arXiv:1106.4712v2 [cond-mat.stat-mech] 1 Aug 2011
Matrix Coordinate Bethe Ansatz: Applications to XXZ and ASEP models N. Crampeab,∗ , E. Ragoucyc,† and D. Simond,‡ a
Universit´e Montpellier 2, Laboratoire Charles Coulomb UMR 5221, F-34095 Montpellier, France b
CNRS, Laboratoire Charles Coulomb, UMR 5221, F-34095 Montpellier c
Laboratoire de Physique Th´eorique LAPTH
CNRS and Universit´e de Savoie. 9 chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France. d
LPMA, Universit´e Pierre et Marie Curie, Case Courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, France
Abstract We present the construction of the full set of eigenvectors of the open ASEP and XXZ models with special constraints on the boundaries. The method combines both recent constructions of coordinate Bethe Ansatz and the old method of matrix Ansatz specific to the ASEP. This ”matrix coordinate Bethe Ansatz” can be viewed as a noncommutative coordinate Bethe Ansatz, the non-commutative part being related to the algebra appearing in the matrix Ansatz.
LAPTH-018/2011 ∗
[email protected] [email protected] ‡
[email protected] †
1
Introduction
Questions of integrability of models in statistical and quantum mechanics have been much more studied for periodic systems than for open systems, for which the numbers of particles and excitations may vary. However, open boundary conditions have become central in nonequilibrium physics, for which exactly solvable models are needed to explore new features. Fast advances have been made in the eighties to generalize Yang-Baxter equations to the open case (see [1]). However, although the boundaries which preserve the integrability have been classified quite easily [2, 3], the computation of the eigenvalues and the eigenvectors for the non-diagonal boundaries is a tricky problem. Indeed, no standard method exists to diagonalize completely Sklyanin-type transfer matrices with non-diagonal boundary matrices. Recently, progresses have been made in this direction [4–10], but the problem is far from been solved. The present paper addresses this question for a simple model that is used both in statistical mechanics and in spin chains. The Asymmetric Simple Exclusion Process (ASEP) is the simplest model of transport of hard-core particles along a one-dimensional lattice, that exhibits non-trivial behaviours. Each site of the lattice may be occupied by one particle or empty. On the boundaries, two reservoirs add or remove particles with different rates and create a non-zero flux particle from one to the other. For a lattice of L sites, the state space is 2L -dimensional and it can be mapped exactly to a system of L spin 1/2, the role of the Markov transition matrix being held by the XXZ spin chain Hamiltonian. The role of the reservoirs corresponds in this case to non-diagonal magnetic fields on the boundaries. Although all the coefficients have very different interpretations and may be complex or real depending on the model, the mathematical structure remains the same and integrability techniques can be applied in both cases. In the present paper, ASEP notations are chosen, without any loss of generality if one assumes that the coefficients may become complex. The detailed model is described in section 2, as well as previous integrability results on this model. Although the models are identical, there exists at least one method that has been developed and applied only to the ASEP case: the Matrix Ansatz (or DEHP method) [11]. The reason for this fact is that it requires the a priori knowledge of one eigenvalue to build the corresponding eigenvector; it is precisely the case for stochastic models for which the existence of invariant measures is known. Attempts to establish parallels between Bethe and Matrix Ans¨atze produced various partial results [12, 13] but never a full understanding of the exact relation between both approaches. The present paper shows how to build coordinate Bethe Ansatz eigenvectors from a Matrix Ansatz-type vacuum state in section 3. The plan of the paper is the following. We start in section 2 with a brief summary on open ASEP models. Then, in section 3, we present our method, that can be viewed as a non-commutative coordinate Bethe Ansatz. The non-commutative part is based on the algebra appearing in the matrix Ansatz. We conclude in section 4. Appendices A and B gather technical results needed in our construction. Finally, appendix C is devoted to the study of finite dimensional representations of the matrix Ansatz algebra.
1
2
ASEP Hamiltonians and constraints on boundaries
The Markov transition matrix for the open ASEP model is given by b 1 + KL + W =K
L−1 X
wj,j+1
(2.1)
j=1
where the indices indicate the spaces on which the following matrices act non trivially 0 0 0 0 −s 0 −q p 0 −δ β −α γe b and K = w= 0 q −p 0 , K = αes −γ δ −β 0 0 0 0
(2.2)
It is well-established [14–16] that this ASEP model is related by a similarity transformation to the so-called integrable open XXZ model [1]. Some care must be taken with the properties of the matrix W . In the case of the ASEP with s = 0, the matrix is stochastic but not hermitian. In the context of the ASEP, s is the parameter of the generating function of the current of particles and, for s 6= 0, the matrix W is even not stochastic. In the general case encompassing both ASEP and XXZ, there is no stochastic nor self-adjoint property for W and left and right eigenvectors are generically different. Among the whole set of integrable boundaries, there is one very special subset of constraints for which the resolution is simplified. This subset, in the notation of the ASEP model, is determined by the following constraints between the boundary parameters α, β, γ, δ and s but also of the bulk parameters p and q [9, 10, 16] L−1−n p s cǫ (α, γ)cǫ′ (β, δ) = e for ǫ, ǫ′ = ± and n ∈ {0, 1, 2, . . . , L − 1} (2.3) q c+ (x, y) = y/x and c− (x, y) = −1 . (2.4) The four possible choices for the signs ǫ, ǫ′ and the corresponding constraint are summarized in table 1, as well as the corresponding notation used for the Markov matrix W when the constraints are satisfied. cǫ′ (β, δ)
cǫ (α, γ) c+ (α, γ) =
γ α
c− (α, γ) = −1 c− (α, γ) = −1 c+ (α, γ) =
γ α
c+ (β, δ) =
=1
L−1−n p q
=1
Wn−+
p q
=1
Wn+−
p q
αβ s e γδ
δ β
c− (β, δ) = −1 c+ (β, δ) =
Constraints L−1−n
es
L−1−n
− βδ es
δ β
c− (β, δ) = −1
− αγ es
p q
W
=1
L−1−n
Wn++ Wn−−
Table 1: Possible values for the parameters and the constraints imposed by (2.3) where n takes integer values between 0 and L − 1. ′ b ǫ,ǫ′ the boundary matrices K and K b when the Similarly to W , we denote by Knǫ,ǫ and K n corresponding constraint (2.3) is satisfied. In the ASEP, the parameters α, β, γ and δ are
2
positive. Then, only the first two constraints in the table 1 may be considered as relevant. However, we treat also the last two sets since they become relevant when we map the ASEP problem to the XXZ model. ′ By numerical investigations in [17], it has been established that the whole spectrum of Wnǫ,ǫ is given by two different types of Bethe equations. In previous papers [9, 10], we succeeded in computing the eigenvalues and the left or right eigenvectors corresponding to only one type of Bethe equation via a generalization of the coordinate Bethe Ansatz [18]. It seems that this generalized coordinate Bethe Ansatz is not enough to obtain the missing cases. An advantage of this method consists in giving an interpretation of the number n entering in the constraint: it is the maximal number of pseudo-excitations in the Ansatz. −− For WL−1 (i.e. es = 1), a second method, now called matrix Ansatz (or DEHP method), has been developed in [11] to find the second part of the spectrum. We give the outlines of the historical method in subsection 3.1. Then, we present the new results in subsection 3.3 which consists in a generalization of the matrix Ansatz. It allows us to obtain the second part of the ′ spectrum for any Wnǫ,ǫ , i.e. for any subset of constraints. In [9,10], we proved also that for a new type of constraints the generalized coordinate Bethe Ansatz also provides one part of the spectrum n p ∗ ∗ s for ǫ, ǫ′ = ± and n ∈ {0, 1, . . . , L − 1} (2.5) cǫ (α, γ)cǫ′ (β, δ) = e q p p − q + v − u ± (p − q + v − u)2 + 4uv ∗ . (2.6) c± (u, v) = 2u The four possible choices for the signs ǫ, ǫ′ and the corresponding constraints are summarized ′ in table 2. We denote by Wnǫ,ǫ the matrix W when one constraint (2.5) is satisfied. ǫ
ǫ′
+
+
− − +
−
− +
Constraints n c∗+ (α, γ)c∗+ (β, δ) = es pq n ∗ ∗ s p c− (α, γ)c− (β, δ) = e q n c∗+ (α, γ)c∗− (β, δ) = es pq n ∗ ∗ s p c− (α, γ)c+ (β, δ) = e q
W Wn++ Wn−− Wn+− Wn−+
Table 2: Possible values for the parameters and the constraints imposed by (2.5), where n is an integer between 0 and L − 1. ′
′
As for the matrix Wnǫ,ǫ , in [10], we computed one part of the spectrum for Wnǫ,ǫ Markov matrices thanks to the generalized coordinate Bethe Ansatz. We believe that a very similar procedure as the one presented in following section 3.3 should hold in this case when we transform the matrix as in [10]. We will not give in detail the computations in the present paper.
3
3 3.1
Matrix coordinate Bethe Ansatz Matrix Ansatz without excitation
In this subsection, we present the method introduced in [11] to get the eigenvector with vanishing eigenvalue (i.e. the steady state or the invariant measure) for the Markov process with s = 0. This state corresponds exactly to the part of eigenspace we do not obtain from the −− coordinate Bethe Ansatz for WL−1 in [9, 10]. A particular care is required since it involves various operators acting on different vector spaces. The physical state space, written all along the paper H, is 2L -dimensional and the canonical basis can be indexed by the occupation numbers τi ∈ {0, 1} (resp. spin si ∈ {− 12 , 12 } in the XXZ language) on each site i. All the vectors of H are written in the ket notation |·i and all the dual vectors are written in the bra form h·|. The matrix Ansatz states that the steady state |Φi of the ASEP with s = 0 has components given by: −→ Y hτ1 τ2 . . . τL |Φi = hhV1| (τi D + (1 − τi )E) |V2 ii , (3.1) 1≤i≤L
where the arrow means that the product has to be build from left to right when the index i increases. One has for example h011001|Φi = hhV1|EDDEED|V2ii for L = 6. The noncommuting matrices D and E act on an abstract auxiliary vector space V, that is different from H. The vector |V1 ii lies in this space V (as any vector written with a ket notation |·ii), whereas the vector hhV2| is in its dual V ∗ (as any vector written with a bra notation hh·|). Eq.(3.1) expresses the components of |Φi ∈ H as a scalar product between vectors on the abstract auxiliary space V. −− One checks that |Φi is an eigenvector of WL−1 with vanishing eigenvalue if the two matrices D et E acting on V and the two boundary vectors satisfy the commutation rules: qED − pDE = D + E , hhV1 |(αE − γD + 1) = 0 , (βD − δE + 1)|V2 ii = 0 .
(3.2a) (3.2b) (3.2c)
These three relations reduce quadratic relations in D and E to linear expressions in D and E in the bulk and linear relations in D and E to scalar expressions on the boundaries; it allows one to determine recursively all the components of the eigenvector |Φi and it does not need an explicit representation of the algebra. An irrelevant minus sign relatively to the standard notation for the matrix Ansatz has been introduced in (3.2) for later convenience. In particular, the matrix Ansatz gives an easy access to correlation functions with standard transfer matrix techniques [11]. −− To prove easily that |Φi is an eigenvalue of WL−1 , we firstly rewrite |Φi as follows E E E |Φi = hhV1 | ⊗ ⊗···⊗ |V2 ii (3.3) D D D The convention used in this notation will be used all through the paper and corresponds to E has operator entries instead of scalar ones (two the following interpretation: a vector D 4
dimensional module over the endomorphisms of V), the tensor product of such elements is an element of a 2L -dimensional module over the endomorphisms of V, whose components are products of size L of operators E and D with the usual tensor rule: ′ aa ′ ab′ a a (3.4) ⊗ ′ ≡ ba′ . b b bb′
Let us emphasize on the order of the non-commuting operators in the vector of the r.h.s. of the previous equation. The right action of |V2 ii ∈ V produces a 2L -dimensional vector whose components are vectors of V, that is further projected through the left action of hhV1 | ∈ V ∗ to a 2L -dimensional vector whose components are complex numbers and is thus identified to H. For example, the following vector ′ hhV |aa |V ii 1 2 ′ hhV1 |ab′ |V2 ii a a (3.5) hhV1 | ⊗ ′ |V2 ii ≡ hhV1 |ba′ |V2 ii b b hhV1 |bb′ |V2 ii
is a 4-component vector with complex entries. Secondly, using definitions (2.2) and equations (3.2), we show that E E E 1 1 E ω ⊗ = ⊗ − ⊗ (3.6) D D D −1 −1 D 1 1 E E −− −− b and KL−1 |V2 ii (3.7) = hhV1| |V2 ii = − KL−1 hhV1 | −1 −1 D D
The first equation encodes four relations on the endomorphisms D and E and each equation on the second line two relations between vectors of V or V ∗ . Since this type of result is central in this paper and to be pedagogical, we explain in detail the computation for the second relation in (3.7) (βD − δE)|V2 ii E|V2 ii −δ β E −− (3.8) = KL−1 |V2 ii = (δE − βD)|V2 ii D|V2ii δ −β D 1 |V2 ii |V2 ii (3.9) =− = − −1 −|V2 ii where to go from (3.8) to (3.9) we used (3.2c). Let us remark that we denote by 1 the identity operator acting on V. b do not act in the auxiliary space V where Finally, remarking that the matrices ω, K and K are hhV1 | and |V2 ii, we obtain telescopic terms that can be simplified to get −− WL−1 |Φi = 0
(3.10)
The goal of this paper consists in generalizing (3.3) to obtain also eigenvalues and eigen′ vectors for all the Wnǫ,ǫ . For that purpose, we add some pseudo-excitations above the previous eigenvector as it is done in usual Bethe Ansatz. Before giving the matrix Ansatz with n pseudoexcitations, we need to introduce some vectors playing the role of these excitations as well as some of their properties. 5
3.2
Pseudo-excitations and properties
Let us introduce the notation:
1−u q−p
[u] = and the vectors E − [u] |ω(u)i = uD + [u]
|t(u)i =
u −1
(3.11)
|t(u)i = (p − q)
0 uD + [u]
(3.12)
where the notation |.i (ket in bold) stands for 2-component vectors in which the entries are operators on V (to differentiate them from vectors |.i ∈ H introduced in section 3.1, which contain complex numbers and are obtained after projection with hhV1| and |V2 ii). If the non-commuting matrices D and E satisfy relation (3.2a), we get the following relations w|ω(u)i ⊗ |ω(u)i =|ω(u)i ⊗ |t(u)i − |t(u)i ⊗ |ω(u)i q q q w|ω(v)i ⊗ |ω(v )i =|ω(v)i ⊗ |t(v )i − |t(v)i ⊗ |ω(v )i p p p q w|ω(u)i ⊗ |ω(v)i = − p|ω(u)i ⊗ |ω(v)i + p|ω(v)i ⊗ |ω(u )i p − |t(u)i ⊗ |ω(v)i + |ω(u)i ⊗ |t(v)i q q w|ω(v)i ⊗ |ω(u )i = − q|ω(v)i ⊗ |ω(u )i + q|ω(u)i ⊗ |ω(v)i p p q q + |ω(v)i ⊗ |t(u )i − |t(v)i ⊗ |ω(u )i p p
(3.13a) (3.13b)
(3.13c)
(3.13d)
where u and v are still arbitrary numbers that will be fixed later. The proof of relations (3.13) is straightforward: one projects each relation on the four components then one uses the definition of the [u] numbers as well as the commuting relation (3.2a). Let us remark that relation (3.13a) for u = 1 is similar to relation (3.6). Relations (3.13) deal with the bulk part of W . We study now these vectors on the boundaries. Let us introduce the following functions λ+ (u, v) = 0
and λ− (u, v) = −u − v .
(3.14)
In the following section, we impose the following relation on the right boundary, for ǫ = ±, K|ω(u)i|V2 ii = λǫ (β, δ)|ω(u)i|V2 ii − |t(u)i|V2 ii
(3.15)
where, as explained previously, |ω(u)i|V2 ii means that |V2 ii is right-applied to each entry of |ω(u)i. This relation provides generally two different constraints (K is a two-by-two matrix) on vectors of V except if u = −1/c−ǫ (β, δ). In that case, the remaining constraint becomes (βD − δE + 1)|V2 ii = 0 for u = −1/c−ǫ (β, δ) . We recover the relation (3.2c) for both values of λǫ (β, δ). 6
(3.16)
Similarly, on the left boundary, we impose the following relation b 1 ||ω(u)i = λǫ (α, γ)hhV1||ω(u)i + hhV1 ||t(u)i KhhV
(3.17)
where the notation hhV1||ω(u)i means that hhV1 | is left-applied to each operator entry of |ω(u)i. With a special choice of u, the two constraints reduce again to only one, given by γ (3.18) hhV1 | αes (E − [u]) − s (uD + [u]) + 1 = 0 for u = −e−s c−ǫ (α, γ) . ue
3.3
Matrix Ansatz with excitations
Let us define the tensor product (for modules defined on the endomorphisms of V) of vectors |ω(u)i over the sites i to j and write it as |Ω(u)iji = |ω(u)ii |ω(u)ii+1 . . . |ω(u)ij .
(3.19)
We now fix an integer n and introduce the state with n − m excitations at the ordered positions 1 ≤ xm+1 < . . . < xn ≤ L defined as the (C2 )⊗L -vector in H with the projections: r (xm+1 −1)+...+(xn −1) q |xm+1 , . . . , xn i = (3.20) p
−1 ×hhV1 | |Ω(um+1 )ix1 m+1 −1 |ω(vm+1 )ixm+1 |Ω(um+2 )ixxm+2 ..|ω(vn )ixn |Ω(un+1 )iLxn +1 |V2 ii . m+1 +1 p The overall factor q/p is introduced only in order to normalize the Bethe roots. The coefficients um and vm are related through the recursions: q q um+1 = um , vm+1 = vm , (3.21) p p
and the initial coefficients u1 and v1 are still arbitrary. These vectors correspond to states where m excitations have left the system (through the left boundary). To clarify the notation, the state with no excitation corresponds to m = n and is given by |∅i = hhV1 | |Ω(un+1 )iL1 |V2 ii ,
(3.22)
and, for un+1 = 1, we recover the state |Φi defined in (3.3). We need also to introduce some other definitions concerning the set on which we are going to sum in our Ansatz. The set Gm is a full set of representatives of the coset BCn /BCm (G0 = BCn , by convention) and BCm is the Bm Weyl group, generated by transpositions σj , j = 1, . . . , m − 1 and the reflection r1 (for details, see appendix A). It acts on a vector k = (k1 , . . . , kn ) of Cn : kr1 = (−k1 , k2 , . . . , kn ) ,
kσj = (k1 , . . . , kj+1, kj , . . . , kn ) .
(3.23)
We introduce also the following truncated vector kg(m) , for 0 ≤ m ≤ n and g ∈ Gm , kg(m) = (kg(m+1) , . . . , kgn ) .
(3.24)
We are now in position to state the main result of this paper which provideseigenvalues n ǫ,ǫ′ and eigenstates of WL−1−n i.e. the matrix W with constraint cǫ (α, γ)cǫ′ (β, δ) = es pq . 7
Theorem 3.1 The vector |Φn i =
n X
X
X
(m)
Ag(m) eikg
.x(m)
m=0 xm+1 0, one has c− (u, v) < 0 and c+ (u, v) > 0 (since for uv > 0 one has (p − q + v − u)2 + 4uv > |p − q + v − u|). Thus, for ASEP models, only the constraints corresponding to τ = τ ′ have to be considered (as it is the case for the first choice). We recognize in the framework of matrix Ansatz the same constraints (2.5) that appear for coordinate Bethe Ansatz, although the relation between both approaches are very different.
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[6] R.I. Nepomechie, Bethe Ansatz solution of the open XX spin chain with nondiagonal boundary terms, J. Phys. A34 (2001) 9993 and arXiv:hep-th/0110081; Solving the open XXZ spin chain with nondiagonal boundary terms at roots of unity, Nucl. Phys. B622 (2002) 615 and arXiv:hep-th/0110116; Functional relations and Bethe Ansatz for the XXZ chain, J. Statist. Phys. 111 (2003) 1363 and hep-th/0211001; R. Murgan, R. Nepomechie and Chi Shi, Exact solution of the open XXZ chain with general integrable boundary terms at roots of unity, JSTAT 0608 (2006) P006 and arXiv:hep-th/0605223; L. Frappat, R. Nepomechie and E. Ragoucy, Complete Bethe Ansatz solution of the open spin-s XXZ chain with general integrable boundary terms, JSTAT 0709 (2007) P09008 and arXiv:0707.0653; R.I. Nepomechie, Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms, J. Phys. A37 (2004) 433 and arXiv:hep-th/0304092. [7] H. Frahm, J. Grelik, A. Seel and T. Wirth, Functional Bethe Ansatz methods for the open XXX chain, J. Phys. A44 (2011) 015001 and arXiv:1009.1081. [8] P. Baseilhac and K. Koizumi, Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory, JSTAT 0709 (2007) P09006 and arXiv:hep-th/0703106; P. Baseilhac, New results in the XXZ open spin chain, proceedings of RAQIS07 (Annecyle-Vieux, France) and arXiv:0712.0452. [9] D. Simon, Construction of a coordinate Bethe Ansatz for the asymmetric exclusion process with open boundaries, J. Stat. Mech. (2009) P07017 and arXiv:0903.4968. [10] N. Crampe, E. Ragoucy and D. Simon, Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions, J. Stat. Mech. (2010) P11038 and arXiv:1009.4119. [11] B. Derrida, M. Evans M, V. Hakim and V. Pasquier, Exact solution of a 1d asymmetric exclusion model using a matrix formulation, J. Phys. A26 (1993) 1493. [12] F. C. Alcaraz and M. J. Lazo, The Bethe Ansatz as a matrix product Ansatz, J. Phys. A37 (2004) L1 and arXiv:cond-mat/0304170; Exact solutions of exactly integrable quantum chains by a matrix product Ansatz, J. Phys. A37 (2004) 4149 and arXiv:cond-mat/0312373. [13] O. Golinelli and K. Mallick, Derivation of a matrix product representation for the asymmetric exclusion process from the algebraic Bethe Ansatz, J. Phys. A39 (2006) 10647 and arXiv:cond-mat/0604338. [14] S. Sandow, Partially Asymmetric Exclusion Process with Open Boundaries, Phys. Rev. E50 (1994) 2660 and arXiv:cond-mat/9405073. [15] F. Essler and V. Rittenberg, Representations of the quadratic Algebra and Partially Asymmetric Diffusion with Open Boundaries, J. Phys A29 (1996) 3375 and cond-mat/9506131. 17
[16] J. de Gier and F. Essler, Exact spectral gaps of the asymmetric exclusion process with open boundaries, J. Stat. Mech. (2006) P12011 and cond-mat/0609645. [17] R.I. Nepomechie and F. Ravanini, Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms, J. Phys. A36 (2003) 11391 and arXiv:hep-th/0307095. [18] H. Bethe, Zur Theorie der Metalle. Eigenwerte und Eigenfunktionen Atomkete, Zeitschrift f¨ ur Physik 71 (1931) 205. [19] C. Enaud and B. Derrida, Large deviation functional of the weakly asymmetric exclusion process, J. Stat. Phys. 114 (2004) 537 and arXiv:cond-mat/0307023. [20] V. Belitsky and G.M. Sch¨ utz Diffusion and scattering of shocks in the partially asymmetric simple exclusion process, Elect. Journ. Prob. 7 (2002) 1. [21] K. Krebs, F.H. Jafarpour, and G.M. Sch¨ utz, Microscopic structure of travelling wave solutions in a class of stochastic interacting particle systems, New J. Phys. 5 (2003) 145 and arXiv:cond-mat/0307680. [22] C. Boutillier, P. Fran¸cois, K. Mallick and S. Mallick, A matrix Ansatz for the diffusion of an impurity in the asymmetric exclusion process, J. Phys. A35 (2002) 9703. [23] M. R. Evans, P. A. Ferrari and K. Mallick, Matrix Representation of the Stationary Measure for the Multispecies TASEP, J. Stat. Phys. 135 (2009) 217 and arXiv:0807.0327. [24] S. Corteel, M. Josuat-Verges and L. K. Williams, The Matrix Ansatz, orthogonal polynomials, and permutations, Advances in Applied Mathematics 46 (2011) 209 and arXiv:1005.2696. [25] K. Mallick and S. Sandow, Finite-dimensional representations of the quadratic algebra: Applications to the exclusion process, J. Phys. A30 (1997) 4513 and cond-mat/9705152.
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arXiv:1106.4712v2 [cond-mat.stat-mech] 1 Aug 2011
Matrix Coordinate Bethe Ansatz: Applications to XXZ and ASEP models N. Crampeab,∗ , E. Ragoucyc,† and D. Simond,‡ a
Universit´e Montpellier 2, Laboratoire Charles Coulomb UMR 5221, F-34095 Montpellier, France b
CNRS, Laboratoire Charles Coulomb, UMR 5221, F-34095 Montpellier c
Laboratoire de Physique Th´eorique LAPTH
CNRS and Universit´e de Savoie. 9 chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France. d
LPMA, Universit´e Pierre et Marie Curie, Case Courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, France
Abstract We present the construction of the full set of eigenvectors of the open ASEP and XXZ models with special constraints on the boundaries. The method combines both recent constructions of coordinate Bethe Ansatz and the old method of matrix Ansatz specific to the ASEP. This ”matrix coordinate Bethe Ansatz” can be viewed as a noncommutative coordinate Bethe Ansatz, the non-commutative part being related to the algebra appearing in the matrix Ansatz.
hal-00603322 arXiv:1106.3264 LAPTH-018/2011 ∗
[email protected] [email protected] ‡
[email protected] †
1
Introduction
Questions of integrability of models in statistical and quantum mechanics have been much more studied for periodic systems than for open systems, for which the numbers of particles and excitations may vary. However, open boundary conditions have become central in nonequilibrium physics, for which exactly solvable models are needed to explore new features. Fast advances have been made in the eighties to generalize Yang-Baxter equations to the open case (see [1]). However, although the boundaries which preserve the integrability have been classified quite easily [2, 3], the computation of the eigenvalues and the eigenvectors for the non-diagonal boundaries is a tricky problem. Indeed, no standard method exists to diagonalize completely Sklyanin-type transfer matrices with non-diagonal boundary matrices. Recently, progresses have been made in this direction [4–10], but the problem is far from been solved. The present paper addresses this question for a simple model that is used both in statistical mechanics and in spin chains. The Asymmetric Simple Exclusion Process (ASEP) is the simplest model of transport of hard-core particles along a one-dimensional lattice, that exhibits non-trivial behaviours. Each site of the lattice may be occupied by one particle or empty. On the boundaries, two reservoirs add or remove particles with different rates and create a non-zero flux particle from one to the other. For a lattice of L sites, the state space is 2L -dimensional and it can be mapped exactly to a system of L spin 1/2, the role of the Markov transition matrix being held by the XXZ spin chain Hamiltonian. The role of the reservoirs corresponds in this case to non-diagonal magnetic fields on the boundaries. Although all the coefficients have very different interpretations and may be complex or real depending on the model, the mathematical structure remains the same and integrability techniques can be applied in both cases. In the present paper, ASEP notations are chosen, without any loss of generality if one assumes that the coefficients may become complex. The detailed model is described in section 2, as well as previous integrability results on this model. Although the models are identical, there exists at least one method that has been developed and applied only to the ASEP case: the Matrix Ansatz (or DEHP method) [11]. The reason for this fact is that it requires the a priori knowledge of one eigenvalue to build the corresponding eigenvector; it is precisely the case for stochastic models for which the existence of invariant measures is known. Attempts to establish parallels between Bethe and Matrix Ans¨atze produced various partial results [12, 13] (see however [14] for the case of periodic boundary conditions) but never a full understanding of the exact relation between both approaches. The present paper shows how to build coordinate Bethe Ansatz eigenvectors from a Matrix Ansatz-type vacuum state in section 3. The plan of the paper is the following. We start in section 2 with a brief summary on open ASEP models. Then, in section 3, we present our method, that can be viewed as a non-commutative coordinate Bethe Ansatz. The non-commutative part is based on the algebra appearing in the matrix Ansatz. We conclude in section 4. Appendices A and B gather technical results needed in our construction. Finally, appendix C is devoted to the study of finite dimensional representations of the matrix Ansatz algebra.
1
2
ASEP Hamiltonians and constraints on boundaries
The Markov transition matrix for the open ASEP model is given by b 1 + KL + W =K
L−1 X
wj,j+1
(2.1)
j=1
where the indices indicate the spaces on which the following matrices act non trivially 0 0 0 0 −s 0 −q p 0 −δ β −α γe b and K = w= 0 q −p 0 , K = αes −γ δ −β 0 0 0 0
(2.2)
It is well-established [15–18] that this ASEP model is related by a similarity transformation to the so-called integrable open XXZ model [1]. Some care must be taken with the properties of the matrix W . In the case of the ASEP with s = 0, the matrix is stochastic but not hermitian. In the context of the ASEP, s is the parameter of the generating function of the current of particles and, for s 6= 0, the matrix W is even not stochastic. In the general case encompassing both ASEP and XXZ, there is no stochastic nor self-adjoint property for W and left and right eigenvectors are generically different. Among the whole set of integrable boundaries, there is one very special subset of constraints for which the resolution is simplified. This subset, in the notation of the ASEP model, is determined by the following constraints between the boundary parameters α, β, γ, δ and s but also of the bulk parameters p and q [9, 10, 17, 18] L−1−n p s cǫ (α, γ)cǫ′ (β, δ) = e for ǫ, ǫ′ = ± and n ∈ {0, 1, 2, . . . , L − 1} (2.3) q c+ (x, y) = y/x and c− (x, y) = −1 . (2.4) The four possible choices for the signs ǫ, ǫ′ and the corresponding constraint are summarized in table 1, as well as the corresponding notation used for the Markov matrix W when the constraints are satisfied. cǫ′ (β, δ)
cǫ (α, γ) c+ (α, γ) =
γ α
c− (α, γ) = −1 c− (α, γ) = −1 c+ (α, γ) =
γ α
c+ (β, δ) =
=1
L−1−n p q
=1
Wn−+
p q
=1
Wn+−
p q
αβ s e γδ
δ β
c− (β, δ) = −1 c+ (β, δ) =
Constraints L−1−n
es
L−1−n
− βδ es
δ β
c− (β, δ) = −1
− αγ es
p q
W
=1
L−1−n
Wn++ Wn−−
Table 1: Possible values for the parameters and the constraints imposed by (2.3) where n takes integer values between 0 and L − 1. ′ b ǫ,ǫ′ the boundary matrices K and K b when the Similarly to W , we denote by Knǫ,ǫ and K n corresponding constraint (2.3) is satisfied. In the ASEP, the parameters α, β, γ and δ are
2
positive. Then, only the first two constraints in the table 1 may be considered as relevant. However, we treat also the last two sets since they become relevant when we map the ASEP problem to the XXZ model. ′ By numerical investigations in [19], it has been established that the whole spectrum of Wnǫ,ǫ is given by two different types of Bethe equations. In previous papers [9, 10], we succeeded in computing the eigenvalues and the left or right eigenvectors corresponding to only one type of Bethe equation via a generalization of the coordinate Bethe Ansatz [20]. It seems that this generalized coordinate Bethe Ansatz is not enough to obtain the missing cases. An advantage of this method consists in giving an interpretation of the number n entering in the constraint: it is the maximal number of pseudo-excitations in the Ansatz. −− For WL−1 (i.e. es = 1), a second method, now called matrix Ansatz (or DEHP method), has been developed in [11] to find the second part of the spectrum. We give the outlines of the historical method in subsection 3.1. Then, we present the new results in subsection 3.3 which consists in a generalization of the matrix Ansatz. It allows us to obtain the second part of the ′ spectrum for any Wnǫ,ǫ , i.e. for any subset of constraints. In [9,10], we proved also that for a new type of constraints the generalized coordinate Bethe Ansatz also provides one part of the spectrum n p ∗ ∗ s for ǫ, ǫ′ = ± and n ∈ {0, 1, . . . , L − 1} (2.5) cǫ (α, γ)cǫ′ (β, δ) = e q p p − q + v − u ± (p − q + v − u)2 + 4uv ∗ . (2.6) c± (u, v) = 2u The four possible choices for the signs ǫ, ǫ′ and the corresponding constraints are summarized ′ in table 2. We denote by Wnǫ,ǫ the matrix W when one constraint (2.5) is satisfied. ǫ
ǫ′
+
+
− − +
−
− +
Constraints n c∗+ (α, γ)c∗+ (β, δ) = es pq n ∗ ∗ s p c− (α, γ)c− (β, δ) = e q n c∗+ (α, γ)c∗− (β, δ) = es pq n ∗ ∗ s p c− (α, γ)c+ (β, δ) = e q
W Wn++ Wn−− Wn+− Wn−+
Table 2: Possible values for the parameters and the constraints imposed by (2.5), where n is an integer between 0 and L − 1. ′
′
As for the matrix Wnǫ,ǫ , in [10], we computed one part of the spectrum for Wnǫ,ǫ Markov matrices thanks to the generalized coordinate Bethe Ansatz. We believe that a very similar procedure as the one presented in following section 3.3 should hold in this case when we transform the matrix as in [10]. We will not give in detail the computations in the present paper.
3
3 3.1
Matrix coordinate Bethe Ansatz Matrix Ansatz without excitation
In this subsection, we present the method introduced in [11] to get the eigenvector with vanishing eigenvalue (i.e. the steady state or the invariant measure) for the Markov process with s = 0. This state corresponds exactly to the part of eigenspace we do not obtain from the −− coordinate Bethe Ansatz for WL−1 in [9, 10]. A particular care is required since it involves various operators acting on different vector spaces. The physical state space, written all along the paper H, is 2L -dimensional and the canonical basis can be indexed by the occupation numbers τi ∈ {0, 1} (resp. spin si ∈ {− 12 , 12 } in the XXZ language) on each site i. All the vectors of H are written in the ket notation |·i and all the dual vectors are written in the bra form h·|. The matrix Ansatz states that the steady state |Φi of the ASEP with s = 0 has components given by: −→ Y hτ1 τ2 . . . τL |Φi = hhV1| (τi D + (1 − τi )E) |V2 ii , (3.1) 1≤i≤L
where the arrow means that the product has to be build from left to right when the index i increases. One has for example h011001|Φi = hhV1|EDDEED|V2ii for L = 6. The noncommuting matrices D and E act on an abstract auxiliary vector space V, that is different from H. The vector |V1 ii lies in this space V (as any vector written with a ket notation |·ii), whereas the vector hhV2| is in its dual V ∗ (as any vector written with a bra notation hh·|). Eq.(3.1) expresses the components of |Φi ∈ H as a scalar product between vectors on the abstract auxiliary space V. −− One checks that |Φi is an eigenvector of WL−1 with vanishing eigenvalue if the two matrices D et E acting on V and the two boundary vectors satisfy the commutation rules: qED − pDE = D + E , hhV1 |(αE − γD + 1) = 0 , (βD − δE + 1)|V2 ii = 0 .
(3.2a) (3.2b) (3.2c)
These three relations reduce quadratic relations in D and E to linear expressions in D and E in the bulk and linear relations in D and E to scalar expressions on the boundaries; it allows one to determine recursively all the components of the eigenvector |Φi and it does not need an explicit representation of the algebra. An irrelevant minus sign relatively to the standard notation for the matrix Ansatz has been introduced in (3.2) for later convenience. In particular, the matrix Ansatz gives an easy access to correlation functions with standard transfer matrix techniques [11]. −− To prove easily that |Φi is an eigenvector of WL−1 , we firstly rewrite |Φi as follows E E E |Φi = hhV1 | ⊗ ⊗···⊗ |V2 ii (3.3) D D D The convention used in this notation will be used all through the paper and corresponds to E has operator entries instead of scalar ones (two the following interpretation: a vector D 4
dimensional module over the endomorphisms of V), the tensor product of such elements is an element of a 2L -dimensional module over the endomorphisms of V, whose components are products of size L of operators E and D with the usual tensor rule: ′ aa ′ ab′ a a (3.4) ⊗ ′ ≡ ba′ . b b bb′
Let us emphasize on the order of the non-commuting operators in the vector of the r.h.s. of the previous equation. The right action of |V2 ii ∈ V produces a 2L -dimensional vector whose components are vectors of V, that is further projected through the left action of hhV1 | ∈ V ∗ to a 2L -dimensional vector whose components are complex numbers and is thus identified to H. For example, the following vector ′ hhV |aa |V ii 1 2 ′ hhV1 |ab′ |V2 ii a a (3.5) hhV1 | ⊗ ′ |V2 ii ≡ hhV1 |ba′ |V2 ii b b hhV1 |bb′ |V2 ii
is a 4-component vector with complex entries. Secondly, using definitions (2.2) and equations (3.2), we show that E E E 1 1 E ω ⊗ = ⊗ − ⊗ (3.6) D D D −1 −1 D 1 1 E E −− −− b and KL−1 |V2 ii (3.7) = hhV1| |V2 ii = − KL−1 hhV1 | −1 −1 D D
The first equation encodes four relations on the endomorphisms D and E and each equation on the second line two relations between vectors of V or V ∗ . Since this type of result is central in this paper and to be pedagogical, we explain in detail the computation for the second relation in (3.7) (βD − δE)|V2 ii E|V2 ii −δ β E −− (3.8) = KL−1 |V2 ii = (δE − βD)|V2 ii D|V2ii δ −β D 1 |V2 ii |V2 ii (3.9) =− = − −1 −|V2 ii where to go from (3.8) to (3.9) we used (3.2c). Let us remark that we denote by 1 the identity operator acting on V. b do not act in the auxiliary space V where Finally, remarking that the matrices ω, K and K are hhV1 | and |V2 ii, we obtain telescopic terms that can be simplified to get −− WL−1 |Φi = 0
(3.10)
The goal of this paper consists in generalizing (3.3) to obtain also eigenvalues and eigen′ vectors for all the Wnǫ,ǫ . For that purpose, we add some pseudo-excitations above the previous eigenvector as it is done in usual Bethe Ansatz. Before giving the matrix Ansatz with n pseudoexcitations, we need to introduce some vectors playing the role of these excitations as well as some of their properties. 5
3.2
Pseudo-excitations and properties
Let us introduce the notation:
1−u q−p
[u] = and the vectors E − [u] |ω(u)i = uD + [u]
|t(u)i =
u −1
(3.11)
|t(u)i = (p − q)
0 uD + [u]
(3.12)
where the notation |.i (ket in bold) stands for 2-component vectors in which the entries are operators on V (to differentiate them from vectors |.i ∈ H introduced in section 3.1, which contain complex numbers and are obtained after projection with hhV1| and |V2 ii). If the non-commuting matrices D and E satisfy relation (3.2a), we get the following relations w|ω(u)i ⊗ |ω(u)i =|ω(u)i ⊗ |t(u)i − |t(u)i ⊗ |ω(u)i q q q w|ω(v)i ⊗ |ω(v )i =|ω(v)i ⊗ |t(v )i − |t(v)i ⊗ |ω(v )i p p p q w|ω(u)i ⊗ |ω(v)i = − p|ω(u)i ⊗ |ω(v)i + p|ω(v)i ⊗ |ω(u )i p − |t(u)i ⊗ |ω(v)i + |ω(u)i ⊗ |t(v)i q q w|ω(v)i ⊗ |ω(u )i = − q|ω(v)i ⊗ |ω(u )i + q|ω(u)i ⊗ |ω(v)i p p q q + |ω(v)i ⊗ |t(u )i − |t(v)i ⊗ |ω(u )i p p
(3.13a) (3.13b)
(3.13c)
(3.13d)
where u and v are still arbitrary numbers that will be fixed later. The proof of relations (3.13) is straightforward: one projects each relation on the four components then one uses the definition of the [u] numbers as well as the commuting relation (3.2a). Let us remark that relation (3.13a) for u = 1 is similar to relation (3.6). Relations (3.13) deal with the bulk part of W . We study now these vectors on the boundaries. Let us introduce the following functions λ+ (u, v) = 0
and λ− (u, v) = −u − v .
(3.14)
In the following section, we impose the following relation on the right boundary, for ǫ = ±, K|ω(u)i|V2 ii = λǫ (β, δ)|ω(u)i|V2 ii − |t(u)i|V2 ii
(3.15)
where, as explained previously, |ω(u)i|V2 ii means that |V2 ii is right-applied to each entry of |ω(u)i. This relation provides generally two different constraints (K is a two-by-two matrix) on vectors of V except if u = −1/c−ǫ (β, δ). In that case, the remaining constraint becomes (βD − δE + 1)|V2 ii = 0 for u = −1/c−ǫ (β, δ) . We recover the relation (3.2c) for both values of λǫ (β, δ). 6
(3.16)
Similarly, on the left boundary, we impose the following relation b 1 ||ω(u)i = λǫ (α, γ)hhV1||ω(u)i + hhV1 ||t(u)i KhhV
(3.17)
where the notation hhV1||ω(u)i means that hhV1 | is left-applied to each operator entry of |ω(u)i. With a special choice of u, the two constraints reduce again to only one, given by γ (3.18) hhV1 | αes (E − [u]) − s (uD + [u]) + 1 = 0 for u = −e−s c−ǫ (α, γ) . ue
3.3
Matrix Ansatz with excitations
Let us define the tensor product (for modules defined on the endomorphisms of V) of vectors |ω(u)i over the sites i to j and write it as |Ω(u)iji = |ω(u)ii |ω(u)ii+1 . . . |ω(u)ij .
(3.19)
We now fix an integer n and introduce the state with n − m excitations at the ordered positions 1 ≤ xm+1 < . . . < xn ≤ L defined as the (C2 )⊗L -vector in H with the projections: r (xm+1 −1)+...+(xn −1) q |xm+1 , . . . , xn i = (3.20) p
−1 ×hhV1 | |Ω(um+1 )ix1 m+1 −1 |ω(vm+1 )ixm+1 |Ω(um+2 )ixxm+2 ..|ω(vn )ixn |Ω(un+1 )iLxn +1 |V2 ii . m+1 +1 p The overall factor q/p is introduced only in order to normalize the Bethe roots. The coefficients um and vm are related through the recursions: q q um+1 = um , vm+1 = vm , (3.21) p p
and the initial coefficients u1 and v1 are still arbitrary. These vectors correspond to states where m excitations have left the system (through the left boundary). To clarify the notation, the state with no excitation corresponds to m = n and is given by |∅i = hhV1 | |Ω(un+1 )iL1 |V2 ii ,
(3.22)
and, for un+1 = 1, we recover the state |Φi defined in (3.3). We need also to introduce some other definitions concerning the set on which we are going to sum in our Ansatz. The set Gm is a full set of representatives of the coset BCn /BCm (G0 = BCn , by convention) and BCm is the Bm Weyl group, generated by transpositions σj , j = 1, . . . , m − 1 and the reflection r1 (for details, see appendix A). It acts on a vector k = (k1 , . . . , kn ) of Cn : kr1 = (−k1 , k2 , . . . , kn ) ,
kσj = (k1 , . . . , kj+1, kj , . . . , kn ) .
(3.23)
We introduce also the following truncated vector kg(m) , for 0 ≤ m ≤ n and g ∈ Gm , kg(m) = (kg(m+1) , . . . , kgn ) .
(3.24)
We are now in position to state the main result of this paper which provideseigenvalues n ǫ,ǫ′ and eigenstates of WL−1−n i.e. the matrix W with constraint cǫ (α, γ)cǫ′ (β, δ) = es pq . 7
Theorem 3.1 The vector |Φn i =
n X
X
X
(m)
Ag(m) eikg
.x(m)
m=0 xm+1 0, one has c− (u, v) < 0 and c+ (u, v) > 0 (since for uv > 0 one has (p − q + v − u)2 + 4uv > |p − q + v − u|). Thus, for ASEP models, only the constraints corresponding to τ = τ ′ have to be considered (as it is the case for the first choice). We recognize in the framework of matrix Ansatz the same constraints (2.5) that appear for coordinate Bethe Ansatz, although the relation between both approaches are very different.
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