On the solutions of the Z n -Belavin model with arbitrary number of sites
Kun Hao, Junpeng Cao, Guang-Liang Li, Wen-Li Yang, Kangjie Shi, Yupeng Wang
aa r X i v : . [ m a t h - ph ] M a y On the solutions of the Z n -Belavin model witharbitrary number of sites Kun Hao a,b , Fakai Wen a,c , Junpeng Cao c,d , Guang-LiangLi e , Wen-Li Yang a,b,f , Kangjie Shi a,b and Yupeng Wang c,d a Institute of Modern Physics, Northwest University, Xian 710069, China b Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xian 710069, China c Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, ChineseAcademy of Sciences, Beijing 100190, China d Collaborative Innovation Center of Quantum Matter, Beijing, China e Department of Applied Physics, Xian Jiaotong University, Xian 710049, China f Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing,100048, China
Abstract
The periodic Z n -Belavin model on a lattice with an arbitrary number of sites N is studied via the off-diagonal Bethe Ansatz method (ODBA). The eigenvalues of thecorresponding transfer matrix are given in terms of an unified inhomogeneous T − Q relation. In the special case of N = nl with l being also a positive integer, the resulting T − Q relation recovers the homogeneous one previously obtained via algebraic BetheAnsatz. PACS:
Keywords : Spin chain; Bethe Ansatz; The Z n -Belavin model; T − Q relation Corresponding author: [email protected] Corresponding author: [email protected]
Introduction
Our understanding to phase transitions and critical phenomena has been greatly enhancedby the study on lattice integrable models [1]. Such exact results provide valuable insightsinto the key theoretical development of universality classes in areas ranging from moderncondensed physics [2, 3] to string and super-symmetric Yang-Mills theories [4, 5, 6]. Amongsolvable models [1, 7, 8], elliptic ones stand out as a particularly important class due to thefact that most others can be reduced to from them by taking trigonometric or rational limits.The Z n -Belavin model [9] is a typical elliptic quantum integrable model, with the celebratedXYZ spin chain as the special case of n = 2.The first exact solution of the Z -model with periodic boundary condition was givenby Baxter [10], where the fundamental equation (the Yang-Baxter equation [1, 11]) wasemphasized and the T − Q method was proposed. Takhtadzhan and Faddeev [12] resolved themodel with the algebraic Bethe Ansatz method [7, 13]. By employing the intertwiners vectors[14] which constitute the face-vertex correspondence between the Z n -Belavin model and theassociated face model, Hou et al [15] generalized Takhatadzhan and Faddeev’s approach tothe Z n -Belavin model with a generic n . In their approach, local gauge transformation playeda central role to obtain local vacuum states (reference states) with which the algebraic BetheAnsatz analysis can be performed. However, such reference states are so far only availablefor some very particular number of lattice sites, namely, N = nl with l being a positiveinteger, but not for the other N . This leads to the fact that the conventional Bethe Ansatzmethods have been quite hard to apply to the latter case for many years. In fact, the lack ofa reference state is a common feature of the integrable models without U (1) symmetry andhad been a very important and difficult issue in the field of quantum integrable models.Recently, a systematic method, i.e., the off-diagonal Bethe Ansatz (ODBA) [16, 17] wasproposed to solve the eigenvalue problem of integrable models without U(1)-symmetry. Theclosed XYZ spin chain (or the Z -model) with arbitrary number of sites [18] and several otherlong-standing models [16, 19, 20, 21] have since been solved. In this paper, we adopt ODBAto solve the eigenvalue problem of the periodic Z n -Belavin model with a generic positiveinteger n ≥ N .The paper is organized as follows. Section 2 serves as an introduction of our notationsand some basic ingredients. The commuting transfer matrix associated with the periodic2 n -Belavin model is constructed to show the integrability of the model. In section 3, basedon some intrinsic properties of the Z n -Belavin’s R -matrix, we construct the fused transfermatrices by anti-symmetric fusion procedure and derive some operator identities and thequasi-periodicities of these matrices. Taking the Z model as a concrete example, we expressthe eigenvalues of the transfer matrix in terms of a nested inhomogeneous T − Q relationand the associated Bethe Ansatz equations (BAEs) in Section 4. Generalization to Z n caseis presented in Section 5. We summarize our results and give some discussions in Section 6.A slightly detailed description about the Z case, which might be crucial to understand theprocedure for n ≥
4, is given in Appendix A. In addition, we discuss the ODBA solution of Z n -Belavin model with twisted boundary condition in Appendix B. Z n -Belavin model with periodic boundary condition Let us fix a positive integer n ≥
2, a complex number τ such that Im ( τ ) > w . For convenience, let us introduce the 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 ) } . 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 V denote an n -dimensional linear space with an orthonormal basis {| i i| i = 1 , · · · , n } ,and g, h be two n × n matrices with the elements h ij = δ i +1 j , g ij = ω ni δ i j , with ω n = e π √− n , i, j ∈ Z n , Our σ -function is the ϑ -function ϑ ( u ) [22]. 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 . g = ω n ω n . . . ω nn − , h = . (2.3)It is easy to verify that the matrices satisfy the relation gh = ω n − hg. (2.4)Associated with any α = ( α , α ), α , α ∈ Z n , one can introduce an n × n matrix I α definedby I α = I ( α ,α ) = g α h α , (2.5)and an elliptic function σ α ( u ) given by σ α ( u ) = θ " + α n + α n ( u, τ ) , and σ (0 , ( u ) = σ ( u ) . The Z n -Belavin R-matrix R ( u ) ∈ End( V ⊗ V ) is given by [9, 23, 14] R ( u ) = X α ∈ Z n σ α ( u + wn ) nσ α ( wn ) I α ⊗ I − α , (2.6)which satisfies the quantum Yang-Baxter equation (QYBE) R ( u − u ) R ( u − u ) R ( u − u ) = R ( u − u ) R ( u − u ) R ( u − u ) , (2.7)and the properties [23],Initial condition : R (0) = P , , (2.8)Unitarity : R ( u ) R ( − u ) = σ ( w + u ) σ ( w − u ) σ ( w ) σ ( w ) × id , (2.9)Crossing-unitarity : R t ( − u − nw ) R t ( u ) = σ ( u ) σ ( − u − nw ) σ ( w ) σ ( w ) × id , (2.10) Z n -symmetry : g g R ( u ) g − g − = R ( u ) , h h R ( u ) h − h − = R ( u ) , (2.11)Fusion conditions : R ( − w ) = P ( − )1 , S ( − )12 , R ( w ) = S (+)12 P (+)1 , . (2.12)4ere R ( u ) = P , R ( u ) P , with P , being the usual permutation operator, P ( ∓ )1 , = { ∓ P , } is anti-symmetric (symmetric) project operator in the tensor product space V ⊗ V , S ( ± )12 are some non-degenerate matrices ∈ End( V ⊗ V ) [23, 14] and t i denotes the transposition inthe i -th space. Here and below we adopt the standard notation: for any matrix A ∈ End( V ), A j is an embedding operator in the tensor space V ⊗ V ⊗ · · · , which acts as A on the j -thspace and as an identity on the other factor spaces; R ij ( u ) is an embedding operator ofR-matrix in the tensor space, which acts as an identity on the factor spaces except for the i -th and j -th ones.As usual, the corresponding “row-to-row” monodromy matrix T ( u ) [7], an n × n matrixwith operator-valued elements acting on ( V ) ⊗ N reads T ( u ) = R N ( u − θ N ) R N − ( u − θ N − ) · · · R ( u − θ ) . (2.13)Here { θ i | i = 1 , · · · , N } are arbitrary free complex parameters which are usually called theinhomogeneous parameters. With the help of the QYBE (2.7), one can show that T ( u )satisfies the Yang-Baxter algebra relation R ( u − v ) T ( u ) T ( v ) = T ( v ) T ( u ) R ( u − v ) . (2.14)Let us introduce the transfer matrix t ( u ) t ( u ) = tr ( T ( u )) = tr ( T ( u )) . (2.15)The Z n -Belavin model [9] with periodic boundary condition is a quantum spin chain describedby the Hamiltonian H = ∂∂u { ln t ( u ) }| u =0 , { θ i } =0 − Constant = N X i =1 H i,i +1 , (2.16)where the local Hamiltonian H i,i +1 is H i,i +1 = ∂∂u { P i,i +1 R i,i +1 ( u ) }| u =0 , (2.17)with the periodic boundary condition, namely, H N,N +1 = H N, . (2.18)The commutativity of the transfer matrices[ t ( u ) , t ( v )] = 0 , follows as a consequence of (2.14). This ensures the integrability of the inhomogeneous Z n -Belavin model with periodic boundary. 5 Relations of the eigenvalues
Following the method developed in [19] (see also Chapter 7 of [17]), we apply the fusiontechniques [24, 25, 26, 27] to study the Z n -Belavin model. Besides the fundamental transfermatrix t ( u ) some other fused transfer matrices { t j ( u ) | j = 1 , · · · , n } (see below (3.8)), whichcommute with each other and include the original one as t ( u ) = t ( u ), are constructedthrough an anti-symmetric fusion procedure with the help of the fusion condition (2.12) ofthe R -matrix. The quasi-periodicity of the σ -function σ ( u + 1) = − σ ( u ) , σ ( u + τ ) = − e − iπ ( u + τ ) σ ( u ) , (3.1)indicates that the R -matrix R ( u ) given by (2.6) possesses the quasi-periodic properties R ( u + 1) = − g − R ( u ) g = − g R ( u ) g − , (3.2) R ( u + τ ) = − e − iπ ( u + wn + τ ) h − R ( u ) h = − e − iπ ( u + wn + τ ) h R ( u ) h − , (3.3)which lead to the quasi-periodicity of the transfer matrix t ( u ) given by (2.15) t ( u + 1) = ( − N t ( u ) , (3.4) t ( u + τ ) = ( − N e − iπ { N ( u + wn + τ ) − P Nl =1 θ l } t ( u ) . (3.5)Let us introduce the usual (or non-deformed) anti-symmetric projectors { P ( − )1 , ··· ,m | m =2 , ..., n } in a tensor space of V defined by the induction relations P ( − )1 , ··· ,m +1 = 1 m + 1 (1 − m +1 X j =2 P ,j ) P ( − )2 , ··· ,m +1 , m = 2 , · · · , n − . Iterating the above relation yields alternative definition of the projectors P ( − )1 , ··· ,m = 1 m ! X κ ∈ S m ( − sign ( κ ) P κ , m = 2 , · · · , n, (3.6)where S m is the permutation group of m indices, P κ is a permutation in the group, and sign ( κ ) is 0 for an even permutation κ and 1 for an odd permutation. With the aboveanti-symmetric projectors, we can construct the fused monodromy matrices T h , ··· ,m i ( u ) = P ( − )1 , ··· ,m T ( u ) · · · T ( u − ( m − w ) P ( − )1 , ··· ,m , m = 2 , · · · , n. (3.7)6he corresponding fused transfer matrices { t j ( u ) | j = 1 , · · · , n } (including the original oneas t ( u ) = t ( u )) are then given by t m ( u ) = tr , ··· ,m (cid:8) T h , ··· ,m i ( u ) (cid:9) , m = 2 , · · · , n. (3.8)The last fused transfer matrix t n ( u ) is the so-called quantum determinant [28] which playsthe role of the generating functional of the centers of the associated quantum algebras [29].For generic values of { θ j } , t n ( u ) is proportional to the identity operator, namely, t n ( u ) = Det q ( T ( u )) × id , Det q ( T ( u )) = N Y l =1 σ ( u − θ l + w ) σ ( w ) n − Y k =1 σ ( u − θ l − kw ) σ ( w ) . (3.9)The QYBE (2.7), the fusion condition (2.12) and the relations (2.14) imply that thesefused matrices commute with each other,[ t i ( u ) , t j ( v )] = 0 , i, j = 1 , · · · , n. (3.10)Using the method (see Chapter 7.2 of [17]), we can show the relations T ( θ j ) T h , ··· ,m i ( θ j − w ) = P ( − )1 , ··· ,m T ( θ j ) T h , ··· ,m i ( θ j − w ) , m = 2 , · · · , n ; j = 1 , · · · , N, which immediately lead to the following recursive relations t ( θ j ) t m ( θ j − w ) = t m +1 ( θ j ) , m = 1 , · · · , n − j = 1 , · · · , N. (3.11)Moreover, the fusion condition (2.12) and the fact that P ( − )12 P (+)12 = 0 = P (+)12 P ( − )12 enable usto derive some zeros for the fused matrices, t m ( θ j + kw ) = 0 , j = 1 , · · · , N ; k = 1 , · · · , m − m = 2 , · · · , n. (3.12)Similarly as deriving the relations (3.4)-(3.5), we have the fused transfer matrices enjoy thefollowing periodicity, t m ( u + 1) = ( − mN t m ( u ) , m = 1 , · · · , n, (3.13) t m ( u + τ ) = ( − mN e − iπ { mN ( u + wn + τ − m − w ) − m P Nl =1 θ l } t m ( u ) , m = 1 , · · · , n. (3.14)Let us evaluate the transfer matrix of the closed chain at some special points. The initialcondition of the R -matrix (2.6) implies that t ( θ j ) = R j j − ( θ j − θ j − ) · · · R j ( θ j − θ ) R j N ( θ j − θ N ) · · · R j j +1 ( θ j − θ j +1 ) . The unitarity relation (2.9) allows us to derive the following identity: N Y l =1 t ( θ l ) = N Y l =1 a ( θ l ) × id , a ( u ) = N Y l =1 σ ( u − θ l + w ) σ ( w ) , d ( u ) = a ( u − w ) . (3.15)7 .2 Functional relations of eigenvalues The commutativity (3.10) of the transfer matrices { t m ( u ) | m = 1 , · · · , n } with different spec-tral parameters implies that they have common eigenstates. Let | Ψ i be a common eigenstateof { t m ( u ) } , which dose not depend upon u , with the eigenvalues Λ m ( u ) (we shall take theconvention: Λ( u ) = Λ ( u )), t m ( u ) | Ψ i = Λ m ( u ) | Ψ i , m = 1 , · · · n. The properties (3.9), (3.11) and (3.12) of the transfer matrices { t m ( u ) | m = 1 , · · · , n } implythat the corresponding eigenvalues { Λ m ( u ) | m = 1 , · · · , n } satisfy the functional relationsΛ( θ j ) Λ m ( θ j − w ) = Λ m +1 ( θ j ) , m = 1 , · · · , n − j = 1 , · · · , N, (3.16)Λ m ( θ j + kw ) = 0 , j = 1 , · · · , N ; k = 1 , · · · , m − m = 2 , · · · , n − , (3.17)Λ n ( u ) = Det q ( T ( u )) = a ( u ) n − Y k =1 d ( u − kw ) , (3.18)where the functions a ( u ) and d ( u ) are given by (3.15). From the definitions (2.6), (2.15) and(3.8) of the R -matrix R ( u ) and the associated transfer matrices { t m ( u ) | m = 1 , · · · , n } , wehave thatΛ m ( u ), as a function of u , is an elliptical polynomial of degree mN , m = 1 , · · · , n − . (3.19)The periodicity (3.13)-(3.14) of these transfer matrices imply that the eigenvalues { Λ m ( u ) } are some elliptic polynomials of the fixed degrees (3.19) with the periodicityΛ m ( u + 1) = ( − mN Λ m ( u ) , m = 1 , · · · , n − , (3.20)Λ m ( u + τ ) = ( − mN e − iπ { mN ( u + wn + τ − m − w ) − m P Nl =1 θ l } Λ m ( u ) , m = 1 , · · · , n − . (3.21)Moreover the product identity (3.15) of the transfer matrix t ( u ) leads to the relation N Y l =1 Λ( θ l ) = N Y l =1 a ( θ l ) , (3.22)which serves as the selection rule [18] for the eigenvalues of the transfer matrix from thesolutions of (3.16)-(3.21)The relations (3.16)-(3.22) allow us to determine the eigenvalues { Λ m ( u ) } of the transfermatrices { t m ( u ) } completely. 8 ODBA solution of the Z case Similarly as that [18] for the eight-vertex model (the Z -case) , we demonstrate that (3.16)-(3.22) enable us to express the eigenvalues { Λ m ( u ) } of the transfer matrices { t m ( u ) } simul-taneously in terms of some inhomogeneous T − Q relations [17].For the Z -Belavin model, the corresponding (3.16)-(3.22) readΛ( θ j )Λ m ( θ j − w ) = Λ m +1 ( θ j ) , m = 1 , , j = 1 , · · · , N, (4.1)Λ ( u ) = a ( u ) d ( u − w ) d ( u − w ) , (4.2)Λ ( θ j + w ) = 0 , j = 1 , · · · , N. (4.3)Λ m ( u + 1) = ( − mN Λ m ( u ) , m = 1 , , (4.4)Λ m ( u + τ ) = ( − mN e − iπ { mN ( u + w + τ − m − w ) − m P Nl =1 θ l } Λ m ( u ) , m = 1 , . (4.5) N Y l =1 Λ( θ l ) = N Y l =1 a ( θ l ) . (4.6)Keeping the fact that Λ m ( u ) is an elliptical polynomial of degree mN in mind, we can expressΛ( u ) and Λ ( u ) in terms of the inhomogeneous T − Q relation [18] as follows. Let us introducesome Q -functions Q ( i ) ( u ) = N Y j =1 σ ( u − λ ( i ) j ) σ ( w ) , i = 1 , · · · , , (4.7)parameterized by 4 N parameters { λ ( i ) j | j = 1 , · · · , N ; i = 1 , · · · , } (the so-called Betheroots) determined later by the associated BAEs (see below (4.15)-(4.23)). Associated withthe above Q -functions, we introduce 5 functions { Z i ( u ) | i = 1 , , } and { X i ( u ) | i = 1 , } as Z ( u ) = a ( u ) e iπl u + φ Q (1) ( u − w ) Q (2) ( u ) , (4.8) Z ( u ) = d ( u ) e iπl u + φ Q (2) ( u + w ) Q (3) ( u − w ) Q (1) ( u ) Q (4) ( u ) , (4.9) Z ( u ) = d ( u ) e − iπ { ( l + l ) u +(2 l + l ) w }− φ − φ Q (4) ( u + w ) Q (3) ( u ) , (4.10) X ( u ) = c a ( u ) d ( u ) e iπl u Q (3) ( u − w ) Q (1) ( u ) Q (2) ( u ) , (4.11) X ( u ) = c a ( u ) d ( u ) e iπl u Q (2) ( u + w ) Q (3) ( u ) Q (4) ( u ) , (4.12)9here { l i | i = 1 , · · · , } are 4 integers, { φ i , c i | i = 1 , } are 4 complex numbers. Then we canintroduce the inhomogeneous T − Q relations,Λ( u ) = Z ( u ) + Z ( u ) + Z ( u ) + X ( u ) + X ( u ) , (4.13)Λ ( u ) = Z ( u ) Z ( u − w ) + Z ( u ) Z ( u − w ) + Z ( u ) Z ( u − w )+ X ( u ) Z ( u − w ) + Z ( u ) X ( u − w ) . (4.14)In order that the above parameterizations of Λ( u ) and Λ ( u ) become a solution to (4.1)-(4.6),the 4( N + 1) parameters { λ ( i ) j | j = 1 , · · · , N ; i = 1 , · · · , } and { φ i , c i | i = 1 , } have to satisfythe associated BAEs e iπl λ (1) j + φ Q (2) ( λ (1) j + w ) Q (2) ( λ (1) j )+ c e iπl λ (1) j a ( λ (1) j ) Q (4) ( λ (1) j ) = 0 , j = 1 , · · · , N, (4.15) e iπl λ (2) j + φ Q (1) ( λ (2) j − w ) Q (1) ( λ (2) j )+ c e iπl λ (2) j d ( λ (2) j ) Q (3) ( λ (2) j − w ) = 0 , j = 1 , · · · , N, (4.16) e − iπ { ( l + l ) λ (3) j +(2 l + l ) w }− φ − φ Q (4) ( λ (3) j + w ) Q (4) ( λ (3) j )+ c e iπl λ (3) j a ( λ (3) j ) Q (2) ( λ (3) j + w ) = 0 ,j = 1 , · · · , N, (4.17) e iπl λ (4) j + φ Q (3) ( λ (4) j − w ) Q (3) ( λ (4) j )+ c e iπl λ (4) j a ( λ (4) j ) Q (1) ( λ (4) j ) = 0 , j = 1 , · · · , N, (4.18) − Θ (1) + Θ (2) − N w = m + l τ, (4.19) − Θ (2) − Θ (3) + Θ (1) + Θ (4) − N w = m + l τ, (4.20) − Θ − Θ (3) + Θ (1) + Θ (2) − N w = m + l τ, (4.21) − Θ − Θ (2) + Θ (3) + Θ (4) + 5 N w = m + l τ, (4.22) N Y j =1 Q (1) ( θ j − w ) Q (2) ( θ j ) = e − iπl Θ − Nφ , (4.23)where { m i | i = 1 , · · · , } are 4 integers andΘ = N X l =1 θ l , Θ ( i ) = N X l =1 λ ( i ) l , i = 1 , · · · , . (4.24)We have checked that the functions Λ( u ) and Λ ( u ) given by the inhomogeneous T − Q relations (4.13)- (4.14) are solutions to (4.1)-(4.6) provided that the 4( N + 1) parameters10 λ ( i ) j | j = 1 , · · · , N ; i = 1 , · · · , } and φ , φ , c and c satisfy the associated BAEs (4.15)-(4.23) for arbitrary fixed integers { l i , m i | i = 1 , · · · , } . Therefore the corresponding Λ( u )becomes an eigenvalue of the transfer matrix t ( u ) given by (2.15). In the homogeneouslimit: { θ j → } , the resulting T − Q relation (4.13) and the associated BAEs (4.15)-(4.23)give rise to the eigenvalue and BAEs of the corresponding homogeneous spin chain (i.e., the Z -Belavin model with periodic boundary condition described by the Hamiltonian (2.16) forthe case of n = 3).Some remarks are in order. The integers { l i , m i | i = 1 , · · · , } appeared in the BAEs(4.19)-(4.22) are due to the quasi-periodicity (3.2)-(3.3) of the R -matrix in terms of thespectral parameter u . Any choice of these integers may give rise to the complete set ofeigenvalues Λ( u ). Numerical solutions of the BAEs (4.15)-(4.23) with random choice of w and τ for some small size imply that the solution (4.13) indeed gives the complete solutions ofthe model. Here we present the numerical solutions of the BAEs for the N = 2 case in Table1; The eigenvalue calculated from (4.13) is the same as that from the exact diagonalization ofthe Hamiltonian (2.16) with periodic boundary condition (2.18). Moreover, for a generic w and an arbitrary site number N , the eigenvalue Λ( u ) should be given by an inhomogeneous T − Q relation such as (4.13) with non-vanishing terms related to X i ( u ). However, when N is some particular number (i.e., N = 3 l for a positive integer l ) or the crossing parameter w takes some particular values (i.e., see below (4.33)-(4.34)), the relation (4.13) is reduced toa homogeneous one [1], which corresponds to the c = c = 0 solutions of (4.15)-(4.23).11able 1: Solutions of BAEs (4.15)-(4.23) for the Z case, N = 2, { θ j } = 0, w = − . τ = i and the parameters m = 1, m = m = m = l = l = l = l = 0. The symbol m indicates the number of the eigenenergy E . λ (1)1 λ (1)2 λ (2)1 λ (2)2 λ (3)1 − . − . i . . i − . . i . − . i − . − . i − . . i . − . i − . − . i . . i − . . i . . i − . − . i − . . i . − . i . − . i − . . i . − . i . − . i . . i . . i − . − . i . . i . . i . − . i . − . i . − . i − . . i . − . i . . i . − . i . − . i − . . i . − . i − . . i − . − . i . − . i − . . i . . i − . − . i − . . i . − . i − . . i . . i . − . i . − . iλ (3)2 λ (4)1 λ (4)2 φ φ . . i . − . i . . i − . . i − . − . i . − . i . . i . − . i − . − . i − . . i . . i . − . i − . . i . . i . − . i . − . i . . i . − . i . − . i − . − . i . . i . − . i . . i . . i − . . i − . . i − . . i . − . i − . − . i − . . i . . i . . i − . − . i . − . i − . . i . − . i . − . i − . . i . . i − . − . i . . i . . i − . − . i − . − . i . . ic c E m − . − . i − . . i − . − . . i − . − . i − . . − . i − . . i − . − . − . i − . . i . − . . i − . − . i . − . . i − . . i . − . . i . . i . − . . i . . i . . . i . − . i . .1 Generic w and τ case It follows from (4.15)-(4.18) that for the solution with c = c = 0, the parameters { λ ( i ) j } have to form the pairs: ( λ (1) j = λ (2) k , or λ (1) j = λ (2) k − w.λ (3) j = λ (4) k , or λ (3) j = λ (4) k − w . (4.25)Without losing generality, let us suppose that λ (1) j = λ (2) j Redef = ¯ λ (1) j , j = 1 , · · · , ¯ M ,λ (1)¯ M + j = λ (2)¯ M + j − w Redef = ¯ λ (1)¯ M + j , j = 1 , · · · , N − ¯ M ,λ (3) j = λ (4) j Redef = ¯ λ (2) j , j = 1 , · · · , ¯ M ,λ (3)¯ M + j = λ (4)¯ M + j − w Redef = ¯ λ (2)¯ M + j , j = 1 , · · · , N − ¯ M , where ¯ M and ¯ M are two non-negative integers. The corresponding T − Q relation (4.13)is reduced toΛ( u ) = a ( u ) e iπl u + φ ¯ Q (1) ( u − w )¯ Q (1) ( u ) + d ( u ) e iπl u + φ ¯ Q (1) ( u + w ) ¯ Q (2) ( u − w )¯ Q (1) ( u ) ¯ Q (2) ( u )+ d ( u ) e − iπ { ( l + l ) u +(2 l + l ) w }− φ − φ ¯ Q (2) ( u + w )¯ Q (2) ( u ) , (4.26)where the reduced Q -functions are given by¯ Q ( i ) ( u ) = ¯ M i Y j =1 σ ( u − ¯ λ ( i ) j ) σ ( w ) , i = 1 , , (4.27)provided that the two non-negative integers ¯ M and ¯ M satisfy the relations ( ( N − ¯ M ) w = m + l τ ( ¯ M − ¯ M − N ) w = m + l τ , (4.28)where m , m , l and l are some integers. • For the case of N = 3 l with a positive integer l . The only solution to (4.28) is m = m = l = l = 0 , and ¯ M = 2 l, ¯ M = l. T − Q relation becomesΛ( u ) = a ( u ) e φ ¯ Q (1) ( u − w )¯ Q (1) ( u ) + d ( u ) e φ ¯ Q (1) ( u + w ) ¯ Q (2) ( u − w )¯ Q (1) ( u ) ¯ Q (2) ( u )+ d ( u ) e − φ − φ ¯ Q (2) ( u + w )¯ Q (2) ( u ) , (4.29)the 3 l parameters { ¯ λ (1) j | j = 1 , · · · , l } and { ¯ λ (2) j | j = 1 , · · · , l } satisfy the associatedBAEs and the selection rule a (¯ λ (1) j ) ¯ Q (2) (¯ λ (1) j ) e φ − φ d (¯ λ (1) j ) ¯ Q (2) (¯ λ (1) j − w ) = − ¯ Q (1) (¯ λ (1) j + w )¯ Q (1) (¯ λ (1) j − w ) , j = 1 , · · · , l, (4.30) e φ +2 φ ¯ Q (1) (¯ λ (2) j + w )¯ Q (1) (¯ λ (2) j ) = − ¯ Q (2) (¯ λ (2) j + w )¯ Q (2) (¯ λ (2) j − w ) , j = 1 , · · · , l, (4.31) N Y j =1 Q (1) ( θ j − w ) Q (1) ( θ j ) = e − Nφ . (4.32)For this case (i.e., N = 3 l ), the algebraic Bethe Ansatz can also be applied to and ourresults recover those obtained in [15, 30]. • N = 3 l case. Since τ and w are generic complex numbers, generally (4.28) can notbe satisfied in this case and the eigenvalue Λ( u ) should be given by an inhomogeneous T − Q relation. w case For some degenerate values of w , c = c = 0 solutions indeed exist for an arbitrary sitenumber N . In this case, the parameters w and τ are no longer independent but related withthe constraint condition: w = 3 m + 3 l τ N − M , for m , l ∈ Z ; ¯ M ∈ Z + , (4.33)and there exists an integer n such that¯ M = ( n + 1) ¯ M − n + 13 N ∈ Z + . (4.34)In this case the relation (4.28) is fulfilled by ( ( N − ¯ M ) w = m + l τ ( ¯ M − ¯ M − N ) w = n ( m + l τ ) . T − Q relation becomesΛ( u ) = a ( u ) e iπl u + φ ¯ Q (1) ( u − w )¯ Q (1) ( u ) + d ( u ) e iπn l u + φ ¯ Q (1) ( u + w ) ¯ Q (2) ( u − w )¯ Q (1) ( u ) ¯ Q (2) ( u )+ d ( u ) e − iπ { ( n +1) l u +( n +2) l w }− φ − φ ¯ Q (2) ( u + w )¯ Q (2) ( u ) . (4.35)The resulting BAEs and selection rule read e iπ (1 − n ) l λ (1) j a (¯ λ (1) j ) ¯ Q (2) (¯ λ (1) j ) e φ − φ d (¯ λ (1) j ) ¯ Q (2) (¯ λ (1) j − w ) = − ¯ Q (1) (¯ λ (1) j + w )¯ Q (1) (¯ λ (1) j − w ) , j = 1 , · · · , ¯ M , (4.36) e iπ { (2 n +1) l λ (2) j +( n +2) l w } + φ +2 φ ¯ Q (1) (¯ λ (2) j + w )¯ Q (1) (¯ λ (2) j ) = − ¯ Q (2) (¯ λ (2) j + w )¯ Q (2) (¯ λ (2) j − w ) ,j = 1 , · · · , ¯ M , (4.37) N Y j =1 Q (1) ( θ j − w ) Q (1) ( θ j ) = e − Nφ − iπl P Nj =1 θ j . (4.38) Z n case In Sections 3, we have obtained the very operator product identities (3.11)-(3.15) for thefused transfer matrices { t j ( u ) | j = 1 , · · · , n } . These identities lead to that the correspondingeigenvalues { Λ j ( u ) | j = 1 , · · · , n } of the transfer matrices satisfy the associated relations(3.16)-(3.22). Similarly as those for the Z case, the relations allow us to determine theeigenvalues of the transfer matrix of the Z n -Belavin model completely.Let us introduce some functions { Q ( i ) | i = 1 , · · · , n − } , { Z i | i = 1 , · · · , n } and { X i | i =15 , · · · , n − } as follows: Q ( i ) ( u ) = N i Y j =1 σ ( u − λ ( i ) j ) σ ( w ) , i = 1 , . . . , n − , (5.1) Z ( u ) = e iπl u + φ a ( u ) Q (1) ( u − w ) Q (2) ( u ) ,Z ( u ) = e iπl u + φ d ( u ) Q (2) ( u + w ) Q (3) ( u − w ) Q (1) ( u ) Q (4) ( u ) , ... Z i ( u ) = e iπl i u + φ i d ( u ) Q (2 i − ( u + w ) Q (2 i − ( u − w ) Q (2 i − ( u ) Q (2 i ) ( u ) , ... Z n − ( u ) = e iπl n − u + φ n − d ( u ) Q (2 n − ( u + w ) Q (2 n − ( u − w ) Q (2 n − ( u ) Q (2 n − ( u ) ,Z n ( u ) = e − iπ P n − k =1 l k ( u +( n − k ) w ) − P n − j =1 φ j d ( u ) Q (2 n − ( u + w ) Q (2 n − ( u ) , (5.2)and X ( u ) = c e iπl n u a ( u ) d ( u ) Q (3) ( u − w ) Q (1) ( u ) Q (2) ( u ) ,X ( u ) = c e iπl n +1 u a ( u ) d ( u ) Q (2) ( u + w ) Q (5) ( u − w ) Q (3) ( u ) Q (4) ( u ) , ... X j ( u ) = c j e iπl n + j − u a ( u ) d ( u ) Q (2 j − ( u + w ) Q (2 j +1) ( u − w ) Q (2 j − ( u ) Q (2 j ) ( u ) , ... X n − ( u ) = c n − e iπl n − u a ( u ) d ( u ) Q (2 n − ( u + w ) Q (2 n − ( u ) Q (2 n − ( u ) , (5.3)where the 2 n − { N i | i = 1 , · · · , n − } are given by (5.9)-(5.12), { l i | i =1 , · · · , n − } are arbitrary integers, { φ i , c i | i = 1 , · · · , n − } are 2 n − n , an extra factor function f n ( u ) should be added to thefunction X n ( u ), namely, X n ( u ) = c n e iπl n − u a ( u ) d ( u ) Q ( n − ( u + w ) Q ( n +1) ( u − w ) Q ( n − ( u ) Q ( n ) ( u ) × f n ( u ) , (5.4)16hich ensures that all the numbers { N i | i = 1 , · · · , n − } are positive integers. The explicitexpression of the function f n ( u ) is given by (5.11) (or (5.13)) below.We are now in position to construct the associated inhomogeneous T − Q relations similarto those given by (4.13)-(4.14). Let us introduce the functions { Y l ( u ) | l = 1 , · · · , n − } , ( Y j − ( u ) = Z j ( u ) , j = 1 , · · · , n,Y j ( u ) = X j ( u ) , j = 1 , · · · , n − . (5.5)We further take the notation: Y ( l ) j ( u ) = Y j ( u − lw ) , l = 1 , · · · , n, j = 1 , · · · , n − . (5.6)Then the eigenvalue { Λ m ( u ) | m = 1 , · · · n − } which satisfy the relations (3.16)-(3.22) canbe given in terms of the inhomogeneous T − Q relationsΛ m ( u ) = X ′ ≤ i
12 ; (5.9)and there is no function X n ( u ); 17 For the case of even n and even N , we have N i − = N i = N n − i ) − = N n − i ) = i ( n − i )2 N, i = 1 , · · · , n f n ( u ) = 1; (5.11) • For the case of even n and odd N , we have N i − = N i = N n − i ) − = N n − i ) = i ( n − i )2 N + i , i = 1 , · · · , n f n ( u ) is given by f n ( u ) = σ ( u ) . (5.13)Moreover, the vanishing condition of the residues of Λ m ( u ) at the points λ ( i ) j gives rise to the18AEs: e iπl λ (1) j + φ Q (2) ( λ (1) j + w ) Q (4) ( λ (1) j ) + c e iπl n λ (1) j a ( λ (1) j ) Q (2) ( λ (1) j ) = 0 , j = 1 , · · · , N , (5.14) e iπl λ (2) j + φ Q (1) ( λ (2) j − w ) + c e iπl n λ (2) j d ( λ (2) j ) Q (3) ( λ (2) j − w ) Q (1) ( λ (2) j ) = 0 , j = 1 , · · · , N , (5.15)... e iπl i +1 λ (2 i − j + φ i +1 Q (2 i ) ( λ (2 i − j + w ) Q (2 i +2) ( λ (2 i − j ) + c i e iπl n + i − λ (2 i − j a ( λ (2 i − j ) Q (2 i − ( λ (2 i − j + w ) Q (2 i ) ( λ (2 i − j ) = 0 ,i = 2 , · · · , n − j = 1 , · · · , N i − , (5.16) e iπl i λ (2 i ) j + φ i Q (2 i − ( λ (2 i ) j − w ) Q (2 i − ( λ (2 i ) j ) + c i e iπl n + i − λ (2 i ) j a ( λ (2 i ) j ) Q (2 i +1) ( λ (2 i ) j − w ) Q (2 i − ( λ (2 i ) j ) = 0 ,i = 2 , · · · , n − j = 1 , · · · , N i , (5.17)... e − iπ P n − k =1 l k ( λ (2 n − j +( n − k ) w ) − P n − l =1 φ l Q (2 n − ( λ (2 n − j + w )+ c n − e iπl n − λ (2 n − j a ( λ (2 n − j ) Q (2 n − ( λ (2 n − j + w ) Q (2 n − ( λ (2 n − j ) = 0 , j = 1 , · · · , N n − , (5.18) e iπl n − λ (2 n − j + φ n − Q (2 n − ( λ (2 n − j − w ) Q (2 n − ( λ (2 n − j ) + c n − e iπl n − λ (2 n − j a ( λ (2 n − j ) Q (2 n − ( λ (2 n − j ) = 0 ,j = 1 , · · · , N n − . (5.19)Further, the periodicities (3.21) of the eigenvalues as well as the selection rule (3.22) give19ise to the associated BAEs:Θ (2) − Θ (1) = ( N − N + Nn ) w + m + l τ, (5.20)Θ (1) − Θ (2) − Θ (3) + Θ (4) = ( N − N + Nn ) w + m + l τ, (5.21)... − Θ (2 i − − Θ (2 i − + Θ (2 i − + Θ (2 i ) = ( N i − − N i − + Nn ) w + m i + l i τ, (5.22)... − Θ (2 n − − Θ (2 n − + Θ (2 n − + Θ (2 n − = ( N n − − N n − + Nn ) w + m n − + l n − τ, (5.23) − Θ + Θ (1) + Θ (2) − Θ (3) + ( n − N wn − N w = l n τ + m n , (5.24)... − Θ − Θ (2 j − + Θ (2 j − + Θ (2 j ) − Θ (2 j +1) + ( N j − − N j +1 + ( n − Nn ) w = l n + j − τ + m n + j − , (5.25)... − Θ − Θ (2 n − + Θ (2 n − + Θ (2 n − + ( N n − + ( n − Nn ) w = l n − τ + m n − , (5.26) N Y j =1 Q (1) ( θ j − w ) Q (2) ( θ j ) = e − iπl Θ − Nφ , (5.27)where { m i | i = 1 , · · · , n − } are arbitrary integers andΘ = N X l =1 θ l , Θ ( i ) = N i X l =1 λ ( i ) l , i = 1 , · · · , n − . (5.28)We have checked that for a generic w and τ but the number of sites N = nl with l beinga positive integer, the inhomogeneous T − Q relations (5.7) can be reduced to homogeneousones which were previously obtained by the algebraic Bethe ansatz [15, 30]. Moreover, it isalso found that when the crossing parameter w takes some discrete values (like (4.33) for the n = 3 case) the resulting T − Q relations can also become the homogeneous ones.20 Conclusions
The periodic Z n -Belavin model with an arbitrary site number N and generic coupling con-stants w and τ described by the Hamiltonian (2.16) and (2.18) is studied via the off-diagonalBethe Ansatz method. The eigenvalues { Λ i ( u ) | i = 1 , · · · , n − } of the corresponding trans-fer matrix and fused ones { t i ( u ) | i = 1 , · · · , n − } given by (3.8) are derived in terms of theinhomogeneous T − Q relations (5.7). In the special case of N = nl with a positive integer l , the resulting T − Q relation is reduced to a homogeneous one (such as (4.29)), whichrecovers the result obtained by the algebraic Bethe Ansatz method [15]. On the other hand,if the crossing parameter w take some special values (such as (4.33) for the n = 3 case), theresulting T − Q relation also becomes a homogeneous one (such as (4.35) for the n = 3 case).We remark that the Z n -symmetry (2.11) of the R -matrix R ( u ) ensures that the Z n -Belavinmodel with the twisted boundary condition given by H N,N +1 = G H N, G − , G = I α = I ( α ,α ) , α i ∈ Z n , (6.1)is also integrable. The corresponding transfer matrix t ( α ) ( u ) can be constructed by [31, 32] t ( α ) ( u ) = tr ( G T ( u )) , G = I α = I ( α ,α ) , α i ∈ Z n . (6.2)The Hamiltonian can be derived the same way as the periodic one (c.f., (2.16)). Using thesimilar method developed in previous sections, we can construct the corresponding ODBAsolution, which is given in Appendix B.The eigenvalues of the transfer matrix for the Z n -Belavin model with periodic (or twisted)boundary condition obtained in this paper might help one to construct the correspondingeigenstates, thus further giving rise to studying correlation functions [7] of the model. Forthis purpose, some particular basis such as the separation of variable (SoV) [33] basis [34, 35]or its higher-rank generalization [36] will play an important role. Acknowledgments
The financial supports from the National Natural Science Foundation of China (Grant Nos.11375141, 11374334, 11434013, 11425522 and 11547045), BCMIIS, the National Program forBasic Research of MOST (Grant No. 2016YFA0300603) and the Strategic Priority ResearchProgram of the Chinese Academy of Sciences are gratefully acknowledged. Two of the21uthors (K. Hao and F. Wen) would like to thank IoP/CAS for the hospitality during theirvisit there. They also would like to acknowledge S. Cui for his numerical helps.
Appendix A: T − Q relations for the Z case In this Appendix, we take the Z case as an example to show the procedure for constructingthe inhomogeneous T − Q relations (5.7). The functions (5.1)-(5.3) now read Q ( i ) ( u ) = N i Y j =1 σ ( u − λ ( i ) j ) σ ( w ) , i = 1 , · · · , , (A.1) Z ( u ) = e iπl u + φ a ( u ) Q (1) ( u − w ) Q (2) ( u ) ,Z ( u ) = e iπl u + φ d ( u ) Q (2) ( u + w ) Q (3) ( u − w ) Q (1) ( u ) Q (4) ( u ) ,Z ( u ) = e iπl u + φ d ( u ) Q (4) ( u + w ) Q (5) ( u − w ) Q (3) ( u ) Q (6) ( u ) ,Z ( u ) = e − iπ P k =1 l k ( u +(4 − k ) w ) − P j =1 φ j d ( u ) Q (6) ( u + w ) Q (5) ( u ) , (A.2)and X ( u ) = c e iπl u a ( u ) d ( u ) Q (3) ( u − w ) Q (1) ( u ) Q (2) ( u ) ,X ( u ) = c e iπl u a ( u ) d ( u ) Q (2) ( u + w ) Q (5) ( u − w ) f ( u ) Q (3) ( u ) Q (4) ( u ) ,X ( u ) = c e iπl u a ( u ) d ( u ) Q (4) ( u + w ) Q (5) ( u ) Q (6) ( u ) . (A.3)22he inhomogeneous T − Q relations (5.7) becomeΛ( u ) = Z ( u ) + Z ( u ) + Z ( u ) + Z ( u ) + X ( u ) + X ( u ) + X ( u ) , (A.4)Λ ( u ) = Z ( u ) Z ( u − w ) + Z ( u ) Z ( u − w ) + Z ( u ) Z ( u − w ) + Z ( u ) Z ( u − w )+ Z ( u ) Z ( u − w ) + Z ( u ) Z ( u − w ) + X ( u )( Z ( u − w ) + Z ( u − w ))+( Z ( u ) + Z ( u )) X ( u − w ) + X ( u ) X ( u − w ) + Z ( u ) X ( u − w )+ X ( u ) Z ( u − w ) , (A.5)Λ ( u ) = Z ( u ) Z ( u − w ) Z ( u − w ) + Z ( u ) Z ( u − w ) Z ( u − w )+ Z ( u ) Z ( u − w ) Z ( u − w ) + Z ( u ) Z ( u − w ) Z ( u − w )+ Z ( u ) Z ( u − w ) X ( u − w ) + Z ( u ) X ( u − w ) Z ( u − w )+ X ( u ) Z ( u − w ) Z ( u − w ) , (A.6)Λ ( u ) = Z ( u ) Z ( u − w ) Z ( u − w ) Z ( u − w ) . (A.7)The positive integers { N i | i = 1 , · · · , } and the function f ( u ) are given as follows: • When N is even, we have N = N = N = N = 32 N, N = N = 2 N, (A.8)and the function f ( u ) is f ( u ) = 1 . (A.9) • When N is odd, we have N = N = N = N = 3 N + 12 , N = N = 2 N + 1 , (A.10)and the functions f ( u ) is f ( u ) = σ ( u ) . (A.11)23he associated BAEs (5.14)-(5.27) become e iπl λ (1) j + φ Q (2) ( λ (1) j + w ) Q (4) ( λ (1) j ) + c e iπl λ (1) j a ( λ (1) j ) Q (2) ( λ (1) j ) = 0 , j = 1 , · · · , N , (A.12) e iπl λ (2) j + φ Q (1) ( λ (2) j − w )+ c e iπl λ (2) j d ( λ (2) j ) Q (3) ( λ (2) j − w ) Q (1) ( λ (2) j ) = 0 , j = 1 , · · · , N , (A.13) e iπl λ (3) j + φ Q (4) ( λ (3) j + w ) Q (6) ( λ (3) j ) + c e iπl λ (3) j a ( λ (3) j ) Q (2) ( λ (3) j + w ) f ( λ (3) j ) Q (4) ( λ (3) j ) = 0 ,j = 1 , · · · , N , (A.14) e iπl λ (4) j + φ Q (3) ( λ (4) j − w ) Q (1) ( λ (4) j ) + c e iπl λ (4) j a ( λ (4) j ) Q (5) ( λ (4) j − w ) f ( λ (4) j ) Q (3) ( λ (4) j ) = 0 ,j = 1 , · · · , N , (A.15) e − iπ P k =1 l k λ (5) j − P t =1 (4 − t ) l t w − P l =1 φ l Q (6) ( λ (5) j + w )+ c e iπl λ (5) j a ( λ (5) j ) Q (4) ( λ (5) j + w ) Q (6) ( λ (5) j ) = 0 , j = 1 , · · · , N , (A.16) e iπl λ (6) j + φ Q (5) ( λ (6) j − w ) Q (3) ( λ (6) j ) + c e iπl λ (6) j a ( λ (6) j ) Q (5) ( λ (6) j ) = 0 , j = 1 , · · · , N , (A.17)Θ (2) − Θ (1) = ( N − N + N w + m + l τ, (A.18)Θ (1) − Θ (2) − Θ (3) + Θ (4) = ( N − N + N w + m + l τ, (A.19)Θ (3) − Θ (4) − Θ (5) + Θ (6) = ( N − N + N w + m + l τ, (A.20) − Θ + Θ (1) + Θ (2) − Θ (3) − N w + 3 N w m + l τ, (A.21) − Θ − Θ (2) + Θ (3) + Θ (4) − Θ (5) + N w − N w + 3 N w m + l τ, (A.22) − Θ − Θ (4) + Θ (5) + Θ (6) + N w + 3 N w m + l τ, (A.23) N Y j =1 Q (1) ( θ j − w ) Q (2) ( θ j ) = e − iπl Θ − Nφ . (A.24)24he main purpose of this Appendix is to show the new features occurred in Z case, whichare crucial to understand the structure of the inhomogeneous T − Q relations (5.7) for general Z n case. Appendix B: Z -Belavin model with twisted boundarycondition The Yang-Baxter algebra relation (2.14) and the Z n symmetry (2.11) properties of Z n -BelavinR-matrix lead to the fact that the transfer matrix t ( α ) ( u ) given by (6.2) with different spectralparameters are mutually commuting [ t ( α ) ( u ) , t ( α ) ( v )] = 0. This ensures the integrability ofthe inhomogeneous Z n -Belavin model with twisted boundary condition.Without loss of generality, we take the Z -model with the twisted boundary matrix G = h as an example to construct the solution. The corresponding transfer matrix then reads t (1 , ( u ) = tr ( h T ( u )) . The invariant relation and operator identities of this transfer matrix t ( α ) ( u ) can be derivedin the same way as in dealing with the su ( n ) spin torus [36]. The properties of this transfermatrix imply that the corresponding eigenvalues { Λ m ( u ) | m = 1 , · · · , } satisfy the followingfunctional relationsΛ( θ j )Λ m ( θ j − w ) = Λ m +1 ( θ j ) , m = 1 , , j = 1 , · · · , N, (B.1)Λ ( u ) = Det q { h } Det q { T ( u ) } = a ( u ) d ( u − w ) d ( u − w ) , (B.2)Λ ( θ j + w ) = 0 , j = 1 , · · · , N. (B.3)The periodicity of the Z n -Belavin R-matrix and commuting relation (2.4) of operators g , h give rise to that the eigenvalues are some elliptic polynomials of the fixed degrees mN withthe periodicityΛ m ( u + 1) = ( − mN e miπ Λ m ( u ) , m = 1 , , (B.4)Λ m ( u + τ ) = ( − mN e − iπ { mN ( u + w + τ − m − w ) − m P Nl =1 θ l } Λ m ( u ) , m = 1 , . (B.5)Moreover, the unitarity relation (2.9) and h = id allow us to derive the following identity ( N Y l =1 Λ( θ l ) ) = ( N Y l =1 a ( θ l ) ) . (B.6)25imilar as the periodic case, the relations (B.1)-(B.6) allow us to determine the eigenval-ues { Λ m ( u ) } of the corresponding transfer matrices completely. We can thus express Λ( u )and Λ ( u ) in terms of an inhomogeneous T − Q relation as follows. Let us introduce some Q -functions Q ( i ) ( u ) = N Y j =1 σ ( u − λ ( i ) j ) σ ( w ) , i = 1 , · · · , , parameterized by 4 N parameters { λ ( i ) j | j = 1 , · · · , N ; i = 1 , · · · , } determined later by theassociated BAEs (see below (B.14)-(B.22)). Associated with the above Q -functions, weintroduce 5 functions { Z i ( u ) | i = 1 , , } and { X i ( u ) | i = 1 , } as Z ( u ) = a ( u ) e iπ ( l + ) u + φ Q (1) ( u − w ) Q (2) ( u ) , (B.7) Z ( u ) = ω d ( u ) e iπ ( l + ) u + φ Q (2) ( u + w ) Q (3) ( u − w ) Q (1) ( u ) Q (4) ( u ) , (B.8) Z ( u ) = ω d ( u ) e − iπ { ( l + l + ) u +(2 l + l +2) w }− φ − φ Q (4) ( u + w ) Q (3) ( u ) , (B.9) X ( u ) = c a ( u ) d ( u ) e iπ ( l + ) u Q (3) ( u − w ) Q (1) ( u ) Q (2) ( u ) , (B.10) X ( u ) = c a ( u ) d ( u ) e iπ ( l + ) u Q (2) ( u + w ) Q (3) ( u ) Q (4) ( u ) , (B.11)where ω = e iπ , { l i | i = 1 , · · · , } are 4 integers, { φ i , c i | i = 1 , } are 4 complex numbers.Then we can introduce the inhomogeneous T − Q relations,Λ( u ) = Z ( u ) + Z ( u ) + Z ( u ) + X ( u ) + X ( u ) , (B.12)Λ ( u ) = Z ( u ) Z ( u − w ) + Z ( u ) Z ( u − w ) + Z ( u ) Z ( u − w )+ X ( u ) Z ( u − w ) + Z ( u ) X ( u − w ) . (B.13)In order that the above parameterizations of Λ( u ) and Λ ( u ) become a solution to (B.1)-(B.6), the 4( N + 1) parameters { λ ( i ) j | j = 1 , · · · , N ; i = 1 , · · · , } and { φ i , c i | i = 1 , } have26o satisfy the associated BAEs ω e iπ ( l + ) λ (1) j + φ Q (2) ( λ (1) j + w ) Q (2) ( λ (1) j ) + c e iπ ( l + ) λ (1) j a ( λ (1) j ) Q (4) ( λ (1) j ) = 0 ,j = 1 , · · · , N, (B.14) e iπ ( l + ) λ (2) j + φ Q (1) ( λ (2) j − w ) Q (1) ( λ (2) j )+ c e iπ ( l + ) λ (2) j d ( λ (2) j ) Q (3) ( λ (2) j − w ) = 0 ,j = 1 , · · · , N, (B.15) ω e − iπ { ( l + l + ) λ (3) j +(2 l + l +2) w }− φ − φ Q (4) ( λ (3) j + w ) Q (4) ( λ (3) j )+ c e iπ ( l + ) λ (3) j a ( λ (3) j ) Q (2) ( λ (3) j + w ) = 0 , j = 1 , · · · , N, (B.16) ω e iπ ( l + ) λ (4) j + φ Q (3) ( λ (4) j − w ) Q (3) ( λ (4) j )+ c e iπ ( l + ) λ (4) j a ( λ (4) j ) Q (1) ( λ (4) j ) = 0 ,j = 1 , · · · , N, (B.17) − Θ (1) + Θ (2) − N w = m + ( l + 23 ) τ, (B.18) − Θ (2) − Θ (3) + Θ (1) + Θ (4) − N w = m + ( l + 23 ) τ, (B.19) − Θ − Θ (3) + Θ (1) + Θ (2) − N w = m + ( l + 23 ) τ, (B.20) − Θ − Θ (2) + Θ (3) + Θ (4) + 5 N w = m + ( l + 23 ) τ, (B.21) N Y j =1 Q (1) ( θ j − w ) Q (2) ( θ j ) = e − iπ ( l + )Θ − Nφ . (B.22)where { m i | i = 1 , · · · , } are 4 integers andΘ = N X l =1 θ l , Θ ( i ) = N X l =1 λ ( i ) l , i = 1 , · · · , . Numerical solutions of the BAEs (B.14)-(B.22) for small size with random choice of w and τ imply that the Bethe ansatz solution (B.12) indeed give the complete solutions of themodel. Here we present the numerical solutions of the BAEs for the N = 2 case in Table 2;The eigenvalue calculated from (B.12) is the same as that from the exact diagonalization ofthe Hamiltonian (2.16) with the twisted boundary condition (6.1) associated with G = h .27able 2: Solutions of BAEs (B.14)-(B.22) for the Z -Belavin model with the twisted boundary condition, N = 2, { θ j } = 0, w = − . τ = i and the parameters m = 1, m = m = m = l = l = l = l = 0. 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