## SciPost Submission Page

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

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

#### - Published as SciPost Phys. 5, 026 (2018)

### Submission summary

As Contributors: | Jean Michel Maillet |

Arxiv Link: | https://arxiv.org/abs/1802.08853v2 (pdf) |

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.Published as SciPost Phys. 5, 026 (2018)

### Author comments upon resubmission

### 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.