N Crampe

K Mallick
email: kirone.mallick@cea.fr

E Ragoucy
email: eric.ragoucy@lapth.cnrs.fr

M Vanicat

Inhomogeneous discrete-time exclusion processes

We study discrete time Markov processes with periodic or open boundary conditions and with inhomogeneous rates in the bulk. The Markov matrices are given by the inhomogeneous transfer matrices introduced previously to prove the integrability of quantum spin chains. We show that these processes have a simple graphical interpretation and correspond to a sequential update. We compute their stationary state using a matrix ansatz and express their normalization factors as Schur polynomials. A connection between Bethe roots and Lee-Yang zeros is also pointed out.

Introduction

The study of systems far from equilibrium has greatly benefited from exact results obtained for interacting stochastic particle systems [START_REF] Bertini | Macroscopic Fluctuation Theory[END_REF][START_REF] Katz | Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors[END_REF][START_REF] Krapivsky | A Kinetic View of Statistical Physics[END_REF][START_REF] Liggett | Stochastic Models of Interacting Systems: Contact, Voter and Exclusion Processes[END_REF][START_REF] Schmittmann | Statistical mechanics of driven diffusive systems[END_REF][START_REF] Spohn | Large Scale Dynamics of Interacting Particles[END_REF] and amongst these, the asymmetric simple exclusion process (ASEP) has become a paradigm. In one-dimension, the ASEP can be solved exactly and many physical quantities (such as its phase diagram, N-point correlations, density and current fluctuations etc...) can be calculated analytically [START_REF] Chou | Non-equilibrium statistical mechanics: From a paradigmatic model to biological transport[END_REF][START_REF] Derrida | An exactly soluble non-equilibrium system: the asymmetric simple exclusion process[END_REF][START_REF] Derrida | Non-equilibrium steady states: fluctuations and large deviations of the density and of the current[END_REF][START_REF] Derrida | Microscopic versus macroscopic approaches to non-equilibrium systems[END_REF][START_REF] Schütz | Exactly Solvable Models for Many-Body Systems Far From Equilibrium[END_REF]. Hence, the ASEP (as well as the closely related Kardar-Parisi-Zhang equation) belongs to the class of integrable models: this exceptional property explains its solvability, its intricate algebraic structure and its relations to many different fields of mathematics (representation theory, random matrices, combinatorics...). A new field, stochastic integrability, at the crossroads of probability and algebra, is indeed emerging, in which the ASEP plays a distinguished role [START_REF] Corwin | The Kardar-Parisi-Zhang equation and universality class[END_REF].

Integrability is a fragile property that is easily lost when one tries to generalize or deform a model: a system has to respect stringent mathematical criteria in order to be integrable (basically, the Yang-Baxter relation in the bulk and reflection identities at the boundaries [START_REF] Faddeev | How Algebraic Bethe Ansatz works for integrable model[END_REF][START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF]). Our aim, in the present work, is to study integrable, one species exclusion processes with inhomogeneous and non local transition rates. We shall consider both cases of periodic (section 1) and open boundary conditions (section 2): using the transfer matrix formalism, we define discrete time stochastic processes, by stating explicitly and representing graphically the updating rules for the dynamics. Then, we determine the stationary states of these models exactly. When the inhomogeneities z i that enter the transfer matrix are set to 1, the stationary measure coincides with the (periodic or open) TASEP one.

For the open system, we construct a matrix product Ansatz [START_REF] Derrida | Exact solution of a 1d asymmetric exclusion model using a matrix formulation[END_REF][START_REF] Blythe | Nonequilibrium steady states of matrix-product form: a solver's guide[END_REF] using the Zamolodchikov-Faddeev [START_REF] Zamolodchikov | Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models[END_REF] and the Ghoshal-Zamolodchikov [START_REF] Ghoshal | Boundary S-matrix and boundary state in twodimensional integrable quantum field theory[END_REF] relations, thanks to the formalism developed in [START_REF] Sasamoto | Stationary states of integrable systems in matrix product form[END_REF][START_REF] Crampe | Integrable approach to simple exclusion processes with boundaries. Review and progress[END_REF][START_REF] Crampe | Open two-species exclusion processes with integrable boundaries[END_REF]. Knowing the invariant measure, we compute analytically some physical observables (such as the local density) that will be expressed in terms of Schur polynomials. Thus, our work interrelates integrability, Matrix Ansatz and combinatorial properties of interacting particle systems and contributes to extend the class of exactly solvable models in non-equilibrium statistical mechanics. We also find an unexpected relation between the Bethe roots (solutions of the Bethe equations found in [START_REF] Crampe | Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries[END_REF]) and the Lee-Yang zeros of the normalization function (see section 2.5).

Markov process on the periodic lattice

This section is devoted to the construction of a discrete time Markov process on the periodic lattice (the site L + 1 is identified with the site 1). After discussing the general framework in section 1.1, we construct in section 1.2 a one-parameter family of commuting operators t(x|z) (using the integrability of the model). Using these operators t(x|z), we define a discrete time Markov process that can be interpreted graphically (see section 1.3). The stochastic updating rules corresponding to this process are given explicitly in section 1.4. Finally, the stationary state of the process and some of its properties are studied in section 1.5.

General framework: discrete time Markov processes

We consider a one-dimensional lattice of size L. Each site of the lattice can either be empty or carry one particle; a configuration of the system can be denoted by a L-tuple C = (τ 1 , τ 2 , . . . , τ L ) where we set τ j = 0 if the site j is empty and τ j = 1 if it is occupied. The configuration space is given by C

2 ⊗L = C 2 ⊗ • • • ⊗ C 2 L times
where the j th component C 2 of the tensor space represents the j th site of the lattice. A configuration of the system thus corresponds to a vector The dynamics is stochastic: the system can evolve from a configuration C ′ to another configuration C with the probability M (C, C ′ ). The Markov property is satisfied: the probability M (C, C ′ ) depends only on the two configurations C and C ′ and not on the previous history of the system. Let P t (C) be the probability for the system to be in the configuration C at time t. Its (discrete) time evolution is given by

P t+1 (C) = C ′ =C M (C, C ′ )P t (C ′ ) +   1 - C ′ =C M (C ′ , C)   P t (C) = C ′ M (C, C ′ )P t (C ′ ) , (1.1)
where the diagonal term

M (C, C) = 1 -C ′ =C M (C ′ , C
), is the probability of remaining in configuration C between t and t + 1. This equation can be rewritten in vector-form as follows: defining

|P t =       
P t ( (0, . . . , 0, 0, 0) ) P t ( (0, . . . , 0, 0, 1) ) P t ( (0, . . . , 0, 1, 0) ) . . .

P t ( (1, . . . , 1, 1, 1) )        = τ 1 ,...,τ L ∈{0,1} P t ( (τ 1 , . . . , τ L ) ) |τ 1 . . . τ L (1.2)
the master equation (1.1) becomes

|P t+1 = M |P t . (1.3)
A fundamental quantity is the stationary probability distribution of the process, i.e., a vector |S such that M |S = |S (the existence of this vector is guaranteed because the entries of each column of M sum to 1). If the process is not reversible, that is if we do not have the detailed balance property M (C ′ , C)S(C) = M (C, C ′ )S(C ′ ), the task of finding an explicit expression for |S can be very hard. However in some cases, in particular when the Markov matrix is integrable, this stationary distribution can be calculated exactly.

Inhomogeneous transfer matrix

We now use the theory of integrable systems to construct a family of operators, t(x|z), called transfer matrices, that will allow us to define stochastic updating rules. In this section, we discuss only about the integrability of the transfer matrix. Its Markov property will be discussed at the end of subsection 1.3. The building block of the transfer matrix is the R-matrix: it acts on two sites and depends on a parameter x, called the spectral parameter. Here, we will consider the following R-matrix

R(x) =     1 0 0 0 0 0 x 0 0 1 1 -x 0 0 0 0 1     . (1.4)
It acts in the space C 2 ⊗ C 2 , where we choose the basis |00 , |01 , |10 , |11 which respectively correspond to the following lattice configurations: (0, 0), (0, 1), (1, 0), [START_REF] Baseilhac | Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory[END_REF][START_REF] Baseilhac | Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory[END_REF]. The R-matrix satisfies the Yang-Baxter equation:

R 12 x 1 x 2 R 13 x 1 x 3 R 23 x 2 x 3 = R 23 x 2 x 3 R 13 x 1 x 3 R 12 x 1 x 2 , (1.5)
where the subscripts denote on which tensor space the matrix R has a non trivial action: for instance

R 12 = R ⊗ 1, R 23 = 1 ⊗ R etc.
.. The R-matrix obeys the regularity relation:

R(1) = P (1.6)
where P is the permutation operator (P v ⊗ w = w ⊗ v). It also satisfies the unitarity relation:

R 12 (x) R 21 1 x = 1, (1.7)
where R 21 (x) = P 12 R 12 (x)P 12 . The R-matrix (1.4) is usually associated with the continuous time TASEP [START_REF] Crampe | Integrable approach to simple exclusion processes with boundaries. Review and progress[END_REF].

From the R-matrix, we build an inhomogeneous periodic transfer matrix as follows

t(x|z) = tr 0 R 0,L x z L R 0,L-1 x z L-1 . . . R 0,1 x z 1 . (1.8)
The parameter x is the spectral parameter and the parameters z 1 , . . . , z L are called inhomogeneities. The homogeneous case (i.e. z i = 1) was studied in [START_REF] Golinelli | Family of Commuting Operators for the Totally Asymmetric Exclusion Process[END_REF].

The main feature of the transfer matrix is that, due to the Yang-Baxter equation (1.5), it commutes for different values of the spectral parameter. This is the key to integrability [START_REF] Faddeev | How Algebraic Bethe Ansatz works for integrable model[END_REF]:

[t(x|z), t(y|z)] = 0 .

We now introduce the following operator

M (x|z) = t(x|z)t(z 1 |z) -1 = tr 0 R 0,L x z L R 0,L-1 x z L-1 . . . R 0,1 x z 1 R 2,1 ( z 2 z 1 ) . . . R L,1 ( z L z 1
). (1.9)

Note that we have normalized t(x|z) using t(z 1 |z), but a different choice t(z j |z), j = 2, 3, ..., L leads to a similar rotated matrix M (x|z). Obviously M (x|z) commutes with t(y|z) and has the same eigenvectors. We choose below to use the "normalized matrix" M (x|z) instead of t(x|z) because it allows one to construct easily local Hamiltonians (see e.g. remark below).

Remark:

The operator M (x|z) is related to the continuous time TASEP. Taking the homogeneous limit z i → 1, we are left with M (x) = t(x)t(1) -1 , where t(x) = tr 0 (R 0,L (x) . . . R 0,1 (x)). By standard computation, we obtain

M ′ (1) = L k=1 P k,k+1 .R ′ k,k+1 (1), (1.10)
with the convention L + 1 ≡ 1. The operator

-P.R ′ (1) =     0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0     (1.11)
is the local jump operator of the continuous time TASEP. Hence -M ′ (1) is the Markov matrix of the continuous time TASEP. Therefore, all the results obtained below for the discrete time process will be valid, when z i → 1, for the continuous time TASEP.

Graphical representation

The R-matrix and the transfer matrices defined above have a useful graphical illustration that we now explain.

R-matrix.

The action of the R-matrix R ij (x/y) can be represented by the vertices given in fig. 1. Each vertex corresponds to a matrix element of the R-matrix. A dashed incoming line (according to the arrows direction) denotes the vector |0 (or equivalently an empty site), whereas a continuous thick line denotes the vector |1 (or equivalently an occupied site). In a similar way, the out-going lines (after the crossing point) represent the states 0| and 1| on which we are contracting the matrix R. The missing vertices in fig. 1 correspond to vanishing matrix elements of R. Remark that the number of particles is conserved: in every non vanishing vertex, the number of incoming continuous thick lines is equal to the number of out-going continuous thick lines.

00|R ij ( x y )|00 10|R ij ( x y )|10 01|R ij ( x y )|10 10|R ij ( x y )|01 11|R ij ( x y )|11
With this graphical interpretation, we will be able to compute efficiently a matrix element of a product of R matrices acting in different components of the tensor product.

Transfer matrix. From expression (1.9) we can deduce a graphical representation for M (x|z). The starting point is the lattice illustrated in fig. 2 that one has to fill according to the matrix element one wants to compute. Instead of explaining it in full generality, we take below a concrete example.

0 1 2 3 i . . . . . . L -1 L x z 1 z 2 z 3 z i z L-1 z L Figure 2:
Graphical representation for M (x|z). Since the lattice is periodic, the two double slash should be considered as linked.

As an example we take L = 4 and use the graphical interpretation to compute the transition rate between the initial configuration (1, 1, 1, 0) and the final configuration (1, 1, 0, 1). The initial (resp. final) configuration fixes the form of the incoming (resp. out-going) external lines (dashed or thick) as in fig. 3. Then, we look for drawings of the form given in fig. 3 where the remaining thin lines have to be replaced by thick or dashed lines in such a way that the weights (as given in fig. 1) of all the vertices do not vanish. The total weight of a given possible drawing is then the product of all these weights. It is easy to see that there are only two possible drawings, given in fig. 4 together with their corresponding weights. Finally the weight of 1101|M (x|z)|1110 is the sum of the weights of the possible drawings.

z 2 z 3 z 4 2 3 4 0 x 1 z 1
Using this graphical interpretation, we are able to compute all the possible rates between any two configurations. We remark in particular that the number of particles is conserved (as mentioned previously each non-vanishing vertex preserves the number of particles). Therefore, we restrict ourselves to a given sector with a fixed number of particles. We can also show that all the rates starting from a given configuration (with at least one particle) sum to one which proves that M (x|z) can be used as a Markov matrix in the master equation (1.3). One has to impose also

0 ≤ z i z 1 ≤ 1 and 0 ≤ x z i ≤ 1 , i = 1, 2, ...L, (1.12)
so that the probabilities are positive and less than 1. The sector with no particle is special: its dimension is 1 and the matrix M (x|z) is reduced to the scalar 1 + L i=1 (1 -x/z i ). Therefore, it cannot be considered as a Markov matrix in the empty sector. From now, we consider only the cases with at least one particle.

z 2 z 3 z 4 2 3 4 0 x 1 z 1 (1 -x z 3 ) z 3 z 1 z 2 z 3 z 4 2 3 4 0 x 1 z 1 (1 -z 3 z 1 ) x z 1

Stochastic updating rules

The Markov process given by M (x|z) can be interpreted as a discrete time process with sequential update. The configuration at the time t + 1 is obtained from the one at time t by the following dynamics:

• Particle update: starting from right to left (i.e. from the site L to the site L -1 and so on), a particle at the site i jumps to the right on the neighboring site with a probability 1 -z i /z 1 provided this site is empty. The particle does not jump with the probability z i /z 1 .

We remind that we are on a periodic lattice, so that the site on the right of the site L is the site 1. Note that a particle located on site 1 does not move.

• Hole update: once the particle update is done, one performs the hole update. Contrarily to the particle update, we do not necessarily start and finish at the sites 1 or L. Let r be the site number of a particle (we recall that we restrict ourselves to the case with at least one particle). Starting from the site r, we go from left to right up to the site r -1, using periodicity and knowing that the site on the right of the site L is the site 1. A hole at the site i jumps to the left on the neighboring site with the probability 1 -x/z i-1 provided the site is occupied. The hole may stay at site i with probability x/z i-1 . By convention, we set

z 0 = z L .
As mentioned previously, we would like to emphasize that, due to the inhomogeneities in the transfer matrix, the rates depend on the site where the particle or the hole is situated. All the probabilities are positive and less than 1 thanks to (1.12). Let us also mention that for the homogeneous case, the update simplifies. Indeed, the first step becomes trivial: the particles do not move. These rules are illustrated in figure 5 for a chain with 4 sites in the configuration (1, 1, 1, 0) at time t. We deduce from this figure the different possible rates between the configurations which correspond to the entries of M (x|z). One gets

z 3 z 1 z 2 z 1 (1 -z 3 z 1 ) (1 -z 2 z 1 )(1 -z 3 z 1 ) x z 3 1 -x z 3 x z 2 1 -x z 2 x z 1 1 -x z 1
M (x|z) (1, 1, 0, 1), (1, 1, 1, 0) = 1101|M (x|z)|1110 = 1 - x z 3 z 3 z 1 + 1 - z 3 z 1 x z 1 (1.13)
in accordance with the calculation done in section 1.3 (see also figure 4).

Justification of the sequential up-date. The sequential update described above can be easily identified when considering the Markov matrix M (x|z) at the special point x = z j . Indeed,

we write M (z j |z) = t(z j |z)C C -1 t(z 1 |z) -1
where C is the cyclic permutation. Then, from the explicit expressions

t(z j |z) = R j,j-1 ( z j z j-1 ) • • • R j,1 ( z j z 1 ) R j,L ( z j z L ) • • • R j,j+1 ( z j z j+1 ) (1.14) t(z 1 |z) -1 = R 2,1 ( z 2 z 1 ) R 3,1 ( z 3 z 1 ) • • • R L,1 ( z L z 1 ) (1.15)
it is easy to see that C -1 t(z 1 |z) -1 corresponds to the particle update, while t(z j |z)C corresponds to the hole update at x = z j . Let us remark that such a simple sequential update is specific to the totally asymmetric exclusion process. For the partially asymmetric case, it would be much more involved.

Stationary State Properties

We now investigate some properties of the stationary state of the Markov chain defined by M (x|z). We construct the steady state in a formal manner, then give an elementary expression for the stationary measure. Finally, we obtain some simple formulas for steady-state correlations.

Construction of the steady state

The building block of the stationary state is the following vector

v(z) = z 1 . (1.16) It satisfies the Zamolodchikov-Faddeev relation [51] R 12 (z 1 /z 2 )v 1 (z 1 )v 2 (z 2 ) = v 1 (z 1 )v 2 (z 2 ) .
(1.17)

The stationary state of the process M (x|z) is constructed from

|S = v 1 (z 1 )v 2 (z 2 ) . . . v L (z L ). (1.18)
Recall that the subscripts denote which component of the tensor space the vector v belongs to.

To be more precise, since the process conserves the number of particles, there are L independent sectors (as mentioned previously we do not consider the empty sector), each corresponding to a given number m of particles in the system. The stationary state is hence degenerate: there is one stationary state for each sector. Moreover, the exact normalization Z (m) of the stationary state depends on the sector we are considering. Then, each stationary state is given by the components of |S corresponding to the sector and correctly normalized.

The following calculation justifies that |S given in (1.18) is the stationary state of the system. Indeed, for i = 1, 2, ..., L we have

t(z i |z)|S = R i,i-1 ( z i z i-1 ) . . . R i,1 ( z i z 1 )R i,L ( z i z L ) . . . R i,i+1 ( z i z i+1 )v 1 (z 1 ) . . . v L (z L ) = v 1 (z 1 )v 2 (z 2 ) . . . v L (z L ) = |S .
The first equality is obtained using the regularity property of the R-matrix (1.6) whereas the second equality is obtained using L -1 times the property (1.17). Clearly, t(x|z) is a polynomial of degree less or equal to L in x. It is possible to show (using the graphical interpretation for instance) that in the sectors where there is at least one particle, the degree of t(x|z) is in fact at most equal to L -1. Hence we can deduce through interpolation arguments that in these sectors, the stationary state is given by the corresponding components of |S . Note that when we take the homogeneous limit z i → 1 we recover the uniform stationary distribution of the continuous time TASEP.

Calculation of steady-state averages

From the knowledge of the stationary distribution, we can calculate various physical quantities. We shall see that some observables can be expressed as symmetric polynomials in the inhomogeneity parameters z 1 , . . . , z L .

Probability weight of a given configuration. The weight of the configuration (τ 1 , . . . , τ L ) is readily obtained using (1.18):

S(τ 1 , . . . , τ L ) = L i=1 (δ 1,τ i + z i δ 0,τ i ) . (1.19)
Normalization factor. In the sector with m particles, the normalization factor of the stationary state is obtained by summing the weights of all the configurations with m particles

Z (m) = I⊂{1,...,L} |I|=L-m i∈I z i = e L-m (z 1 , . . . , z L ) , (1.20)
where e L-m is the elementary symmetric homogeneous polynomial of degree L -m. The normalization factor can be written as a Schur polynomial:

Z (m) = s A 1 L-m (z 1 , . . . , z L ), where 1 L-m = (1, ..., 1 L-m , 0, ..., 0 m ). (1.21)
We remind the definition of the Schur polynomial (of type A) associated with a partition λ = (λ 1 , . . . , λ L ) with

λ 1 ≥ • • • ≥ λ L ≥ 0: s A λ (z 1 , . . . , z L ) = det (z j ) L-i+λ i i,j det z L-i j i,j . (1.22)
This expression of the partition function in terms of the Schur polynomial allows us to relate the value of Z (m) in the homogeneous limit (z i → 1) with the dimension of the representation π A (λ) of sl(L) labeled by the Young tableau [λ]. Indeed, using the Weyl character formula (see e.g. [START_REF] Varadarajan | Lie Groups, Lie Algebras, and Their Representations[END_REF][START_REF] Frappat | Dictionary on Lie algebras and superalgebras[END_REF] for a review), we obtain

s A λ (1, . . . , 1) = 1≤j≤i≤L λ j -λ i + i -j i -j = dim(π A (λ)) . (1.23)
In particular, we have

Z (m) z 1 =•••=z L =1 = dim(π A (1 L-m )) = L m (1.24)
in accordance with a direct computation starting from (1.20).

Density. In the sector with m particles, the particle density at site i is obtained by summing the weights of all the configurations with m particles, one of them being at site i

n i = e L-m (z 1 , . . . , z i-1 , z i+1 , . . . , z L ) e L-m (z 1 , . . . , z L ) . (1.25)
We can show that

L i=1 n i = m as expected.
Higher correlation functions. The higher correlation functions take also a very simple form. The correlations between the sites

i 1 < i 2 < • • • < i ℓ is given by n i 1 n i 2 . . . n i ℓ = e L-m (z 1 , . . . , z i 1 -1 , z i 1 +1 , . . . , z i 2 -1 , z i 2 +1 , . . . , z i ℓ -1 , z i ℓ +1 , . . . , z L ) e L-m (z 1 , . . . , z L ) . (1.26)
For ℓ > m, the correlation functions vanish as expected since the number of particles is m and the correlation functions for more than m particles has no meaning.

Markov process on the open lattice with boundaries

We now consider a model on a finite lattice with two boundaries that play the role of reservoirs from which particles can enter the system or exit from it. In order to construct an integrable system, one needs to solve simultaneously the Yang-Baxter equation in the bulk, leading to the R-matrix, and Sklyanin's reflection equations at the boundaries [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF], which define two boundary operators K and K. These operators allow us to construct an inhomogeneous transfer matrix in section 2.1, that we illustrate graphically in section 2.2 and interpret as a stochastic process in section 2.3. The stationary state of this process is computed using the matrix product technique [START_REF] Derrida | Exact solution of a 1d asymmetric exclusion model using a matrix formulation[END_REF][START_REF] Blythe | Nonequilibrium steady states of matrix-product form: a solver's guide[END_REF] that allows us to calculate stationary observables (section 2.4). Using the explicit expression of the normalisation function, we relate Bethe roots and Lee-Yang zeros in section 2.5.

Inhomogeneous transfer matrix for the open system

The building blocks of the transfer matrices for the open system are the R matrix defined in section 1.4 and the following boundary matrices:

K(x) = (a+x)x xa+1 0 1-x 2 xa+1 1 and K(x) = 1 xb+1 1 xb+1 0 xb xb+1
.

(2.1)

These matrices act in the space C 2 and we choose the basis |0 , |1 corresponding to the lattice configurations (0), (1) respectively. These K-matrices are associated with the continuous time TASEP with boundaries as will be explained below.

The matrix K obeys the reflection equation [45]

R 12 ( x 1 x 2 ) K 1 (x 1 ) R 21 (x 1 x 2 ) K 2 (x 2 ) = K 2 (x 2 ) R 12 (x 1 x 2 ) K 1 (x 1 ) R 21 ( x 1 x 2 ) . (2.2)
The matrix K is related to an auxiliary K matrix, defined as K(x) = tr 0 K 0 ( 1 x )R 01 ( 1 x 2 )P 01 , which obeys the reflection equation (2.2) associated with the matrix R12

(x) = R 21 (1/x): R12 ( x 1 x 2 ) K1 (x 1 ) R21 (x 1 x 2 ) K2 (x 2 ) = K2 (x 2 ) R12 (x 1 x 2 ) K1 (x 1 ) R21 ( x 1 x 2
) .

(2.

3)

The explicit form of K is

K(x) =     1 (x 2 -1) x(x + b) 0 xb + 1 x(x + b)     . (2.4)
We can now construct the inhomogeneous open transfer matrix as follows:

t(x|z) = tr 0 K 0 (x) R 0,L ( x z L ) . . . R 0,1 ( x z 1 ) K 0 (x) R 1,0 (xz 1 ) . . . R L,0 (xz L ) . (2.5)
It is shown in [START_REF] Crampe | Integrable approach to simple exclusion processes with boundaries. Review and progress[END_REF] that these transfer matrices commute for different values of the spectral parameter: [t(x|z), t(y|z)] = 0. Note that the derivation of this property cannot be obtained directly from the formalism developed in [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF], because the matrix R does not satisfy the crossing unitarity. We use the operator t(x|z) to define the following discrete time Markov process

|P t+1 = t(x|z)|P t . (2.6)
The parameters must satisfy the following constraints

0 ≤ xz i ≤ 1, 0 ≤ x z i ≤ 1 and ax, bx ≥ 0 (2.7)
to ensure that the entries of t(x|z) are probabilities. We can also show that the entries on each column of t(x|z) sum to one, which guarantees the conservation of the probability |P t .

Continuous time limit. In the homogeneous limit z i → 1, we are left with the operator

t(x) = t(x|z)| z i =1 . A straightforward computation gives - 1 2 t ′ (1) = - 1 2 K ′ 1 (1) - L-1 k=1 P k,k+1 .R ′ k,k+1 (1)
+ 1 2 K′ L (1) . (2.8)
where the prime ( . ) ′ stands for the derivative with respect to the spectral parameter. Then, introducing the parameters α = 1/(a + 1) and β = 1/(b + 1), we obtain

- 1 2 K ′ (1) = -α 0 α 0 , -P.R ′ (1) =     0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0     , 1 2 K′ (1) = 0 β 0 -β . (2.9)
Hence -1/2 t ′ (1) is nothing but the continuous time Markov matrix of the open TASEP. Therefore, the stationary state of the continuous time TASEP computed in [START_REF] Derrida | Exact solution of a 1d asymmetric exclusion model using a matrix formulation[END_REF] can be obtained by taking the homogeneous limit of the one computed here.

Graphical representation of the transition rates

In fig. 6, each matrix element of the K-matrix is drawn as a left-reflection diagram. The incoming lines (according to the arrows direction) before the reflection point denote the vector the matrix K is acting on. A dashed line corresponds to the vector |0 (or equivalently to an empty site), whereas a continuous thick line corresponds to |1 (or equivalently to an occupied site). In a similar way, the out-going lines (after the reflection point) represent the state of the vector with which we are contracting to the left the matrix K.

Similarly, the graphical representation for the matrix K is given in fig. 7.

We can now draw diagrams to represent the action of the full transfer matrix. The general picture is displayed in fig. 8.

1/x 1/x 1/x 1/x x x x x (a+x)x xa+1 1-x 2 xa+1 0 1 0|K(x)|0 1|K(x)|0 0|K(x)|1 1|K(x)|1
Figure 6: Graphical representation of the K-matrix. As an example we can compute graphically for L = 1 the transition rate 0|t(x|z)|1 between the initial configuration (1) and the final configuration (0) (see fig. 9). The sum of both contributions drawn in fig. 9 gives

1/x 1/x 1/x 1/x x x x x 1 xb+1 0 1 xb+1 xb xb+1 0| K(x)|0 0| K(x)|1 1| K(x)|0 1| K(x)|1
0|t(x|z)|1 = x(a + x)(1 -xz 1 ) (ax + 1)(bx + 1) + xz 1 (1 -x z 1 ) bx + 1 = (1 -x 2 )
x(a + z 1 ) (ax + 1)(bx + 1) .

(2.10) x x

x x 1/x 1/x z 1 z 2 z i z L-1 z L 1 2 i L -1 L . . . . . .
1/x 1/x z 1 1 0 0 z 1 1 0 0 x(a+x)(1-xz 1 ) (ax+1)(bx+1) x x 1/x 1/x z 1 1 0 0 xz 1 (1-x z

Dynamical rules

The transfer matrix t(x|z) defines a discrete time Markov process on the finite size lattice with open boundaries. The corresponding stochastic dynamics can be described explicitly using a sequential update: starting from a given configuration at time t, the configuration at time t + 1 is obtained by the following stochastic rules

• Initialisation:

-The left boundary is replaced by an additional site (the site 0 with inhomogeneity parameter 1) occupied by a particle with probability 1 ax+1 and unoccupied with probability ax ax+1 .

-The right boundary is replaced by an additional site (the site L + 1 with inhomogeneity parameter 1) occupied by a particle with probability bx bx+1 and unoccupied with probability 1 bx+1 .

• Particle update: starting from right to left (from site L to site 0), a particle at site i can jump to the right on the site i + 1 (provided that the site is empty) with probability 1 -xz i and stay at the same place with probability xz i .

• Hole update: once arrived at site 0, we go the other way starting from left to right (from site 1 to site L + 1): a hole at site i can jump to the left on the site i -1 (provided that the site is occupied) with probability 1 -x z i-1 and stay at the same place with probability x z i-1 .

• Summation: Then we have to drop the additional sites 0 and L+1 and to sum the weights corresponding to the same final configuration.

An example of such sequential update is given in fig. 10 for L = 1. From this figure we can compute the transition rates 0|t(x|z)|1 = ax (ax + 1)(bx + 1)

xz 1 1 - x z 1 + 1 -xz 1 + 1 (ax + 1)(bx + 1) (1 -xz 1 )x 2 + xz 1 1 - x z 1 (2.11) = (1 -x 2 ) x(a + z 1 ) (ax + 1)(bx + 1) , (2.12) 1|t(x|z)|1 = x(x + b) bx + 1 + (1 -x 2 )(1 -xz 1 ) (ax + 1)(bx + 1) . ( 2

.13)

The equation (2.12) is, of course, identical to the expression (2.10) derived using the graphical representation in fig. 9.

Stationary state and observables

We now determine the stationary state of the inhomogeneous Markov chain, by using a matrix product representation, which will be constructed following a method developed in [START_REF] Crampe | Integrable approach to simple exclusion processes with boundaries. Review and progress[END_REF]. We shall then compute some physical characteristics of the steady-state.

(ax+1)(bx+1) abx 2 (ax+1)(bx+1) 1 (ax+1)(bx+1) bx (ax+1)(bx+1) xz 1 1 -xz 1 1 (1 -xz 0 )(1 -xz 1 ) xz 0 (1 -xz 1 ) xz 1 1 x z 1 1 -x z 1 1 1 1 -x z 0 x z 0 1 -x z 1 x z 1 1

Matrix Product Ansatz

The stationary state is expressed as a matrix product. Indeed, the probability of the configuration C = (τ 1 , . . . , τ L ) can be written as

S(τ 1 , . . . , τ L ) = 1 Z L W | L i=1 ((1 -τ i )E(z i ) + τ i D(z i ))|V , (2.14)
where the normalization factor is

Z L = W |C(z 1 ) . . . C(z L )|V , with C(z) = E(z) + D(z).
The algebraic elements E(z) and D(z) can be gathered in a vector

A(z) = E(z) D(z) .
(2.15)

Using this vector, the stationary state can be written in a more compact way:

|S = 1 Z L W |A 1 (z 1 )A 2 (z 2 ) . . . A L (z L )|V . (2.16)
We now use the formalism first introduced in [START_REF] Sasamoto | Stationary states of integrable systems in matrix product form[END_REF] and developed in [START_REF] Crampe | Integrable approach to simple exclusion processes with boundaries. Review and progress[END_REF][START_REF] Crampe | Open two-species exclusion processes with integrable boundaries[END_REF] to determine the algebra generated by the matrix elements of A(z) and by the boundary vectors. The quadratic algebra satisfied in the bulk by E and D will be obtained using the Zamolodchikov-Faddeev relations; the boundary Ghoshal-Zamolodchikov relations show how the vectors W | and |V are coupled to E and D.

The vector A(z) allows us to write easily the exchange relations between E(z) and D(z):

R 12 (z 1 /z 2 )A 1 (z 1 )A 2 (z 2 ) = A 2 (z 2 )A 1 (z 1 ).
(2.17)

This Zamolodchikov-Faddeev relation [START_REF] Zamolodchikov | Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models[END_REF] defines an associative algebra because the R-matrix obeys the Yang-Baxter equation. Written explicitly it gives four relations

           [E(z 1 ), E(z 2 )] = 0 [D(z 1 ), D(z 2 )] = 0 z 1 z 2 D(z 1 )E(z 2 ) = D(z 2 )E(z 1 ) D(z 1 )E(z 2 ) = z 2 z 2 -z 1 [E(z 2 )D(z 1 ) -E(z 1 )D(z 2 )] (2.18)
Let us remark that the third relation is implied by the fourth. The relations between the boundary vectors W |, |V and the algebraic elements E(z), D(z) are given by the Ghoshal-Zamolodchikov relations

W |K(z)A(1/z) = W |A(z), K(z)A(1/z)|V = A(z)|V . (2.19)
These boundary relations are consistent with the bulk exchange relations thanks to the reflection equation. Remark that the boundary vector |V (resp. W |) in fact defines a class of representations (resp. dual representations) for the ZF algebra, where |V (resp. W |) exists and is non-vanishing. We prove below that these classes are not empty by providing an explicit (dual) representation that belongs to them.

Written explicitly we obtain four relations

   W |E(z) = W | (a+z)z za+1 E(1/z) W |E(z) = z(a+z) z 2 -1 W |(D(1/z) -D(z)) and D(z)|V = zb+1 z(b+z) D(1/z)|V D(z)|V = bz+1 z 2 -1 (E(z) -E(1/z))|V
(2.20) Let us remark that for each boundary, the first relation is implied by the second one.

We now prove that the vector |S is the stationary state of the transfer matrix. Following what was done in [START_REF] Crampe | Integrable approach to simple exclusion processes with boundaries. Review and progress[END_REF], one picks up the vector A i (z i ) and moves it completely to the right using L -i times the Zamolodchikov-Faddeev relation (2.17). Next, the vector A i (z i ) can be reflected against the right boundary vector |V using the Ghoshal-Zamolodchikov relation (2.19). After that, one can push A i (1/z i ) to the left through all the A j (z j ) using L -1 times (2.17). Then, one can reflect A i (1/z i ) against the left boundary vector W | using (2.19). Finally, one can move back A i (z i ) to its initial position to complete the cycle. Thus, we are left with: The vertices that do not enter a thick line have degree 0 in x. Thus, the total degree of the matrix element is the sum of the degree of each thick line, plus the degree coming from the boundaries that we look at separately. We first consider lines that are in the 'bulk' (that do not include a boundary). Looking at the weights of the vertices in fig. 1, one can see that the incoming part of a line always carry a degree 1. Moreover, a line corresponding to a degree n must cross at least n exit lines, the first on the left being always of degree 0. This implies that the bulk part is at most of degree L. Figure 7 shows that the right boundary does not change this counting. The left boundary can add at most a degree 2, as it can be seen on figure 6. Altogether, this leads to a total degree L + 2. Explicit representation. We now construct an explicit representation of the algebraic elements E(z) and D(z). We introduce the shift operators ǫ and δ such that δǫ = 1, which are well known in the context of the continuous time open TASEP since they enter the construction of the original matrix ansatz [START_REF] Derrida | Exact solution of a 1d asymmetric exclusion model using a matrix formulation[END_REF]. We define

t(z i |z)|S = |S . ( 2
E(z) = z + ǫ and D(z) = 1/z + δ , (2.23)
and the boundary vectors such that

W | ǫ = a W | and δ| V = b | V . (2.24)
It is readily verified that the Zamolodchikov-Faddeev relation (2.17) and the Ghoshal-Zamolodchikov relations (2.19) are satisfied. From the results of [START_REF] Derrida | Exact solution of a 1d asymmetric exclusion model using a matrix formulation[END_REF] giving explicit forms for ǫ, δ, W | and | V , we deduce that W | and | V can be chosen such that W | V = 0 which guarantees the non-vanishing of |S .

Steady-state probabilities

The matrix Ansatz allows us to calculate the stationary probability of any given configuration. Let 1 ≤ j 1 < • • • < j r ≤ L be integers and C(j 1 , . . . , j r ) be the configuration (τ 1 , . . . , τ L ) with τ i = 1 if i = j k and τ i = 0 otherwise. We define the (non-normalized) weight of the word with D(j 1 ), . . . , D(j r ) at positions j 1 , . . . , j r as

W L (j 1 , . . . , j r ) = Z L × S(C(j 1 , . . . , j r )) (2.25) = W | . . . D(z j 1 ) . . . D(z j 2 ) . . . D(z jr ) . . . |V , (2.26)
where the dots stand for E(z i ) operators.

The same quantity can be calculated using the explicit representation (2.23):

W L (j 1 , . . . , j r ) = W | . . . δ + 1 z j 1 . . . δ + 1 z j 2 . . . δ + 1 z jr . . . | V . (2.27)
The weight W L (j 1 , . . . , j r ) computed from the relations (2.18) and (2.20) is proportional to the weight W L (j 1 , . . . , j r ) computed using the explicit representation (thanks to the unicity of the steady-state guaranteed by the Perron-Frobenius theorem). Thus, we have

W L (j 1 , . . . , j r ) = f (z 1 , . . . , z L ) W L (j 1 , . . . , j r ). (2.28)
The multiplicative coefficient is obtained by comparing the weights of the empty configuration:

f (z 1 , . . . , z L ) = W |E(z 1 ) . . . E(z L )|V (a + z 1 ) . . . (a + z L ) W |V ; (2.29)
f (z 1 , . . . , z L ) is symmetric under the permutation of the z i thanks to (2.18) but also under the transformation z i → 1/z i thanks to (2.20). The group generated by these transformations is denoted by BC L in reference to the Weyl group of the root system of the Lie algebra sp(2L).

The expression of W L (j 1 , . . . , j r ) is given by

W L (j 1 , . . . , j r ) = z 1 . . . z L z j 1 . . . z jr r p=0 b p L q r-p =j r-p 1 z q r-p q r-p -1 q r-p-1 =j r-p-1 1 z q r-p-1 • • • q 2 -1 q 1 =j 1 1 z q 1 × j 1 -1 l 0 =1 1 + a z l 0 j 2 -1 l 1 =q 1 +1 1 + a z l 1 • • • j r-p+1 -1 l r-p =q r-p +1 1 + a z l r-p (2.30)
By convention when p = r in the first sum, there is no summation over the q i and the formula reduces to b r j 1 -1

l 0 =1 1 + a z l 0
. We also set j r+1 = L + 1 in the last product when p = 0. The proof of this formula is obtained by induction on the size L, using the identity

D(z jr ) E(z jr+1 ) = 1 z jr + δ (z jr+1 + ǫ) = 1 z jr (z jr+1 + ǫ) + z jr+1 1 z jr+1 + δ (2.31) = 1 z jr E(z jr+1 ) + z jr+1 D(z jr+1 ) . (2.32)
Another important step is to compute the normalization factor of the probability distribution. We define C(z) = E(z) + D(z) and C(z) = E(z) + D(z). The normalization factors are thus given by Z L (z 1 , . . . , z L ) = W |C(z 1 ) . . . C(z L )|V , and Z L (z 1 , . . . , z L ) = W | C(z 1 ) . . . C(z L )| V for the explicit representation. Thanks to the property (2.28), we have

Z L (z 1 , . . . , z L ) = f (z 1 , . . . , z L ) Z L (z 1 , . . . , z L ). (2.33)
Z L and Z L are symmetric under BC L . Our goal is to get an analytic expression for the normalization factor. For this purpose, for any sequence of complex numbers u = (u 1 , u 2 , ...), we define the shifted product by

(z|u) k = (z -u 1 )(z -u 2 ) • • • (z -u k ) , k > 0, 1 if k = 0, (2.34)
We have the result (see proof in appendix A):

Z L (z 1 , . . . , z L ) = det (z j |v) L+2-i -(1/z j |v) L+2-i i,j det z L+1-i j -(1/z j ) L+1-i i,j
with v = (-a, -b, 0, . . . , 0). (2.35) When a, b = 0, we recognize in the L.H.S. of equation (2.35), the expression of the Schur polynomial of type C associated to the partition 1 L = (1, ...., 1). We remind that the Schur polynomial of type C associated with the partition λ = (λ 1 , . . . , λ L ) with λ 1 ≥ . . . λ L ≥ 0 is defined by:

Z L (z 1 , . . . , z L ) a=b=0 = s C λ (z 1 , . . . , z L ) = det (z j ) L+1-i+λ i -(1/z j ) L+1-i+λ i i,j det z L+1-i j -(1/z j ) L+1-i i,j
.

(2.36)

As in the periodic case where the normalization factor is linked to the Schur polynomial of type A and to the representation of the Lie algebra sl(L) (see (1.23) and (1.24)), the normalization factor (2.36) is given in terms of the Schur polynomial of type C and is associated to representation of the Lie algebra sp(2L). These observations will allow us to use results of the Lie algebra theory to take the homogeneous limit z i → 1 (see the remark below). When a, b are arbitrary, we need to use some generalizations of the Schur polynomial, called shifted (or factorial) Schur polynomials. A shifted Schur polynomial of type A, is defined, for any sequence of complex numbers u = (u 1 , u 2 , ...) [START_REF] Koike | Young-diagrammatic methods for the representation theory of the classical groups of type B n , C n , D n[END_REF], as follows:

s A λ (z 1 , . . . , z L |u) = det (z j |u) L-i+λ i i,j det z L-i j i,j
.

(2.37)

Similarly, we define the shifted (or factorial) Schur polynomial of type C as5

s C λ (z 1 , . . . , z L |u) = det (z j |u) L+1-i+λ i -(1/z j |u) L+1-i+λ i i,j det z L+1-i j -(1/z j ) L+1-i i,j . (2.38)
Thus, the normalization factor (2.35) is the shifted Schur polynomial of type C associated to the partition 1 L and to the sequence v = (-a, -b, 0, 0, . . . ). For this particular partition, the shifted Schur polynomial of type C can be expanded on the usual type C Schur polynomials as (see proof in appendix B)

Z L (z 1 , . . . , z L ) = s C 1 L (z 1 , . . . , z L |v) = L+1 n=1 a n -b n a -b s C 1 L+1-n (z 1 , . . . , z L ). (2.39)
Using the explicit representation, we can also determine the particle density at site i from the stationary measure:

n i (z 1 , . . . , z L ) = b + 1 z i Z L-1 (z 1 , . . . , z i-1 , z i+1 , . . . , z L ) Z L (z 1 , . . . , z L ) + (1 -ab) × L-i-1 k=0 Z i-1+k (z 1 , . . . , z i-1 , z i+1 , . . . , z i+k ) b=0 Z L-i-1-k (z i+2+k , . . . , z L ) a=0 Z L (z 1 , . . . , z L ) .
Remark: In the homogeneous limit z i → 1, s C 1 L+1-n (1, . . . , 1) is equal to the dimension of the sp(2L) representation associated to the partition 1 L+1-n :

s C 1 L+1-n (1, . . . , 1) = n L + 1 2L + 2 L + 1 -n = dim π C (1 L+1-n ) . (2.40)
This expression is obtained through the general formula (see [START_REF] Frappat | Dictionary on Lie algebras and superalgebras[END_REF] for instance):

dim π C (λ) = 1≤i<j≤L λ i -λ j + j -i j -i λ i + λ j + 2L + 2 -j -i 2L + 2 -j -i L i=1 λ i + L + 1 -i L + 1 -i .
Then from (2.39), we get .42) The equality between these two expressions is ensured by the identity

Z L (1, 1, ..., 1) = g L (a) -g L (b) a -b with g L (x) = L n=0 n + 1 L + 1 2L + 2 L -n x n+1 (
Z L = h L ( 1 α ) -h L ( 1 β ) 1 α -1 β with h L (x) = L p=1 p 2L -p 2L -p L x p+1 . ( 2
g L (x -1) = h L (x) - 1 L 2L L -1 (2.43)
and recalling that a = 1 α -1 and b = 1 β -1. In particular, when a = b = 0 (i.e. α = β = 1), Z L is equal to the dimension of the representation 1 L , which is the Catalan number C L+1 = 1 L+2 2L + 2 L + 1 .

Partition function and Baxter Q-operator

The models considered here have non-diagonal boundary matrices (see (2.1)) and cannot be solved by the usual integrability methods. In the last few years, various specific techniques have been developed for solving such problems, based on the functional Bethe ansatz [START_REF] Nepomechie | Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms[END_REF][START_REF] Murgan | Bethe ansatz derived from the functional relations of the open XXZ chain for new special cases[END_REF][START_REF] Frahm | Functional Bethe ansatz methods for the open XXX chain[END_REF], the coordinate Bethe ansatz [START_REF] Crampe | Generalized coordinate Bethe ansatz for non diagonal boundaries[END_REF][START_REF] Crampe | Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions[END_REF][START_REF] Crampe | Matrix Coordinate Bethe Ansatz: Applications to XXZ and ASEP models[END_REF][START_REF] Simon | Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries[END_REF], the separation of variables [START_REF] Frahm | Separation of Variables in the open XXX chain[END_REF][START_REF] Niccoli | Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators[END_REF], the q-Onsager approach [START_REF] Baseilhac | Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory[END_REF] and the algebraic Bethe ansatz [START_REF] Belliard | Algebraic Bethe ansatz for open XXX model with triangular boundary matrices[END_REF][START_REF] Belliard | Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz[END_REF]. Recently, a generalization [START_REF] Cao | Off-diagonal Bethe ansatz and exact solution a topological spin ring[END_REF][START_REF] Nepomechie | Inhomogeneous T-Q equation for the open XXX chain with general boundary terms: completeness and arbitrary spin[END_REF] of the TQ relations expresses the eigenvalues of problems in terms of solutions of a new type of Bethe equations, called inhomogeneous Bethe equations.

In this section, we shall find an unexpected relation between the 'partition function' Z L and the Baxter Q operator appearing in the T Q-relations.

In [START_REF] Crampe | Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries[END_REF], the Bethe equations corresponding to the eigenvalues and the eigenvectors of the transfer matrix (2.5) have been computed using the modified algebraic Bethe ansatz. In this context, the eigenvalue of (2.5) corresponding to the stationary state Λ(x) is given by Λ

(x) = x L+1 b + x bx + 1 L k=1 xu k -1 u k -x - (x 2 -1) (bx + 1) L j=1 (x -z j )(x - 1 z j ) L k=1 u k u k -x , (2.44)
where u k are called Bethe roots and are solutions of the following Bethe equations

L p=1 (u j -z p )(u j z p -1) u j z p = (u j + b) L k=1 k =j u j - 1 u k , for j = 1, 2, . . . , L . (2.45)
Let us stress that here the Bethe roots are the ones corresponding to the steady state. The Bethe roots corresponding to other states obey different Bethe equations (see [START_REF] Crampe | Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries[END_REF], or [START_REF] De Gier | Bethe Ansatz Solution of the Asymmetric Exclusion Process with Open Boundaries[END_REF] for the homogeneous case). We now introduce the following function: .46) This function (up to a coefficient x L ) is linked to the polynomial Q of Baxter [START_REF] Baxter | Exactly solved models in statistical mechanics[END_REF]: its zeros are the Bethe roots. The eigenvalue (2.44) can be written in terms of the Q function as follows

Q(x) = L k=1 1 u k - 1 x . ( 2
Λ(x) = x(b + x)Q(1/x) (bx + 1)Q(x) - (x 2 -1) (bx + 1)Q(x) L j=1 (x -z j )( 1 xz j -1) . ( 2

.47)

For the steady-state eigenvector, we know that Λ(x) = 1 (see section 2.4). Hence Q(x) satisfies

x(x + b)Q(1/x) -(1 + bx)Q(x) = (x 2 -1) L j=1 (x -z j )( 1 xz j -1) . (2.48)
This equation, called TQ relation, allows us to compute explicitly the function Q: for a given L, it can be shown that equation (2.48) has a unique solution of the form (2.46). For the model studied here, the function Q and the normalization factor Z(z 1 , . . . , z L ) are closely related. Namely, we get

Q(x) = Z(z 1 , . . . , z L ) a→-1/x = L+1 n=1 (-1/x) n -b n -1/x -b s C 1 L+1-n (z 1 , . . . , z L ) (2.49) = L p=0 - 1 x p L-p n=0 b n s C 1 L-n-p (z 1 , . . . , z L ) . (2.50)
where we have used the explicit form of Z given in (2.39). To prove this result, we remark that Q(x) given by (2.50) has the form (2.46) and we show that it satisfies the TQ relation (2.48). We have

x(x + b)Q(1/x) -(1 + bx)Q(x) = -x L+1 n=1 ((-x) n -(-1/x) n ) s C 1 L+1-n (z 1 , . . . , z L ) (2.51) = (x 2 -1)s C 1 L (z 1 , . . . , z L |x, 1/x, 0, . . . , 0) . (2.52)
Finally, we readily check that s C 1 L (z 1 , . . . , z L |x, 1/x, 0, . . . , 0) vanishes at the points x = z j and x = 1/z j which allows us to conclude that (2.52) is equal to the L.H.S. of (2.48). Relation (2.49) means that the Bethe roots (zeros of the function Q) are linked to the zeros of the steady-state normalization factor in the complex plane of the transition rate a. These zeros appeared previously (for the homogeneous case) in [START_REF] Blythe | Lee-Yang zeros and phase transitions in nonequilibrium steady states[END_REF][START_REF] Bena | Statistical Mechanics of Equilibrium and Nonequilibrium Phase Transitions: The Yang-Lee Formalism[END_REF] as Lee-Yang zeros and allows the generalization of the Lee-Yang theory for the phase transition of non equilibrium system. Therefore, relation (2.49) expresses an unexpected relation between two objects arising from very different contexts.

Conclusion

In this paper, we study discrete time Markov processes with periodic or open boundaries and with inhomogeneities. The transfer matrices associated with quantum spin chains are used to construct the Markov matrix. We provide a graphical representation of these processes and an interpretation in terms of sequential updates. We also study their stationary states using a matrix ansatz developed in [START_REF] Sasamoto | Stationary states of integrable systems in matrix product form[END_REF][START_REF] Crampe | Integrable approach to simple exclusion processes with boundaries. Review and progress[END_REF] and express their normalization in terms of Schur polynomials. Finally, in the case with open boundaries, we find a connection between the Bethe roots and the Lee-Yang zeros.

We believe that numerous generalizations of these works are possible and that connections with other approaches will be fruitful. For the moment, we studied the models on general grounds, without specifying the values of the inhomogeneities. However, it would be interesting to take particular choices inspired by physical problems. In these cases, we may be able to go further in the computations of the physical quantities. For example, one can choose the inhomogeneities randomly, according to a certain law, and study the behavior of the density and of the current.

To compute the stationary state, we use a generalization of the matrix ansatz but other methods have been developed to study such problems. For example, the study of qKZ equations [START_REF] Francesco | Around the Razumov-Stroganov conjecture: proof of a multi-parameter sum rule[END_REF][START_REF] Cantini | qkz equations and ground state of the o(1) loop model with open boundary conditions[END_REF] seems to be closely related to the approach developed here. A common point of both methods is the use of different types of symmetric polynomials (or their deformation). Let us also mention the combinatorial interpretation of the matrix ansatz developed in [START_REF] Corteel | Tableaux combinatorics for the asymmetric exclusion process[END_REF] which may be generalized to the case with inhomogeneities.

The method presented here can be used for other models: for example, in a recent work [START_REF] Crampe | Open two-species exclusion processes with integrable boundaries[END_REF], we classified integrable boundary conditions for the open two species exclusion process generalizing the model introduced in [START_REF] Uchiyama | Two-Species Asymmetric Simple Exclusion Process with Open Boundaries[END_REF] (see also [START_REF] Cantini | Asymmetric Simple Exclusion Process with open boundaries and Koornwinder polynomials[END_REF][START_REF] Corteel | Macdonald-koornwinder moments and the two-species exclusion process[END_REF]). Thus, discrete time inhomogeneous two species exclusion processes can be constructed and studied following the lines presented here. The generalization to N -species also seems possible [START_REF] Cantini | Matrix-product formula for Macdonald polynomials[END_REF].

Finally, the correspondence between the Bethe roots and the Lee-Yang zeros deserves further investigation. We do not know if it is valid only for the TASEP or if it is a general feature. Such connection may shed light on the Bethe equations and on the Lee-Yang theory for out-ofequilibrium models [START_REF] Blythe | Lee-Yang zeros and phase transitions in nonequilibrium steady states[END_REF][START_REF] Blythe | Nonequilibrium steady states of matrix-product form: a solver's guide[END_REF].

B Expansion of shifted Schur polynomials of type C

We show that the type C shifted Schur polynomial (2.35) can be expanded on the regular type C Schur polynomials as follows Define the matrix M (z 1 , . . . , z L ) = (z j |v) L+2-i -(1/z j |v) L+2-i i,j . Let R i (z 1 , . . . , z L ) be the i-th row of M (z 1 , . . . , z L ). The expansion (B.2), we can rewritten as R i (z 1 , . . . , z L ) = P i-1 (z 1 , . . . , z L ) + (a + b)P i (z 1 , . . . , z L ) + abP i+1 (z 1 , . . . , z L ), (B.3) with P k (z 1 , . . . , z L ) the row vector (z L+1-k j -1/z L+1-k j ) j . Remark that P L+1 (z 1 , . . . , z L ) = 0. We develop det (M (z 1 , . . . , z L )) using the expansion (B.3) and the multi-linearity of the determinant. Since we have only L + 1 different row vectors P i , antisymmetry of the determinant imposes that we need to choose L distinct row vectors P i among P 0 , . . . , P L in order to get a non vanishing determinant (hence there is just one P i missing in each non vanishing determinant). Then, the expansion of det(M (z 1 , . . . , z L )) reads det(M (z 1 , . . . , z L )) = In opposition to the previous cases, there is some freedom in this diagram. Indeed, although the first rows R 1 , . . . , R L+1-n uniquely fix which P i 's are chosen (written in bold), the row R L+2-n corresponds to two choices:

• we choose P L+2-n (which comes with the factor a + b). Then, the remaining rows just correspond to the computation of C n-1 .

• we choose P L+3-n (which comes with the factor ab). In this case, we have to choose P L+2-n in the row R L+3-n (which comes with the factor 1). Then, the remaining rows correspond to the computation of C n-2 (except that the order of P L+2-n and P L+3-n is inverted, hence a factor (-1) by antisymmetry of the determinant).

Altogether, we get a recursion relation for the coefficients C n :

C n = (a + b) C n-1 -ab C n-2 . (B.

|τ 1 .

 1 . . τ L = |τ 1 ⊗• • •⊗|τ L , where we have defined the basis vectors |0 =

Figure 1 :

 1 Figure 1: Non-vanishing vertices associated with R.

Figure 3 :

 3 Figure 3: Starting point for the computation of 1101|M (x|z)|1110 .

Figure 4 :

 4 Figure 4: The two different drawings involved in the computation of the transition rate 1101|M (x|z)|1110 with their respective weights.

Figure 5 :

 5 Figure 5: An example of sequential update corresponding to the Markov matrix M (x|z). The first line is the configuration at time t and the third line shows the possible configurations at time t + 1. The second line corresponds to the intermediate configurations after the update of the particles and the hole update still to be done. The label of the arrows provides the rate of the corresponding change of configurations.

Figure 7 :

 7 Figure 7: Graphical representation of the K-matrix.

0 0Figure 8 :

 08 Figure 8: Graphical representation of the transfer matrix.

1 )Figure 9 :

 19 Figure 9: Graphical computation of the transition rate 0|t(x|z)|1 for L = 1. The two different contributions are represented with their respective weights.

  ax

Figure 10 :

 10 Figure 10: An example of sequential update corresponding to the Markov matrix t(x|z). The first line is the configuration at time t. The second line represents the possible configurations after adding the two supplementary sites corresponding to the boundaries. The third line corresponds to the intermediate configurations after the updates of the particles. The fourth line represents the possible configurations after the updates of the holes. The label of the arrows provides the rate of the corresponding change of configurations. The last line represents the final configurations at time t + 1 after the summation step.

Figure 12 :

 12 Figure 12: Decomposition of a matrix element into thick continuous lines.

2 . 41 )

 241 in accordance with the results known for continuous time open TASEP[START_REF] Derrida | Exact solution of a 1d asymmetric exclusion model using a matrix formulation[END_REF]

s C 1 L

 1 (z 1 , . . . , z L |v) = L+1 n=1 a n -b n a -b s C 1 L+1-n (z 1 , . . . , z L ), (B.1)where v = (-a, -b, 0, . . . , 0).Proof. Starting froms C 1 L (z 1 , . . . , z L |v) = det (z j |v) L+2-i -(1/z j |v) L+2-i i,j det z L+1-i j -(1/z j ) L+1-i i,jthe general idea is to use the multi-linearity of the determinant in the numerator to expand it and recognize non shifted Schur polynomials of type C. First we rewrite (z j |v) L+2-i -(1/z j |v) L+2-i = z L+2-

CP 0

 0 n u n (z 1 , . . . , z L ), where u L+1-k (z 1 , . . . , z L ) = det (z 1 , . . . , z L ). . .

P k- 1

 1 (z 1 , . . . , z L ) P k+1 (z 1 , . . . , z L ) . . .P L (z 1 , . . . , z L )

8 )

 8 It is straightforward to check that C n = a n -b n a-b verifies (B.8), C 1 = 1 and C 2 = a + b. We complete the proof of (B.1) by remarking that u n (z 1 , . . . , z L )/ det z L+1-i j -(1/z j ) L+1-i i,j = s C 1 L+1-n (z 1 , . . . , z L ).

It is easy to see that s C λ (z1, . . . , zL|u) is indeed a polynomial in the variables z1, . . . , zL.

Acknowledgments

It is a pleasure to thank A. Molev for discussions and references on Schur polynomials. M. Vanicat thanks the Laboratoire Charles Coulomb for hospitality during his stay.

A Normalization factor and Schur polynomials of type C

The normalization factor can be expressed in terms of a shifted Schur polynomial of type C Z(z 1 , . . . , z L ) = s C 1 L (z 1 , . . . , z L |v) (A.1)

Proof Let us show be induction on L the previous formula.

• For L = 1, we have

• Let L ≥ 1 such that the property holds for L-1. Consider Z L (z 1 , . . . , z L ) and s C 1 L (z 1 , . . . , z L |v) as Laurent polynomials in the variable z L . It is easy to see that

We have also

Z L-1 (z 1 , . . . , z L-1 )/z L . Using the determinant formula for s C 1 L (z 1 , . . . , z L |v), we can develop the determinants in the numerator and denominator with respect to the last column to show that s

Using the induction hypothesis we deduce that

Hence Z L (z 1 , . . . , z L ) and s C 1 L (z 1 , . . . , z L |v) are Laurent polynomials in the variable z L of degree 1 in z L and 1/z L and they have the same coefficients in front of z L and 1/z L . So we can write

with f independent of z L . But Z L (z 1 , . . . , z L ) and s C 1 L (z 1 , . . . , z L |v) are symmetric under the permutation of the z i 's, so f cannot depend of the z i 's and hence is constant. We compute this constant by evaluating the previous expression at the particular point where all the z i 's are equal to 1. On one hand the value of s C 1 L (1, . . . , 1|v) can be computed using (2.39) and (2.40). On the other hand the value of Z L (1, . . . , 1) is equal to the value of the usual continuous time TASEP normalization factor, which matches with the previous value (see (2.43)). Hence we have Z L (1, . . . , 1) = s C 1 L (1, . . . , 1|v), so f = 0. This concludes the proof. and C n are some coefficients to be determined. Let us first compute C 1 (which corresponds to the term with missing P L in (B.4)). There is only one possibility to get the row vector P 0 : it comes necessarily from the row R 1 . Then P 1 appears in the rows R 1 and R 2 , but since we have already chosen P 0 for the row R 1 , P 1 has to be taken in row R 2 . One shows by induction that for all i ∈ {0, . . . L -1}, P i has to be taken in row R i+1 . It is summarized on the following diagram.

Each line in (B.5) corresponds to the expansion (B.3) of a row vector R i . The coefficients have been omitted, but can be easily recovered from (B.3). The bold P i 's in the diagram are the ones chosen in the expansion to obtain the term u 1 (z 1 , . . . , z L ). The shaded P i 's are the ones that do not appear in u 1 (z 1 , . . . , z L ). In the case considered here, all the chosen P i 's correspond to the first term in (B.3) and hence come with a factor 1. We deduce that C 1 = 1. The computation of C 2 (with P L-1 missing) looks similar. There is again only one choice for the P i 's to get u 2 (z 1 , . . . , z L ). It is given by


Keihann Yavari

Wylliam Delaunay

Nicolas De Rycke

Thibault Reynaldo

Paul Aillard

Monika Srebro-Hooper

Victoria Y Chang

Gilles Muller

Denis Tondelier

Bernard Geffroy

Jeanne Crassous
email: jeanne.crassous@univ-rennes1.fr

Arnaud Voituriez
email: arnaud.voituriez@cnrs.fr

Angela Marinetti
email: angela.marinetti@cnrs.fr

Muriel Hissler
email: muriel.hissler@univ-rennes1.fr

Phosphahelicenes: from chiroptical and photophysical properties to OLED applications

Keywords: phosphahelicenes, circular dichroism, luminescence, OLED, quantum-chemical calculations

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Helicenes are non-planar polycyclic aromatic hydrocarbons (PAHs) formed of ortho-condensed aromatic rings. peculiar helical and π-conjugated structure, they chiroptical properties such as huge values of (OR) parameters, intense electronic circular dichroism ( and substantial circularly polarized luminescence (CPL Accordingly, they have revealed appealing applications in many domains, particularly in optoelectronic carbohelicenes generally display low fluor yields (ΦFL) because their non-planar π-electron system triggers spin-orbit coupling (SOC) that accelerates intersystem crossing (ISC) pathways (ΦISC:∼0.9). [3] An appealing strategy to emission properties of helicenes consists heteroatom(s) or heterocycle(s) in the helic example, significant improvements of helicenes' properties have been recently achieved via heterocycles as carbazole, [2b,4a] maleimide oxaborine, [4d] or azaborole. [4e] Among : from chiroptical and photophysical Wylliam Delaunay, [a] Nicolas De Rycke, [a,c] Thibault Reynaldo, Victoria Y. Chang, [e] Gilles Muller, [e] Denis Tondelier, Angela Marinetti,* [b] Muriel Hissler,* [a] and Jeanne Crassous physico-chemical and properties of a series of phosphahelicenes, focusing on electronic circular dichroism and circularly polarized luminescence.

absorption and ECD spectra chemical calculations allowed highlight general features of these helicenic phosphines.

electrochemical properties and thermal as emitters in organic r polycyclic aromatic hydrocarbons condensed aromatic rings. Due to their conjugated structure, they display unique values of optical rotation circular dichroism (ECD), and substantial circularly polarized luminescence (CPL). [1] hey have revealed appealing applications in many optoelectronics. [2] However, helicenes generally display low fluorescence quantum electron system triggers orbit coupling (SOC) that accelerates intersystem crossing An appealing strategy to modify consists in introducing in the helical structure. For ignificant improvements of helicenes' fluorescence introduction of such maleimide, [2e,4b] siloles, [4c] Among other possible heterocycles, the phosphole ring tuning emission properties of helicenes derivatives are known to be efficient organic light-emitting diodes (OLEDs introduction of a phosphole ring within a helicenic structure has enabled to construct a new class of helical systems phosphahelicenes. [7,8] Some of these helically chiral compounds typified in Figure 1, appeared highly promising in enantioselective catalysis, [9] both as catalyzed enantioselective enyne cyclo organocatalysts in asymmetric [3+2] cyclization allenes with electron-poor olefins. photophysical properties of these heterohelicenes their potential application in OLEDs, date.

In this work, we describe the photophysical ( absorption and fluorescence) and ECD and OR), and examine the phosphahelicenes including phosphahelicene and phosphahelicene sulfide 5 experimental data are supported density functional calculations. The physical properties rec in solution showed that the P features expected for a material that can be inserted as active emitter in OLEDs. The results for demonstrate the potential of these compounds for optoelectronic applications.

[6]-P-endo-1a

[7]-P-endo-3

[8]-P-endo

photophysical properties

Thibault Reynaldo, [a] Paul Aillard, [b] [f] Bernard Geffroy, [f] Jeanne Crassous* [a] ring is an interesting subunit for tuning emission properties of helicenes since phosphole efficient fluorescent emitters for emitting diodes (OLEDs). [5,6] Recently, the introduction of a phosphole ring within a helicenic structure has enabled to construct a new class of helical systems, i.e.

these helically chiral compounds, appeared highly promising in both as chiral ligands for goldcatalyzed enantioselective enyne cycloisomerizations, [7d-f] and as in asymmetric [3+2] cyclizations of γ-substituted poor olefins. [7g,h] However, the ies of these heterohelicenes, determining their potential application in OLEDs, have not been reported to , we describe the photophysical (i.e. UV/Vis and chiroptical properties (i.e. CPL activity [4,[START_REF] Field | CPL active helicenes: a)[END_REF] of a series of phosphahelicene oxides 1-4, [7b,d] 5 shown in Figure 1. The experimental data are supported and analyzed with the help of

The physical properties recorded in solution showed that the P-containing helicenes exhibit material that can be inserted as active for a series of built OLED devices potential of these compounds for optoelectronic

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Results and Discussion

UV/Vis absorption and emission of 1-5

Compounds 1-4 have been prepared as described in our previous papers. [7b,d] The phosphahelicene sulfide compound. It has been synthesized from the corresponding oxide 1b, as reported in the Supporting Information (SI)

Compounds 1a and 1b are benzooxophosphole that display carbo [6]helicene units meta-fused oxophosphole ring containing a pendant phenyl ring at position 5. [7d] In both systems the menthyl group is directed toward the inner groove of the helix (endo isomer). compounds can be considered as pseudo they demonstrate reverse stereochemistry of the helix of the P atom (RP/SP) but unchanged stereochemistry chiral l-menthyl group. Consequently, as Figure 2a, 1a,b display almost identical UV consisting of sets of very strong absorption bands ( 1 cm -1 ) between 250-350 nm (i.e. at 276, 323, 345 nm, with the one around 276 nm reaching ε ~ 50x10 3 M -1 bands of medium and low intensity (2.7x10 20x10 3 M -1 cm -1 ) between 350-450 nm. It is also worth noting that the phosphahelicene sulfide 5, which structure and stereochemistry as 1b, displays phosphole derivatives fused with one terminal pendant phenyl ring at position In both systems the menthyl group is directed toward the isomer). These epimeric pseudo-enantiomers since stereochemistry of the helix (P/M) and stereochemistry of the can be seen from identical UV/Vis spectra bands (ε > 20x10 3 M - at 276, 323, 345 nm, with the 1 cm -1 ), and several intensity (2.7x10 3 M -1 cm -1 < ε < It is also worth noting which has the same displays only slightly modified UV/Vis spectrum compared Figure S25 in the SI). This shows function (P=O in 1b vs. P=S influence the corresponding absorption properties that these properties are predominantly determin helical π-electron system of such compounds Compound 2 possesses a similar structure as carbo [6]helicenes 1a,b but displays terminal unit; [7b] accordingly it belongs to phosphafluorenes oxides (dibenzophosphole spectrum shows very intense absorption cm -1 ) between 250-350 nm (i.e. at one at 281 nm reaching ε ~ 6 medium intensity at 370 nm (ε ṽery weak bands at 414 and 438 the low-energy tail in 2 is ipso compared to 1a,b and much less intense.

Compound 3 is a phospha additional pendant phenyl ring at position 5 of the phosphole unit, [7d] while 4 corresponds to an oxidized phosphafluorene molecule with eight ortho-condensed rings; structurally classified as a phospha[8]helicene. It can be clearly seen from Figure 2 that the phosphahelicene oxides display very similar UV/Vis spectr is (a) and ECD (b) spectra of phosphahelicenes 1-4 (in CH2Cl2, r.t., c ~ 10 -5 ystems 1'-4' (see Figure 3). No spectral shift has been applied. Calculated excitation strengths indicated as 'stick' spectra. Numbered excitations correspond to those analyzed in detail. See also SI.

2 compared to compound 1a (see . This shows that the nature of the Pin 5, Figure S25) does not absorption properties and implies predominantly determined by the of such compounds. possesses a similar structure as displays a fused benzophosphole it belongs to a family of dibenzophosphole oxides). Its UV/Vis absorption bands (ε > 20x10 3 M -1 . at 281, 309, 343 nm, with the 0x10 3 M -1 cm -1 ), one band of ~ 17x10 3 M -1 cm -1 ) followed by 8 nm (ε < 10 3 M -1 cm -1 ). Overall ipsochromically (blue) shifted as much less intense. is a phospha[7]helicene oxide with an additional pendant phenyl ring at position 5 of the phosphole corresponds to an oxidized phosphafluorene-type condensed rings; [7b] it can be therefore a phospha[8]helicene. It can be clearly phosphahelicene oxides 3 and 4 is spectra. Indeed, both 3 and 4 mainly

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show a strong band at ~ 283 nm (ε ~ 45-52x10 a set of three other intense bands at ~ 320, 335 and 355 nm (ε ~ 17-25x10 3 M -1 cm -1 ), and one of moderate intensity at 395 nm (ε ~4-7x10 3 M -1 cm -1 ); compound additional low-energy tail above 400 nm (ε <5x10 Overall, adding an ortho-fused phenyl ring, as in influence much the UV/Vis absorption when going from a phospha [7]-(3) to a phospha[8]helicene (4) and appears to be more extended when a phenyl group is grafted at position 5 of the phosphole ring (i.e. in 3). In order to provide insight into experimentally observed features of the oxo-phosphahelicene derivati dependent) DFT calculations [11] were then truncated systems with the l-menthyl and the substituents replaced by methyls, referred to as following and presented in Figure 3. The same set of calculations were also carried out for the full molecule available X-ray crystallographic data [7b] as a starting point in DFT geometry optimization. Results obtained for the two systems, i.e. 2' and 2, appear similar, which supports adopted models. A full account of computational details and results of additional calculations are provided in the SI.

As can be seen in Figures 2a and2c, the calculated BPE0*/SV(P)) [12] UV/Vis absorption spectra agree well with the experimental ones. In particular, the increase in intensity of the low-energy tail of 1 and 3 vs. 2 and reproduced. Computational analysis shows that this increase is linked with a spatially more extended π-conjugat 'pendant-phenyl' compounds compared to their 'fused analogues, which is clearly visible in the frontier molecular orbitals (see Figure 4 and SI position of the phosphole ring relative to the helicene core (meta vs. ortho) seems not to affect a qualitative picture of frontier MOs in the examined systems, unlike 52x10 3 M -1 cm -1 ) and ands at ~ 320, 335 and 355 nm and one of moderate intensity at ); compound 3 displays an <5x10 3 M -1 cm -1 ). , as in 4, does not is absorption when going from a and π-conjugation more extended when a phenyl group is grafted optimized molecular structures of the systems 1'-4' studied as models for the phosphahelicene derivatives 1-4.

In order to provide insight into experimentally observed derivatives 1-4, (timethen performed on menthyl and the n-propyl referred to as 1'-4' in the he same set of calculations were also carried out for the full molecule 2 using as a starting point in Results obtained for the two , appear similar, which supports the omputational details and s are provided in the SI.

, the calculated (LCis absorption spectra agree well with the experimental ones. In particular, the increase in intensity and 4 is correctly reproduced. Computational analysis shows that this increase conjugation in the compared to their 'fused-phenyl' in the corresponding frontier molecular orbitals (see Figure 4 and SI). Note that the the helicene core to affect a qualitative picture of frontier MOs in the examined systems, unlike the nature and substitution of their terminal ring for the 5-phenyl-substituted phosphole derivatives polarization of the electron density towards the phenyl moiety can be noticed in the isosurfaces of HOMO and LUMO, while for their benzo-fused analogues almost evenly spread out over Accordingly, the lowest-energy UV be assigned predominantly as electron system extended over the dangling phenyl substituent (Table 1 corresponding excitation of 3' involves mainly the HOMO/LUMO excitations with significant oscillator strengths in this spectral region -no. 1 that correspond respectively 1,HOMO/LUMO and HOMO,HOMO contributions from the HOMO/LUMO+1 and HOMO pairs in the lowest-energy excitations of phosphole-phenyl→helicene and helicene phenyl charge-transfer (CT) excitation with sizeable oscillator strength in calculated at respectively 352 and 363 nm, is noticeably blue shited and generally less intense as compared aforementioned for 1' and 3'. For both has a large contribution from HOMO/LUMO pair that represents π-π* transition within delocalized across the whole molecule contribution from HOMO-1/LUMO+1 (for 4') π-π* transition mainly observed blue-shift of the low correlates well with the increased HOMO vs. 1', 3' originating mostly from an increase of the LUMO energy level that is seen fused terminal phenyl ring. involvement of P(O)R group was revealed in the low excitations of 1'-4' implying that molecule, increase the size of the its π-electron conjugation up to the terminal phosphole rings. Its effect is thus very similar to that of BR recently reported azaborahelicene derivatives.

The emission properties of phosphahelicene and the phosphahelicene sulfide CH2Cl2 solutions at room temperature and in 2 methyltetrahydrofuran (MeTHF) at 77K at concentration of 10 5 M. At room temperature, all compounds behave as blue fluorescent emitters, with emission wavelengths around 450 nm (Table 2). Beside compounds show highly structured emission coming from the vibronic C=C stretching modes, typical for structures. The main effect of the helicenic core emission properties of 1-5 is temperature quantum yields ( emission spectra (see SI). Indeed, while terminal phosphole oxide units reasonable Φf (7-11%), a drastical decrease in its value is observed for the benzo-fused phospha[8]helicene = 2%) and the phosphahelicene highest quantum yield is obtained for the phosph type phospha [6]helicene oxide 77K, the emission spectra of the rigid phosphafluorene 3 terminal ring (pendant vs. fused). Indeed, substituted phosphole derivatives 1' and 3', polarization of the electron density towards the phenyl moiety can be noticed in the isosurfaces of HOMO and LUMO, while analogues 2' and 4' those orbitals are spread out over the whole molecule.

energy UV/Vis band in 1' and 3' can π-π* transitions within helical πextended over terminal phosphole ring and the dangling phenyl substituent (Table 1 andFigure 4). The 3' (no. 1 calculated at 384 nm) LUMO pair. In the case of 1', two oscillator strengths were computed (376 nm) and no. 2 (367 nm) respectively to HOMO/LUMO+1, HOMOand HOMO,HOMO-1/LUMO transitions. The contributions from the HOMO/LUMO+1 and HOMO-1/LUMO energy excitations of 1' afford a partial helicene and helicene→phosphole-(CT) character. The lowest-energy with sizeable oscillator strength in 2' and 4', no. 2 calculated at respectively 352 and 363 nm, is noticeably blueand generally less intense as compared to those For both 2' and 4' the excitation has a large contribution from HOMO/LUMO pair that * transition within π-electron system delocalized across the whole molecule, and a smaller 1/LUMO+1 (for 2') / HOMO-1/LUMO mainly within helicene core. The of the low-energy UV/Vis absorption increased HOMO-LUMO gap for 2', 4' originating mostly from an increase (destabilization) that is seen for the systems with fused terminal phenyl ring. Note that no noticeable of P(O)R group was revealed in the low-energy implying that its main role is to rigidify the the helical structure and extend up to the terminal phosphole-phenyl Its effect is thus very similar to that of BR2 moiety in our recently reported azaborahelicene derivatives. [4e] The emission properties of phosphahelicene oxides 1-4 sulfide 5 were then examined in solutions at room temperature and in 2methyltetrahydrofuran (MeTHF) at 77K at concentration of 10 - M. At room temperature, all compounds behave as blue emission wavelengths around 450 compounds 3 and 4, systems 1, 2 and 5 show highly structured emission coming from the vibronic C=C stretching modes, typical for helicoidal π-conjugated structures. The main effect of the helicenic core on the is however seen in the room temperature quantum yields (Φf) and their low temperature see SI). Indeed, while helicenes with oxide units, i.e. 1a,b and 3, display %), a drastical decrease in its value is phospha[8]helicene oxide 4 (Φf helicene sulfide 5 (Φf = 1%). The highest quantum yield is obtained for the phosphafluorenephospha [6]helicene oxide 2 (Φf = 32%). Furthermore, at 77K, the emission spectra of the rigid phosphafluorene-based

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compounds 2 and 4 demonstrate fluorescence emission accompanied by phosphorescence which is typical for helicenic structures. [4e] For compounds 1, display pendant phenyl rings at position 5 of the phosphole ring, only fluorescent emission is observed indicating that such freely rotating group promotes non-radiative deactivation

Accepted Manuscript

Chemistry -A European Journal

This article is protected by copyright. All rights reserved. The electrochemical behavior of 1-5 was cyclic voltammetry (CV). The CV curves were recorded in degassed CH2Cl2 solutions that contained 0.2 M of tetrabutylammonium hexafluorophosphate (Bu supporting electrolyte and 10 -3 M of phospha corresponding redox properties are summarized compounds 1, 3 and 5 demonstrate similar oxidized and reduced at around +0.95 V and respectively, however only 3 presents reversible redox processes. The nature of the P-function seems to minor role. The phosphahelicene sulfide 5 oxidize and slightly more difficult to reduce energetic destabilization of the HOMO and the LUMO The electrochemical HOMO-LUMO gap remain for all three compounds bearing the pendant phenyl substituent The phosphafluorene-type compounds 2 and irreversible oxidation process at around +0.95 V '. Values listed in the parentheses are the corresponding orbital energies, in eV was investigated by cyclic voltammetry (CV). The CV curves were recorded in solutions that contained 0.2 M of ntetrabutylammonium hexafluorophosphate (Bu4NPF6) as the phosphahelicene. The summarized in Table 2. The similar behavior. They are and -2.30 V (vs. Fc/Fc + ), presents reversible redox seems to play rather a is slightly easier to more difficult to reduce which implies an of the HOMO and the LUMO levels. LUMO gap remains however similar the pendant phenyl substituent. and 4 exhibit only an oxidation process at around +0.95 V (vs. Fc/Fc + ); no reduction is seen in the electrochemical windows show that a different redox behavior can be observed depending directly on the organophosphorus substructure . All values referenced to the reversible formal potential of the ferrocene/ferrocenium couple. Measured by thermogravimetric analysis at 10% decomposition.

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Chiroptical properties of 1-5

The electronic circular dichroism spectra and SI) and optical rotation parameters ( compounds 1-5 revealed features considered as classical helicene derivatives. The two epimeric M and oxides 1a,b display mirror-image ECD spectra used as an example, showing several strong negative 260, 274, 304, and 315 nm (∆ε = -200, -235, cm -1 , respectively), several intense positive bands at 346, 355, 372, and 392 nm (∆ε = +180, +140, +150, respectively), and a very weak one at 418 nm ( 1). The benzo-fused helicene [6]-M-endo generally similar ECD spectrum to its phenylanalogue [6]-M-endo-1b but overall more intense low-energy bands: ∆ε = +82, +317, -155, -285, 1 at 258, 276, 329, 343, and 366 nm, respectively P-endo-3 and [8]-P-endo-4 display almost identical with negative intensity centered at around 260, 280, ~ -50, -240, and -95 M -1 cm -1 , respectively) and a positive around 355 nm (∆ε ~ +215 to +240 M -1 cm vibronic structure and higher intensity than the corresponding and emission data for phosphahelicenes 1-5 along with their electrochemical properties and decomposition temperature Emission r.t. [a] λmax / nm Φf r.t. [b] Emission 77K [c] λmax / nm E ox / V [d] 449, 476, 509 0.10 F: 432, 443, 465, 501, 533 +0.96 [e] 455, 481, 514 0.07 F: 439, 446, 471, 504, 541 +0.95 [e] 439, 465, 493 0.32 430,458,488 P: 548, 594, 645 +0.96 [e] 456 [e] 456, 479 516 0.01 F: 443, 472, 505 +0.93 [e] at room temperature. [b] Measured relative to quinine sulfate (H2SO4, 0.1 M), Φf ref = 0.546 phosphorescence. [d] Obtained during cyclic voltammetric investigations in 0.1 M Bu referenced to the reversible formal potential of the ferrocene/ferrocenium couple. % decomposition. spectra (ECD, Figure 2b s (ORs, Table 3) of considered as classical for and P phosphahelicene CD spectra with [6]-P-endo-1a, several strong negative bands at 235, -116, and -110 M -1 positive bands at 346, 355, and +103 M -1 cm -1 , and a very weak one at 418 nm (∆ε = +17 M -1 cm - endo-2 demonstrates a -substituted helicene overall more intense and with less 285, and -177 M -1 cm - respectively. Helicenes [7]display almost identical ECD spectra, around 260, 280, and 305 (∆ε , respectively) and a positive band cm -1 ) with unresolved vibronic structure and higher intensity than the corresponding band in [6]-P-endo-1a. Finally, n sulfide [6]-M-endo-5 displays ECD spectrum of its oxide analogue [6]-M-endo medium-energy (> 330 nm) spectral confirming minor influence of the P absorption properties of such compounds

F:
The experimental ECD envelopes for by the TDDFT calculations performed truncated model systems 1'-4' (see Figure 2b,d). Analysis of MO pair contributions to dominant excitations in the low medium-energy regions of the simulated Figure 4, and SI) demonstrates predominant the main positive / negative band indeed, negligible involvement of P(O)R group MOs their lower orbital energies. For phospholes 2' and 4' the band originates no. 3 calculated at 327 and 341 nm, involves HOMO,HOMO-1/LUMO+1 and HOMO/LUMO+1, HOMO-1,HOMO correspond to π-π* transitions within helicene benzophosphole→helicene and helicene signature due to the fact that the engaged MOs are centered in 6 and decomposition temperature values. See E red / V [d] Tdecomp. / °C [f] -2.30 [e] 332 -2.28 [e] 334 -352

-2.20 335 -344 -2.33 [e] = 0.546 ± 15%. [c] Measured in MeTHF at c etric investigations in 0.1 M Bu4NPF6 in CH2Cl2. Platinum electrode referenced to the reversible formal potential of the ferrocene/ferrocenium couple. [e] Irreversible process. [f] inally, note that the phosphahelicene ECD spectrum very similar to that endo-1b, especially in the low-and energy (> 330 nm) spectral regions (see Figure S26), confirming minor influence of the P-function on the (chir)optical absorption properties of such compounds (see above).

The experimental ECD envelopes for 1-4 are well reproduced performed for the corresponding (see Figure 2b,d). Analysis of MOpair contributions to dominant excitations in the low-and simulated ECD spectra (Table 1, demonstrates predominant π-π* character of band in P-1, P-3, P-4 / M-2 with, involvement of P(O)R group MOs likely due to For the isoelectronic benzo-fused the band originates mainly from excitation and 341 nm, respectively. The excitation UMO+1, HOMO-1/LUMO pairs for 2' 1,HOMO-2/LUMO for 4' that all * transitions within helicene π-electron core with helicene and helicene→benzophosphole CT the engaged MOs are centered in

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different parts of the compound's π-system. For the phenylsubstituted phosphahelicenes 1' and 3' two excitations with sizeable rotatory strength values were found to be responsible for the experimentally observed main positive ECD intensity: nos. 2 and 3 for 1' and nos. 3 and 4 for 3'. The higher-energy (shorter-wavelength) excitations, no. 3 for 1' and no. 4 dor 3', calculated for both systems at around 336 nm, have assignment similar to that described above for 2' and 4' (see Table 1, Figure 4, and SI). The lower-energy excitations, no. 2 for 1' and no. 3 for 3', with contributions from HOMO,HOMO-1/LUMO and HOMO/LUMO+2, HOMO-2/LUMO pairs, respectively, correspond to π-π* transitions involving π-system delocalized generally across the whole molecule with non-negligible CT component. Due to its predominant HOMO/LUMO character the excitation no. 2 of 1' has lower energy (367 nm) than the excitation no. 3 of 3' (347 nm) and in consequence it appears responsible for additional low-energy ECD band observed in the experimental spectrum of 1. The above assignment of the simulated ECD spectra of 1'-4' confirms that the primary effect of 1-oxophosphole unit in 1-5 is to extend helicoidal π-electron system up to the phenyl substituent of the phosphole ring that can be involved then in electronic π-π*-type transitions. [4e] The experimental specific (resp. molar) optical rotations of phosphahelicenes 1-5 are gathered in Table 3. Their magnitudes all fall within the same range, i.e. from +2145° cm 3 dm -1 g -1 (resp. +15000° cm 2 dmol -1 ) for [7]-P-endo-3 to +2670 (resp. +17950) for [8]-P-endo-4. Although it is interesting to note that while 3 and 4 display almost identical ECD spectra they show the lowest and highest optical rotation values of the whole series, no visible relation between structures of the examined phosphahelicenes and their ORs is observed. This can be however due to experimental errors in the measurements that are estimated to be around ±5-10%. Calculations performed for the truncated model systems 1'-4' predict qualitatively consistent OR parameters for all the molecules but they give noticeably larger values than the corresponding experimental ones for 1-4 (Table 3 andSI). The overestimation is slightly higher for values obtained using the LC-PBE0* functional employing an 'optimally tuned' range-separation parameter for a helicene [12] than for those calculated with BHLYP and it seems to be predominantly linked with the replacement of the chiral menthyl group in 1-4 by the achiral methyl in 1'-4'. Indeed, computations for the full system 2 leads to visibly lower OR values. This indicates that although the menthyl substituent seems not to determine to any significant degree ECD spectra of phosphahelicenes 1-5 (see SI), it clearly affects their optical rotations. Note that n-propyl groups present in 1-5 can also indirectly -by affecting molecular structures of these systems -influence the corresponding OR values. [13] Circularly polarized luminescence (CPL) activity, which combines emission and chirality, is a particularly appealing chiroptical property of organic chiral chromophores such as helicenic derivatives. [START_REF] Field | CPL active helicenes: a)[END_REF] There is indeed a growing interest in CPL, [14] and CPL-emitters have recently been incorporated into OLEDs to obtain circularly polarized (CP) OLEDs. [2h] The CPL activity of phosphahelicenes 1-5 was therefore also examined via measurements in degassed CH2Cl2 solutions at concentrations around 1x10 -3 M. The epimeric helicenes [6]-P-endo-1a and [6]-M-endo-1b display mirror-image CPL spectra (Figure 5) with luminescence anisotropy factor glum = +8x10 -4 and -7x10 -4 at 452 nm (excitation at 404-416 nm) for P-1a and M-1b, respectively. These values are in the same range as for organic helicene derivatives, [1] and very similar to glum determined for triphenylenebased phospha[7]helicene and benzopicene-based phospha [9]helicene whose CPL activity has been very recently reported. [8d] As for other phosphahelicenes 2-5, no measurable CPL was detected. These compounds may either undergo decompositon when subjected to the Xenon lamp of the CPL experimental set-up or display too low CPL signal to be measured.

System

Expt. [a] Calc. [b] BHLYP LC-PBE0* [a] Measured in CH2Cl2 at c ~ 10 -5 M. [b] Values for P-1', M-2' and M-2 (in parentheses), P-3', and P-4'. See also SI.

OLED application

All the compounds 1-5 present decomposition temperature above 330°C ( Since phosphahelicene 2 presents irreversible redox processes, it was used as a dopant in an m-CP matrix (doping rate of 3% and 4% w for device A and B, respectively; Table 4). As evidenced by the CIE coordinates, a blue fluorescent electroluminescence was obtained, spectrum of which is not affected by the increase of the current density and matches the emission of 2 in the solid state (Figure S36). The emission of m-CP is only slightly observed at 408 nm, which indicates an almost quantitative energy transfer from the m-CP layer to the emitter 2. For devices A and B, luminance and external quantum efficiency are moderate likely due to a low charge transport. The current efficiency (CE) reaches a constant plateau for current densities above 10 mA/cm 2 (Figures S37-38). The maximum CE value is 0.94 and 0.75 cd/A for device A and B, respectively. By increasing the dopant concentration in the m-CP matrix from 3% in device A to 4% in B the efficiency of the OLED drops dramatically from 94% to 52%. Finally, with the aim of preparing CP-OLEDs, i.e. OLEDs emitting a circularly polarized light, the polarization of the light was measured by using a quater-wave polarizer. However, no circular polarization could be detected with this type of device. This result is probably due to the very low glum of compound 2, which could not even be measured in our CPL apparatus. Further studies are currently performed towards optimizing the OLEDs structure in order to produce CP light. Von [a] (V) EQE [b] (%) CE [b] (cd/A) PE [b] (lm/W)

CIE color coordinates

Conclusions

In conclusion, careful analysis of experimental and theoretical photophysical and chiroptical features of phosphahelicenes 1-5 enabled us to reveal factors that influence properties of such compounds. We have highlighted, for instance, the role of the pendant phenyl substituent of the terminal phosphole unit, compared to the fused benzo-group, in the overall extended π-conjugation of phosphahelicene derivatives.

The pendant phenyl group also strongly influences the emissive properties of such systems, both at room temperature and at 77K, with the occurence of deactivating processes that are not observed for the more rigid benzo-fused derivatives. It was also showed that stereochemistry of the menthyl subsituent present at the P-atom has no strong influence on the ECD spectra of phosphahelicenes 1-5, but may affect their corresponding optical rotation values. On the other hand, the nature of the P-function (P=O vs. P=S) was found to have overall negligible influence on all photophysical and chiroptical properties of such compounds. Finally, we showed that the phosphahelicene derivative [6]-Mendo-2 which demonstrated the best performance in terms of emission, electrochemical properties and thermal stability among considered here systems, could be used as a blue-emitting chiral dopant in an OLED device. Further studies are ongoing toward the manufacture of phosphahelicene-based OLEDs displaying circularly polarized electroluminescence.

Experimental Section

Experimental and computational details are described in the Supplementary Information.

Figure 1 .

 1 Figure 1. Chemical structures of the phosphahelicenes work. Men* = l-menthyl.

Figure 2 .

 2 Figure 2. Comparison of the experimental UV/Vis (a) and ECD (b) spectra of phosphahelicenes (d) spectra of their corresponding truncated model systems strengths indicated as 'stick' spectra. Numbered excitations correspond to those analyzed in detail

Figure 3 .

 3 Figure 3. BP/SV(P)-optimized molecular structures of the systems studied as models for the phosphahelicene derivatives

Figure 4 .

 4 Figure 4. Isosurfaces (0.03 au) of frontier MOs of 1'-4'

Figure 5 .

 5 Figure 5. CPL (top, solid lines added to show luminescence spectral line shape) and total luminescence (bottom) spectra of P-1a (top red) and M-1b (top black) in degassed dichloromethane solution (c = 10 -3 M) at 295K, upon excitation at 404 and 416 nm, respectively.

  This article is protected by copyright. All rights reserved. a critical issue for OLED device stability and lifetime. However, taking into account their (photo)physical properties, only 2 could be of potential use as emitting material (EM) in OLEDs. To investigate its performance in OLEDs, we built a series of OLED devices with structures of indium tin oxide (ITO) / copper phthalocyanine (CuPc) (10 nm) / N,N'-di(1-naphtyl)-N,N'diphenyl-(1,1'-biphenyl)-4,4'diamine (α-NPB) (40 nm) / tris(4carbazoyl-9-ylphenyl)amine (TCTA) (10 nm) / 1,3-bis(Ncarbazolyl)benzene (m-CP) : 2 (4 or 3%) (20 nm) / 2,2′,2"-(1,3,5benzinetriyl)-tris(1-phenyl-1-H-benzimidazole) (TPBi) (50 nm) / LiF(1.2nm) / Al (100 nm) where CuPc is the hole injection layer, α-NPB and TCTA are the hole transporting layers, and TPBi is the electron transporting layer (see SI for details).

  CE = current efficiency, and PE = power efficiency recorded at 30 mA/cm 2 corresponding to the maximum efficiency of the devices.

Table 1 .

 1 Selected dominant excitations and occupied (occ) derivatives 1'-4'. This observation shows choice of a phosphole subs benzophosphole, since different deactivation pathways of the excited state are noticed.

							10.1002/chem.201806140
	demonstrate fluorescence emission accompanied by phosphorescence which is typical for , 3 and 5, which at position 5 of the phosphole fluorescent emission is observed indicating that radiative deactivation processes. Selected dominant excitations and occupied (occ) -unoccupied (unocc) MO pair contributions (greater than 10%) of the processes. This observation shows the importance of a choice of a phosphole substructure, phosphafluorene vs. benzophosphole, since different deactivation pathways of the unoccupied (unocc) MO pair contributions (greater than 10%) of the model phosphahelicene
	Excitation	E / eV	λ / nm / nm	f	R / 10 -40 cgs	occ no.	unocc no. unocc no.	%
	P-1'
	1	3.29	376 376	0.089	51.06	136	138	30.8
						135	137	28.2
						136	137	19.9
	2	3.38	367 367	0.366	424.46	136	137	65.1
						135	137	16.8
	3	3.70	335 335	0.219	452.64	135	138	43.8
						136	138	20.0
						134	137	11.7
	M-2'
	2	3.53	352 352	0.102	-116.97	129	130	76.5
						128	131	12.2
	3	3.79	327 327	0.424	-992.61	129	131	36.8
						128	131
						128	130	21.6
	P-3'
	1	3.23	384 384	0.182	27.78	136	137	83.4
	3	3.57	347 347	0.081	415.27	136	139	55.2
						134	137	18.1
	4	3.68	337 337	0.179	483.04	135	137	40.8
						136	138	14.4
						135	139	12.5
						135	138	10.7
	P-4'
	2	3.41	363 363	0.029	4.54	129	130	68.3
						128	130	12.3
	3	3.64	341 341	0.213	1026.96	129	131	37.3
						128	130	22.8
						127	130	21.6
								4

Table 2 .

 2 UV/Vis absorption and emission data for phosphahelicenes also SI.

	Compound	Absorption λmax / nm (ε / M -1 cm -1 ) [a]	Emission λ
		278 (55700), 317 (37200),
	1a	384 (17300), 415 (7500), 347 (29100), 375 (22800),	44
		436 (3700)
		278 (55100), 320 (37000),
	1b	376 (24000),395 (17200), 334 (30500), 360 (23300),	4
		414 (7800), 438 (4100)
	2	281 (60100), 309 (29400), 343 (29100), 370 (16700),	43
		382 (8500), 407 (1180),
		430 (770)
	3	284 (52600), 322 (26900), 338 (25000), 356 (18700),	456, 472
		435 (1900)
		282 (45000), 317 (21700),
	4	334 (19800), 357 (17500), 383 (8400), 408 (1100),	448, 465
		435 (800)
	5	278 (53500), 320 (42800), 345 (35400), 380 (26600),	456, 479
		440 (4100)

Table 3 .

 3 Experimental and calculated optical rotation parameters of the phosphahelicene derivatives 1-5 studied in this work. Specific OR in deg cm 3 dm -1 g -1 and molar OR in deg cm 2 dmol -1 .

Table 2

 2

) indicating good thermal stability which is

Table 4 .

 4 Electroluminescent performance of OLED devices incorporating compound 2 in emitting layer.

		Doping
	Device	rate of 2
		(%)

Acknowledgements

We thank the Ministère de l'Education Nationale, de la Recherche et de la Technologie, the Centre National de la Recherche Scientifique, and the Agence Nationale de la Recherche (HELPHOS, ANR-15-CE29-0012-03) for financial support. G.M. aknowledges the National Institute of Health, Minority Biomedical Research Support (grant 1 SC3 GM089589-08) and the Henry Dreyfus Teacher-Scholar Award.

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Keihann Yavari, [b] Wylliam Delaunay, [a] Nicolas De Rycke, [a,c] Thibault [a] Paul Aillard, [b] Monika Hooper,* [d] Victoria Y. Chang, [e] Gilles Muller, [e] Denis Tondelier, [f] Bernard Geffroy, [f] Arnaud Voituriez,* [b] Angela Marinetti,* [b] Muriel Hissler,* [a] Jeanne Crassous* [a] Page No.

Phosphahelicenes: from chiroptical and photophysical properties to OLED applications 10.1002/chem.201806140 Accepted Manuscript