forbids the use of quantum Monte Carlo. Furthermore, the possibility to easily implement SU(N ) symmetry at the elementary tensor level makes them particularly relevant to address QSL phases. It can also be used when only discrete symmetries are broken, such as time-reversal T and parity inversion P. We based our PEPS on the AKLT and RVB wavefunctions, but many other generalization are possible. Indeed, after selection of a virtual space and classification according to their symmetries, many tensors can be tested for a given Hamiltonian.

An exciting perspective is to consider virtual spaces V that allow for chargeconjugation C symmetry breaking. This was not possible in our discussion because we only considered staggered or self-conjugate V but it can be achieved with other choices for V . Indeed, a brand new family of tensors can be constructed for the irrep 6 of SU(4) on the square lattice with D = 9 and a virtual space V = 4 ⊕ 4 ⊕ 1. These tensors allows to construct wavefunctions that break every combinations of C, P and T but the product CPT .

From a technical point of view, our CTMRG algorithm can still be improved in many aspects. The bottleneck of our current implementation is tensor contraction and to some extend memory use. One solution could be to bypass BLAS to contract noncontiguous tensors using ad hoc loops in a fast language. Using sparse matrices instead of dense tensors may also grants computational boosts, especially for the elementary tensor E that symmetry requirements make very sparse. In a second time, the U(1) symmetry is not restricted to RVB tensors and it should be possible to implement it without downgrading it to Z 2 . A better understanding of the origin of the Z N -1 2 symmetry may allow to use a larger subgroup of SU(N ) in the diagonalization process. Independently from the CTMRG, the optimization algorithm we handled in section 5.3 could be improved by computing the gradient with algorithmic differentiation instead of numerical estimation.

Résumé en français †

Ce résumé reprend le plan du manuscrit en anglais, avec cinq sections correspondant à chacun des chapitres. On ne reprendra cependant que certaines sous-parties en se focalisant sur les définitions des objets manipulés. La première section est une introduction au contexte général de la physique SU(N ) en matière condensée. On détaille comment le Hamiltonien ab initio de la matière condensée se simplifie en Hamiltonien de systèmes de spins sur réseaux. On montre que cette physique de spins se généralise à SU(N ) et s'avère réalisée expérimentalement dans des systèmes d'atomes froids. La deuxième section présente certains éléments de théorie de représentation des groupes de Lie, qui est le cadre mathématique de la description quantique du spin et explique sa généralisation à SU(N ). Dans la troisième section, on décrit les algorithmes de réseaux de tenseurs, en se concentrant sur la famille des états projetés sur des paires intriquées (PEPS) et leur contraction. Ces méthodes permettent de construire numériquement des fonctions d'ondes de systèmes quantiques, on s'attachera en particulier à implémenter la symétrie SU(N ) au niveau du tenseur élémentaire. Les deux dernières sections présentent les résultats obtenus avec ces méthodes. La quatrième section est consacrée à la généralisation à SU(3) sur réseau carré de la fonction d'onde Affleck-Kennedy-Lieb-Tasaki (AKLT). Dans la cinquième et dernière section, on s'intéresse aux états de type liens de valence résonnants (RVB), que l'on généralise à SU(N ). On discute en particulier le cas d'un système de deux fermions SU(4) par site sur réseau carré, pour lequel on construit une phase topologique Z 2 .

1 Physique des systèmes SU(N )

Systèmes de spins quantiques

La physique SU(N ) est une généralisation de la physique des spins et du magnétisme, on s'attachera donc dans un premier temps à la description des systèmes de spins. La physique du solide est la branche de la physique qui étudie les propriétés de basses énergies d'un grand nombre de particules, lorsque les noyaux atomiques s'ordonnent autour d'une configuration stable et fixe. La totalité des phénomènes observés dans les matériaux, qu'ils soient isolants ou conducteurs, magnétiques, supraconducteurs ou autres provient des mêmes constituants fondamentaux : les noyaux atomiques, les électrons et l'interaction électromagnétique entre eux. Ce sont les comportements collectifs de ces constituants qui génèrent une telle diversité dans les phénomènes observés. Décrire ces comportements collectifs à partir des principes premier s'avère illusoire, aussi une théorie réaliste repose sur l'utilisation de Hamiltoniens effectifs. Un modèle effectif a la même physique que le système considéré initialement dans une limite donnée mais est bien plus simple. En fonction du phénomène que l'on veut comprendre, différents modèles effectifs peuvent être dérivés d'un même Hamiltonien, l'objectif étant de ne conserver que les termes pertinents pour le phénomène considéré et d'ignorer les autres.

Dans le cas du magnétisme des isolants, on considère que les noyaux atomiques forment un réseau cristallin parfait et statique qui sert de cadre fixe aux autres constituants. Les électrons de coeur sont fortement liés aux noyaux et leur rôle peut être limité à un simple écrantage statique qui renormalise le potentiel des ions. La physique du système est régie par les électrons des couches externes et leurs mouvements au sein du potentiel cristallin. On décompose l'espace de Hilbert sur une base produit d'orbitales atomiques locales. Pour simplifier encore plus, on se limite à une seule orbitale par site et on se place au demi-remplissage, avec en moyenne un électron par site. Les électrons peuvent passer d'un site à l'autre avec une amplitude t qui correspond à leur énergie cinétique, on restreint ce saut aux seuls premiers voisins. Sous l'effet de l'écrantage, l'interaction de Coulomb entre électrons devient de courte portée et peut être approximée par une interaction effective locale U . Le système est décrit par le modèle de Hubbard, dont le Hamiltonien s'écrit :

Ĥ2 = -t i,j ,σ ĉ † i,σ ĉj,σ + U i ni↑ ni↓ . ( 1
)
Ce modèle admet deux limites bien définies : pour t/U 1, l'interaction effective entre électrons est faible et les électrons sont faiblement corrélés. Le terme cinétique domine, les électrons se délocalisent sur le réseau et forment des bandes de conduction : le matériau est un métal conducteur. La transition de Mott sépare ce régime métallique du régime t/U 1, où les électrons sont fortement corrélés et le coût en énergie d'une double occupation est élevé. Les électrons se localisent sur les sites du réseau et le matériau est un isolant dit isolant de Mott, par opposition aux isolants de bandes.

Aux basses températures, deux électrons ne peuvent pas occuper le même site mais ils peuvent toujours minimiser leur énergie par des sauts virtuels d'un site à l'autre. À cause du principe de Pauli, ces sauts virtuels ne sont possibles que si les électrons de deux sites voisins ont des spins opposés, ce qui favorise un ordre antiferromagnétique. En traitant ce mécanisme par la théorie de perturbations au second ordre en t/U , un Hamiltonien effectif de spin est obtenu comme la limite aux fortes interactions du modèle de Hubbard :

Ĥ3 = J i,j Ŝi • Ŝj , (2)
avec un couplage antiferromagnétique J = 4t 2 /U . Dans ce système, les électrons et les ions sont gelés, les seuls degrés de liberté sont les spins localisés sur le réseau. Le système est un isolant magnétique décrit par une fonction d'onde de spin :

|Ψ = c i 1 ...i N |i 1 ...i N . ( 3
)
Ce déroulé est une simple esquisse des phénomènes à l'oeuvre. Le point le plus notable est que le magnétisme n'est pas dû aux interactions magnétiques dans la matière. En effet, sans être systématiquement négligeables, celles-ci sont bien trop faibles pour induire un ordre magnétique à des températures de plusieurs centaines de Kelvins. C'est la répulsion coulombienne, associée au principe de Pauli, qui est à l'origine du magnétisme, comme on le voit ici avec la constante effective de couplage de spins J qui est induite par U . D'autres interactions magnétiques effectives existent, en particulier l'interaction d'échange et le super-échange, ils sont construits à partir des mêmes ingrédients. En fonction des détails microscopiques, le signe et l'amplitude de l'interaction spin-spin varie ainsi que sa portée. Un Hamiltonien de spin plus général et longue portée pertinent est donc :

Ĥ4 = i,j J ij Ŝi • Ŝj ; (4)
C'est le Hamiltonien de Heisenberg, qui est à la base du magnétique quantique. On observe qu'il ne privilégie aucune direction et que l'énergie d'un état ne change pas si on effectue une rotation sur tous les spins du système : c'est la marque de la symétrie SU(2), le premier pas vers SU(N ). Le magnétique quantique est un domaine très riche qui donne naissance à de nombreuses phases différentes. À haute température, le système sera toujours désordonné et non-magnétique, ne brisant aucune symétrie. Aux basses températures cependant, différentes phases ordonnées apparaissent sous l'effet des interactions. Lorsque toutes les constantes de couplage J ij sont négatives, le système s'ordonne selon un ordre ferromagnétique, avec tous les spins pointant vers la même direction (voir figure 1 (a)). Le système n'est plus invariant par rotation et la symétrie SU(2) est spontanément brisée. Lorsque les constantes J ij sont de signe alterné sur un réseau bipartite, le système se scinde spontanément en deux sous-réseaux d'aimantations opposées : c'est la phase antiferromagnétique de 1 (b). Cet état est le plus courant comme fondamental des isolants de Mott, par exemple chez les cuprates supraconducteurs aux faibles dopages.

Ces différentes phases correspondent à des états faiblement intriqués que l'on peut imaginer à partir d'une image classique simple. Ce n'est plus le cas pour une nouvelle catégorie de phases purement quantiques dans lesquelles l'intrication joue un rôle majeur. Ces phases se retrouvent dans des systèmes où le magnétisme est frustré et sont en général des singulets de spin, avec S tot = 0. La description de ces phases repose sur le concept de lien de valence, ou dimère, dans lequel deux spins sont maximalement intriqués et forment un singulet. Un état quantique singulet peut se décomposer sur l'ensemble des états produits de singulet du système, qui est une famille de vecteurs génératrice mais pas libre. On parle d'état cristal de liaisons covalentes (valence bond crystal, VBC) lorsque les liens de valence ou une combinaison sous forme de plaquette s'ordonnent selon un motif régulier recouvrant le réseau, comme en figure 1 (c).

Le dernier état qui nous intéresse est présenté en figure 1 (d). Cet état ne brise aucune symétrie et apparaît comme désordonné, avec des corrélations spin-spin de faible portée : on parle de liquide de spin quantique (QSL), par analogie avec les liquides. C'est une phase exotique avec des propriétés surprenantes, comme des excitations fractionaires, les spinons. Le premier exemple proposé d'un tel liquide est l'état liaisons covalentes résonnantes (resonnant valence bond, RVB).

Physique SU(2)

La description quantique des spins repose sur l'étude du groupe SU(2). Elle unifie les moments cinétiques orbitaux et de spins dans un même formalisme qui impose des contraintes fortes sur la structure de l'espace de Hilbert. Le point de départ est la relation de commutation des composantes S α des observables de moment cinétique :

[ Ŝα , Ŝβ ] = i αβγ Ŝγ , (5)
où = 1 et αβγ est le symbole de Levi-Civita. En imposant cette relation, toute la structure de l'espace de Hilbert est fixée. On la dérive en définissant l'opérateur spin total Ŝ2 = Ŝx2 + Ŝy2 + Ŝz2 ainsi que les opérateurs d'échelle :

Ŝ+ = Ŝx + i Ŝy , Ŝ-= Ŝx -i Ŝy . ( 6
)
On décompose l'espace de Hilbert en blocs de sous-espaces propres de Ŝ2 associés à la valeur propre s(s + 1). Comme S z et Ŝ2 commutent, on peut diagonaliser l'opérateur S z dans ces blocs et on montre que tous ses vecteurs propres peuvent être générés à partir du vecteur de plus grande valeur propre. Les valeurs propres m de S z prennent toutes les valeurs -s, -s+1, . . . , s-1, s. La valeur propre s est entière ou demi-entière et l'espace propre de Ŝ2 associé à la valeur propre s(s + 1) est de dimension 2s + 1. On appelle spin chacun de ces différents sous-espaces caractéristiques. Lorsque plusieurs particules sont prises en compte, l'espace de Hilbert du système est l'espace produit tensoriel des différentes variables locales. On peut lui définir un moment cinétique total :

Ŝ = i Ŝi . ( 7
)
qui obéit aux même relations de commutation. On peut donc également décomposer cet espace produit en différents spins indépendants. La décomposition du produit tensoriel en somme directe de spins est totalement contrainte par les spins locaux pris en compte, on parle de règles de fusion des spins. Dans le cas de deux spins 1/2, l'espace total se décompose comme la somme directe d'un singulet et d'un triplet, ce qui donne la règle de fusion :

2 ⊗ 2 = 1 ⊕ 3. (8)
On vérifie que la dimension totale de l'espace est bien la même des deux côtés. Un Hamiltonien est dit SU(2)-symétrique s'il commute avec les trois composantes S x , S y et S z . Cette symétrie impose que le spectre du Hamiltonien soit compatible avec la décomposition en spins de l'espace de Hilbert : tous les états d'un même spin doivent avoir la même énergie et le spectre est fortement dégénéré. On peut donc décomposer le Hamiltonien sur la base des projecteurs sur les différents spins et le nombre de degrés de liberté d'un Hamiltonien SU(2) symétrique est fortement réduit.

La physique SU(N ) est la généralisation de cette physique en remplaçant les commutateurs des spins par d'autres relations de commutation. Les deux couleurs ↑ et ↓ de SU(2) sont remplacés par N couleurs et les opérateurs S x , S y et S z par N 2 -1 opérateurs différents. Cette physique est réalisée en certains points discrets de Hamiltoniens symétriques, lorsque plusieurs spins différents se retrouvent dégénérés. Bien que de tels points sont discrets et donc inatteignable pour des systèmes réels, leur seule présence a des conséquences importantes pour le diagramme de phase associé, avec des phases étendues qui gardent certaines caractéristiques du point de symétrie étendu.

Sur le plan expérimental, les systèmes avec une symétrie SU(N ) n'existent donc pas naturellement en matière condensée. Il est cependant possible d'en créer à partir de gaz d'atomes froids. Des atomes froids refroidis et piégés dans un réseau optique permettent de réaliser des Hamiltoniens de matière condensée, en particulier le modèle de Hubbard, avec un contrôle total des différents paramètres du système. Certains atomes, en particulier les alkalins, les alkalino-terreux et certains atomes à la structure électronique proche comme Yb peuvent être utilisés pour réaliser des Hamiltoniens SU(N ) symétriques, en profitant de la dégénérescence du spin nucléaire I. De tels systèmes ont une symétrie renforcée avec N = 2I + 1 couleurs et permettent de tester des prédictions théoriques au-delà de la matière habituelle.

2 Théorie de représentation de SU(N )

Définitions et formalisme

On rappelle qu'un groupe est un ensemble muni d'une loi de composition interne admettant un élément neutre tel que tout élément admet un inverse. Le groupe est dit abélien lorsque la loi de composition est commutative. On dit qu'un groupe G agit sur un ensemble X s'il existe une application :

G × X → X, (g, x) → g • x telle que : ∀x ∈ X, e • x = x et ∀g, h ∈ G, (hg) • x = h • (g • x).
Cette action peut également être vue comme un morphisme de groupes de G vers le groupe des bijections de X. La théorie de représentation est la branche des mathématiques qui étudie le cas où X a une structure d'espace vectoriel et où l'action de groupe préserve cette structure. Ainsi on définit une représentation de G comme l'action linéaire de G sur un espace E, ou de manière équivalente un morphisme de groupes :

ρ : G → GL(E), g → ρ(g). (9)
À chaque élément du groupe on associe donc une matrice de GL(E) qui respecte les règles de produit du groupe, c'est à dire que ∀g, h ∈ G, ρ(g)ρ(h) = ρ(gh). Cette théorie est très riche et possède d'innombrables applications, mais elle comporte également de nombreuses difficultés pour peu que le groupe G ait des propriétés topologiques désagréables. On ne traitera pas ces cas problématiques et on supposera toujours que le groupe G est compact, potentiellement fini. On se limitera également aux seules représentations de dimension finie. On définit la représentation triviale d'un groupe par E = C, ρ(g) = Id. L'intérêt de la théorie de représentation est qu'être une représentation non-triviale d'un groupe donné est une contrainte forte qui donne beaucoup d'informations sur la structure de l'espace vectoriel considéré. En physique quantique, on est naturellement amené à considérer des espaces de Hilbert qui réalisent des représentations de différents groupes de symétrie, en particulier des symétries spatiales (rotation, réflexions, translations).

Si E est un espace de Hilbert muni d'un produit scalaire, on dit que la représentation ρ est unitaire si son action préserve les angles :

∀g ∈ G, ∀u, v ∈ E ρ(g) • u | ρ(g) • v = u|v , (10)
c'est à dire que ρ prend ses valeurs dans le groupe unitaire de E. On a alors le théorème :

Théorème 1 Toute représentation de dimension finie d'un groupe compact peut être rendue unitaire.

Quitte à redéfinir le produit scalaire, l'action d'un groupe ne change pas la norme d'un vecteur. Sans perte de généralité, on considérera donc toujours les représentations rencontrées comme unitaires. On dit qu'une représentation est irréductible si elle ne peut être pas décomposée en une somme directe de sous-espaces vectoriels stables sous l'action du groupe. On parle alors d'irrep pour irreducible representation, les irreps sont les briques fondamentales de la théorie de représentation. Une représentation est complètement réductible si elle peut se décomposer en une somme directe d'irreps. On a alors un deuxième résultat majeur :

Théorème 2 Toute représentation de dimension finie d'un groupe compact est totalement réductible.

En particulier, le produit tensoriel de deux représentations de G est lui-même une représentation de G puisqu'on peut agir avec G sur chacune des composantes du produit. Ce théorème impose donc que tout produit tensoriel peut se décomposer de manière unique en une somme directe d'irreps. Les règles de fusion sont en fait le dictionnaire explicitant cette décomposition.

Si (ρ, E) et (τ, F ) sont deux représentations de G, on définit un opérateur d'entrelacement comme une application linéaire T : E → F telle que :

∀g ∈ G ∀v ∈ E T (ρ(g) • v) = τ (g) • T (v).
Deux représentations sont dites équivalentes s'il existe un opérateur d'entrelacement inversible entre elles. Dans les faits, on traitera deux représentations équivalentes comme étant identiques et on s'intéressera aux différentes classes de représentations non-équivalentes. Le lemme de Schur stipule qu'il n'existe pas d'opérateur d'entrelacement entre deux irreps complexes non équivalentes et que pour deux irreps équivalentes tous les opérateurs d'entrelacement sont identiques modulo multiplication par un scalaire.

Dans le cas des groupes finis, il existe un nombre fini de représentations irréductibles distinctes, le nombre exact étant le nombre de classes de conjugaison de G. Toutes la théorie de représentation d'un groupe est contenue dans sa table de caractère, qui liste chaque irrep ainsi que la trace des matrices représentant chacun des éléments du groupe dans cette irrep.

Représentation des groupes de Lie

La mécanique quantique amène à s'intéresser à l'action de groupes continus sur l'espace de Hilbert, en premier lieu le groupe des rotations de l'espace en trois dimensions SO(3). De tels groupes sont dit différentiables, c'est à dire qu'ils ont une structure de variété différentielle en plus de leur structure de groupe. Ces groupes sont appelés groupes de Lie et leurs représentations sont bien comprises. On détaillera ici les principaux concepts de la théorie de représentation et on les appliquera afin de caractériser les représentations irréductibles de SU(N ).

Le point de départ de l'étude des groupes de Lie est d'utiliser leur structure différentielle afin de déterminer leurs représentations. Le groupe est un objet compliqué, on s'intéressera donc non pas à lui directement mais à son espace tangent, que l'on munit d'une structure d'algèbre : l'algèbre de Lie. On va étendre la notion de représentation à cette algèbre, caractériser ses représentations et de là, en déduire celles du groupe de Lie.

Dans toute la suite, on supposera que G est un groupe de Lie matriciel compact, c'est à dire un sous-groupe compact de GL N (C). En plus de la représentation triviale, on peut automatiquement définir une nouvelle représentation de G, la représentation fondamentale, qui correspond au produit matriciel des éléments de G sur les vecteurs de C N . Cette représentation est irréductible et de dimension N , on la notera donc N. Pour toute représentation de ρ de G, on peut définir sa conjuguée par τ : g → -ρ(g) * . Physiquement, cette opération correspond à la conjugaison de charge C. Cette nouvelle représentation est de même dimension que l'originale, on la désignera par une barre au-dessus de la dimension. Une représentation équivalente à sa conjuguée est dite réelle, ou auto-adjointe. On désignera la représentation conjuguée de la fondamentale comme étant juste la conjuguée, ou N.

On rappelle que SU(N ) est le groupe spécial unitaire de taille N , autrement dit :

SU(N ) = {U ∈ M N (C) | U † U = I et det U = 1}. ( 11
(X, Y ) → [X, Y ] = XY -Y X. ( 13
)
Le crochet est une application bilinéaire anti-symétrique qui laisse stable l'algèbre de Lie. Il vérifie l'identité de Jacobi :

[X, [Y, Z]] + [Z, [X, Y ]] + [Y, [Z, X]] = 0. (14)
L'algèbre de Lie encode les propriétés locales du groupe, elle est son espace tangent en l'identité. Elle est caractérisée de manière unique par les relations de commutation des éléments d'une base de matrices T α :

[T α , T β ] = f αβγ T γ . ( 15
)
Les éléments f αβγ sont appelés les constantes de structure. L'algèbre de Lie agit sur elle-même via les crochets de Lie et définit donc une représentation irréductible ayant sa dimension, dite représentation adjointe. On obtient l'algèbre de Lie à partir du groupe en prenant la différentielle de la définition du groupe : dans le cas de SU(N ), on obtient l'algèbre de Lie :

su(N ) = {M ∈ M N (C) | M † + M = 0 et Tr M = 0}, (16)
soit le sous-espace des matrices anti-hermitiennes de trace nulle, de dimension N 2 -1. Par convention on met un facteur i devant les matrices de l'algèbre et on considère des matrices hermitiennes. On vérifie que pour N = 2, on retrouve les relations de commutation des observables de spin. Le concept de spin comme représentation irréductible du groupe SU(2) prend alors son sens. On définit une représentation d'une algèbre de Lie comme une application linéaire qui conserve les crochets de Lie, c'est à dire un morphisme d'algèbre. Les notions de représentations unitaires et irréductibles s'y étendent naturellement. La caractérisation des irreps de SU(N ) se fait de manière semblable à celle de SU(2) : on détermine l'algèbre de Lie, ce qu'on a fait. On détermine ensuite toutes les représentations de l'algèbre et leurs caractéristiques. Enfin, on applique l'exponentielle sur ces représentations pour passer d'une représentation de l'algèbre à une représentation du groupe. Comme SU(N ) est simplement connexe, chaque représentation du groupe correspond à une et une seule représentation de l'algèbre.

La structure des représentations irréductibles de su(N ) est semblable à celles de su(2). On détermine une base d'une irrep donnée en codiagonalisant N -1 opérateurs de l'algèbre qui commutent. Chaque vecteur de cette base est indexé par un ensemble de N -1 entiers (on divise par un facteur 2 pour retrouver les conventions usuelles des spins), appelé poids. On définit une relation d'ordre entre ces poids et toute irrep est caractérisée par son unique vecteur de plus haut poids. On applique ensuite les N -1 opérateurs d'échelle de manière successive pour générer tous les états de la base de l'irrep. On montre une représentation graphique de cette construction en figure 2 pour la représentation de spin 1 de SU(2) et la représentation à deux fermions 6 de SU(4).

Tableaux de Young

On utilise le formalisme des tableaux de Young afin de désigner les irreps de SU(N ). Un tableau de Young est un arrangement de boîtes triées par lignes de longueurs décroissantes de haut en bas. Un tableau de Young donné est représenté par r entiers entre crochets correspondant aux longueurs de chacune des r lignes du tableau. La représentation fondamentale de SU(N ) est une seule boîte :

= [1] = N. (17)
Ajouter des boîtes sur une ligne signifie symétriser les états et en ajouter dans une colonne signifie anti-symétriser les états. À chaque irrep de SU(N ) correspond un et un seul tableau, qui a au plus N -1 lignes. Le singulet, qui correspond à une colonne de hauteur N est marqué par un boulet •. Le plus haut poids correspond exactement au nombre de colonnes d'une taille donnée, et on passe facilement du plus haut poids au tableau de Young. On donne ici l'exemple de SU(4), avec les représentations fondamentales, conjuguée, singulet, adjointe, à deux bosons et à deux

(2) (0) (-2) S - S - (0,1,0) (1,-1,1) (-1,0,1) (1,0,-1) (-1,1,-1) (0,-1,0) S 2- S 1- S 3- S 3- S 1- S 2-
F i g u r e 2 -Diagramme de poids de deux irreps de SU(N ). (gauche) Irrep de spin-1 de SU(2), de plus haut poids (2). Il y a un unique opérateur de Cartan défini comme 2S z . (droite) Représentations à deux fermions de SU(4) (0, 1, 0) = 6. On compte trois opérateurs de Cartan et trois opérateurs d'échelle S 1-, S 2-et S 3-associé aux racines simples (2, -1, 0), (1, -2, 1) et (0, -1, 2) respectivement.

fermions :

= [1] = (1, 0, 0) = 4 = [1, 1, 1] = (0, 0, 1) = 4 = • = (0, 0, 0) = 1 = [2, 1, 1] = (1, 0, 1) = 15 = [2] = (2, 0, 0) = 10 = [1, 1] = (0, 1, 0) = 6. (18)
Un algorithme simple de combinatoire permet de calculer la dimension de l'irrep associée à un tableau de Young donné. Un deuxième un peu plus compliqué permet de calculer la décomposition du produit tensoriel de deux irreps en somme directe d'irreps et permet donc de générer toutes les règles de fusion de SU(N ).

Méthodes de réseaux de tenseurs

De nombreuses méthodes numériques ont été développées afin de simuler des systèmes quantiques. Les plus utilisées sont :

-la diagonalisation exacte, qui comme son nom l'indique consiste à diagonaliser la matrice du Hamiltonien et calculer ses vecteurs propres. Elle nécessite beaucoup de mémoire et ne peut être utilisée que sur des systèmes de très petite taille ; -le Monte Carlo quantique, qui consiste à réaliser un échantillonnage préférentiel de la fonction de partition à température finie avec une probabilité égale au poids de Boltzmann. Il permet de prendre en compte des systèmes nettement plus grands et a permis d'obtenir des résultats remarquables dans les cas où il est utilisable. Il est limité par le problème de signe, qu'on retrouve dans F i g u r e 3 -Représentation diagrammatique de tenseurs de différents rangs : scalaire or tenseur de rang 0 x sans patte, vecteur ou tenseur de rang 1 Vi avec une patte i, matrice Mij avec deux pattes i et j et tenseur de rang 3 T ijk .

les systèmes magnétiques frustrés ou les fermions en interaction et qui s'avère NP-difficile ; -le Monte Carlo variationnel, qui part d'une ansatz de la fonction d'onde avec un petit nombre de paramètres ajustables. Ces paramètres sont optimisés en explorant l'espace de Hilbert par des méthodes stochastiques ; -les méthodes de réseaux de tenseurs, qui sont l'objet de cette section.

Description tensorielle d'un état quantique

Mathématiquement, les tenseurs sont une généralisation des matrices aux plus grandes dimensions : un tenseur de rang p est un tableau de scalaires de p dimensions, c'est à dire doté de p indices. Un scalaire est donc un tenseur de rang 0, un vecteur est un tenseur de rang 1 et une matrice est un tenseur de rang 2. Chaque indice est doté d'une dimension propre, c'est à dire l'intervalle dans lequel il peut varier et la taille totale du tenseur est le produit de toutes ses dimensions. Le produit matriciel se généralise en produit tensoriel : si A i 1 ,...,ip et B ip,j 2 ,...,jq sont deux tenseurs de rang p et q, on définit le tenseur C de rang p + q -2 obtenu en contractant l'indice commun i p par :

C[i 1 , . . . , i p , j 2 , . . . , j q ] = ip A[i 1 , . . . , i p ] * B[i p , j 2 , . . . , j q ]
On peut également contracter plusieurs indices à la fois du moment que les dimensions coïncident. La trace d'une matrice se généralise comme la contraction de deux indices d'un même tenseur. La notation indicielle T ijk devient peu lisible lorsque le rang et le nombre des tenseurs augmente. On la remplace donc par un représentation graphique : un tenseur est une boîte avec des pattes, chaque patte est dotée d'une dimension et représente un indice (voir figure 3). Le produit tensoriel se dessine en reliant les pattes de deux tenseurs, la trace en reliant deux pattes d'un même tenseur (voir figure 4).

Ce formalisme tensoriel peut être utilisé pour décrire des systèmes quantiques. On considère un système décrit par p degrés de liberté discrets, typiquement des spins sur réseau, chaque degré de liberté étant décrit par un indice entier i k . La base canonique du système est la base i {↑, ↓} et tout état de l'espace de Hilbert se décompose dans cette base : F i g u r e 5 -La fonction d'onde |Ψ d'un système quantique de variables discrètes i k peut être vue comme un tenseur de grande taille. Le rang du tenseur correspond au nombre de variables et sa dimension est la dimension de l'espace produit tensoriel k d k . Ce tenseur peut être remodelé comme un produit de tenseurs locaux n'impliquant qu'une seule variable physique i k au prix de l'introduction de nouvelles variables virtuelles α k . Cette décomposition en état produit de matrices est exacte si la dimension virtuelle est suffisamment grande, sinon c'est une approximation.

|Ψ = c i 1 ,...,i N |i 1 . . . i N avec i k =↑, ↓
Le principe fondamental des méthodes de tenseurs est de réduire le nombre de coefficients à manipuler sans dénaturer la physique de la fonction d'onde sous-jacente. Pour cela, on décompose le tenseur c comme un produit de tenseurs plus petits, chaque tenseur décrivant chacun un site. Le cas unidimensionnel est présenté en figure 5 : la fonction d'onde d'un système de quatre variables est décomposée comme un produit de matrices et ses coefficients sont calculés en prenant la trace :

c i 1 i 2 i 3 i 4 = Tr A i 1 B i 2 C i 3 D i 4 . ( 19
)
Pour cela, on fait donc apparaître de nouvelles variables α 1 , α 2 , . . . α N reliant les différents sites, dites variables virtuelles, par opposition aux variables physiques i 1 . . . i N , et on note D leur dimension. Il existe de nombreux types de réseaux de tenseurs. On se limitera ici aux tenseurs de type états projetés sur des paires intriquées (PEPS) . De tels tenseurs consistent à contracter les variables virtuelles de deux tenseurs sur un état singulet, ce qui maximise l'entropie d'intrication. Cela permet également d'utiliser des tenseurs SU(N) symétriques, puisqu'on conserve la symétrie lors de la contraction de deux tenseurs. Pour D suffisamment grand, cette décomposition est exacte : on a juste réécrit le tenseur avec plus d'indices. Par contre, si D est assez petit, on peut manipuler le tenseur en mémoire tout en ayant une description approchée de c. Considérons le cas le plus simple D = 1. Chaque site est décrit pas deux scalaires a ↑ et a ↓ et chaque coefficient est donnée par c i 1 ,...,i N = a 1 (i 1 ) . . . a N (i N ). On a donc un état produit, comme en champ moyen. Un état de ce type n'est pas intriqué, chaque site est indépendant des autres. Or certains états ne peuvent être décrits par des états produit : typiquement, pour N = 2, l'état singulet (| ↑↓ -| ↓↑ )/ √ 2 ne peut pas être obtenu pour D = 1.

Si on prend D = 2, on peut définir les matrices A ↑ et A ↓ comme : La valeur de D est donc le paramètre déterminant d'un réseau de tenseurs : pour des petits D, on ne peut construire qu'un faible nombre d'états, les états faiblement intriqués, tandis que de grands D permettent de construire un plus grand nombre d'états mais font exploser la mémoire nécessaire.

A ↑ = 0 1 0 0 , A ↓ = 0 0 1 0 , B ↑ = 0 -1 0 0 , B ↓ = 0 0 1 0 (
Les réseaux de tenseurs fournissent donc une description raisonnable de la physique d'un système si et seulement si ce système est faiblement intriqué. Un état aléatoire de l'espace de Hilbert est presque sûrement fortement intriqué, il n'est donc pas accessible par cette méthode. Cependant, la physique de la matière condensée est une physique 3. Méthodes de réseaux de tenseurs 103 de basse énergie, qui s'intéresse aux propriétés de l'état fondamental et des premiers états excités d'un système. Or ces états de basse énergie sont soumis à la loi des aires.

Un Hamiltonien physique raisonnable ne peut contenir que des interactions à faible portée, plus précisément qui décroissent exponentiellement avec la distance. La loi des aires stipule que pour tout hamiltonien à faible portée, l'entropie d'intrication d'un sous-système dans l'état fondamental croît comme l'aire du sous-système : l'entropie n'est pas extensive, soit en O(L d ), mais en O(L d-1 ). Cette propriété reste valable pour les premiers états excités. Ainsi les réseaux de tenseurs échouent à décrire un état quelconque de l'espace de Hilbert, mais sont adaptés pour traiter la variété différentielles des états de basse énergie.

On utilisera exclusivement des tenseurs de type états projetés sur des paires intriquées (projected entangled pair states, PEPS). Ces tenseurs sont construits pour des réseaux de nombre de coordination z avec z variables virtuelles de dimension D et une variable physique de dimension d. Sur chaque arête du réseau, les pattes des deux tenseurs reliés sont projetées sur l'état maximalement intriqué, ce qui maximise l'entropie d'intrication.

Tenseurs SU(N ) symétriques

Le propre de cette thèse est de construire et manipuler des tenseurs SU(N ) symétriques, qui permettent de construire et décrire des phases liquides de spin. Cela est réalisé en imposant à l'espace virtuel V d'être une représentation de SU(N ). Un tenseur SU(N ) symétrique doit obéir aux règles de fusion et donc projeter le produit tensoriel de z variables sur la variable physique, c'est à dire que le tenseur est luimême un singulet de SU(N ). À chaque projecteur correspond un tenseur possible, dont les coefficients sont donc obtenus à partir des vecteurs propres de Hamiltoniens SU(N ) agissant sur z variables symétriques, en sélectionnant l'irrep voulue parmi celles autorisées par le produit tensoriel. Cette contrainte est très forte car le nombre de projecteurs possibles est très petit par rapport au nombre de ses coefficients. Le nombre de degré de liberté du système est donc très fortement réduit et on a un bon ansatz avec peu de paramètres à ajuster. On peut encore raffiner cette classification en utilisant le groupe de symétrie ponctuel du réseau. Sur le réseau carré par exemple, on va imposer au tenseur d'être à la fois dans une représentation de SU(N ) et dans une représentation de C 4v .

Algorithme de renormalisation de la matrice de coin

La contraction d'un réseau de tenseurs est un problème difficile qui demande une très grande quantité de mémoire. Celle-ci croît de manière exponentielle avec la taille du système, des algorithmes de contraction approximés sont donc nécessaires. On présente ici l'algorithme de renormalisation de la matrice de coin (corner transfer matrix renormalization group, CTMRG), tel que proposé par Orús et Vidal.

On travaille sur le réseau carré directement dans la limite thermodynamique, donc avec un réseau infini, et avec le tenseur à double couche E. L'idée est de construire un ensemble de tenseurs de bord qui simulent un environnement infini pour le tenseur E. Cet environnement est approximé par un tenseur de bord T de dimension χ×D 2 ×χ et une matrice de coin C de dimension χ × χ, où χ est la dimension de l'environnement prise aussi grande que possible. On se limite au cas d'un tenseur E invariant par rotation dans la représentation A 1 + iA 2 du groupe ponctuel C 4v . En conséquence, les quatre coins et les quatre bords sont identiques, comme présenté en figure 6 et de plus la matrice de coin est hermitienne : elle peut facilement être diagonalisée. La matrice C est obtenue par ajouts successifs de tenseur E dans le coin. À chaque étape, la nouvelle matrice obtenue est diagonalisée et seules les χ plus grandes valeurs propres sont conservées. Un nouveau tenseur T est ensuite calculé à l'aide de la matrice de diagonalisation U , comme présenté en figure 7. On itère le processus jusqu'à convergence du spectre de C en valeur absolue. Une fois la convergence atteinte, n'importe quelle observable locale (non topologique) peut être calculée en entourant les tenseurs des observables par l'environnement convergé.

Fonction d'onde AKLT SU(3)

L'état Affleck-Kennedy-Lieb-Tasaki (AKLT) a été une étape majeure dans la compréhension de la physique des chaînes de spin entier. On généralise ici la construction de cet état à deux dimensions et à SU(3) et on étudie les propriétés d'intrication de cet état. Cette section est adaptée de l'article [START_REF] Gauthé | Entanglement properties of the twodimensional SU(3) Affleck-Kennedy-Lieb-Tasaki state[END_REF].

Physique AKLT

Le théorème Lieb-Schultz-Mattis impose que dans la limite thermodynamique, toute chaîne de spin demi-entier est soit critique, avec des corrélations longue portée, soit a un fondamental dégénéré, ce qui nécessite une symétrie brisée ou en 2D un ordre topologique. En 1983, Haldane a monté que la chaîne bilinéaire de spin 1 avec des conditions au bord périodique a un unique fondamental avec un trou spectral. Comme ce fondamental est unique et que le Hamiltonien est SU(2) invariant, ce fondamental doit être un singulet et ne brise pas l'invariance par rotation. La phase associée est appelée la phase de Haldane, elle est caractérisée par son trou spectral, sa symétrie et ses excitations de bord, qui sont fractionaires avec un spin-1/2 par extrémité.

Afin de comprendre cette phase, Affleck, Kennedy, Lieb et Tasaki ont proposé en 1987 un état paradigmatique exact pour la chaîne de spin 1 qui appartient à cette phase. Le principe est de décomposer chaque spin 1 comme deux spins 1/2, intriquer deux spins 1/2 sur les liens dans un singulet et projeter deux les deux spins 1/2 d'un même site sur l'état physique de spin 1. Une représentation visuelle de cette construction est montrée en figure 8 (a). Le point fort de cette construction est que le Hamiltonien parent peut être facilement déterminé et peut être relié à la chaîne de Haldane. En effet, le Hamiltonien SU(2) symétrique au plus proche voisin le plus général pour un système de spin 1 s'écrit :

H 1BB = 1 2 i [S i • S i+1 + β(S i • S i+1 ) 2 + 2/3], (21)
et on retrouve la chaîne de Haldane pour β = 0. Pour β = 1/3, ce Hamiltonien peut être réécrit comme une somme de projecteurs sur les états de spins 2 de deux sites voisins :

H 1D SU(2) = H 1BB (β = 1/3) = i P S=2 i,i+1 . ( 22
)
Puisque le Hamiltonien H 1D SU(2) est une somme de projecteur, il est positif. Or il annule l'état AKLT car deux sites sont constitués de quatre spins 1/2 et après en avoir projeté deux dans un singulet il n'est plus possible de faire un spin 2. L'état AKLT est donc le fondamental du Hamiltonien 21 pour β = 1/3. Il a été montré que dans la limite d'un système infini, ce fondamental est unique. Cet état n'est pas le fondamental de la chaîne de Haldane, mais on peut montrer par des simulations numériques que ce dernier et l'état AKLT sont adiabatiquement connectés et appartiennent à la même phase. Avec des conditions au bord ouvertes, les spins 1/2 de chaque extrémité sont libres et le fondamental est quatre fois dégénéré. Ces excitations fractionaires sont la marque d'une phase topologique protégée par symétrie.

État AKLT SU(3)

Cette construction s'étend naturellement à 2D. On considère un réseau carré avec une variable de spin 2 sur chaque site. Chaque spin 2 est décomposé en quatre spins 1/2 et deux spins 1/2 voisins directs sont projetés sur un singulet, comme montré en figure 8 (b). Cette construction peut être vue comme un PEPS de variable physique le spin 2 de dimension d = 5 et comme variable virtuelle les spins 1/2 avec une matrice de transfert propageant un état de la gauche vers la droite. Cette matrice agit sur un état extrémal qui s'exprime en fonction des variables virtuelles, comme montré en figure 9. Cette matrice de transfert permet de calculer la longueur de corrélation du système à partir de son trou spectral ∆. Le résultat est tracé sur le panneau (a) de la figure 10. On trouve une toute petite longueur de corrélation qui est bien inférieure à la circonférence du tenseur : les résultats ne sont donc pas trop sensibles aux effets de taille finie.

Propriétés d'intrication

Le vecteur propre dominant de la matrice de transfert permet de construire le Hamiltonien d'intrication du système. On le remodèle comme une matrice frontière de dimension D Nv × D Nv qui agit sur les variables virtuelles. On peut transformer cette matrice en la matrice densité réduite ρ du demi-cylindre via une isométrie : la formule

H = E 0 + i =j J(|i -j|) S i • S j + H rest , (23)
avec E 0 = Tr(H). Les termes d'ordres supérieurs H rest sont des corrections de poids bien plus faible -seulement 5% (6%) de la norme euclidienne de H -E 0 pour N v = 8 (N v = 6) -et on s'attend à ce qu'elles jouent un rôle négligeable. Les coefficients J ij de cette décomposition sont tracés en figure 10 (b), on observe qu'ils suivent une décroissance exponentielle. Il a été montré que ce spectre est adiabatiquement connecté à la chaîne de Heisenberg 3 -3 au plus proche voisin.

Fonctions d'onde RVB SU(N )

La fonction d'onde RVB a été proposée par Anderson pour un système de spins. Elle consiste en la somme sur toutes les configurations de recouvrement de dimère au premier du système avec le même poids pour toutes, comme présenté en figure 12 (a La généralisation à SU(N ) nécessite une précaution. Il faut pouvoir projeter l'espace virtuel sur un singulet au niveau des liens, ce qui n'est pas possible pour un espace virtuel quelconque. On doit soit considérer un réseau bipartite avec conjugaison sur les sites B comme dans la section précédente ou considérer une représentation auto-conjuguée dans l'espace virtuel.

Fonction d'ondes RVB N -N

On considère dans un premier temps un système sur réseau carré de représentations alternées fondamentale-conjuguée, qui respecte ces conditions. Le tenseur pour la À nouveau le tenseur est le même sur les deux sous-réseaux et le réseau de tenseurs est donc invariant par translation, mais la fonction d'onde ne l'est pas avec une conjugaison alternée. L'algorithme de CTMRG permet de contracter le système de manière efficace et de calculer différentes observables. On trace les fonctions de corrélation en figure 13. On observe une décroissance exponentielle des fonctions de corrélation de "spins" S • S, comme attendu pour une phase qui ne brise aucune symétrie. Les fonctions de corrélation dimère-dimère sont, elles, critiques et ont une décroissance algébrique bien reconnaissable malgré une chute exponentielle qui est un artefact de l'algorithme employé.

Les exposants critiques obtenus pour ces fonctions de corrélation sont en bon accord avec les données de la littérature obtenues par d'autres méthodes : on peut être raisonnablement confiant en l'algorithme employé.

Liquide de spin topologique RVB SU(4)

On applique maintenant nos algorithmes à un système de variables dans la représentation à deux fermions 6 de SU(4) sur réseau carré. Cette représentation est auto-conjuguée et permet donc de décrire un système invariant par translation avec un PEPS, au lieu de devoir imposer une conjugaison alternée. La variable physique est donc d = 6 et l'espace virtuel est la représentation de SU(4) V = 6 ⊕ 1. Sur réseau carré, on a la règle de fusion :

⊕ • ⊗4 = 16 ⊕ . . . ( 24
)
et seize tenseurs SU(4) sont donc possibles avec cette décomposition. En les classifiant en fonction de leur représentation du groupe ponctuel C 4v , on se restreint à trois tenseurs A 1 notés T 0 , T 1 et T 2 et un tenseur A 2 . Le tenseur T 0 correspond à l'état RVB canonique au premier voisin, les autres sont une généralisation. On applique l'algorithme de CTMRG sur ces tenseurs : ils correspondent à des liquides de spin critiques. L'algorithme permet de calculer la longueur de corrélation ξ du système ainsi que son entropie d'intrication S Env , qui est celle de l'environnement défini par le CTMRG. En utilisant la relation :

S Env = S 0 + c/6 ln ξ, ( 25
)
on peut déterminer la charge centrale de la théorie conforme associée. On montre les ajustement en figure 14 : on trouve une charge centrale de 1 dans les quatre cas.

Pour le tenseur T 1 , il est nécessaire de prendre en compte une correction d'ordre supplémentaire. Chacun de ces tenseurs pris individuellement a une une symétrie supplémentaire U(1) liée à la conservation du nombre de dimères que le tenseur permet de construire. Lorsqu'on considère une combinaison linéaire de T 0 et d'un des trois autres tenseurs, cette symétrie U(1) est brisée et seule la parité du nombre de dimères est conservée, correspondant à une symétrie Z 2 . On s'attend à ce que cette brisure de symétrie en un groupe discret génère un ordre topologique dans le système, avec deux topologies différentes correspondant à des parités différentes. Un autre algorithme de réseau de tenseurs, le groupe de renormalisation des tenseurs (Tensor renormalization group, TRG) est utilisé afin de calculer les matrices modulaires du système. Ces matrices modulaires permettent de prouver la présence d'un ordre topologique car un état trivial et un état topologique ont des matrices modulaires bien différentes. On montre le résultat en figure 15 pour un tenseur combinaison linéaire de T 0 et T 2 . On a donc bien construit une phase topologique Z 2 pour un système SU(4).

Tensor Network Methods for SU(N ) Spin Systems

The study of strongly correlated electron systems is one of the most challenging target of modern condensed matter physics. Beyond the Mott transition, these systems are magnetic insulators that can be described by a spin wavefunction. This concept can be generalized by replacing the spin variable by an irreducible representation of the group SU(N ), which is relevant in some cold atomic gases experiments. This thesis aims to determine the physical properties of paradigmatic wavefunctions of condensed matter systems ruled by SU(N ) symmetry using tensor network algorithms. These methods have already proven to be efficient to tackle problems with discrete variables on a lattice. Here, the formalism of Projected Entangled Pair States (PEPS) is used to design elementary tensors with intrinsic SU(N ) symmetry that describe quantum spin liquid phases. This method is first applied to the generalization to SU(3) symmetry group and in two dimensions on the square lattice of the Affleck-Kennedy-Lieb-Tasaki (AKLT) wavefunction. It is shown to belong to the class of symmetry protected topological phases. Subsequently, the generalization to SU(N ) of resonant valence bond (RVB)-like states on the square lattice is investigated, first for staggered fundamental-conjugate representations. A system of two SU(4) fermions per site is then considered and described with generalized RVB wavefunctions. These states are shown to represent a Z 2 topological quantum spin liquid, possibly chiral, that does not break any spatial symmetry. A reasonable, short-range Hamiltonian able to stabilize this phase is proposed.

Méthodes de réseaux de tenseurs pour les systèmes de spins SU(N )

L'étude des systèmes fortement corrélés est un des champs de recherche les plus stimulant de la physique de la matière condensée. Au-delà de la transition de Mott, ces systèmes sont des isolants magnétiques qui peuvent être décrits par une fonction d'onde de spins. On peut généraliser ce concept en remplaçant la variable de spin par une représentation irréductible du groupe SU(N ), ce qui s'avère pertinent dans certaines expériences d'atomes froids. Cette thèse vise à déterminer les propriétés physiques de fonctions d'onde paradigmatiques de systèmes de matière condensée régis par la symétrie SU(N ) à l'aide d'algorithmes de réseaux de tenseurs. Ces méthodes se sont avérées remarquablement efficaces pour traiter des problèmes de variables discrètes sur réseau. On emploie ici le formalisme des états projetés sur des paires intriquées (PEPS) afin de concevoir des tenseurs avec une symétrie SU(N ) intrinsèque décrivant des phases liquides de spins quantiques. Cette méthode est d'abord appliquée à la généralisation à SU(3) sur réseau carré de la fonction d'onde Affleck-Kennedy-Lieb-Tasaki (AKLT). On montre qu'elle appartient à la classe des phases topologiques protégées par symétrie. On s'intéresse ensuite à la généralisation à SU(N ) des états de type liens de valence résonnants (RVB) sur réseau carré, dans un premier temps pour des représentations alternées fondamentale-conjuguée. On considère dans un second temps un système à deux fermions SU(4) par site que l'on décrit par des fonctions d'onde RVB généralisées. On montre que ces états correspondent à une phase liquide de spin quantique avec ordre topologique Z 2 , potentiellement chirale, qui ne brise aucune symétrie spatiale. On propose un Hamiltonien raisonnable de courte portée pouvant stabiliser cette phase.

Figure 1 . 1 -

 11 Figure 1.1 -Sketches of different magnetic phases on the square lattice. Spins inside an orange envelop form a singlet state. (a) Ferromagnetic state. (b) Antiferromagnetic state. (c) Valence bond crystal state. (d) Quantum spin liquid.

Chapter 1 .

 1 Figure 1.2 -Realization of the Hubbard model with cold atoms. Two different types of atoms are trapped in the minima of a periodic optical potential. They can tunnel to another lattice node with amplitude t and interact with each other with an effective on-site interaction U .

Figure 1 . 3 -

 13 Figure 1.3 -Quantum gas microscope images of a Bose-Einstein condensate and a Mott insulator.

Figure 1

 1 Figure 1.4 -(a) Integer quantum Hall effect in the presence of a magnetic field B showing a chiral mode on the edge. The same physics is observed in the Haldane model [Hal88] with an external field vanishing on average but locally inhomogeneous. (b) In the quantum spin Hall effect [KM05], two distinct modes with opposite spins (red and blue) and opposite directions appear at the edges of the system. The edge modes are protected by time-reversal symmetry.

Figure 2 . 1 -

 21 Figure 2.1 -Elements of the point group C4v.

Chapter 2 .

 2 Figure 2.2 -The Lie algebra g is the tangent vector space of the Lie group G at identity, where G is seen as a smooth manifold.

Figure 3 . 1 -

 31 Figure 3.1 -Diagrammatic representations of tensors of different rank: scalar or rank-0 tensor x with no leg, vector or rank-1 tensor Vi with leg i, matrix Mij with legs i and j and rank-3 tensor T ijk .

Figure 3

 3 Figure 3.5 -(a) A bi-layer E = A ⊗ A * tensor is constructed by contracting the physical index of the bra and ket tensors. (b) A bi-layer tensor is also defined for a one-site vector observable O by inserting the matrix of the observable between the bra and the ket. The extra leg corresponds to the vector index.

Figure 3 . 6 -

 36 Figure 3.6 -SU(2)-invariant tensors with a spin-1/2 physical variable and virtual space V = 2 ⊕ 1 on the square lattice. We indicate the irreps of C4v matching the SU(2) symmetry. (a) Occupation numbers {1, 3}. (b) Occupation numbers {3, 1}.

3. 4 .

 4 Figure 3.8 -CTMRG algorithm for a rotation invariant tensor E. A unique corner matrix C and a unique side tensor T are renormalized by adding E iteratively. (a) Renormalization of the corner matrix C. (b) Renormalization of the edge tensor T .

Chapter 3 .

 3 Figure 3.9 -Computation of the dimer-dimer observable expectation value (S1 • S2)(S4 • S5) .

Chapter 4 .

 4 Figure 4.1 -The SU(2) spin-1 and spin-2 AKLT spin liquids in 1D (a) and 2D (b). Virtual spin-1/2 (orange circles) are entangled into singlets (ellipses). Dashed circles represent projectors on the largest spin irrep.

Chapter 4 .

 4 SU(3) AKLT state

Figure 4

 4 Figure 4.2 -(a,b) The AKLT SU(3) wave function is defined similarly to the SU(2) case: four virtual states in the fundamental (anti-fundamental) irrep of SU(3) of dimension D = 3, are attached on even (odd) sites and projected onto the fully symmetric 15 (15) irrep. Virtual states of all neighboring sites are projected on SU(3) singlets to form a tensor network. (c) By contracting two identical site tensors on their physical indices one gets a new tensor E of dimension D 2 = 9.

Figure 4 . 3 -

 43 Figure 4.3 -The fixed-point boundary state is defined as the leading eigenvector of the transfer matrix. The latter is defined by contracting the local E tensor along a circle, leaving the left and right legs open.

Figure 4

 4 Figure 4.4 -(a) Bulk gap of an infinite AKLT SU(N ) cylinder vs circumference Nv. The extrapolated Nv → ∞ values of ξ = 1/∆ are shown on the plot. (b) Coefficients of the effective entanglement Hamiltonian (decomposed in term of Heisenberg-like operators) for SU(2) and SU(3) AKLT wavefunctions vs site separation (semi-log plot). Straight lines are fits according to an exponential behavior J(r) = J0 exp(-r/λ). Data for SU(2) are taken from reference [Cir+11].

Figure 4

 4 Figure 4.5 -Entanglement spectra on infinite cylinders of finite circumference N v . (a) SU(2) AKLT wavefunction computed with N v = 12, irreps are indexed by their spin. (b) SU(3) AKLT wavefunction computed with N v = 8, irreps are indexed according to their Young tableaux. (c) Comparison of the low-energy part of the two spectra superposed on the same graph (only trivial and adjoint irrep are kept, with new symbols for the SU(2) spectrum). The SU(2) spectrum is rescaled to match the first singlet excitation (at k = π) of the two spectra. Lines are sinusoidal fits of the edge of the 2-spinon continuum.

Figure 5

 5 Figure 5.1 -(a) Transfer matrix T = T ⊗ T . (b) |Ψ Env infinite quantum chain obtained by contracting E TN on half the 2D plane (here from top to bottom). (c) Leading eigenvector Σ of the transfer matrix.

Figure 5 . 2 -

 52 Figure 5.2 -Comparison of the results of the CTMRG algorithm for the RVB SU(N ) N -N wavefunctions on the square lattice. (a) Decay of the spectrum s of the corner transfer matrix. The value of the corner dimension χ is 400 for SU(2), 385 for SU(3), 503 for SU(4), 501 for SU(5) and 450 for SU(6), the behavior is nearly identical with the same exponent for all N . (b) Maximum correlation length ξ of the system as a function of χ. (c) Environment entanglement entropy S Env plotted as a function of the logarithm of the correlation length ξ. Linear fits enable to extract the expected central charge c = 1.

Figure 5 . 3 -

 53 Figure 5.3 -Correlation functions of the RVB SU(N ) N -N wavefunctions on the square lattice.

. 3 )

 3 Figure 5.4 -(a) nearest neighbor RVB tensor T0 with only one virtual 6 state and three holes. (b)

Figure 5

 5 Figure 5.5 -RVB states defined as a resonance state between SU(4) singlet bond configurations on a 2D square lattice. Singlets (shown as ellipses) are made from two antisymmetric 6-representations (a) canonical RVB state: equal-weight sum of all nearest neighbor dimer coverings. (b) Extended RVB state, with dimers beyond nearest neighbor.

Figure 5 . 6 -Figure 5 . 7 -Figure 5

 56575 Figure5.6 -Decay of the spectrum s of the corner transfer matrix for the four critical wavefunctions defined by T0, T1, T2 and T3. The black points are identical to those in figure5.2 (a). The plateaux correspond to SU(4) multiplets of the spectra, which are larger and more visible for A = T0. The eigenvalues do not decay all with the same exponent any more and the decay is extremely slow for T1, which requires very large χ.

Figure 5 . 9 -

 59 Figure5.9 -The dimer-dimer correlation functions are plotted for the four elementary tensors T0, T1, T2, and T3. We observe a critical behavior with algebraic decay which is cut at large distances by an exponential fall-off. The range of the algebraic behavior extends as χ grows (and would become infinite for χ → ∞).

Figure 5 . 10 -Figure 5

 5105 Figure5.10 -Environment entanglement entropy S Env plotted as a function of the logarithm of the correlation length ξ. Linear fits enable to extract the central charge c of the wavefunctions defined by T0, T2 and T3; a higher order correction a ξ -b is included for T1 . We find c 1 in all cases.

Figure 5 .

 5 Figure 5.13 -Approximation of the σ b boundary operator on a Nv = 3 ring obtained from the CTMRG edge tensor T . The D 2 degrees of freedom on each site have been reshaped as D × D.

Figure 5

 5 Figure 5.14 -Renyi entanglement entropies Sq versus the half-cylinder circumference Nv for

Figure 5 . 15 -

 515 Figure 5.15 -Entanglement spectrum of a semi-infinite cylinder of circumference Nv = 6 with respect to the wavevector along the edge for the three wavefunctions (a), (b) and (c) (see figure (5.12))for the location in the parameter space). The two topological even and odd sectors are normalized separately.

Chapter 5 .

 5 Figure 5.16 -Tensor renormalization group algorithm. The tensor is reshaped into a rank-3 tensor, with two possible geometries. These tensors are truncated with a singular value decomposition (SVD), keeping only the χ largest singular values, then the truncated tensors are re-assembled into a new one. Starting from the initial tensor E, the process is iterated until convergence.

Figure 5

 5 Figure 5.17 -(a) Modular matrix S of a critical or a trivial gapped phase. (b) Modular matrix S of a topological phase. (c) Trace of the modular matrix S obtained by TRG for A(θ, 0) versus θ.The value 2 corresponds to a topological phase while 4 implies a trivial phase. We show two sets of data obtained for two ranges of the cut parameter χ and we note that larger χ are needed when the correlation length grows.

θ

  Figure 5.19 -Zoom of figure 5.18 around θ = -0.7π. We note a change in the slope linked to the level crossing of tensors T0 and T1. The Z2 topological phase is a good ground state candidate in this region.

1.

  Physique des systèmes SU(N ) 93 F i g u r e 1 -Esquisse de différentes phases magnétiques sur le réseau carré. Les spins au sein d'une enveloppe orange forment un état singulet. (a) État ferromagnétique. (b) État antiferromagnétique. (c) État cristal de liaisons covalentes (VBC). (d) Liquide de spin quantique.

F i g u r e 4 -

 4 Représentation diagrammatique de la contraction de tenseurs. (a) produit scalaire de deux vecteurs i XiYi. (b) Produit matrice-vecteur j MijXj. (c) Produit matrice-matrice k M ik N kj . (d) Trace de la matrice M : Tr M = i Mii. (e) Contraction de tenseur plus générale : la contraction du tenseur de rang 4 d A et du tenseur de rang 3 B selon deux pattes définit un tenseur de rang 3.

  20) et l'état |Ψ αβ = Tr A α B β est exactement l'état singulet (non-normalisé). On comprend ici le rôle des variables virtuelles : elles propagent l'intrication dans le système. Pour D = 1, il n'y a pas d'intrication et on a un état produit, pour D → ∞, on peut décrire des états fortement intriqués. L'intrication est quantifiée par l'entropie d'intrication, qui mesure à quel point une région A de l'espace réel est intriquée avec son complémentaire B. On construit la matrice densité réduite ρ A de A à partir la matrice densité ρ du système tout entier en prenant la trace sur les degrés de liberté de B : ρ A = Tr B ρ. L'entropie d'intrication de A est définie comme l'entropie de von Neumann de ρ A : S A = -Tr ρ A ln ρ A .

F i g u r e 6 -

 6 On construit un environnement approximatif autour de toute région carrée qui simule un réseau infini de tenseurs E autour. Puisqu'on se limite à un réseau de tenseurs invariant par translation et rotation, il y a un seul coin C et un seul bord T . Cet environnement peut ensuite être utilisé pour calculer n'importe quelle observable.

F

  i g u r e 7 -Algorithme de CTMRG pour un tenseur E invariant par rotation. Une unique matrice de coin C et un unique tenseur de bord T sont renormalisés en ajoutant des tenseurs E de manière itérative. (a) Renormalisation de la matrice de coin C. (b) Renormalisation du tenseur de bord T .

  u r e 10 -(a) Trou spectral d'un cylindre AKLT SU(N ) en fonction de la circonférence Nv. On précise les valeurs extrapolées Nv → ∞ de la longueur de corrélation ξ = 1/∆. (b) Coefficients du Hamiltonien d'intrication effectif (décomposé sur la base des opérateurs de type Heisenberg) pour les fonctions d'ondes AKLT SU(2) et SU(3) en fonction de la distance. Les lignes droites sont des ajustements selon une loi exponentielle J(r) = J0 exp(-r/λ). Les données pour SU(2) proviennent de la référence [Cir+11].

F

  ρ = U † (Σ b ) 2 U . On définit alors le Hamiltonien d'intrication agissant sur le bord de l'espace virtuel par (Σ b ) 2 = exp(-H) avec une température arbitraire fixée à 1. Li et Haldane ont émis la conjecture que le spectre de H, ou spectre d'intrication, est en parfaite correspondance avec les modes de bord physique du système. On trace ce spectre pour les fonctions d'onde SU(2) et SU(3) en figure 11. On observe qu'ils sont similaires, avec un fondamental singulet à k = 0 et des premières excitations sinusoïdales caractéristiques d'un continuum à deux spinons. Afin de mieux comprendre ce spectre, on décompose le Hamiltonien d'intrication sur la base des opérateurs de Heisenberg S i • S j pour toutes les distances i -j selon 108 i g u r e 11 -Spectres d'intrication de cylindres infinis de circonférence finie N v . (a) Fonction d'onde AKLT SU(2) calculée pour N v = 12, les irreps sont indexées par leur spin. (b) Fonction d'onde AKLT SU(3) calculée pour N v = 8, les irreps sont indexées par leurs tableaux de Young. (c) Comparaison des secteurs de basses énergies des deux spectres superposés sur le même graphique. Seules les irreps triviales et adjointes sont conservées en utilisant de nouveaux symboles pour SU(2). Le spectre SU(2) est ré-échelonné de manière à faire coïncider la première excitation singulet des deux spectres à k = π. Les lignes sont des ajustements sinusoïdaux de la limite basse du continuum à deux spinons.

F i g u r e 12 -

 12 États RVB définis comme un état résonnant de recouvrement de dimères sur le réseau carré. (a) État RVB canonique : somme même poids de toutes les configurations de recouvrement de dimères. (b) État RVB généralisé, avec des dimères à plus longues distances et des configurations de poids différents.

  ). Cet état s'écrit naturellement en terme de tenseurs, avec une variable physique d et un espace virtuel V = d ⊕ • de dimension D = d + 1.

  α = 1.57 SU(5) α = 1.66 SU(6) α = 1.72 F i g u r e 13 -Fonctions de corrélations des fonctions d'onde RVB SU(N ) N -N sur réseau carré.(gauche) Longueurs de corrélations de "spins" S • S. On observe une décroissance exponentielle avec une longueur de corrélation de "spin" ξS. (droite) Fonctions de corrélations dimère-dimère. On observe une décroissance algébrique avec un exposant critique α jusqu'à une décroissance exponentielle induite par le CTMRG à grande distance. fonction d'onde RVB N -N au premier voisin s'écrit simplement :∀ i, 1 ≤ i ≤ d, A i [i, D, D, D] = 1 A i [D, i, D, D] = 1 A i [D, D, i, D] = 1 A i [D, D, D, i] = 1.

F

  i g u r e 14 -Entropie d'intrication de l'environnement S Env tracé en fonction du logarithme de la longueur de corrélation ξ. Les ajustements linéaires permettent de déterminer la charge centrale c des fonctions d'onde définies par les tenseurs T0, T2 et T3 ; une correction d'ordre supérieur a ξ -b est prise en compte pour T1 . On trouve c 1 dans tous les cas.

F i g u r e 15 -

 15 (a) Matrice modulaire S d'un état critique ou trivial avec un trou spectral. (b)Matrice modulaire S caractéristique d'une phase topologique. (c) Trace de la matrice modulaire S obtenue pour un tenseur A = cos θT2 + sin θT0. La valeur 2 correspond à une phase topologique alors que la valeur 4 implique une phase triviale. On montre deux jeux de données différents correspondant à différentes valeurs du paramètre de coupure χ. Lorsque la longueur de corrélation du système augmente, donc près des points critiques θ = 0 et θ = π, de plus grands χ sont nécessaires pour détecter la phase topologique.

Table 2 .

 2 1 -Character table for the point group C4v.

  state FM. This state is the most symmetric and any pair of sites in the lattice is projected in the most symmetric irrep 20. In this state, H(θ) 20 = cos(θ) + sin θ 4 therefore the energy per site is Plaquette states P. On the square lattice we can also construct states where four sites in a square form a singlet and cover the lattice with these plaquettes. This state spontaneously breaks the translation invariance of the lattice. To construct it, we have to consider the projector ⊗4 → •. Three independent singlets can be made on the square, they can easily be obtained by diagonalizing the quadratic Casimir operator on four sites. The point group C 4v naturally acts on this space and we can decompose the states in term of its irreducible representations: two singlets have A 1 symmetry and the last one has B 2 symmetry. Although those state are symmetric, we must be careful that the bond operators are not. Each site belongs to one vertical bond, one horizontal bond and two uncorrelated bonds leading to other plaquettes. Exact diagonalization gives the energies per site Figure5.18 -Energies of different wavefunctions under the Hamiltonian H(θ). The energy obtained by tensor network computations are taken after extrapolation χ → ∞. The red points are obtained by optimization of the four parameters ai for a given θ. We remark two main regions of crossing for θ ≈ π/4 and θ ≈ -7π/10 (see zoom in figure5.19).

	5.3. Host Hamiltonian								77
									T 0
	4								T 1
	2								T 2 T 3
	-4 -2 0 θ								FM U D P A 1 P B 2
	-π -3π/4 -π/2 -π/4	0 θ	π/4		π/2	3π/4	π
				e F M = 2 cos θ +	1 2	sin θ	(5.10)
	Uncorrelated state U. In this state, each site is totally uncorrelated from its
	neighbors, which means H(θ) unc = Tr(H)/36 = 5 12 sin θ. Hence the energy per site
	is				e U =	5 6	sin θ	(5.11)
	Dimerized state D. In a fully dimerized state, each site belongs to one singlet of
	energy H(θ) 1 = -5 cos(θ)+ 25 4 sin θ. All the other neighbors are totally uncorrelated.
	Every dimer covering states have the same energy per site
				e D = -	5 2	cos θ +	15 4	sin θ	(5.12)
	e A 1 P = -	3 2	cos θ +	29 12	sin θ ±	√ 8 2	85 -13 cos(2θ) -76 sin(2θ)	(5.13)
				e B 2 P = -2 cos θ +	19 6	sin θ	(5.14)

  décrivant l'état du spin k. L'état du système est donc fixé par l'objet c i 1 ,...,i N , qui est vu comme un tenseur de rang N et de dimension 2 N . Construire numérique la fonction d'onde Ψ se comprend donc comme déterminer les coefficients du tenseur c. La difficulté du problème est la taille du système : la taille de c est la dimension de l'espace de Hilbert du problème, qui croit exponentiellement avec la taille du système. Pour un carré de 8 × 8 = 64 spins 1/2, le simple stockage du tenseur requiert plus de 10 8 disques durs de 1 TiB : stocker tous ces coefficients est illusoire.

They may however have a hidden order, see section 1.4.2.

Some internal symmetries of the tensor may also be used, including the action of the permutation group S4 on the virtual legs.

Difficulties may arise when a given weight appears more than once in the irrep d. A gauge freedom appears in the definition of the tensor and a more subtle approach is required to lift it.

The construction itself does not require the representation to be irreducible, but a reducible one makes little sens physically.

Private communication for the values beyond the boundary of figure 31.

K. Totsuka, private communication. Similarly, a non-degenerate featureless gapped ground state is not possible on the square lattice with SU(4) fundamental and antifundamental representation on the two sublattices[START_REF] Jian | Lieb-Schultz-Mattis theorem and its generalizations from the perspective of the symmetry-protected topological phase[END_REF].

†. French summary, as required by the doctoral school rules.

Résumé en français F i g u r e 8 -Fonctions d'onde AKLT SU(2) pour une chaînes de spin 1 (a) et un réseau carré de spin 2 (b). Les spin-1/2 virtuels (cercles oranges) sont intriqués au sein de singulets (ellipses). Les cercles en pointillés représentent les projecteurs sur l'irrep de plus haut spin.

F i g u r e 9 -Le point fixe de l'extrémité est défini comme le vecteur propre dominant de la matrice de transfert. Celle-ci est obtenue en contractant le tenseur local E sur un cercle, en laissant les pattes droites et gauches ouvertes.

une dimension virtuelle D = 2. Le tenseur est obtenu en prenant la combinaison totalement symétrique des quatre spins 1/2.

Cette construction peut encore se généraliser à SU(3), à condition de scinder le réseau en deux sous-réseaux A et B conjugués l'un de l'autre : chaque spin 1/2 est remplacé par un quark sur les sites A et par un antiquark sur les sites B. Sur les liens, les variables virtuelles sont projetées sur un méson, qui est un singulet de SU(3). La variable physique des sites A est la représentation la plus symétrique obtenue à partir de quatre quarks, soit la représentation 15, et c'est sa conjuguée 15 sur les sites B. Le tenseur de départ A est à nouveau obtenu en prenant la combinaison la plus symétrique des quarks pour chaque état de couleur donnée. Le résultat est un réseau de tenseurs de dimension physique d = 15 et de dimension virtuelle D = 3. Le tenseur de base est le même sur les deux sous-réseaux : ainsi le réseau de tenseurs est invariant par translation, mais l'état physique ne l'est pas puisque les deux sous-réseaux sont conjugués.

On contracte ce tenseur A avec son conjugué selon la patte physique et on groupe les pattes virtuelles pour obtenir le tenseur élémentaire E de dimension D 2 = 9 pour chacune des quatre pattes. C'est avec ce tenseur que les calculs sont réalisés. On construit un cylindre infini en imposant des conditions au bord périodiques selon une direction avec une circonférence N v . Chaque rangée du cylindre peut être vue comme