Exact spectrum of the XXZ open spin chain from theq-Onsager algebra representation theory

Abstract

The transfer matrix of the XXZ open spin-1 2 chain with general integrable boundary conditions and generic anisotropy parameter (q is not a root of unity and |q| = 1) is diagonalized using the representation theory of the q−Onsager algebra. Similarly to the Ising and superintegrable chiral Potts models, the complete spectrum is expressed in terms of the roots of a characteristic polynomial of degree d = 2 N . The complete family of eigenstates are derived in terms of rational functions defined on a discrete support which satisfy a system of coupled recurrence relations. In the special case of linear relations between left and right boundary parameters for which Bethe-type solutions are known to exist, our analysis provides an alternative derivation of the results of Nepomechie et al. and Cao et al.. In the latter case the complete family of eigenvalues and eigenstates splits in two sets, each associated with a characteristic polynomial of degree d < 2 N . Numerical checks performed for small values of N support the analysis.

Off-diagonal Bethe ansatz solution of the XXX spin chain with arbitrary boundary conditions

“…Since Yang and Baxter's pioneering works [4,5,1], the quantum Yang-Baxter equation (QYBE), which define the underlying algebraic structure, has become a cornerstone for constructing and solving the integrable models. There are several well-known methods for deriving the Bethe ansatz (BA) solution of integrable models: the coordinate BA [6,1,7,8,9], the T-Q approach [1,10], the algebraic BA [11,12,13], the analytic BA [14], the functional BA [15] and others [16,17,18,19,20,21,22,23,24,25,26,27,28,29].…”

Separation of variables in the open XXX chain

“…At the same time various problems concerning systems with open boundaries are still not solved completely. Even for the prototype spin-1 2 XXZ chain with general open boundary conditions techniques for the solution of the spectral problem have been developed only recently [1,4,[11][12][13][14]. This model, apart from being the simplest starting point for studies of boundary effects in a correlated system, allows to investigate the approach to a stationary state in one-dimensional diffusion problems for hard-core particles [7,8] and transport through one-dimensional quantum systems [5].…”

“…For generic values of the anisotropy the spectral problem has been formulated as a T Q-equation assuming that the large j-limit of the transfer matrices with spin j in auxiliary space exists [19]. No such constraints are needed in the derivation of a different set of recursion relations for the diagonalization of (1.1) based on the representation theory of the q-Onsager algebra [1]. Very recently, Galleas has formulated another functional approach to determine the eigenvalues of (1.1) in the generic case without a reference state [11].…”

Determinant formula for the partition function of the six-vertex model with a non-diagonal reflecting end

“…Then such an explicit determinant representation was re-derived [23,24] by using F-basis of the closed XXZ chain. However, it is well known that to obtain exact solution of open spin chain with non-diagonal boundary terms is very non-trivial [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41] comparing with that of the one with simple diagonal boundary terms. In this paper, we will investigate the determinant representation of the DW partition function of the six-vertex model with a non-diagonal reflection end which is specified by a generic non-diagonal K-matrix found in [42,43].…”