We have recently used unitarity and cross-unitarity properties of an symmetric R-matrix to construct the transfer matrix t(u) for an N-site open spin chain. Here we give a fusion procedure for such a chain, and we prove that the fused transfer matrix is commutative with the original transfer matrix t(u).
The authors have generalized Sklyanin's approach of constructing open integrable quantum spin chains to the case of PT-invariant R matrices. They formulate a fusion procedure for such chains. In particular, they show that the fused transfer matrix can be expressed in terms of products of the original transfer matrix and products of certain quantum determinants which can be explicitly evaluated. Applications of these results include constructing open integrable higher-spin chains, as well as obtaining functional equations for transfer-matrix eigenvalues, which may be solved by an analytical Bethe ansatz.
A general fusion procedure for open gl(N) spin chains: Application to the ABJM spin chain
We formulate a general fusion procedure for open gl(N) spin chains. We construct the fused boundary reflection matrices and the corresponding fused reflection equations. By using the intertwining relation between the fused reflection matrices and the fusion operator, we identify the invariant subspace of the fused reflection matrices carrying the irreducible representations of gl(N). We also construct the fused transfer matrix and evaluate it explicitly in the total tensor product space and the invariant subspaces. Finally, we demonstrate that the ABJM spin chain model originates from such fusion procedure and derive three classes of boundary reflection matrices solutions on the anti-fundamental representation space of su(4).
In this paper, we studied the exact solution of the C2(1) invariant quantum spin chain with off-diagonal open boundary condition. We obtain a solution of the reflection equation where the all matrix element of reflection matrix are nonzeros. By using the technique of fusion, we construct the fused transfer matrix and find the closed recursive relations among the transfer matrices. Based on the algebraic analysis, we obtain the eigenvalue of the system and express it as the inhomogeneous T−Q relation.
Antisymmetric fusions of the Belavin model with integrable open boundary conditions are studied. The centre of the corresponding operator algebra, the Sklyanin determinant, is constructed. We prove that the transfer matrices of the fusion hierarchies mutually commute. Utilizing the factorized L operators, this commuting family is expressed as mutually commuting difference operators acting on a function space. This gives Ruijsenaars-Macdonald-type commuting difference operators, which describe the conservation laws of an integrable system.