# The spectrum of a vertex model and related spin one chain sitting in a genus five curve

## TL;DR

This paper analyzes a complex vertex model with spectral parameters on a genus five curve, deriving eigenvalues, Bethe equations, and exploring phase transitions between gapped and massless excitations.

## Contribution

It introduces a novel vertex model with weights on a genus five curve and provides analytical and numerical insights into its spectral properties and phase behavior.

## Key findings

- Eigenvalues expressed via meromorphic functions on elliptic curves
- Low-energy excitations can be gapped or massless depending on interaction strength
- Critical point coincides with degeneration of the spectral curve into genus one components

## Abstract

We derive the transfer matrix eigenvalues of a three-state vertex model whose weights are based on a $\mathrm{R}$-matrix not of difference form with spectral parameters lying on a genus five curve. We have shown that the basic building blocks for both the transfer matrix eigenvalues and Bethe equations can be expressed in terms of meromorphic functions on an elliptic curve. We discuss the properties of an underlying spin one chain originated from a particular choice of the $\mathrm{R}$-matrix second spectral parameter. We present numerical and analytical evidences that the respective low-energy excitations can be gapped or massless depending on the strength of the interaction coupling. In the massive phase we provide analytical and numerical evidences in favor of an exact expression for the lowest energy gap. We point out that the critical point separating these two distinct physical regimes coincides with the one in which the weights geometry degenerate into union of genus one curves.