An Integrable Nineteen Vertex Model Lying on a Hypersurface

TL;DR

This paper introduces a new exactly solvable nineteen vertex model with a rich geometric structure, including a K3 surface, expanding the understanding of integrable models and their underlying algebraic varieties.

Contribution

The authors construct a family of integrable nineteen vertex models with weights on a degree seven algebraic threefold, revealing novel geometric structures like a K3 surface in the physical submanifold.

Findings

Model's weights lie on a degree seven algebraic threefold

The physical submanifold is governed by a K3 surface

Parameterization depends on three independent spectral parameters

Abstract

We have found a family of solvable nineteen vertex model with statistical configurations invariant by the time reversal symmetry within a systematic study of the respective Yang-Baxter relation. The Boltzmann weights sit on a degree seven algebraic threefold which is shown birationally equivalent to the three-dimensional projective space. This permits to write parameterized expressions for both the transition operator and the $\mathrm{R}$-matrix depending on three independent affine spectral parameters. The Hamiltonian limit tells us that the azimuthal magnetic field term is connected with the asymmetry among two types of spectral variables. The absence of magnetic field defines a physical submanifold whose geometrical properties are remarkably shown to be governed by a quartic $\mathrm{K}3$ surface. This expands considerably the class of irrational manifolds that could emerge in the theory of quantum integrable models.