Multiple reference states and complete spectrum of the Z n Belavin model with open boundaries
aa r X i v : . [ h e p - t h ] J un Multiple reference states and complete spectrum ofthe Z n Belavin model with open boundaries
Wen-Li Yang , a,b and Yao-Zhong Zhang b a Institute of Modern Physics, Northwest University, Xian 710069, P.R. China b Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia
Abstract
The multiple reference state structure of the Z n Belavin model with non-diagonalboundary terms is discovered. It is found that there exist n reference states, each ofthem yields a set of eigenvalues and Bethe Ansatz equations of the transfer matrix.These n sets of eigenvalues together constitute the complete spectrum of the model. Inthe quasi-classic limit, they give the complete spectrum of the corresponding Gaudinmodel. PACS:
Keywords : Algebraic Bethe Ansatz; Z n Belavin model; Gaudin model.
Introduction
Our understanding of quantum phase transitions and critical phenomena has been greatlyenhanced by the study of exactly solvable models (integrable models) [1]. Such exact resultsprovide valuable insights into the key theoretical development of universality classes in areasranging from modern condensed physics [2] to string and super-symmetric Yang-Mills the-ories [3]. Among solvable models, elliptic ones stand out as a particularly important classdue to the fact that most other ones can be obtained from them by some trigonometric orrational limits. In this paper, we focus on the elliptic Z n Belavin model [4] with integrableboundary conditions, with the celebrated XYZ spin chain as the special case of n = 2.Two-dimensional integrable models have traditionally been solved by imposing periodicboundary conditions. For such bulk systems, the quantum Yang-Baxter equation (QYBE) R ( u − u ) R ( u − u ) R ( u − u ) = R ( u − u ) R ( u − u ) R ( u − u ) , (1.1)leads to families of commuting row-to-row transfer matrix which may be diagonalized by thequantum inverse scattering method (QISM) (or algebraic Bethe Ansatz) [5].Not all boundary conditions are compatible with integrability in the bulk. The bulkintegrability is only preserved when one imposes certain boundary conditions. In [6], Sklyanindeveloped the boundary QISM, which may be used to describe integrable systems on a finiteinterval with independent boundary conditions at each end. This boundary QISM uses anew algebraic structure, the reflection equation (RE) algebra. The solutions to the REand its dual are called boundary K-matrices which in turn give rise to boundary conditionscompatible with the integrability of the bulk model [6, 7, 8].The boundary QISM has been successfully applied to diagonalize the double-row transfermatrices of various integrable models with non-trivial boundary conditions mostly corre-sponding to the diagonal K-matrices. The problem of diagonalizing the double-row transfermatrix for most general non-diagonal K-matrices by the algebraic Bethe Ansatz has beenlong-standing due to the difficulty of finding suitable reference states (or pseudo-vacuumstates). Recently, much progress has been made for the open XXZ spin chain. Bethe Ansatzsolutions for non-diagonal boundary terms where the boundary parameters obey some con-straints have been proposed by various approaches [9, 10, 11, 12, 13, 14, 15, 16]. It has been Solutions with arbitrary boundary parameters were recently proposed by functional Bethe Ansatz [17]and q-Onsager algebra [18]. However, it seems highly non-trivial to rederive these results in the frameworkof algebraic Bethe Ansatz . needed [19, 15], in contrast with thediagonal boundary case [6]. This suggests that in the framework of algebraic Bethe Ansatzthere should exist two reference states corresponding to the two sets of Bethe Ansatz equa-tions and eigenvalues. Such multiple reference state structure was confirmed in our recentwork [20].However, for models related to higher rank algebras [21, 22, 23, 24, 25, 26] only onereference state, and consequently only one set of eigenvalues and Bethe Ansatz equationsof their transfer matrices, have been constructed so far. It is natural to expect that thereexist extra reference states, giving rise to extra sets of eigenvalues and associated BetheAnsatz equations for such models. In this paper we investigate the multiple reference statestructure for an elliptic model related to A n − algebra - the Z n Belavin model with generalnon-diagonal boundary terms. It is found that there actually exist n reference states forsuch a model. Each of these reference states yields a set of eigenvalues and correspondingBethe Ansatz equations of the transfer matrix. The n sets of eigenvalues together constitutethe complete spectrum of the model. In the quasi-classic limit, they give the correspondingspectrum of the associated Gaudin model [27].The paper is organised as follows. In section 2, we briefly review the Z n Belavin modelwith integrable open boundary conditions, which also serves as an introduction to our notionand basic ingredients. In section 3, we introduce the intertwiner vectors and the correspond-ing face-vertex correspondence relations which will play key roles in transforming the modelin “vertex picture” to the one in the “face picture”. In section 4, after finding the n referencestates, we use the algebraic Bethe Ansatz to obtain the corresponding n sets of eigenvaluesand the associated n sets of Bethe Ansatz equations of the transfer matrix of the model. Insection 5, we take the quasi-classic limit to extract the spectrum of the associated Gaudinoperators. Section 6 is for conclusion. The Appendix provides the definitions of some ele-mentary functions appeared in section 4 and 5.3 Z n Belavin model with open boundaries
Let us fix a positive integer n ≥
2, a complex number τ such that Im ( τ ) > w . Introduce the following elliptic functions θ (cid:20) ab (cid:21) ( u, τ ) = ∞ X m = −∞ exp (cid:8) √− π (cid:2) ( m + a ) τ + 2( m + a )( u + b ) (cid:3)(cid:9) , (2.1) θ ( j ) ( u ) = θ (cid:20) − jn (cid:21) ( u, nτ ) , σ ( u ) = θ (cid:20) (cid:21) ( u, τ ) , (2.2) ζ ( u ) = ∂∂u { ln σ ( u ) } . (2.3)Among them the σ -function satisfies the following identity: σ ( u + x ) σ ( u − x ) σ ( v + y ) σ ( v − y ) − σ ( u + y ) σ ( u − y ) σ ( v + x ) σ ( v − x )= σ ( u + v ) σ ( u − v ) σ ( x + y ) σ ( x − y ) . Let g, h, be n × n matrices with the elements h ij = δ i +1 j , g ij = ω i δ i j , with ω = e π √− n , i, j ∈ Z n . For any α = ( α , α ), α , α ∈ Z n , one introduces an n × n matrix I α by I α = I ( α ,α ) = g α h α , and an elliptic function σ α ( u ) by σ α ( u ) = θ (cid:20) + α n + α n (cid:21) ( u, τ ) , and σ (0 , ( u ) = σ ( u ) . The Z n Belavin R-matrix is [4, 28] R B ( u ) = σ ( w ) σ ( u + w ) X α ∈ Z n σ α ( u + wn ) nσ α ( wn ) I α ⊗ I − α . (2.4)The R-matrix satisfies the QYBE (1.1) and the properties [28],Unitarity : R B ( u ) R B ( − u ) = id , (2.5)Crossing-unitarity : ( R B ) t ( − u − nw )( R B ) t ( u ) = σ ( u ) σ ( u + nw ) σ ( u + w ) σ ( u + nw − w ) id , (2.6)Quasi-classical property : R B ( u ) | w → = id . (2.7) Our σ -function is the ϑ -function ϑ ( u ) [29]. It has the following relation with the Weierstrassian σ -function denoted by σ w ( u ): σ w ( u ) ∝ e η u σ ( u ), η = π ( − P ∞ n =1 nq n − q n ) and q = e √− τ . Consequently,our ζ -function (2.3) is different from the Weierstrassian ζ -function by an additional term − η u . R B ( u ) = P R B ( u ) P with P being the usual permutation operator and t i denotesthe transposition in the i -th space. Here and below we adopt the standard notation: forany matrix A ∈ End( C n ), A j is an operator embedded in the tensor space C n ⊗ C n ⊗ · · · ,which acts as A on the j -th space and as an identity on the other factor spaces; R ij ( u ) is anembedding operator of R-matrix in the tensor space, which acts as an identity on the factorspaces except for the i -th and j -th ones. The quasi-classical properties (2.7) of the R-matrixenables one to introduce the associated classical Z n r-matrix r ( u ) [30] R B ( u ) = id + w r ( u ) + O ( w ) , when w −→ ,r ( u ) = 1 − nn ζ ( u ) + X α ∈ Z n − (0 , σ ′ (0) σ α ( u ) nσ ( u ) σ α (0) I α ⊗ I − α , σ ′ (0) = ∂∂u σ ( u ) | u =0 . (2.8)In the above equation, the elliptic ζ -function is defined in (2.3).One introduces the “row-to-row” monodromy matrix T ( u ) [5], which is an n × n matrixwith elements being operators acting on ( C n ) ⊗ N T ( u ) = R B ( u + z ) R B ( u + z ) · · · R B N ( u + z N ) . (2.9)Here { z i | i = 1 , · · · , N } are arbitrary free complex parameters which are usually called inho-mogeneous parameters. With the help of the QYBE (1.1), one can show that T ( u ) satisfiesthe so-called “RLL” relation R B ( u − v ) T ( u ) T ( v ) = T ( v ) T ( u ) R B ( u − v ) . (2.10)An integrable open chain can be constructed as follows [6]. Let us introduce a pair ofK-matrices K − ( u ) and K + ( u ). The former satisfies the RE R B ( u − u ) K − ( u ) R B ( u + u ) K − ( u )= K − ( u ) R B ( u + u ) K − ( u ) R B ( u − u ) , (2.11)and the latter satisfies the dual RE R B ( u − u ) K +1 ( u ) R B ( − u − u − nw ) K +2 ( u )= K +2 ( u ) R B ( − u − u − nw ) K +1 ( u ) R B ( u − u ) . (2.12)For the models with open boundaries, instead of the standard “row-to-row” monodromymatrix T ( u ) (2.9), one needs the “double-row” monodromy matrix T ( u ) T ( u ) = T ( u ) K − ( u ) T − ( − u ) . (2.13)5sing (2.10) and (2.11), one can prove that T ( u ) satisfies R B ( u − u ) T ( u ) R B ( u + u ) T ( u ) = T ( u ) R B ( u + u ) T ( u ) R B ( u − u ) . (2.14)Then the double-row transfer matrix of the inhomogeneous Z n Belavin model with openboundary is given by τ ( u ) = tr ( K + ( u ) T ( u )) . (2.15)The commutativity of the transfer matrices[ τ ( u ) , τ ( v )] = 0 , (2.16)follows as a consequence of (1.1)-(2.6) and (2.11)-(2.12). This ensures the integrability ofthe inhomogeneous Z n Belavin model with open boundary.In this paper, we consider a non-diagonal
K-matrix K − ( u ) which is a solution to the RE(2.11) associated with the Z n Belavin R-matrix [31] K − ( u ) st = n X i =1 σ ( λ i + ξ − u ) σ ( λ i + ξ + u ) φ ( s ) λ,λ − w ˆ ı ( u ) ¯ φ ( t ) λ,λ − w ˆ ı ( − u ) . (2.17)The corresponding dual K-matrix K + ( u ) which is a solution to the dual RE (2.12) has beenobtained in [32]. With a particular choice of the free boundary parameters with respect to K − ( u ), we introduce the corresponding dual K-matrix K + ( u ) K + ( u ) st = n X i =1 (Y k = i σ (( λ i − λ k ) − w ) σ ( λ i − λ k ) ) σ ( λ i + ¯ ξ + u + nw ) σ ( λ i + ¯ ξ − u − nw ) × φ ( s ) λ,λ − w ˆ ı ( − u ) ˜ φ ( t ) λ,λ − w ˆ ı ( u ) . (2.18)In (2.17) and (2.18), φ, ¯ φ, ˜ φ are intertwiners which will be specified in section 4. We considerthe generic { λ i } such that λ i = λ j ( modulo Z + τ Z ) for i = j . This condition is necessary forthe non-singularity of K ± ( u ). It is convenient to introduce a vector λ = P ni =1 λ i ǫ i associatedwith the boundary parameters { λ i } , where { ǫ i , i = 1 , · · · , n } is the orthonormal basis of thevector space C n such that h ǫ i , ǫ j i = δ ij . Moreover, in the following we always assume thatthe suffix index of the parameter λ i takes value in Z n cyclic group, namely, λ i ± n = λ i , i = 1 , . . . , n. (2.19)6 A (1) n − SOS R-matrix and face-vertex correspondence
The A n − simple roots { α i } can be expressed in terms of the orthonormal basis { ǫ i } as: α i = ǫ i − ǫ i +1 , i = 1 , · · · , n − , and the fundamental weights { Λ i | i = 1 , · · · , n − } satisfying h Λ i , α j i = δ ij are given byΛ i = i X k =1 ǫ k − in n X k =1 ǫ k . Set ˆ ı = ǫ i − ǫ, ǫ = 1 n n X k =1 ǫ k , i = 1 , · · · , n, then n X i =1 ˆ ı = 0 . (3.1)For each dominant weight Λ = P n − i =1 a i Λ i , a i ∈ Z + (the set of non-negative integers), thereexists an irreducible highest weight finite-dimensional representation V Λ of A n − with thehighest vector | Λ i . For example the fundamental vector representation is V Λ .Let h be the Cartan subalgebra of A n − and h ∗ be its dual. A finite dimensional diagonal-izable h -module is a complex finite dimensional vector space W with a weight decomposition W = ⊕ µ ∈ h ∗ W [ µ ], so that h acts on W [ µ ] by x v = µ ( x ) v , for any x ∈ h , v ∈ W [ µ ].For example, the fundamental vector representation V Λ = C n , the non-zero weight spaces W [ˆ ı ] = C ǫ i , i = 1 , · · · , n .For a generic m ∈ C n , define m i = h m, ǫ i i , m ij = m i − m j = h m, ǫ i − ǫ j i , i, j = 1 , · · · , n. (3.2)Let R ( u, m ) ∈ End ( C n ⊗ C n ) be the R-matrix of the A (1) n − SOS model [33], R ( u, m ) = n X i =1 R iiii ( u, m ) E ii ⊗ E ii + X i = j (cid:8) R ijij ( u, m ) E ii ⊗ E jj + R jiij ( u, m ) E ji ⊗ E ij (cid:9) , (3.3)where E ij is the matrix with elements ( E ij ) lk = δ jk δ il . The coefficient functions are R iiii ( u, m ) = 1 , R ijij ( u, m ) = σ ( u ) σ ( m ij − w ) σ ( u + w ) σ ( m ij ) , i = j, (3.4) R j iij ( u, m ) = σ ( w ) σ ( u + m ij ) σ ( u + w ) σ ( m ij ) , i = j, (3.5)7nd m ij are defined in (3.2). The R-matrix satisfies the dynamical (modified) QYBE R ( u − u , m − wh (3) ) R ( u − u , m ) R ( u − u , m − wh (1) )= R ( u − u , m ) R ( u − u , m − wh (2) ) R ( u − u , m ) , (3.6)and the quasi-classical property R ( u, m ) | w → = id . (3.7)We adopt the notation: R ( u, m − wh (3) ) acts on a tensor v ⊗ v ⊗ v as R ( u, m − wµ ) ⊗ id if v ∈ W [ µ ]. Moreover, the R-matrix satisfies the unitarity and the modified crossing-unitarityrelation.Let us introduce n intertwiner vectors which are n -component column vectors φ m,m − w ˆ ( u )labelled by ˆ ( j = 1 , . . . , n ). The k -th element of φ m,m − w ˆ ( u ) is given by φ ( k ) m,m − w ˆ ( u ) = θ ( k ) ( u + nm j ) . (3.8)We remark that the n intertwiner vectors φ m,m − w ˆ ( u ) are linearly independent for a generic m ∈ C n .Using the intertwining vector, one derives the following face-vertex correspondence rela-tion [33] R B ( u − u ) φ m,m − w ˆ ı ( u ) ⊗ φ m − w ˆ ı,m − w (ˆ ı +ˆ ) ( u )= X k,l R ( u − u , m ) klij φ m − w ˆ l,m − w (ˆ l +ˆ k ) ( u ) ⊗ φ m,m − w ˆ l ( u ) . (3.9)Then the QYBE (1.1) of the Z n Belavin’s R-matrix R B ( u ) is equivalent to the dynamicalYang-Baxter equation (3.6) of the A (1) n − SOS R-matrix R ( u, m ). For a generic m , we mayintroduce other types of intertwiners ¯ φ, ˜ φ satisfying the conditions, n X k =1 ¯ φ ( k ) m,m − w ˆ µ ( u ) φ ( k ) m,m − w ˆ ν ( u ) = δ µν , (3.10) n X k =1 ˜ φ ( k ) m + w ˆ µ,m ( u ) φ ( k ) m + w ˆ ν,m ( u ) = δ µν , (3.11)from which one can derive the relations, n X µ =1 ¯ φ ( i ) m,m − w ˆ µ ( u ) φ ( j ) m,m − w ˆ µ ( u ) = δ ij , (3.12) n X µ =1 ˜ φ ( i ) m + w ˆ µ,m ( u ) φ ( j ) m + w ˆ µ,m ( u ) = δ ij . (3.13)8ith the help of (3.10)-(3.13), we obtain the following relations from the face-vertex corre-spondence relation (3.9): (cid:16) ˜ φ m + w ˆ k,m ( u ) ⊗ id (cid:17) R B ( u − u ) (cid:16) id ⊗ φ m + w ˆ ,m ( u ) (cid:17) = X i,l R ( u − u , m ) klij ˜ φ m + w (ˆ ı +ˆ ) ,m + w ˆ ( u ) ⊗ φ m + w (ˆ k +ˆ l ) ,m + w ˆ k ( u ) , (3.14) (cid:16) ˜ φ m + w ˆ k,m ( u ) ⊗ ˜ φ m + w (ˆ k +ˆ l ) ,m + w ˆ k ( u ) (cid:17) R B ( u − u )= X i,j R ( u − u , m ) klij ˜ φ m + w (ˆ ı +ˆ ) ,m + w ˆ ( u ) ⊗ ˜ φ m + w ˆ ,m ( u ) , (3.15) (cid:16) id ⊗ ¯ φ m,m − w ˆ l ( u ) (cid:17) R B ( u − u ) (cid:16) φ m,m − w ˆ ı ( u ) ⊗ id (cid:17) = X k,j R ( u − u , m ) klij φ m − w ˆ l,m − w (ˆ k +ˆ l ) ( u ) ⊗ ¯ φ m − w ˆ ı,m − w (ˆ ı +ˆ ) ( u ) , (3.16) (cid:16) ¯ φ m − w ˆ l,m − w (ˆ k +ˆ l ) ( u ) ⊗ ¯ φ m,m − w ˆ l ( u ) (cid:17) R B ( u − u )= X i,j R ( u − u , m ) klij ¯ φ m,m − w ˆ ı ( u ) ⊗ ¯ φ m − w ˆ ı,m − w (ˆ ı +ˆ ) ( u ) . (3.17)The face-vertex correspondence relations (3.9) and (3.14)-(3.17) will play an important rolein translating formulas in the “vertex form” into those in the “face form”.Corresponding to the vertex type K-matrices (2.17) and (2.18), one introduces the fol-lowing face type K-matrices K and ˜ K [32] K ( λ | u ) ji = X s,t ˜ φ ( s ) λ − w (ˆ ı − ˆ ) , λ − w ˆ ı ( u ) K ( u ) st φ ( t ) λ, λ − w ˆ ı ( − u ) , (3.18)˜ K ( λ | u ) ji = X s,t ¯ φ ( s ) λ, λ − w ˆ ( − u ) ˜ K ( u ) st φ ( t ) λ − w (ˆ − ˆ ı ) , λ − w ˆ ( u ) . (3.19)Through straightforward calculations, we find the face type K-matrices simultaneously have diagonal forms K ( λ | u ) ji = δ ji k ( λ | u ; ξ ) i , ˜ K ( λ | u ) ji = δ ji ˜ k ( λ | u ) i , (3.20)where functions k ( λ | u ; ξ ) i , ˜ k ( λ | u ) i are given by k ( λ | u ; ξ ) i = σ ( λ i + ξ − u ) σ ( λ i + ξ + u ) , (3.21)˜ k ( λ | u ) i = ( n Y k = i,k =1 σ ( λ ik − w ) σ ( λ ik ) ) σ ( λ i + ¯ ξ + u + nw ) σ ( λ i + ¯ ξ − u − nw ) . (3.22) As will be seen below (see (4.27)), the spectral parameter u and the boundary parameter ξ of the reduceddouble-row monodromy matrices constructed from K ( λ | u ) will be shifted in each step of the nested BetheAnsatz procedure [21]. Therefore, it is convenient to specify the dependence on the boundary parameter ξ of K ( λ | u ) in addition to the spectral parameter u . K ( λ | u ) and ˜ K ( λ | u ) satisfy the SOS type reflectionequation and its dual, respectively [32]. Although the K-matrices K ± ( u ) given by (2.17) and(2.18) are generally non-diagonal (in the vertex form), after the face-vertex transformations(3.18) and (3.19), the face type counterparts K ( λ | u ) and ˜ K ( λ | u ) become simultaneously diagonal. This fact enables the authors to apply the generalized algebraic Bethe Ansatzmethod developed in [21] for SOS type integrable models to diagonalize the transfer matrix τ ( u ) (2.15). By means of (3.12), (3.13), (3.19) and (3.20), the transfer matrix τ ( u ) (2.15) can be recastinto the following face type form: τ ( u ) = tr ( K + ( u ) T ( u ))= X µ,ν tr (cid:16) K + ( u ) φ λ − w (ˆ µ − ˆ ν ) ,λ − w ˆ µ ( u ) ˜ φ λ − w (ˆ µ − ˆ ν ) ,λ − w ˆ µ ( u ) T ( u ) φ λ,λ − w ˆ µ ( − u ) ¯ φ λ,λ − w ˆ µ ( − u ) (cid:17) = X µ,ν ¯ φ λ,λ − w ˆ µ ( − u ) K + ( u ) φ λ − w (ˆ µ − ˆ ν ) ,λ − w ˆ µ ( u ) ˜ φ λ − w (ˆ µ − ˆ ν ) ,λ − w ˆ µ ( u ) T ( u ) φ λ,λ − w ˆ µ ( − u )= X µ,ν ˜ K ( λ | u ) µν T ( λ | u ) νµ = X µ ˜ k ( λ | u ) µ T ( λ | u ) µµ . (4.1)Here we have introduced the face-type double-row monodromy matrix T ( λ | u ), T ( λ | u ) νµ = ˜ φ λ − w (ˆ µ − ˆ ν ) ,λ − w ˆ µ ( u ) T ( u ) φ λ,λ − w ˆ µ ( − u ) ≡ X i,j ˜ φ ( j ) λ − w (ˆ µ − ˆ ν ) ,λ − w ˆ µ ( u ) T ( u ) ji φ ( i ) λ,λ − w ˆ µ ( − u ) . (4.2)This face-type double-row monodromy matrix can be expressed in terms of the face typeR-matrix R ( u, λ ) (3.3) and the K-matrix K ( λ | u ) (3.18) [21]. Moreover from (2.14), (3.9) and(3.13) one may derive the following exchange relations among T ( m | u ) νµ : X i ,i X j ,j R ( u − u , m ) i j i j T ( m + w (ˆ + ˆ ı ) | u ) i i × R ( u + u , m ) j i j i T ( m + w (ˆ + ˆ ı ) | u ) j j = X i ,i X j ,j T ( m + w (ˆ + ˆ ı ) | u ) j j R ( u + u , m ) i j i j × T ( m + w (ˆ + ˆ ı ) | u ) i i R ( u − u , m ) j i j i . (4.3)10ollowing [21] and motivated by our recent work [20] for the open XXZ chain, we introduce n sets of operators {A ( s ) , B ( s ) , C ( s ) , D ( s ) } , labelled by s = 1 , . . . , n , as follows: A ( s ) ( m | u ) = T ( m | u ) ss , B ( s ) i ( m | u ) = T ( m | u ) si σ ( m is ) , C ( s ) i ( m | u ) = T ( m | u ) is σ ( m si ) , i = s, (4.4) D ( s ) ji ( m | u ) = σ ( m js − δ ij w ) σ ( m is ) (cid:8) T ( m | u ) ji − δ ji R (2 u, m + w ˆ s ) j ss j A ( s ) ( m | u ) (cid:9) , i, j = s. (4.5)Some remarks are in order. Among the n sets of operators, the first set (corresponding to s = 1) is the very one which was used in [21, 23] to construct the algebraic Bethe Ansatz.Such algebraic Bethe Ansatz based on the first set of operators only gives rise to the firstset of eigenvalues and associated Bethe Ansatz equations. In order to find the complete setsof eigenvalues, we find that the whole n sets of operators {A ( s ) , B ( s ) , C ( s ) , D ( s ) | s = 1 , . . . , n } are needed.After tedious calculations analogous to those in [21], we have found the commutationrelations among A ( s ) ( m | u ), D ( s ) ( m | u ) and B ( s ) ( m | u ) from (4.3). Here we give those whichare relevant for our purpose A ( s ) ( m | u ) B ( s ) i ( m + w (ˆ ı − ˆ s ) | v )= σ ( u + v ) σ ( u − v − w ) σ ( u + v + w ) σ ( u − v ) B ( s ) i ( m + w (ˆ ı − ˆ s ) | v ) A ( s ) ( m + w (ˆ ı − ˆ s ) | u ) − σ ( w ) σ (2 v ) σ ( u − v ) σ (2 v + w ) σ ( u − v − m si + w ) σ ( m si − w ) B ( s ) i ( m + w (ˆ ı − ˆ s ) | u ) A ( s ) ( m + w (ˆ ı − ˆ s ) | v ) − σ ( w ) σ ( u + v + w ) X α = s σ ( u + v + m αs +2 w ) σ ( m αs + w ) B ( s ) α ( m + w ( ˆ α − ˆ s ) | u ) D ( s ) αi ( m + w (ˆ ı − ˆ s ) | v ) ,i = s, (4.6) D ( s ) ki ( m | u ) B ( s ) j ( m + w (ˆ − ˆ s ) | v )= σ ( u − v + w ) σ ( u + v + 2 w ) σ ( u − v ) σ ( u + v + w ) × ( X α ,α ,β ,β = s R ( u + v + w, m − w ˆ ı ) k β α β R ( u − v, m + w ˆ ) β α j i × B ( s ) β ( m + w (ˆ k + ˆ β − ˆ ı − ˆ s ) | v ) D ( s ) α α ( m + w (ˆ − ˆ s ) | u ) o − σ ( w ) σ (2 u + 2 w ) σ ( u − v ) σ (2 u + w ) ( X α,β = s σ ( u − v + m sα − w ) σ ( m sα − w ) R (2 u + w, m − w ˆ ı ) k βα i × B ( s ) β ( m + w (ˆ k + ˆ β − ˆ ı − ˆ s ) | u ) D ( s ) αj ( m + w (ˆ − ˆ s ) | v ) o σ ( w ) σ (2 v ) σ (2 u + 2 w ) σ ( u + v + w ) σ (2 v + w ) σ (2 u + w ) × (X α = s σ ( u + v + m sj ) σ ( m sj − w ) R (2 u + w, m − w ˆ ı ) k αj i × B ( s ) α ( m + w (ˆ k + ˆ α − ˆ ı − ˆ s ) | u ) A ( s ) ( m + w (ˆ j − ˆ s ) | v ) o ,i, j, k = s, (4.7) B ( s ) i ( m + w (ˆ ı − ˆ s ) | u ) B ( s ) j ( m + w (ˆ ı + ˆ − s ) | v )= X α,β = s R ( u − v, m − w ˆ s ) β αj i B ( s ) β ( m + w ( ˆ β − ˆ s ) | v ) B ( s ) α ( m + w ( ˆ α + ˆ β − s ) | u ) ,i, j = s. (4.8)For the special case of s = 1, the above commutation relations (4.6)-(4.8) recover those in[21, 23].In order to apply the algebraic Bethe Ansatz method, in addition to the relevant com-mutation relations (4.6)-(4.8), one needs to construct a reference state associated with each s , which is the common eigenstate of the operators A ( s ) , D ( s ) ii and is annihilated by theoperators C ( s ) i . In contrast to the trigonometric and rational cases with diagonal K ± ( u ) [6],the usual highest-weight state ⊗ · · · ⊗ , is no longer the pseudo-vacuum state. However, after the face-vertex transformations (3.18)and (3.19), the face type K-matrices K ( λ | u ) and ˜ K ( λ | u ) simultaneously become diagonal.This suggests that one can translate the Z n Belavin model with non-diagonal K-matricesinto the corresponding SOS model with diagonal
K-matrices K ( λ | u ) and ˜ K ( λ | u ) given by(3.18)-(3.19). Then one can construct the pseudo-vacuum in the “face language” and usethe algebraic Bethe Ansatz method to diagonalize the transfer matrix.One of the reference states corresponding to the case of s = 1 (or the first one of the n reference states (4.9) below) was found in [21] and yields the first set of eigenvalues of thetransfer matrix. Here, we give the complete n reference states {| Ω ( s ) ( λ ) i| s = 1 , . . . , n } . For12ach s , we propose | Ω ( s ) ( λ ) i = φ λ − ( N − w ˆ s,λ − Nw ˆ s ( − z ) ⊗ φ λ − ( N − w ˆ s,λ − ( N − w ˆ s ( − z ) · · · ⊗ φ λ,λ − w ˆ s ( − z N ) . (4.9)These states ( s = 1 , . . . , n ) depend on the boundary parameters { λ i } and the inhomogeneousparameters { z j } , but not on the boundary parameters ξ and ¯ ξ . We find that the states givenby (4.9) are exactly the reference states in the following sense, A ( s ) ( λ − N w ˆ s | u ) | Ω ( s ) ( λ ) i = k ( λ | u ; ξ ) s | Ω ( s ) ( λ ) i , (4.10) D ( s ) ij ( λ − N w ˆ s | u ) | Ω ( s ) ( λ ) i = δ ij f ( s ) ( u ) k ( λ | u + w ξ − w j × ( N Y k =1 σ ( u + z k ) σ ( u − z k ) σ ( u + z k + w ) σ ( u − z k + w ) ) | Ω ( s ) ( λ ) i ,i, j = s (4.11) C ( s ) i ( λ − N w ˆ s | u ) | Ω ( s ) ( λ ) i = 0 , i = s (4.12) B ( s ) i ( λ − N w ˆ s | u ) | Ω ( s ) ( λ ) i 6 = 0 , i = s. (4.13)Here f ( s ) ( u ) is given by f ( s ) ( u ) = σ (2 u ) σ ( λ s + u + w + ξ ) σ (2 u + w ) σ ( λ s + u + ξ ) . (4.14)In order to apply the algebraic Bethe Ansatz method to diagonalize the transfer matrix,we need to assume N = n × l with l being a positive integer [21]. For convenience, let usintroduce a set of integers: N i = ( n − i ) × l, i = 0 , , · · · , n − , (4.15)and n ( n − l complex parameters { v ( i ) k | k = 1 , , · · · , N i +1 , i = 0 , , · · · , n − } . As in the usualnested Bethe Ansatz method, the parameters { v ( i ) k } will be used to specify the eigenvectorsof the corresponding reduced transfer matrices. They will be constrained later by BetheAnsatz equations. For convenience, we adopt the following convention: v k = v (0) k , k = 1 , , · · · , N . (4.16) Such states played an important role in constructing extra centers of the elliptic algebra at roots of unity[34].
13e will seek the common eigenvectors (i.e. the so-called Bethe states) of the transfer matrixby acting the creation operators B ( s ) i on the reference state | Ω ( s ) ( λ ) i| v , · · · , v N i ( s ) = X i , ··· ,i N = s F i ,i , ··· ,i N B ( s ) i ( λ + w (ˆ ı − ˆ s ) | v ) B ( s ) i ( λ + w (ˆ ı +ˆ ı − s ) | v ) × · · · B ( s ) i N ( λ + w N X k =1 ˆ ı k − wN ˆ s | v N ) | Ω ( s ) ( λ ) i . (4.17)The indices in the above equation should satisfy the following condition: the number of i k = j , denoted by j ), is l , i.e. j ) = l, j = s. (4.18)Then (3.1) and the above restriction (4.18) imply λ + w N X k =1 ˆ ı k − wN ˆ s = λ + wl n X j = s ˆ − wN ˆ s = λ − w ( l + N )ˆ s = λ − wN ˆ s, (4.19)which is crucial for the diagonalization of the transfer matrix in the remaining part of thepaper.With the help of (4.1), (4.4) and (4.5) we rewrite the transfer matrix (2.15) in terms ofthe operators A ( s ) and D ( s ) ii τ ( u ) = n X ν =1 ˜ k ( λ | u ) ν T ( λ | u ) νν = ˜ k ( λ | u ) s A ( s ) ( λ | u ) + X i = s ˜ k ( λ | u ) i T ( λ | u ) ii = ˜ k ( λ | u ) s A ( s ) ( λ | u ) + X i = s ˜ k ( λ | u ) i R (2 u, λ + w ˆ s ) issi A ( s ) ( λ | u )+ X i = s ˜ k ( λ | u ) i (cid:16) T ( λ | u ) ii − R (2 u, λ + w ˆ s ) issi A ( s ) ( λ | u ) (cid:17) = n X i =1 ˜ k ( λ | u ) i R (2 u, λ + w ˆ s ) issi A ( s ) ( λ | u )+ X i = s ˜ k (1) ( λ | u + w i σ ( λ is − w ) σ ( λ is ) (cid:16) T ( λ | u ) ii − R (2 u, λ + w ˆ s ) issi A ( s ) ( λ | u ) (cid:17) = α (1) s ( u ) A ( s ) ( λ | u ) + X i = s ˜ k (1) ( λ | u + w i D ( s ) ii ( λ | u ) . (4.20)Here we have used (4.5) and introduced the function α (1) s ( u ), α (1) s ( u ) = n X i =1 ˜ k ( λ | u ) i R (2 u, λ + w ˆ s ) issi , (4.21)14nd the reduced K-matrix ˜ K (1) ( λ | u ) with elements given by˜ K (1) ( λ | u ) ji = δ ji ˜ k (1) ( λ | u ) i , i, j = s (4.22)˜ k (1) ( λ | u ) i = ( n Y k = i,s σ ( λ ik − w ) σ ( λ ik ) ) σ ( λ i + ¯ ξ + u + ( n − w ) σ ( λ i + ¯ ξ − u − ( n − w ) , i = s. (4.23)Using the technique developed in [21], after tedious calculations, we find that with thecoefficients F i ,i , ··· ,i N in (4.17) properly chosen, the Bethe state | v , · · · , v N i ( s ) becomes theeigenstate of the transfer matrix (2.15), τ ( u ) | v , · · · , v N i ( s ) = Λ s ( u ; ξ, { v k } ) | v , · · · , v N i ( s ) , (4.24)provided that the parameters { v ( i ) k | k = 1 , , · · · , N i +1 , i = 0 , , · · · , n − } satisfy the followingBethe Ansatz equations: β (1) s ( v j ) σ (2 v j + w ) σ (2 v j + 2 w ) N Y k = j,k =1 σ ( v j + v k ) σ ( v j − v k − w ) σ ( v j + v k + 2 w ) σ ( v j − v k + w )= N Y k =1 σ ( v j + z k ) σ ( v j − z k ) σ ( v j + z k + w ) σ ( v j − z k + w ) Λ (1) s ( v j + w ξ − w , { v (1) a } ) , (4.25) β ( i +1) s ( v ( i ) j ) σ (2 v ( i ) j + w ) σ (2 v ( i ) j + 2 w ) N i +1 Y k = j,k =1 σ ( v ( i ) j + v ( i ) k ) σ ( v ( i ) j − v ( i ) k − w ) σ ( v ( i ) j + v ( i ) k + 2 w ) σ ( v ( i ) j − v ( i ) k + w )= N i Y k =1 σ ( v ( i ) j + z ( i ) k ) σ ( v ( i ) j − z ( i ) k ) σ ( v ( i ) j + z ( i ) k + w ) σ ( v ( i ) j − z ( i ) k + w ) Λ ( i +1) s ( v ( i ) j + w ξ ( i ) − w , { v ( i +1) a } ) ,i = 1 , · · · , n − . (4.26)Here { β ( i ) s ( u ) | i = 1 , . . . , n − } are functions given in Appendix A, { Λ ( i ) s ( u ; ξ, { v ( i ) k } ) | i =1 , . . . , n − } are the eigenvalues (given in Appendix A) of the reduced transfer matrices inthe nested Bethe Ansatz process, and the reduced boundary parameters { ξ ( i ) } and inhomo-geneous parameters { z ( i ) k } are given by ξ ( i +1) = ξ ( i ) − w , z ( i +1) k = v ( i ) k + w , i = 0 , · · · , n − . (4.27)In the above we have adopted the convention: ξ = ξ (0) , z (0) k = z k . The correspondingeigenvalue Λ s ( u ; ξ, { v k } ) is given byΛ s ( u ; ξ, { v k } ) = β (1) s ( u ) σ ( λ s + ξ + u + w ) σ ( λ s + ξ + u ) N Y k =1 σ ( u + v k ) σ ( u − v k − w ) σ ( u + v k + w ) σ ( u − v k )15 σ (2 u ) σ ( λ s + u + w + ξ ) σ (2 u + w ) σ ( λ s + u + ξ ) N Y k =1 σ ( u − v k + w ) σ ( u + v k + 2 w ) σ ( u − v k ) σ ( u + v k + w ) × N Y k =1 σ ( u + z k ) σ ( u − z k ) σ ( u + z k + w ) σ ( u − z k + w ) Λ (1) s ( u + w ξ − w , { v (1) i } ) . (4.28)It is easy to check that the first set of eigenvalues Λ ( u ; ξ, { v ( i ) k } ) and the correspondingBethe Ansatz equations are exactly those found in [21]. However, the rest n − { Λ s ( u ; ξ, { v k } ) | s = 2 , . . . , n − } and the associated Bethe Ansatz equations arenew ones. For the special case of n = 2, which corresponds to the open XYZ spin chain,we find that the two sets of eigenvalues { Λ s ( u ; ξ, { v k } ) | s = 1 , } , after rescaling of an overallfactor due to different normalizations of the R- and K-matrices, recover those in [35] obtainedby directly solving the T - Q relation (or functional Bethe Ansatz). As shown in [35], these twosets of eigenvalues give the complete spectrum of the transfer matrix of the open XYZ spinchain. As a consequence, the corresponding two sets of Bethe states {| v , · · · , v N i ( s ) | s = 1 , } together constitute the complete eigenstates of the transfer matrix of the open XYZ model.Therefore, it is expected that the n sets of eigenvalues { Λ s ( u ; ξ, { v k } ) | s = 1 , . . . , n } (4.28)[resp. Bethe states {| v , · · · , v N i ( s ) | s = 1 , . . . , n } (4.17) and (4.25)-(4.26)] together give riseto the complete spectrum [resp. the complete eigenstates] of the transfer matrix (2.15) ofthe open Z n Belavin model.
As will be seen from the definitions of the intertwiners (3.8), (3.10) and (3.11), specialized to m = λ , φ λ,λ − w ˆ ı ( u ) and ¯ φ λ,λ − w ˆ ı ( u ) do not depend on w while ˜ φ λ,λ − w ˆ ı ( u ) does. Consequently,the K-matrix K − ( u ) does not depend on the crossing parameter w , but K + ( u ) does. So weuse the convention: K ( u ) = lim w → K − ( u ) = K − ( u ) . (5.1)We further assume that the parameter ¯ ξ has the following behavior as w → ξ = ξ + wδ + O ( w ) , (5.2) In [12, 23, 20], a special case of ¯ ξ = ξ was studied. The generalization to the case with nonvanishing δ is straightforward. δ . It implies that lim w → ¯ ξ = ξ. Then the K-matrices satisfy the following relationlim w → { K + ( u ) K − ( u ) } = lim w → { K + ( u ) } K ( u ) = id . (5.3)Let us introduce the elliptic Gaudin operators { H j | j = 1 , , · · · , N } associated with theinhomogeneous Z n Belavin model with open boundaries specified by the generic K-matrices(2.17) and (2.18): H j = Γ j ( z j ) + N X k = j r kj ( z j − z k ) + K − j ( z j ) ( N X k = j r jk ( z j + z k ) ) K j ( z j ) , (5.4)where Γ j ( u ) = ∂∂w { ¯ K j ( u ) }| w =0 K j ( u ), j = 1 , · · · , N, with ¯ K j ( u ) = tr (cid:8) K +0 ( u ) R B j (2 u ) P j (cid:9) .Here { z j } are the inhomogeneous parameters of the inhomogeneous Z n Belavin model and r ( u ) is given by (2.8). For a generic choice of the boundary parameters { λ , · · · , λ n , ¯ ξ } ,Γ j ( u ) is a non-diagonal matrix.Following [36, 37], the elliptic Gaudin operators (5.4) are obtained by expanding thedouble-row transfer matrix (2.15) at the point u = z j around w = 0: τ ( z j ) = τ ( z j ) | w =0 + wH j + O ( w ) , j = 1 , · · · , N, (5.5) H j = ∂∂w τ ( z j ) | w =0 . (5.6)The relations (2.7) and (5.3) imply that the first term τ ( z j ) | w =0 in the expansion (5.5) isequal to an identity, namely, τ ( z j ) | w =0 = id . (5.7)Then the commutativity of the transfer matrices { τ ( z j ) } (2.16) for a generic w implies[ H j , H k ] = 0 , i, j = 1 , · · · , N. (5.8)Thus the elliptic Gaudin system defined by (5.4) is integrable. Moreover, the relation (5.6)between { H j } and { τ ( z j ) } and the fact that the first term on the r.h.s. of (5.5) is iden-tity operator enable us to extract the eigenstates of the elliptic Gaudin operators and thecorresponding eigenvalues from the results obtained in the previous section.17sing (4.27), (A.3), (A.4) and (A.6)-(A.8), the Bethe Ansatz equations (4.25) and (4.26)become, respectively, β ( i +1) s ( v ( i ) j ) σ (2 v ( i ) j + w ) σ (2 v ( i ) j + 2 w ) N i +1 Y k = j,k =1 σ ( v ( i ) j + v ( i ) k ) σ ( v ( i ) j − v ( i ) k − w ) σ ( v ( i ) j + v ( i ) k + 2 w ) σ ( v ( i ) j − v ( i ) k + w )= β ( i +2) s ( v ( i ) j + w σ ( λ i + s +1 + ξ ( i +1) + v ( i ) j + w ) σ ( λ i + s +1 + ξ ( i +1) + v ( i ) j + w ) × N i Y k =1 σ ( v ( i ) j + v ( i − k + w ) σ ( v ( i ) j − v ( i − k − w ) σ ( v ( i ) j + v ( i − k + w ) σ ( v ( i ) j − v ( i − k + w ) × N i +2 Y k =1 σ ( v ( i ) j + v ( i +1) k + w ) σ ( v ( i ) j − v ( i +1) k − w ) σ ( v ( i ) j + v ( i +1) k + w ) σ ( v ( i ) j − v ( i +1) k + w ) ,i = 0 , · · · , n − , (5.9) β ( n − s ( v ( n − j ) σ (2 v ( n − j + w ) σ (2 v ( n − j +2 w ) N n − Y k = j,k =1 σ ( v ( n − j + v ( n − k ) σ ( v ( n − j − v ( n − k − w ) σ ( v ( n − j + v ( n − k +2 w ) σ ( v ( n − j − v ( n − k + w )= σ ( λ s − + ¯ ξ + v ( n − j + w ) σ ( λ s − + ξ ( n − − v ( n − j − w ) σ ( λ s − + ¯ ξ − v ( n − j − w ) σ ( λ s − + ξ ( n − + v ( n − j + w ) × N n − Y k =1 σ ( v ( n − j + v ( n − k + w ) σ ( v ( n − j − v ( n − k − w ) σ ( v ( n − j + v ( n − k + w ) σ ( v ( n − j − v ( n − k + w ) . (5.10)Here we have used the convention: v ( − k = z k , k = 1 , · · · , N . The quasi-classical property(3.7) of R ( u, m ), (A.1) and (A.2) lead to the following relations β ( i +1) s ( u, ¯ ξ,
0) = 1 , ∂∂u β ( i +1) s ( u, ¯ ξ,
0) = 0 , i = 0 , · · · , n − . (5.11)Noticing the restriction (5.2), one may introduce some functions { γ ( i +1) s ( u ) } associated with { β ( i +1) s ( u, ¯ ξ, w ) } γ ( i +1) s ( u ) = ∂∂w β ( i +1) s ( u, ¯ ξ, w ) | w =0 + δ ∂∂ ¯ ξ β ( i +1) s ( u, ¯ ξ, | ¯ ξ = ξ , i = 0 , · · · , n − . (5.12)Using (5.5), we can extract n sets of eigenvalues { h ( s ) j | s = 1 , . . . , n } (resp. the correspondingBethe Ansatz equations) of the Gaudin operators H j (5.4) from the expansion around w = 0for the first order of w of the eigenvalues (4.28) of the transfer matrix τ ( u = z j ) (resp. theBethe Ansatz equations (5.9) and (5.10) ). Finally, the eigenvalues of the Z n elliptic Gaudinoperators are h ( s ) j = γ (1) s ( z j ) + ζ ( λ s + ξ + z j ) − N X k =1 { ζ ( z j + x k ) + ζ ( z j − x k ) } , (5.13)18here ζ -function is defined in (2.3). The n ( n − l parameters { x ( i ) k | k = 1 , , · · · , N i +1 , i =0 , , · · · , n − } (including x k as x k = x (0) k , k = 1 , · · · , N ) are determined by the followingBethe Ansatz equations γ ( i +1) s ( x ( i ) j ) − ζ (2 x ( i ) j ) − N i +1 X k = j,k =1 n ζ ( x ( i ) j + x ( i ) k ) + ζ ( x ( i ) j − x ( i ) k ) o = γ ( i +2) s ( x ( i ) j ) − N i X k =1 n ζ ( x ( i ) j + x ( i − k ) + ζ ( x ( i ) j − x ( i − k ) o + ζ ( λ i + s +1 + ξ + x ( i ) j ) − N i +2 X k =1 n ζ ( x ( i ) j + x ( i +1) k )+ ζ ( x ( i ) j − x ( i +1) k ) o ,i = 0 , · · · , n − , (5.14) γ ( n − s ( x ( n − j ) − ζ (2 x ( n − j ) − N n − X k = j,k =1 n ζ ( x ( n − j + x ( n − k )+ ζ ( x ( n − j − x ( n − k ) o = (cid:16) δ + n (cid:17) ζ ( λ s − + ξ + x ( n − j )+ (cid:18) − n − δ (cid:19) ζ ( λ s − + ξ − x ( n − j ) − N n − X k =1 n ζ ( x ( n − j + x ( n − k )+ ζ ( x ( n − j − x ( n − k ) o . (5.15)Here we have used the convention: x ( − k = z k , k = 1 , · · · , N in (5.14). Then the n setsof eigenvalues { h ( s ) j | s = 1 , . . . , n } given by (5.13)-(5.15) (c.f. [23]) together constitute thecomplete spectrum of the Gaudin operators H j (5.4). We have discovered the multiple reference state structure of the Z n Belavin model withboundaries specified by the non-diagonal K-matrices K ± ( u ), (2.17) and (2.18). It is foundthat there exist n reference states {| Ω ( s ) ( λ ) i| s = 1 , . . . , n } (4.9), which lead to n sets of Bethestates | v , · · · , v N i ( s ) (4.17). These Bethe states give rise to n sets of Bethe Ansatz equations(4.25)-(4.26) and eigenvalues (4.28), labelled by s = 1 , . . . , n . The fist set of them, whichcorresponds to the s = 1 case, gives the results found in [21]. It is expected that these n setsof eigenvalues { Λ s ( u ; ξ, { v k } ) | s = 1 , . . . , n } (4.28) together give rise to the complete spectrumof the transfer matrix τ ( u ) (2.15) for the Z n Belavin model with generic boundaries. In thequasi-classical limit (i.e. w → n sets of eigenvalues { h ( s ) j | s = 1 , . . . , n } givenby (5.13)-(5.15) together constitute the complete spectrum of the Gaudin operators H j (5.4).19aking the scaling limit [38] of our general results, for the special n = 2 case, we recoverthe results obtained in [20] for the open XXZ spin chain. It is believed that such structureof multiple reference states also exists for the open A (1) n − trigonometric vertex model studiedin [24]. Acknowledgements
The financial support from Australian Research Council is gratefully acknowledged.
Appendix A: Definitions of the α -, β -, Λ - functions In this appendix, we give the definitions of the functions α ( i ) s ( u ), β ( i ) s ( u ) and Λ ( i ) ( u ; ξ, { v ( i ) k } ),which appeared in the expressions of the eigenvalues and the Bethe Ansatz equations (4.25)-(4.28).In order to carry out the nested Bethe Ansatz for the Z n Belavin model with the genericopen boundary conditions, one needs to introduce a set of reduced K-matrices ˜ K ( r ) ( λ | u )which include the original one ˜ K ( λ | u ) = ˜ K (0) ( λ | u ):˜ K ( r ) ( λ | u ) ji = δ ji ˜ k ( r ) ( λ | u ) i , i, j = r + 1 , · · · , n, r = 0 , · · · , n − , ˜ k ( r ) ( λ | u ) i = ( n Y k = i,k = r +1 σ ( λ ik − w ) σ ( λ ik ) ) σ ( λ i + ¯ ξ + u + ( n − r ) w ) σ ( λ i + ¯ ξ − u − ( n − r ) w ) . Moreover we introduce a set of functions { α ( r ) ( u ) | r = 1 , · · · , n − } related to the reducedK-matrices ˜ K ( r ) ( λ | u ) α ( r ) ( u ) = n X i = r R (2 u, λ + w ˆ r ) irri ˜ k ( r − ( λ | u ) i , r = 1 , · · · , n − , (A.1)and an associated set of functions { β ( i ) ( u, ¯ ξ, w ) | i = 1 , . . . , n − } β ( i +1) ( u, ¯ ξ, w ) ≡ β ( i +1) ( u ) = α ( i +1) ( u ) σ ( λ i +1 + ξ − u − i w ) σ ( λ i +1 + ξ + u + w − i w ) , i = 0 , · · · , n − . (A.2)In the process of carrying out the nested Bethe Ansatz [21], one needs to introduce a setof functions { Λ ( i ) ( u ; ξ, { v ( i ) k } ) | i = 0 , . . . , n − } which correspond to the eigenvalues of thereduced transfer matrices. The functions { Λ ( i ) ( u ; ξ, { v ( i ) k } ) } are given by the following recur-rence relationsΛ ( i ) ( u ; ξ ( i ) , { v ( i ) k } ) = β ( i +1) ( u ) σ ( λ i +1 + ξ ( i ) + u + w ) σ ( λ i +1 + ξ ( i ) + u ) N i +1 Y k =1 σ ( u + v ( i ) k ) σ ( u − v ( i ) k − w ) σ ( u + v ( i ) k + w ) σ ( u − v ( i ) k )20 σ (2 u ) σ ( λ i +1 + u + w + ξ ( i ) ) σ (2 u + w ) σ ( λ i +1 + u + ξ ( i ) ) N i +1 Y k =1 σ ( u − v ( i ) k + w ) σ ( u + v ( i ) k +2 w ) σ ( u − v ( i ) k ) σ ( u + v ( i ) k + w ) × N i Y k =1 σ ( u + z ( i ) k ) σ ( u − z ( i ) k ) σ ( u + z ( i ) k + w ) σ ( u − z ( i ) k + w ) Λ ( i +1) ( u + w ξ ( i ) − w , { v ( i +1) j } ) ,i = 1 , · · · , n − , (A.3)Λ ( n − ( u ; ξ ( n − ) = σ ( λ n + ¯ ξ + u + w ) σ ( λ n + ξ ( n − − u ) σ ( λ n + ¯ ξ − u − w ) σ ( λ n + ξ ( n − + u ) . (A.4)The reduced boundary parameters { ξ ( i ) } and inhomogeneous parameters { z ( i ) k } are given by(4.27). It is remarked that all the functions { α ( i ) ( u ) } , { β ( i ) ( u ) } and { Λ ( i ) ( u ; ξ, { v ( i ) k } ) } are indeed the functions of the boundary parameters { λ i } .Since that the suffix index of the boundary parameters { λ i } takes value in Z n , one canintroduce a Z n cyclic operator P (i.e. P n = id), which acts on the space of functions of { λ , . . . , λ n } . On any function f ( λ , . . . , λ n ) the action of the operator P is given by P ( f ( λ , . . . , λ n − , λ n )) = f ( λ , . . . , λ n , λ n +1 ) = f ( λ , . . . , λ n , λ ) . (A.5)Then we introduce the following α -, β -,Λ-functions: α ( i ) s ( u ) = P s − ( α ( i ) ( u )) , s = 1 , . . . n, i = 1 , . . . n − , (A.6) β ( i ) s ( u ) = P s − ( β ( i ) ( u )) , s = 1 , . . . n, i = 1 , . . . n − , (A.7)Λ ( i ) s ( u ; ξ, { v ( i ) k } ) = P s − Λ ( i ) ( u ; ξ, { v ( i ) k } ) , s = 1 , . . . n, i = 1 , . . . n − . (A.8)Direct calculation shows that the two definitions of α (1) s , (4.21) and (A.6), coincide with eachother. References [1] R. J. Baxter,
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