Off-diagonal Bethe Ansatz solution of the $\tau_2$-model

The generic quantum $\tau_2$-model (also known as Baxter-Bazhanov-Stroganov (BBS) model) with periodic boundary condition is studied via the off-diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix (solutions of the recursive functional relations in $\tau_j$-hierarchy) with generic site-dependent inhomogeneity parameters are given in terms of an inhomogeneous T-Q relation with polynomial Q-functions. The associated Bethe Ansatz equations are obtained. Numerical solutions of the Bethe Ansatz equations for small number of sites indicate that the inhomogeneous T-Q relation does indeed give the complete spectrum.


Introduction
Among quantum integrable models, the τ 2 (BBS)-model [1] plays a special role for its unique properties, e.g., it is one of the simplest quantum integrable models associated with cyclic representation of the Weyl algebra; it allows to include multiple inhomogeneity parameters on each single site without breaking the integrability of the model; and more interestingly, the τ 2 -model under certain parameter constraint is highly related to some other integrable models such as the chiral Potts model [2,3,4,5,6,7] and the relativistic quantum Toda chain model [8]. Many papers have appeared in literature for such connections and many efforts have been made to obtain the solutions of chiral Potts model by solving the τ 2 -model with a recursive functional relation [9,10,11,12]. However, it was found that only in the superintegrable sub-sector [2] the algebraic Bethe Ansatz method can be applied on this model to obtain Baxter's T − Q [13] solutions and Bethe Ansatz equations, while for the generic τ 2 -model, though its integrability [1] was proven, there is no simple Q-operator solution in terms of Baxter's T − Q relation. The Q-operator is in fact a very complicated function defined in high genus space and its concrete form is still hard to be derived.
In this paper, we adopt the off-diagonal Bethe Ansatz method [14] (for comprehensive introduction, see [15]) to study the quantum τ 2 -model. It seems that the situation of the generic τ 2 -model is quite similar to the quantum XYZ model with an odd number of sites [14,15], in which there is also no simple polynomial solutions of the Q-function in terms of Baxter's T − Q relation. However, by including an extra off-diagonal term in the T − Q relation (i.e., the inhomogeneous T −Q relation), we show that the eigenvalues of the generic τ 2 transfer matrix can be expressed explicitly in terms of a trigonometric polynomial Q function and thus a proper set of Bethe Ansatz equations can be derived.
The structure of the paper is the following. In the subsequent section, we give a brief introduction of the τ 2 transfer matrix. In section 3, we study the fundamental properties of the transfer matrix and its fusion hierarchy. In section 4, we give the eigenvalues of the transfer matrix and the associated Bethe Ansatz equations. Concluding remarks are given in section 5 and the detailed proofs about the inhomogeneous T − Q relation and its degenerate case are given in Appendices A & B.
Let V denote a p-dimensional linear space (i.e. the local Hilbert space) with an orthonormal basis {|m |m ∈ Z p }. X and Z are two p × p matrices acting on the basis as follows: Here and below we adopt the standard notations: for any matrix A ∈ End(V), A n is an embedding operator in the tensor space V ⊗ V ⊗ · · ·, which acts as A on the n-th space and as identity on the other factor spaces. Then the embedding operators {X n , Z n |n = 1, · · · , N} satisfy the ultra-local Weyl algebra: The τ 2 -model can be described by an quantum spin chain [1]. With each site n 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 [1] L n (u) = e u d (+) where d are some parameters associated with the n-th site. These parameters are subjected to two constraints: It was shown [1] that the L-operators satisfy the relations: where the R-matrix R(u) is given by (2.1). The corresponding one-row monodromy matrix T (u) is thus defined as: which satisfies the quadratic relation known as the Yang-Baxter algebra The transfer matrix t(u) of the τ 2 -model with periodic boundary condition is then given by the partial trace of the monodromy matrix T (u) in the auxiliary space, namely, The quadratic relation (2.11) leads to the fact that the transfer matrices with different spectral parameters are mutually commutative [16], i.e., [t(u), t(v)] = 0, which guarantees the integrability of the model by treating t(u) as the generating functional of the conserved quantities.
The aim of this paper is to construct the eigenvalues Λ(u) of the transfer matrix t(u) for generic inhomogeneity parameters {d n , h (±) n |n = 1, · · · , N} obeying the constraints (2.8).

3
Properties of the transfer matrix 3

.1 Asymptotic behaviors and average values
Following [18,19], let us introduce the operator Q which commutes with the transfer matrix The explicit expression (2.7) of the L-operator and the definition (2.10) of the monodromy matrix T (u) imply that the transfer matrix t(u) given by (2.12) enjoys the asymptotic behavior: where D (±) and F (±) are four constants related to the inhomogeneous parameters as follows: Moreover, (2.7) allows us to derive the quasi-periodicity which leads to the quasi-periodicity of the transfer matrix t(u) The above relation implies that the transfer matrix t(u) can be expressed in terms of e u as a Laurent polynomial of the form where {t n |n = 0, 1 · · · , N} form the N + 1 conserved charges. In particular, t N and t 0 are given by where the constants D (±) and F (±) are given by (3.3).
The property (2.3) of the R-matrix and the relation (2.11) enables one to introduce the quantum determinant [20,21] of the associated Yang-Baxter algebra Direct calculation shows that it is proportional to the identity operator and has the factorized form: where D (±) and F (±) are given by (3.3).
Let us define the average value O(u) of the matrix elements of the monodromy matrix T (u) (or the L-operators L n (u)) using the averaging procedure [22]: |n = 1, · · · , N}. It was shown [22] that where the average value of each L-operator is given by and n = 1, · · · , N. It is remarked that the average values of the matrix elements are Laurent polynomials of e pu , which implies T (u + η) = T (u), L n (u + η) = L n (u), n = 1, · · · , N, (3.14) where the constants D (±) and F (±) are given by (3.3).
When the crossing parameter η takes the special values (2.2), which correspond to the case of the root of unity, the spin-p 2 transfer matrix satisfy the truncation identity [1,22,18] where the functions A(u) and D(u) are the average values of the operators A(u) and D(u), and are given by (3.12)-(3.13). It is remarked that p−1 2 is an integer and the functions A(u) and D(u) are invariant under shifting with η (3.14).
In the following part of the paper, we shall show that the asymptotic behaviors (3.2), the determinant representation (3.19) of the transfer matrix t ( p 2 ) (u) and the truncation identity (3.20) completely determine the eigenvalues of the fundamental transfer matrix t(u) given by (2.12). Then with the help of (3.19) we can obtain eigenvalues of all the others higher spin-j transfer matrices t (j) (u).

4
Eigenvalues of the fundamental transfer matrix

Functional relations of eigenvalues
The commutativity (3.17) of the fused transfer matrices {t (j) (u)} with different spectral parameters implies that they have common eigenstates. Let |Ψ be a common eigenstate of these fused transfer matrices with the eigenvalues Λ (j) (u) The relation (3.1) allows us to decompose the whole Hilbert space H into p subspaces, i.e., according to the action of the operator Q: The commutativity of the transfer matrices and the operator Q implies that each of the subspace is invariant under t (j) (u). Hence the whole set of eigenvalues of the transfer matrices can be decomposed into p series, denoted by Λ (j) k (u) respectively. The eigenstates corresponding to Λ (j) k (u) belong to the subspace H (k) . The quasi-periodicity (3.5) of the transfer matrix t(u) implies that the corresponding eigenvalue Λ k (u) satisfies the property The asymptotic behavior (3.2) of the transfer matrix t(u) gives rise to the fact that the corresponding eigenvalue Λ k (u) enjoys the behavior The analyticity of the L-operator (2.7), the quasi-periodicity (4.2) and (4.3) imply that the eigenvalue Λ k (u) possesses the following analytical property Λ k (u), as a function of e u , is a Laurent polynomial of degree N like (3.6).
The fusion hierarchy relation (3.18) and the determinant representation (3.19) of the fused transfer matrices allows one to express all the eigenvalues Λ where the functions a(u) and d(u) are given by (3.9) and (3.10). For example, the first three ones are given by The truncation identity (3.20) of the spin-p 2 transfer matrix leads to the fact that the corresponding eigenvalue Λ where the functions A(u) and D(u) are given by (3.12)-(3.13).

T-Q relation 4.2.1 Generic case
Following the method developed in [14] (or for details we refer the reader to [15]), let us introduce the following inhomogeneous T − Q relation where φ k is a generic complex number 3 , the functions a(u) and d(u) are given by (3.9) and (3.10), the function F k (u) is given by 9) and the function Q(u) is a trigonometric polynomial of degree (p − 1)N Here the (p − 1)N + 1 parameters c k and {λ j |j = 1, · · · , (p − 1)N} satisfy the associated Bethe Ansatz equations (BAEs) Here the constants D (±) and F (±) are given by (3.3) and G (±) and H (±) read (4.14) Notice that for a given φ k , either  Table 1&2 (N = 2) and Table 3&4 (N = 3) respectively. The Λ(u) curves calculated from exact diagonalization and from the T − Q relation coincide exactly ( Figure 1&2), which imply that the inhomogeneous T − Q relation does indeed give the complete and correct spectrum of the generic τ 2 transfer matrix.
The curves calculated from exact diagonalization coincide with those derived from the inhomogeneous T − Q relation. Table 2: The Bethe roots solved from the Bethe Ansatz equations (4.11)-(4.13) for p = 3, N = 2 and φ k = 1 with the inhomogeneity parameters d         n , h (±) n |n = 1, · · · , N} obeying the constraint (2.8), the inhomogeneous term in the T − Q relation (4.7) does not vanish. In this subsection we consider some special case such that the inhomogeneous term vanishes.

Conclusions
The most general cyclic representations of the quantum τ 2 -model (also known as Baxter-Bazhanov-Stroganov (BBS) model) with periodic boundary condition has been studied via the off-diagonal Bethe Ansatz method [15]. Based on the the truncation identity (3.20) of the fused transfer matrices, we construct the inhomogeneous T − Q relation (4.7) of the eigenvalue of the fundamental transfer matrix t(u) and the associated BAEs (4.11)-(4.13).
It should be noted that for generic inhomogeneity parameters {d n , h (±) n |n = 1, · · · , N} obeying the constraint (2.8), the inhomogeneous term (i.e., the third term) in the T − Q relation (4.7) does not vanish, as long as one takes a polynomial Q function. However, if these inhomogeneity parameters satisfy the further constraints (4.15) and (4.17) (or (4.16) and (4.17)), the corresponding T − Q relation reduces to the conventional one (4.18) (or (4.19)).
Let us consider the function F k (u) given by (4.8). For generic inhomogeneity parameters n , h (±) n |n = 1, · · · , N} satisfying the constraint (2.8), we know that, as a function of e u , F k (u) is a Laurent polynomial of degree pN with the form where {z j mod (2iπ)|j = 1, · · · , pN} are the zeros of F k (u) which are all different from each other and the constant F n , h (±) n |n = 1, · · · , N}. Moreover, it follows that Let us introduce the function g(u) which is given by k (u) and Λ ( p−2 2 ) k (u) are given by the determinant representation (4.5) with Λ k (u) given by (4.7). From the above definition, one knows that the function g(u) as a function of e u is a Laurent polynomial of degree pN of similar form as (A.1). Hence g(u) is uniquely determined by its pN +1 points values such as +∞ (or −∞) and {z j |j = 1, · · · , pN}. Thanks to the property (A.3), we have m ∈ Z, j = 1, · · · , pN.
Substituting the above relations into (4.5) and noting the fact pη = 2iπ, after some tedious calculation, we have In deriving the second equality, we have used the fact: F k (z j ) = 0. Then (A.6) implies that the function g(u) vanishes at the points z j , namely, The BAEs (4.12)-(4.13) imply that the functions given by the T −Q relation (4.7) also satisfy  Suppose that the inhomogeneous T − Q relation (4.7) can be reduced to the conventional one, namely, where the Q-function isQ and M is a non-negative integer to be specified by (4.15) (or (4.16)). The asymptotic behavior n , h (±) n |n = 1, · · · , N} satisfy the constraints (2.8), (4.15) and (4.17). Similarly we can prove that the reduced T − Q relation (4.19) satisfies (4.3) and (4.6) provided that the inhomogeneity parameters obey the constraints (2.8), (4.16) and (4.17).