Exact solution of the trigonometric SU(3) spin chain with generic off-diagonal boundary reflections

The nested off-diagonal Bethe ansatz is generalized to study the quantum spin chain associated with the $SU_q(3)$ R-matrix and generic integrable non-diagonal boundary conditions. By using the fusion technique, certain closed operator identities among the fused transfer matrices at the inhomogeneous points are derived. The corresponding asymptotic behaviors of the transfer matrices and their values at some special points are given in detail. Based on the functional analysis, a nested inhomogeneous T-Q relations and Bethe ansatz equations of the system are obtained. These results can be naturally generalized to cases related to the $SU_q(n)$ algebra.


Introduction
Exact solution is a very important issue in studies of statistical mechanics, condensed matter physics, quantum field theory and mathematical physics [1,2] since those results can provide important benchmarks for understanding physical effects in a variety of systems. The coordinate Bethe ansatz and the algebraic Bethe ansatz are two powerful methods to obtain the exact solution of the integrable systems [3,4,5,6,7]. With these methods, many interesting exactly solvable models, such as the one-dimensional Hubbard model, supersymmetric t − J model, Heisenberg spin chain and the δ-potential quantum gas model, were exactly solved.
For integrable systems with U(1) symmetry, it is easy to find a reference state and these conventional Bethe ansatz can be applied to. Indeed, most of the previous studies focus on periodic or diagonal open boundary conditions without breaking the U(1) symmetry. However, there exists another kind of integrable systems which does not have the U(1) symmetry, such as the integrable systems with generic off-diagonal boundary reflections. Because the reference state of this kind of integrable system is absent, the conventional Bethe ansatz methods are failed. On the other hand, many interesting phenomena arise in this kind of systems, such as the topological elementary excitations in the spin-1/2 torus [8], spiral phase in the Heisenberg model with unparallel boundary magnetic field [9] and stochastic process in non-equilibrium statistical mechanics [10,11,12]. Motivated by these important applications, many interesting methods such as the q-Onsager algebra [13,14,15], the modified algebraic Bethe ansatz [16,17,18,19] and the Sklyanin's separation of variables (SoV) [20,21,22,23,24] were also applied to some integrable models without U(1) symmetry.
Recently, a new approach, i.e., the off-diagonal Bethe ansatz (ODBA) [8] was proposed to obtain exact solutions of generic integrable models either with or without U(1) symmetry.
For comprehensive introduction to this method we refer the readers to [37]. In order to study the high rank integrable models, the nested version of ODBA has been proposed for the isotropic (or rational) models [33]. In this paper, we study the anisotropic rank-2 spin model with generic integrable boundary conditions. Here the R-matrix is the trigonometric one associated with the SU q (3) algebra and the boundary reflection matrices are the most generic reflection matrices which have non-vanishing off-diagonal elements. Because the off-diagonal elements of the reflection matrices break the U(1) symmetry, the exact solution of the system has been missing even its integrability was known for many years ago. By using the fusion technique and nested ODBA, we successfully obtain the closed operator identities, the values at the special points and the asymptotic behaviors. Based on them, we construct the nested inhomogeneous T − Q relation and obtain the eigenvalue of the transfer matrix thus the energy spectrum of the system. These results can be generalized to multiple components spin chains related to more higher rank algebra cases.
The paper is organized as follows. Section 2 is the general description of the model. The SU q (3) R-matrix and corresponding generic integral non-diagonal boundary reflection matrices are introduced. In Section 3, by using the fusion technique, we derive the closed operator identities for the fused transfer matrices and the quantum determinant. The asymptotic behaviors of the fused transfer matrix and their values at special points are also obtained.
In section 4, we list some necessary functional relations which are used to determine the eigenvalues. Section 5 is devoted to the construction of the nested inhomogeneous T − Q relation and the Bethe ansatz equations. In section 6, we summarize our results and give some discussions. Some results related to the other types of the general off-diagonal boundary reflections are given in Appendix.

The model
Throughout, V denotes a three-dimensional linear space and let {|i , i = 1, 2, 3} be an orthonormal basis of it. We shall adopt the standard notations. 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 identity on the other factor spaces. For B ∈ End(V ⊗ V), B ij is an embedding operator of B in the tensor space, which acts as identity on the factor spaces except for the i-th and j-th ones.
The R-matrix R(u) ∈ End(V ⊗ V) used in this paper is the trigonometric one associated with the SU q (3) algebra, which was first proposed by Perk and Shultz [38] and further studied in [39,40,41,42,43], where the matrix elements are The R-matrix satisfies the quantum Yang-Baxter equation (QYBE) (2.4) and possesses the following properties, Initial condition : R 12 (0) = sinh ηP 12 , (2.5) Unitarity relation : Crossing Unitarity relation : PT-symmetry : R 21 (u) = R t 1 t 2 12 (u), (2.8) Here R 21 (u) = P 12 R 12 (u)P 12 with P 12 being the usual permutation operator and t i denotes transposition in the i-th space. The functions ρ 1 (u), ρ 2 (u) and the crossing matrix M are given by It is easy to check that the R-matrix (2.1) also has the following properties Let us introduce the reflection matrix K − (u) and its dual one K + (u). The former satisfies the reflection equation (RE) (2.14) and the latter satisfies the dual RE In this paper we consider the generic non-diagonal K-matrices K − (u) found in [44,45,46].
There are three kinds of reflecting K-matrix: with the constraint Thus the four boundary parameters c, c 1 c 2 and ζ are not independent with each other.
with the constraint with the constraint The dual non-diagonal reflection matrix K + (u) is given by In order to construct the model's Hamiltonian of the system, we first introduce the "rowto-row" (or one-row) monodromy matrices T 0 (u) andT 0 (u) where {θ j , j = 1, · · · , N} are the inhomogeneous parameters and N is the number of sites.
The one-row monodromy matrices are the 3 × 3 matrices in the auxillary space 0 and their elements act on the quantum space V ⊗N . For the system with open boundaries, we need to define the double-row monodromy matrix T 0 (u) Then the transfer matrix of the system is constructed as [7] t(u) = tr 0 {K + 0 (u)T 0 (u)}.

Fusion
Following [33], we apply the fusion technique [47,48,49] to study the present model. The fusion procedure will lead to the desired operator identities to determine the spectrum of the transfer matrix t(u) given by (2.23). For this purpose, let us introduce the following vectors in the tensor space V ⊗ V similarly as [36] |Φ (1) 12 in the tensor space V ⊗ V and Direct calculation shows that the R-matrix given by (2.1) at some degenerate points are proportional to the projectors, where the diagonal matrices S 12 and S 123 are given by The fused transfer matrices are defined as 14) and the notation t 1 (u) = t(u) is used. By repeatedly using the QYBE (2.4), the RE (2.14), the dual RE (2.15) and the definition (3.8), one can prove that all these fused transfer matrices are commutative with each other Thus they have the common eigenstates. Furthermore, we find that the transfer matrix given by (3.8) satisfies the following operator production identities We note that the fused transfer matrix t 3 (u) equals to its quantum determinant multiplying the unity matrix. Thus the operators production identities (3.16) are closed. The explicit form of the fused transfer matrix t 3 (u) reads The quantum determinant of the reflecting matrix (I) given by (2.16) is The quantum determinant of the reflecting matrix (II) given by (2.17) is The quantum determinant of the reflecting matrix (III) given by (2.18) is The quantum determinant of the dual reflecting matrices K + (u) can be obtained by the Then the equation (3.18) can be proved easily based on the facts Form the definition of fused transfer matrices (3.8), the corresponding asymptotic behaviors can be calculated directly. Obviously, different reflection parameters will give different asymptotic behaviors. Without losing the generality, we consider the case corresponding to the reflection matrices K ± (u) given by (2.16) and (2.19) and the details for the results for the other cases will be presented in Appendix. Then the asymptotic behaviors read where the operator Q (1) is In the derivation, the relations c(c − e ζ ) = c 1 c 2 and c ′ (c ′ − e ζ ′ ) = c ′ 1 c ′ 2 are used. It is remarked that the non-diagonal K-matrices (given by (2.16) and (2.19)) only break two of the original three U(1)-symmetries for the diagonal K-matrices or periodical case, and that the system still has a remaining U(1) -symmetry which is generated by the operator Q (1) .
The fused transfer matrices t m (u) have other useful properties. For example, their values at some special points can be calculated directly by using the properties of the R-matrix and the reflection matrices K ± . We list them in the following where the notations ρ − K (u) and ρ + K (u) are defined as In the derivation, we have used the relations Again, we use the notation Λ 1 (u) = Λ(u), which represents the eigenvalue of transfer matrix t(u) given by (2.23). The Λ(u), as an entire function of u, is a trigonometric polynomial of degree 2N + 4, which can be completely determined by 2N + 5 conditions. The Λ 2 (u), as an entire function of u, is a trigonometric polynomial of degree 4N + 12, which can be completely determined by 2N + 13 conditions 3 .
From the operator production identities (3.16) and (3.33), we have The values of Λ(u) at the special points should be the same as those given by (3.34)-(3.37) of the transfer matrix t(u). At the same time, the values of Λ 2 (u) at the special points should be the same as those given by (3.38)-(3.46) of the fused transfer matrix t 2 (u).
The asymptotic behaviors of Λ m (u) can be obtained by acting the operators in (3.28)- 3 It is noted that the relations (3.17) give the other 2N conditions.

Nested inhomogeneous T − Q relation
Now we construct the eigenvalues Λ m (u) of the fused transfer matrices t m (u). For simplicity, we define some functions b 0 (u) = N j=1 sinh(u − θ j ) sinh(u + θ j ), a 0 (u) = b 0 (u + η), (5.1) where L 1 and L 2 are non-negative integers. Due to the survived U(1) conserved charge Q (1) in the system, the number of one kind of Bethe roots can be chosen as M, which is similar as the algebraic Bethe ansatz. Without losing generality, we put L 2 = M. In order to construct the eigenvalues of the fused transfer matrices, we introduce threez(u) functions Here z m (u) is defined as with the notations Q (0) (u) = b 0 (u), Q (3) (u) = 1 and x 1 (u) is defined as where K (m) (u) are the decompositions of the quantum determinant and f 1 (u) is a function which will be determined later.
The nested functional T − Q ansatz is expressed as obtained from these two points also should be the same, which requires (5.10) We note that Λ 3 (u) is a trigonometric polynomial automatically. The fact that Λ 3 (u) should be the quantum determinant requires The consistency of Bethe ansatz equations also require that the function f 1 (u) has the crossing symmetry Furthermore, the eigenvalues Λ m (u) should satisfy the functional relations (4.2). This gives other constraints of the function f 1 (u). Considering all the above requirements, we parameterize the function f 1 (u) as ) 3 h sinh(2u) sinh 2 (2u + η) sinh(2u + 2η) sinh(2u − η) sinh(2u + 3η), (5.13) where h is a constant which is determined by the asymptotic behaviors of the Λ m (u). Now, we are ready to give the Bethe ansatz equations as following , l = 1, . . . , M + N + 6, (5.14) sinh(2λ (2) k + 3η) sinh(2λ give the different behaviors, let us consider them one by one. For the reflection matrices K ± given by (2.16) and (2.19), the decomposition K (i) (u) can be chosen as Then one can check that the values of ansatz (5.6)-(5.7) at the special points (4.5)-(4.6) are the same as those of the corresponding fused transfer matrices, and we finish our construction.
Taking the homogeneous limit {θ j = 0, j = 1, . . . , N}, we conclude that the T −Q relation Λ(u) given by (5.6) is the eigenvalue of the transfer matrices t(u) of the trigonometric SU (3) open spin chain with the most general off-diagonal integrable boundary conditions. The energy of the Hamiltonian (2.24) reads where the Bethe roots should satisfy the Bethe ansatz equations (5.14)-(5.15). Above results can be reduced to the diagonal boundaries ones obtained by the algebraic Bethe ansatz [50,51,52].
Numerical solutions of the BAEs (5.14)-(5.15) for small size 5 with a random choice of η imply that the BAEs indeed give the complete solutions of the model. Here we present the result for the N = 2 case: the numerical solutions of the BAEs for the N = 2 case are shown in Table 1, while the calculated Λ(u) curves for the case of N = 2 are shown in Figure 1.

Diagonal boundary case
When the parameters c, c 1 , c 2 , c ′ , c ′ 1 , c ′ 2 in the reflection matrices K ± (u) given by (2.16) and (2.19) vanish, the corresponding K-matrices become diagonal ones. Let us denote them bȳ Then the corresponding T − Q relations (5.6)-(5.7) become the usual homogeneous ones and now are given by Here the corresponding Q-functions arē The resulting homogeneous relation (6.6) recovers that obtained by the algebraic Bethe ansatz method [52], while the reference state is chosen as

Conclusions
In this paper, we study the exact solution of the anisotropic quantum spin chain with generic open boundary conditions and associated with SU q (3) algebra. After giving the off-diagonal reflection matrixes, by using the fusion technique, we obtain some closed operator identities among the transfer matrices, the degenerate points and the corresponding asymptotic behaviors. Based on them, we construct the nested inhomogeneous T − Q relations and the Bethe ansatz equations. These results can be generalized to the higher rank case. Moreover, when the boundary parameters take special values corresponding to the diagonal reflection matrices, our results recover those previously obtained by the conventional Bethe ansatzs. (2) +(c 1 c ′ 2 + c ′ 1 c 2 e 4η ) e N η e −ηQ (2)