Analytic theory of the eight-vertex model

TL;DR

This paper introduces a new continuous field parameter in the eight-vertex model, enabling a comprehensive analytic review of its functional relations and revealing that different Bethe Ansatz solutions are connected through analytic continuation.

Contribution

The paper uncovers an unnoticed arbitrary field parameter in the eight-vertex model, facilitating a unified analytic approach and understanding of the model's spectrum and solutions.

Findings

Identified a continuous spectrum in the unrestricted solid-on-solid model.

Showed that eigenvalues are different branches of the same multivalued function.

Developed techniques for analyzing Bethe Ansatz equations.

Abstract

We observe that the exactly solved eight-vertex solid-on-solid model contains an hitherto unnoticed arbitrary field parameter, similar to the horizontal field in the six-vertex model. The parameter is required to describe a continuous spectrum of the unrestricted solid-on-solid model, which has an infinite-dimensional space of states even for a finite lattice. The introduction of the continuous field parameter allows us to completely review the theory of functional relations in the eight-vertex/SOS-model from a uniform analytic point of view. We also present a number of analytic and numerical techniques for the analysis of the Bethe Ansatz equations. It turns out that different solutions of these equations can be obtained from each other by analytic continuation. In particular, for small lattices we explicitly demonstrate that the largest and smallest eigenvalues of the transfer matrix of the eight-vertex model are just different branches of the same multivalued function of the field parameter.