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

by J. M. Maillet, G. Niccoli, B. Pezelier

Submission summary

As Contributors: Jean Michel Maillet
Arxiv Link: https://arxiv.org/abs/1802.08853v2
Date accepted: 2018-08-20
Date submitted: 2018-06-29
Submitted by: Maillet, Jean Michel
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Mathematical Physics
Approach: Theoretical

Abstract

This article is a direct continuation of [1] where we begun the study of the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax operator. There we addressed this problem for the case where one of the K-matrices describing the boundary conditions is triangular. In the present article we consider the most general integrable boundary conditions, namely the most general boundary K-matrices satisfying the reflection equation. The spectral analysis is developed by implementing the method of Separation of Variables (SoV). We first design a suitable gauge transformation that enable us to put into correspondence the spectral problem for the most general boundary conditions with another one having one boundary K-matrix in a triangular form. In these settings the SoV resolution can be obtained along an extension of the method described in [1]. The transfer matrix spectrum is then completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions and equivalently as the set of solutions to an analogue of Baxter's T-Q functional equation. We further describe scalar product properties of the separate states including eigenstates of the transfer matrix.

Ontology / Topics

See full Ontology or Topics database.

Algebraic Bethe Ansatz (ABA) Baxter T-Q equation Bazhanov-Stroganov model Boundary K-matrices Integrable boundary conditions Lax operators Reflection algebra Separation of Variables (SoV) method Spectral analysis Transfer matrix

Published as SciPost Phys. 5, 026 (2018)



Author comments upon resubmission

We first would like to express our thanks to the referees for their attentive reading, the useful help given in finding some typos, their suggestions and the interest shown in our paper. Our reply to referees questions and comments have been already given in our author reply to each of the three reports. Therefore we do not reproduce them again here.

List of changes

1. To carry out the legitimate requirements of clarifications of symbols definition pointed out by the referees, we have added all along the manuscript several recalling to the original definitions of symbols whenever needed (essentially whenever the definition was given several pages ago). Also href links have been provided whenever they exist.

2. Concerning the comparison of our results with the ones given in [60], we have modified the sentence at the end of our introduction, where we refer to the paper [60], to give a more detailed comparison, and we have added some more technical comments about this just after our Theorem 5.1.

3. Concerning the comparison of the SoV approach with the algebraic Bethe ansatz one, in particular concerning the role of separate states with respect to off-shell Bethe states, we have added the proof of the rewriting in Corollary 5.1 and referred to the papers [61] and [104], where it was first argued that this type of proof is model independent. Finally, we have added also a comment on the derivation given in [56] by the different logic of the modified Bethe ansatz for the XXZ spin-1/2 chain.

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