Exact solution of the Izergin-Korepin model with general non-diagonal boundary terms

The Izergin-Korepin model with general non-diagonal boundary terms, a typical integrable model beyond A-type and without U(1)-symmetry, is studied via the off-diagonal Bethe ansatz method. Based on some intrinsic properties of the R-matrix and the K-matrices, certain operator product identities of the transfer matrix are obtained at some special points of the spectral parameter. These identities and the asymptotic behaviors of the transfer matrix together allow us to construct the inhomogeneous T-Q relation and the associated Bethe ansatz equations. In the diagonal boundary limit, the reduced results coincide exactly with those obtained via other methods.

1 Introduction [9] and self-avoiding walks at a boundary [41]. The IK model with U(1)-symmetry, i.e. with periodic boundary condition or with diagonal boundaries have been extensively studied [9,42,43,44,45,46,47,48,49,50,51,52]. Even the most general integrable boundary condition (corresponding to the non-diagonal reflection matrix) has been known almost for 20 years [44,49,51], its exact solution is still missing. The purpose of the present paper is to solve this interesting problem with ODBA.
The paper is organized as follows. Section 2 serves as an introduction to the model and our notations. In Section 3, we derive certain operator product identities for the transfer matrix of the model with general non-diagonal boundary terms by using some intrinsic properties of the R-matrix and K-matrices. The asymptotic behaviors of the transfer matrix are also obtained. Section 4 is devoted to the construction of the inhomogeneous T − Q relation and the corresponding Bethe ansatz equations. Section 5 is attributed to the reduction to case of diagonal boundaries. It is found that the reduced results coincide exactly with those obtained by other Bethe ansatz methods. In section 6, we summarize our results and give some discussions. Some detailed technical proof is given in Appendix A.

Transfer matrix
Throughout, V denotes a three-dimensional linear space and let {|i |i = 1, 2, 3} be an orthonormal basis of it. The R-matrix R(u) ∈ End(V ⊗ V) of the IK model is given by [38] R 12 (u) = The R-matrix satisfies the quantum Yang-Baxter equation (QYBE) and possesses the following properties, Initial condition : Unitarity relation : Crossing relation : Periodicity : 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 function ρ 1 (u) and the crossing matrix V are given by (2.10) Throughout this paper we 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; R ij (u) is an embedding operator of R-matrix in the tensor space, which acts as identity on the factor spaces except for the i-th and j-th ones.
The unitarity property (2.5) and crossing relation (2.6) of the R-matrix and expressions (2.9)-(2.10) of the function ρ 1 (u) and the V -matrix imply that the R-matrix satisfies the crossing-unitarity relation It is easily to check that the R-matrix also satisfies the following relation (2.14) Let us introduce the "row-to-row" (or one-row ) monodromy matrices T (u) andT (u), which are 3 × 3 matrices with operator-valued elements acting on V ⊗N , Here {θ j |j = 1, · · · , N} are arbitrary free complex parameters which are usually called as inhomogeneous parameters. The transfer matrix can be constructed as follows [20,53].
Let us introduce further a pair of K-matrices K − (u) and K + (u). The former satisfies the reflection equation (RE) (2.17) and the latter satisfies the dual RE For open boundaries, one needs to consider the double-row monodromy matrix T(u) The double-row transfer matrix t(u) is thus given by t(u) = tr(K + (u)T(u)). In this paper we consider the generic non-diagonal K-matrix K − (u) found in [44] and was classified as type II 4 in [49,51] and non-diagonal K + (u) given by Besides the crossing parameter η, the corresponding transfer matrix t(u) given by (2.20) has four other free parameters {ǫ, σ, ǫ ′ , σ ′ } describing the boundary fields.

Operator identity of the transfer matrix
Following [15] we apply the fusion technique to study the the present model. In this case, we need to use the fusion techniques both for R-matrices [54] and for K-matrices [46,55].
Similar as that in [15], the fusion procedure will lead to the desired operator identities to determine the spectrum of the transfer matrix t(u) given by (2.20). For this purpose, let us introduce the following vectors in the tensor space V ⊗ V (3.4) and the associated projectors 5 P The generalization to the non-diagonal K-matrices of type I in [49,51] is straightforward. 5 In contrast to most of rational models, here P Direct calculation shows that the R-matrix given by (2.1) at some degenerate points is proportional to the projectors, where S (i) 12 are some non-degenerate matrices ∈ End(V ⊗ V). After some calculations, we find where the function ρ 3 (u) is The above relations imply that the associated one-row monodromy matrices satisfy the following relations Now let us construct the corresponding fusion procedures for the K-matrices following [46,55]. The properties (3.6) of the R-matrix at the degenerate points and the RE (2.17) and its dual RE (2.18) allow us to construct the fused K-matrices from the origin ones (2.21) and (2.22). After some tedious calculations, we find that where the functions f ± (u) and Det q (K ± (u)) are Following the method in [15] and using the relations (3.12)-(3.19), after a long calculation, we find the transfer matrix given by (2.20) satisfies the following operator identities Here the functions δ 1 (u) and δ 2 (u) are Following the method in [56] and using the the explicit expressions (2.21) and (2.22) of the K-matrices, we find that the transfer matrix (2.20) possesses the crossing symmetry The proof of the above relation is given in Appendix A. We remark that the very operator identities (3.24)-(3.25) at the points −θ j is guaranteed by their equalities at the points θ j and the crossing symmetry (3.28) of the transfer matrix. Therefore, in the following we will use the operator identities (3.24)-(3.25) at the points θ j and the crossing symmetry (3.28) to determine the eigenvalues of the transfer matrix. Now let us derive some properties of the transfer matrix. Firstly the expressions of the K-matrices K ± (u) given by (2.21) and (2.22) imply that K ± (u) have the following periodicity This leads to the periodicity of the transfer matrix t(u) Moreover the explicit expressions of the R-matrix and the K-matrices allow us to have the following asymptotic behaviors of the transfer matrix The explicit expressions of the K-matrices given by (2.21) and (2.22) also imply that K ± (u) satisfy the following properties The unitarity (2.5) and the crossing unitarity (2.11) of the R-matrix imply that the one-row monodromy matrices T (u) given by (2.15) andT (u) given by (2.16) satisfy the following The above relations allow us to work out the transfer matrix at special points 0, iπ, 6η, 6η+iπ as follows

Functional relations and the T-Q relation
The property [t(u), t(v)] = 0 implies that the eigenstates of t(u) are independent of u.
Suppose |Ψ is an eigenstate of t(u) with an eigenvalue Λ(u), namely, The very operator identities (3.24)-(3.25) of the transfer matrix at the points θ j imply the corresponding eigenvalue Λ(u) satisfies the similar relations where the functions δ i (u) are given by (3.26) The periodicity (4.4) and the asymptotic behavior

T-Q ansatz
Let us introduce the following T − Q ansatz where the functions c(u), b(u) and d(u) are the elements of the R-matrix given by (2.2) and the functions Q i (u) and the constant c are given by ,N = 4(N + l 1 ) + 2(l 2 − 1), (4.10) , (4.12) Here l 1 and l 2 are arbitrary non-negative integers 6 . It is easily to check that the functions Q i (u) possess the following properties Q i (u + 2iπ) = Q i (u), for i = 1, 2, and Q 2 (u) = Q 1 (−u + 4η). (4.14) With the help of the above relations, we have checked that the T − Q ansatz (4.9) indeed satisfies the relations (4.3)-(4.6). Moreover the special choice of the constant c given by (4.12) implies that the ansatz also satisfies the asymptotic behavior (4.7). The explicit expressions of the functions b(u), c(u) and d(u) given by (2.1)-(2.2) imply that These properties give rise to the fact that the T − Q ansatz (4.9) satisfies the very functional relations (4.1)-(4.2). This means that our ansatz (4.9) actually satisfies the relations (4.1)-(4.7) except the analytic properties (4.8).
From the explicit expression (4.9) of Λ(u), one may find that the T − Q ansatz (4.9) might have some apparent simple poles at the following points: 2η, 4η, 3η + i π 2 , mod(iπ), (4.16) and λ j + 2η, −λ j + 2η, λ j + 4η + iπ, −λ j + 4η + iπ, mod(2iπ), j = 1, . . . ,N. (4.17) Direct calculation shows that the residues of the T − Q anstz (4.9) at the points given by (4.16) vanishes, or the ansatz has no singularity at these points. Moreover we have checked that the T − Q anstz (4.9) also has no singularity at the points given by (4.17) provided that theN parameters {λ j |j = 1, . . . ,N} satisfy the following BAEs The results of anisotropic spin-1 2 chains [30] strongly suggest that fixed l 1 and l 2 might give a complete set of eigenvalues of the transfer matrix. In such a sense, different l 1 and l 2 might only give different parameterizations of the eigenvalues but not different states.
where the function Q 2 (u) is given by (4.11). The non-singular property of the T − Q anstz (4.9) at the points (4.17) can be verified by directly calculating the residues of the anstz at these points.

Reduction to the conventional T − Q ansatz
It follows from (4.12) that the parameter c does depend on not only the boundary parameters but also the parameters {λ j } (such a dependence also appeared in the anisotropic spin-1 2 chains with arbitrary boundary fields [30]). The vanishing condition of c, i.e. c = 0, will lead to the constraint (see (4.23) below) among the boundary parameters, in this case one might find a proper "local vacuum" to apply the conventional Bethe ansatz [11,57]. The Bethe ansatz equations (4.18) imply that for this case the parameters {λ j } have to form two types of pairs: Suppose the number of the first type pairs is M (M being a non-negative integer such that 0 ≤ M ≤N 2 ), the resulting T − Q relation (4.9) becomes the conventional one [1] where the resulting function Q(u) is The resulting BAEs become On the other hand, the form of the pairs (4.19) implies that The relation (4.12) and the definition (4.10) ofN give rise to the following constraint between the boundary parameters 7 cosh(σ ′ − σ + 2η) = cosh(4Mη − 4Nη + 2η), 0 ≤ M. If the boundary parameters σ and σ ′ satisfy the following constraint σ ′ − σ = −4kη mod(2iπ), 1 − N ≤ k ≤ N, and k ∈ Z, (4.26) then M can take the two allowed values denoted by M ± respectively, If the boundary parameters σ and σ ′ satisfy M can take only one allowed value, For the case that the boundary parameters satisfy the relation (4.24) (or (4.28)), the eigenvalue of the transfer matrix is characterized by a fixed M in (4.25) (or (4.29)). This Λ(u) itself might give the complete set of the eigenvalues of the transfer matrix. However, if the boundary parameters obey the relation (4.26), there exist two Λ ± (u) corresponding to the two different allowed M in (4.27). Similar as that of the anisotropic spin-1 2 chain with arbitrary boundary fields [11,26,58,59], these two Λ ± (u) together might constitute the complete set of the eigenvalues of the transfer matrix.

For some degenerate η
Similar as that in the closed XYZ chain and the anisotropic spin-1 2 chain with arbitrary boundary fields [30], if the isotropic (or crossing) parameter η takes the following discrete value the constraint (4.23) is automatically satisfied for arbitrary boundary parameters ǫ, ǫ ′ , σ and σ ′ . Thus in this case the eigenvalue of the transfer matrix is given by (4.20) and the corresponding BAEs are given by (4.22). It should be emphasized that these degenerate points (4.30) become dense in the thermodynamic limit (N, m, M → ∞). This enables one to obtain the thermodynamic properties (up to the order of O(N −2 )) of the model for generic values of η via the conventional thermodynamic Bethe ansatz methods [60,61]. This method has been proven to be very successful in the derivation of the surface energy of the XXZ spin chain with arbitrary boundary fields [62].

Reduction to the case with diagonal K-matrices
Now let us consider the diagonal K-matrices (i.e., taking the limits ǫ, ǫ ′ → +∞ of (2.21) and (2.22)). The resulting K-matrices read where the matrix M is given by (2.12). The T − Q ansatz (4.20) is reduced to the one [46] obtained by analytic Bethe ansatz method where the function Q(u) is still given by (4.21). In particular, in this case the vanishing condition of c is automatically satisfied thanks to the relation (4.12). In this case the U(1)symmetry is recovered and M can take any of the following values 8 , For each of the above allowed values, the resulting BAEs become

Conclusions
The Izergin-Korepin model with general non-diagonal boundary terms specified by the most general non-diagonal K matrices given by (2.21) and (2.22) has been studied by the offdiagonal Bethe ansatz method. Based on some intrinsic properties of the R-matrix and K-matrices, we derive the very functional relations (3.24) and (3.25) of the transfer matrix. These relations, together with other properties, allow us to construct an off-diagonal (or inhomogeneous) T − Q relation (4.9) of the eigenvalue of the transfer matrix and the associated BAEs (4.18). When the boundary parameters satisfy one constraint (4.23), the resulting T − Q relation is reduced to the conventional one (4.20), which might allow one to use the method developed in [62] to study the thermodynamic properties (up to the order of O(N −2 )) of the model for generic values of η via the conventional thermodynamic Bethe ansatz methods [60,61]. Taking the limit ǫ, ǫ ′ → +∞, the corresponding K-matrices become diagonal ones and the resulting T − Q relation is recovered to that in [46].