Integrable three-state vertex models with weights lying on genus five curves

TL;DR

This paper explores new integrable three-state vertex models with weights on genus five algebraic curves, revealing bielliptic structures and non-additive R-matrices that satisfy the Yang-Baxter equation.

Contribution

It introduces two families of regular Lax operators with nineteen weights on genus five curves, expanding the understanding of integrable models with complex algebraic structures.

Findings

Weights lie on genus five algebraic curves

Curves admit degree two morphisms onto elliptic curves

R-matrices satisfy Yang-Baxter equation and are non-additive

Abstract

We investigate the Yang-Baxter algebra for $\mathrm{U}(1)$ invariant three-state vertex models whose Boltzmann weights configurations break explicitly the parity-time reversal symmetry. We uncover two families of regular Lax operators with nineteen non-null weights which ultimately sit on algebraic plane curves with genus five. We argue that these curves admit degree two morphisms onto elliptic curves and thus they are bielliptic. The associated $\mathrm{R}$-matrices are non-additive in the spectral parameters and it has been checked that they satisfy the Yang-Baxter equation. The respective integrable quantum spin-1 Hamiltonians are exhibited.