Abstract
We derive exact inversion identities satisfied by the transfer matrix of inhomogeneous interaction-round-a-face (IRF) models with arbitrary boundary conditions using the underlying integrable structure and crossing properties of the local Boltzmann weights. For the critical restricted solid-on-solid (RSOS) models these identities together with some information on the analytical properties of the transfer matrix determine the spectrum completely and allow to derive the Bethe equations for both periodic and general open boundary conditions.
Highlights
Functional relations between the transfer matrices of integrable models together with the knowledge of their analytical properties provide a powerful basis for the solution of their spectral problem
Generalized inversion relations for the restricted solid-on-solid (RSOS) model have been obtained from the fusion hierarchy [6,7] and have been used to identify the low energy effective theory of the critical model through solution of nonlinear integral equations [8] or to study their surface critical behavior [9]
Using (2.10) or (3.3) together with some information on the analytical properties of the transfer matrix the solution of the spectral problem is possible: from (3.1) we find that the transfer matrix is periodic in u with period π and both T(u) and its eigenvalues can be written as Fourier polynomials
Summary
Functional relations between the transfer matrices of integrable models together with the knowledge of their analytical properties provide a powerful basis for the solution of their spectral problem. Adapting Sklyanin’s separation of variables the eigenvalues of the transfer matrix of the dynamical six-vertex model on a lattice with odd number of sites and with antiperiodically twisted boundary conditions have been shown to satisfy quadratic equations for a discrete set of spectral parameters [29] In another approach functional relations have been derived from the dynamical Yang–Baxter equation to determine the partition function of the SOS model with domain wall boundaries [30,31]. In this paper we derive inversion identities for the transfer matrix of general IRF models directly in the face formulation of the Yang–Baxter algebra using unitarity and crossing properties of the local Boltzmann weights For this we consider inhomogeneous face models subject to periodic and generic integrable open boundary conditions. The eigenvalues of the transfer matrix are parametrized in terms of the solution to Bethe equations which allow to study properties of finite chains and to perform the thermodynamic limit
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