Bethe ansatz solution of the $\tau_2$-model with arbitrary boundary fields

The quantum $\tau_2$-model with generic site-dependent inhomogeneity and arbitrary boundary fields is studied via the off-diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix are given in terms of an inhomogeneous T-Q relation, which is based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices. Moreover, the associated Bethe Ansatz equations are also obtained.


Introduction
The finite-size inhomogeneous τ 2 -model also known as the Baxter-Bazhanov-Stroganov model (BBS model) [1,2,3,4] is a N-state spin lattice model, which is intimately related to some other integrable models under certain parameter constraints such as the chiral Potts model [5,6,7,8,9,10] and the relativistic quantum Toda chain model [11]. Lots of papers have appeared to explain such connections and many efforts have been made to calculate the eigenvalues of the chiral Potts model by solving the τ 2 -model with a recursive functional relation [4,12,13]. The τ 2 -model is a simple quantum integrable models associated with cyclic representation of the Wely algebra. Although its integrability has been proven [3] for decades, there is still no effective method to solve the model completely due to lack of a simple Q-operator solution in terms of Baxter's T − Q relation. In fact, the Q-operator is a very complicated function defined in high genus space and its concrete expression is hard to be derived. Very recently, Paul Fendley had found a "parafermionic" way to diagonalise a simple solvable Hamiltonian associated with the chiral Potts model [14]. Subsequently, this method was generalised to solve the τ 2 -model with particular open boundaries [15,16,17].
Until very recently, the Bethe Ansatz solution of the periodic τ 2 -model with generic sitedependent inhomogeneity was obtained by constructing an inhomogeneous T − Q relation with polynomial Q-functions (i.e. off-diagonal Bethe Ansatz method (ODBA) [18,19,20]), which provides a perspective to investigate the τ 2 -model with generic open boundary condition. Moreover, such a solution allows the authors [21] further to retrieve the corresponding The outline of this paper is as follows. In Section 2, we begin with a brief introduction of the fundamental transfer matrix. In Section 3, we study the properties of the transfer matrix and employing the so-called fusion procedure [22,23,24] to construct the higherspin transfer matrices, which obey an infinite fusion hierarchy. In Section 4, we obtain the truncation identity for the fused transfer matrices when the bulk anisotropy value takes the special case η = 2iπ p and the exact functional relations of the fundamental transfer matrix. In Section 5, we give the eigenvalues of the transfer matrix in terms of some inhomogeneous T − Q relation and the associated Bethe Ansatz equation. In the last Section, we summarize our results and give some discussions. Some detailed technical calculations are given in appendices A and B.

Transfer matrix
Let us fix an odd integer p such that p ≥ 3, and let V be a p-dimensional vector space (i.e. the local Hilbert space) with an orthonormal basis {|m |m ∈ Z p }. Define two p × p matrices X and Z which act on the basis as where q ≡ e −η is a p-root of unity (i.e., q p = 1). The embedding operators {X n , Z n |n = 1, · · · , N} denote the generators of the ultra-local Weyl algebra: It has been shown that the τ 2 -model can be described by a quantum integrable spin chain [3]. In order to construct the monodromy matrix, one need to introduce the L-operators for each site of the quantum chain. The associated L-operator L n (u) ∈ End(C 2 ⊗ V) defined in the most general cyclic representation of U q (sl 2 ), is given by [3] L n (u) = e u d (+) are some parameters associated with each site. These parameters are subjected to two constraints, which ensure that the above L-operator L n satisfies the Yang-Baxter algebra [3], The associated R-matrix R(u) ∈ End(C 2 ⊗ C 2 ) is the well-known six-vertex R-matrix given by with the crossing parameter η taking the special values η = 2iπ/p, p = 2l + 1, l = 1, 2, · · · . (2.7) The R-matrix satisfies the quantum Yang-Baxter equation (QYBE) [25,26] and becomes some projectors when the spectral parameter u takes some special values as Symmetric-fusion conditions : R(η) = 2 sinh η Diag(cosh η, 1, 1, cosh η) P (+) , (2.9) where P (+) (P (−) ) is the symmetric (anti-symmetric) projector of the tensor space C 2 ⊗ C 2 .
Associated with the local L-operators {L n (u)|n = 1, . . . , N} given by (2.3), let us introduce the one-row monodromy matrix T (u) The local relations (2.5) imply that the monodromy matrix T (u) also satisfies the Yang-Baxter algebra which ensures the integrability of the τ 2 -model with the periodic boundary condition [3].
Integrable open chain can be constructed as follows [27]. Let us introduce a pair of K-matrices K − (u) and K + (u). The former satisfies the reflection equation (RE) [28] (2.12) and the latter satisfies the dual RE In order to construct the associated open spin chain, let us introduce theL(u) in the form 14) It is easy to check that L n (u) enjoys the crossing property L n (u) = σ yLt n (−u − η)σ y , n = 1, . . . , N, (2.15) and the inverse relation where the function Det q {L n (u)} is the quantum determinant (which will be given by below In this paper, we consider the most generic non-diagonal K − (u) matrix found in Refs. [29,30], which is in the form of , where α − , β − , and θ − are three free boundary parameters. The most generic non-diagonal K-matrix K + (u) is given by We note that the two K-matrices possess the following properties where σ α with α = x, y, z are the Pauli matrices.
where D (±) and F (±) are four constants related to the inhomogeneous parameters as follows, Moreover, we can calculate the special values of the associated transfer matrix at u = 0, iπ 2 with the help of the relations (2.22), namely, The expressions of the L-operators (2.3) and (2.14) allows us to derive their quasi-periodicities The quasi-periodicity of K-matrices (2.21) enables us to obtain the associated periodicity property of the transfer matrix t(u) The above relation implies that the transfer matrix t(u) can be expressed in terms of e 2u as a Laurent polynomial of the form where {t n |n = N +2, N +1, · · · , −(N +2)} form the 2N +5 conserved charges. In particular, t N +2 and t −(N +2) are given by where the constants D (±) and F (±) are determined by (3.2).
Following the method in [31,32] and using the crossing relation of the L-matrix (2.15) and the explicit expressions of the K-matrices (2.19) and (2.20), we verify that the corresponding transfer matrix t(u) satisfies the following crossing relation We can define the average value O(u) of the matrix elements of the monodromy matrices T (u) andT (u) (or the L-operators L n (u) and theL-operatorsL n (u)) by using the averaging procedure [33]: Ref. [33] that (3.11) and the average values of each L-operator andL-operator are given by with n = 1, · · · , N. Note that the average values of the matrix elements are Laurent polynomials of e pu , which implies

Fusion hierarchy
The main tool adopted in this paper to solve the open τ 2 -model is the so-called fusion technique, by which high-dimensional representations can be obtained from the low-dimensional ones. The fusion technique was first developed in Refs. [22,23,24] for R−matrices, and then generalised for K−matrices in Refs. [34,35,36,37]. In recent years, this technique has been extensively used in solving lots of integrable models [38,39]. Following the procedure in Ref. [23], we introduce the projectors where S m is the permutation group of m indices, and P σ is the permutation operator in the (1 + P 23 P 12 + P 12 P 23 + P 12 + P 23 + P 13 ).
We introduce further the fused spin-j monodromy matrices T The fused transfer matrices t (j) (u) which correspond to a spin-j auxiliary space can be constructed by the fused monodromy matrices and K-matrices as The double-row transfer matrix t(u) given by (2.17) corresponds to the fundamental case . Also, the fused transfer matrices constitute commutative families These transfer matrices also satisfy the so-called fusion hierarchy [34,35,36,40] with the conventions t (− 1 2 ) (u) = 0 and t (0) = id. The coefficient δ(u), the so-called quantum determinant [22,41,42,43], is given by where , (3.36) Let us introduce the functions a(u) and d(u) as follows: Similar as the definitions (3.2), the constants G (±) and H (±) are Then it is easy to check that the quantum determinant (3.29) can be expressed in terms of the above functions as Similar to the τ 2 -model with the periodic condition [20], we can also use the recursive relation (3.28) and the coefficient function (3.41) to express the fused transfer matrix t (j) (u) in terms of the fundamental one t ( 1 2 ) (u) with a 2j-order functional relation which can be expressed as the determinant of some 2j × 2j matrix [44], namely, (3.42)

Truncation identity
We now proceed to formulate the desired operator identities to determine the spectrum of the transfer matrix t(u) given by (2.17). For this purpose, we first derive separate truncation identities for the monodromy matrices and K-matrices. We recall that the fusion approach described in the previous section. When the crossing parameters η takes the special values η = 2iπ p , one can find that the spin-p 2 fused monodromy matrices mentioned in (3.24), (3.25), all take the block-lower triangular forms [33] where and Here {g(u), h(u),ĝ(u),ĥ(u)} are some operators, which are irrelevant to the transfer matrices.
Moreover, through tedious calculations, it is found that the general non-diagonal boundary fused K − -matrices (3.21) for η = 2iπ p and j = p 2 take the following form like (4.1) where the functions and the (p − 1) . (4.13) The explicit expressions of the elements of the fused monodromy matrices and the K-matrices for the cases p = 3 are given in Appendix A.
Hence, we are finally in position to formulate the truncation identity for the fused transfer matrices t (j) (u) defined in (3.26). Based on the results of (4.1) and (4.2) for the fused monodromy matrices and those of (4.6) and (4.13) for the fused K-matrices, we obtain where the coefficientsÃ(u) andD(u) are given by 3 .
It is remarked that the functions K

(u) and the average values of each monodromy matrices are invariant under shifting with η.
Combining the fusion hierarchy (3.28) and the closing relation (4.14) for η = 2iπ p , we arrive at the functional relation for the fundamental transfer matrix straightforward. Here we give an example of the functional relations for p = 3:

Functional relations of eigenvalues
The commutativity (3.27) of the fused transfer matrices {t (j) (u)} with different spectral parameters implies that they have common eigenstates. One can set |Ψ to be a common eigenstate of these fused transfer matrices with eigenvalues Λ (j) (u), i.e., The asymptotic behavior (3.1) and the special values at u = 0, iπ 2 of the transfer matrix t(u) enables us to derive that the corresponding eigenvalue Λ(u) have the following functional According to the fusion hierarchy relation (3.28) and the determinant representation (3.42) of the fused transfer matrices, the eigenvalues Λ (j) (u) give some similar representation in terms of the fundamental one Λ(u) = Λ ( 1 2 ) (u) where the functions a(u) and d(u) are given by (3.37) and (3.38). The truncation identity (4.14) of the spin-p 2 transfer matrix leads to the fact that the corresponding eigenvalue Λ ( p 2 ) (u) satisfies the relation where the functionsÃ(u) andD(u) are given by (4.15)-(4.16). For example, the functional relation of the eigenvalue for p = 3 is It is believed [1,3,45] that the relations (5.1)-(5.5), the analytic property (5.6) and the truncation identity (5.8) allow us to completely determine the eigenvalues Λ(u) of the fundamental transfer matrix t(u) given by (2.17).

Degenerate case
The third term of the T − Q relation (5.9) does not vanish when the generic inhomogeneous parameters {d n , h (±) n |n = 1, · · · , N} (only obey the constraint (2.4) which ensure the integrability of the model) and the boundary parameters are absence of restriction.

Conclusion
In this paper, we have studied the most general cyclic representation of the quantum τ 2model (also known as Baxter-Bazhanov-Stroganov (BBS) model) with generic integrable boundary conditions via the ODBA method [19]. Based on the truncation identity (4.14) of the fused transfer matrices obtained from the fusion technique, we construct the corresponding inhomogeneous T − Q relation (5.9) and the associated BAEs Note added: After this paper was completed we became aware of the recent results reported in [50]. The authors use the Sklyanin's separation of variables (SoV) method [51,52] to study the spectral problem for the open τ 2 -model with some constrains on inhomogeneous parameters and also on the boundary parameters.

Acknowledgments
We would like to thank Prof. Y. Wang for his valuable discussions and continuous encour-

A Specific cases of the fused K-matrices
In this appendix we present the explicit expressions of the matrix elements of the fused Kmatrices K ±( p 2 ) (u) given in (4.6) and (4.13) for the p = 3 case as an example. In this case, the corresponding matrix elements are are independent of the site (i.e., n), which corresponds to the homogeneous model. In the both cases, the average values of L-operators {L n (u)|n = 1, 2, · · · , N} given in (3.12) (or {L n (u)|n = 1, 2, · · · , N} given in (3.13)) with different sites commute with each other. Thus one can diagonalize them simultaneously. Then the average values of the elements of monodromy matrices (3.10) and (3.11) can be easy to obtain.