On the solutions of the $Z_n$-Belavin model with arbitrary number of sites

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 the corresponding 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 Bethe Ansatz.


Introduction
Our understanding to phase transitions and critical phenomena has been greatly enhanced by the study on lattice integrable models [1]. Such exact results provide valuable insights into the key theoretical development of universality classes in areas ranging from modern condensed physics [2,3] to string and super-symmetric Yang-Mills theories [4,5,6]. Among solvable models [1,7,8], elliptic ones stand out as a particularly important class due to the fact 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 celebrated XYZ spin chain as the special case of n = 2.
The first exact solution of the Z 2 -model with periodic boundary condition was given by Baxter [10], where the fundamental equation (the Yang-Baxter equation [1,11]) was emphasized and the T −Q method was proposed. Takhtadzhan and Faddeev [12] resolved the model 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 the associated face model, Hou et al [15] generalized Takhatadzhan and Faddeev's approach to the Z n -Belavin model with a generic n. In their approach, local gauge transformation played a central role to obtain local vacuum states (reference states) with which the algebraic Bethe Ansatz analysis can be performed. However, such reference states are so far only available for some very particular number of lattice sites, namely, N = nl with l being a positive integer, but not for the other N. This leads to the fact that the conventional Bethe Ansatz methods have been quite hard to apply to the latter case for many years. In fact, the lack of a reference state is a common feature of the integrable models without U(1) symmetry and had 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] was proposed to solve the eigenvalue problem of integrable models without U(1)-symmetry. The closed XYZ spin chain (or the Z 2 -model) with arbitrary number of sites [18] and several other long-standing models [16,19,20,21] have since been solved. In this paper, we adopt ODBA to solve the eigenvalue problem of the periodic Z n -Belavin model with a generic positive integer n ≥ 2 and an arbitrary lattice number N.
The paper is organized as follows. Section 2 serves as an introduction of our notations and some basic ingredients. The commuting transfer matrix associated with the periodic Z n -Belavin model is constructed to show the integrability of the model. In section 3, based on some intrinsic properties of the Z n -Belavin's R-matrix, we construct the fused transfer matrices by anti-symmetric fusion procedure and derive some operator identities and the quasi-periodicities of these matrices. Taking the Z 3 model as a concrete example, we express the eigenvalues of the transfer matrix in terms of a nested inhomogeneous T − Q relation and the associated Bethe Ansatz equations (BAEs) in Section 4. Generalization to Z n case is presented in Section 5. We summarize our results and give some discussions in Section 6.
A slightly detailed description about the Z 4 case, which might be crucial to understand the procedure 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 ) > 0 and a generic complex number w. For convenience, let us introduce the elliptic functions Among them the σ-function 3 satisfies the following identity: Let V denote an n-dimensional linear space with an orthonormal basis {|i |i = 1, · · · , n}, and g, h be two n × n matrices with the elements 3 Our σ-function is the ϑ-function ϑ 1 (u) [22]. It has the following relation with the Weierstrassian σfunction denoted by σ w (u): σ w (u) ∝ e η1u 2 σ(u), η 1 = π 2 ( 1 6 − 4 namely, It is easy to verify that the matrices satisfy the relation Associated with any α = (α 1 , α 2 ), α 1 , α 2 ∈ Z n , one can introduce an n × n matrix I α defined by and an elliptic function σ α (u) given by The Z n -Belavin R-matrix R(u) ∈ End(V ⊗ V) is given by [9,23,14] which satisfies the quantum Yang-Baxter equation (QYBE) and the properties [23], Initial condition : Unitarity : Crossing-unitarity :

10)
Z n -symmetry : Fusion conditions : Here R 21 (u) = P 1,2 R 12 (u)P 1,2 with P 1,2 being the usual permutation operator, P (∓) 1,2 = 1 2 {1 ∓ P 1,2 } 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 in the 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-th space and as an identity on the other factor spaces; R ij (u) is an embedding operator of R-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 matrix with operator-valued elements acting on (V) ⊗N reads (2.13) Here {θ i |i = 1, · · · , N} are arbitrary free complex parameters which are usually called the inhomogeneous parameters. With the help of the QYBE (2.7), one can show that T (u) satisfies the Yang-Baxter algebra relation (2.14) Let us introduce the transfer matrix t(u) t(u) = tr 0 (T 0 (u)) = tr (T (u)) .

(2.15)
The Z n -Belavin model [9] with periodic boundary condition is a quantum spin chain described by the Hamiltonian with the periodic boundary condition, namely, The commutativity of the transfer matrices follows as a consequence of (2.14). This ensures the integrability of the inhomogeneous Z n -Belavin model with periodic boundary.

Relations of the eigenvalues
Following the method developed in [19] (see also Chapter 7 of [17]), we apply the fusion techniques [24,25,26,27] to study the Z n -Belavin model. Besides the fundamental transfer matrix t(u) some other fused transfer matrices {t j (u)|j = 1, · · · , n} (see below (3.8)), which commute with each other and include the original one as t 1 (u) = t(u), are constructed through an anti-symmetric fusion procedure with the help of the fusion condition (2.12) of the R-matrix.
Iterating the above relation yields alternative definition of the projectors 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 above anti-symmetric projectors, we can construct the fused monodromy matrices The corresponding fused transfer matrices {t j (u)|j = 1, · · · , n} (including the original one as t 1 (u) = t(u)) are then given by t m (u) = tr 1,··· ,m T 1,··· ,m (u) , m = 2, · · · , n. (3.8) The last fused transfer matrix t n (u) is the so-called quantum determinant [28] which plays the role of the generating functional of the centers of the associated quantum algebras [29].
Let us evaluate the transfer matrix of the closed chain at some special points. The initial condition of the R-matrix (2.6) implies that The unitarity relation (2.9) allows us to derive the following identity:

Functional relations of eigenvalues
The commutativity (3.10) of the transfer matrices {t m (u)|m = 1, · · · , n} with different spectral parameters implies that they have common eigenstates. Let |Ψ be a common eigenstate of {t m (u)}, which dose not depend upon u, with the eigenvalues Λ m (u) (we shall take the convention: Λ(u) = Λ 1 (u)), The properties (3.9), (3.11) and (3.12) of the transfer matrices {t m (u)|m = 1, · · · , n} imply that the corresponding eigenvalues {Λ m (u)|m = 1, · · · , n} satisfy the functional relations where the functions a(u) and d(u) are given by (3.15). From the definitions (2.6), (2.15) and 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 periodicity Moreover the product identity (3.15) of the transfer matrix t(u) leads to the relation
• N = 3l case. Since τ and w are generic complex numbers, generally (4.28) can not be satisfied in this case and the eigenvalue Λ(u) should be given by an inhomogeneous T − Q relation.

Degenerate w case
For some degenerate values of w, c 1 = c 2 = 0 solutions indeed exist for an arbitrary site number N. In this case, the parameters w and τ are no longer independent but related with the constraint condition: and there exists an integer n 1 such that In this case the relation (4.28) is fulfilled by .
The 2n − 2 positive integers {N i |i = 1, · · · , 2n − 2} and the function f n 2 (u) in (5.4) are given as follows: • For the case of odd n, we have and there is no function X n 2 (u); • For the case of even n and even N, we have (5.10) and f n 2 (u) = 1; (5.11) • For the case of even n and odd N, we have the function f n 2 (u) is given by Moreover, the vanishing condition of the residues of Λ m (u) at the points λ (i) j gives rise to the BAEs: . . .
Further, the periodicities (3.21) of the eigenvalues as well as the selection rule (3.22) give rise to the associated BAEs: . . .
where {m i |i = 1, · · · , 2n − 2} are arbitrary integers and We have checked that for a generic w and τ but the number of sites N = nl with l being a positive integer, the inhomogeneous T − Q relations (5.7) can be reduced to homogeneous ones which were previously obtained by the algebraic Bethe ansatz [15,30]. Moreover, it is also 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.

Conclusions
The periodic Z n -Belavin model with an arbitrary site number N and generic coupling constants w and τ described by the Hamiltonian (2.16) and (2.18) is studied via the off-diagonal Bethe Ansatz method. The eigenvalues {Λ i (u)|i = 1, · · · , n − 1} of the corresponding transfer matrix and fused ones {t i (u)|i = 1, · · · , n − 1} given by (3.8) are derived in terms of the inhomogeneous 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)), which recovers 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), the resulting 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 -Belavin model with the twisted boundary condition given by is also integrable. The corresponding transfer matrix t (α) (u) can be constructed by [31,32] t (α) (u) = tr 0 (G 0 T 0 (u)) , G = I α = I (α 1 ,α 2 ) , α i ∈ Z n . (6. 2) The Hamiltonian can be derived the same way as the periodic one (c.f., (2.16)). Using the similar method developed in previous sections, we can construct the corresponding ODBA solution, 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 corresponding eigenstates, thus further giving rise to studying correlation functions [7] of the model. For this 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.
authors (K. Hao and F. Wen) would like to thank IoP/CAS for the hospitality during their visit there. They also would like to acknowledge S. Cui for his numerical helps.
Appendix A: T − Q relations for the Z 4 case In this Appendix, we take the Z 4 case as an example to show the procedure for constructing the inhomogeneous T − Q relations (5.7). The functions (5.1)-(5.3) now read , i = 1, · · · , 6, (A.1) and The inhomogeneous T − Q relations (5.7) become The positive integers {N i |i = 1, · · · , 6} and the function f 2 (u) are given as follows: • When N is even, we have and the function f 2 (u) is • When N is odd, we have and the functions f 2 (u) is The associated BAEs (5.14)-(5.27) become = 0, j = 1, · · · , N 6 , (A.17) −Θ − Θ (4) + Θ (5) + Θ (6) The main purpose of this Appendix is to show the new features occurred in Z 4 case, which are crucial to understand the structure of the inhomogeneous T −Q relations (5.7) for general Z n case.

Appendix B: Z 3 -Belavin model with twisted boundary condition
The Yang-Baxter algebra relation (2.14) and the Z n symmetry (2.11) properties of Z n -Belavin R-matrix lead to the fact that the transfer matrix t (α) (u) given by (6.2) with different spectral parameters are mutually commuting [t (α) (u), t (α) (v)] = 0. This ensures the integrability of the inhomogeneous Z n -Belavin model with twisted boundary condition.
Without loss of generality, we take the Z 3 -model with the twisted boundary matrix G = h as an example to construct the solution. The corresponding transfer matrix then reads The invariant relation and operator identities of this transfer matrix t (α) (u) can be derived in the same way as in dealing with the su(n) spin torus [36]. The properties of this transfer matrix imply that the corresponding eigenvalues {Λ m (u)|m = 1, · · · , 3} satisfy the following functional relations Λ(θ j )Λ m (θ j − w) = Λ m+1 (θ j ), m = 1, 2, j = 1, · · · , N, (B.1) The periodicity of the Z n -Belavin R-matrix and commuting relation ( to satisfy the associated BAEs j ) = 0, j = 1, · · · , N, (B.14)  Table 2; The eigenvalue calculated from (B.12) is the same as that from the exact diagonalization of the Hamiltonian (2.16) with the twisted boundary condition (6.1) associated with G = h.