Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I

TL;DR

This paper analyzes the spectral problem of transfer matrices for cyclic representations of the 6-vertex reflection algebra using separation of variables, providing a complete characterization of eigenvalues and eigenstates.

Contribution

It introduces a novel method to characterize transfer matrix spectra via polynomial solutions and Baxter-like equations for cyclic representations with boundary constraints.

Findings

Complete spectral characterization in terms of polynomial solutions

Eigenstates expressed in algebraic Bethe ansatz form

Method applicable to lattice sine-Gordon model with open boundaries

Abstract

We study the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax operator. The results apply as well to the spectral analysis of the lattice sine-Gordon model with integrable open boundary conditions. This spectral analysis is developed by implementing the method of separation of variables (SoV). The transfer matrix spectrum (both eigenvalues and eigenstates) is completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions. Moreover, we prove an equivalent characterization as the set of solutions to a Baxter's like T-Q functional equation and rewrite the transfer matrix eigenstates in an algebraic Bethe ansatz form. In order to explain our method in a simple case, the present paper is restricted to representations containing one constraint on the boundary parameters and on the parameters of the Bazhanov-Stroganov Lax operator. In a next article, some more technical tools (like Baxter's gauge transformations) will be introduced to extend our approach to general integrable boundary conditions.