Quantized six-vertex model on a torus

TL;DR

This paper investigates the integrability of a quantized six-vertex model on a torus, establishing new algebraic relations and connecting it to known integrable systems like dimer models and Toda chains.

Contribution

It introduces a generalized integrable framework for the quantized six-vertex model on a torus, including new tetrahedron equations and inversion relations.

Findings

Proves commutativity of transfer matrices for admissible graphs.

Derives a family of commuting quantum Hamiltonians.

Reformulates the model in terms of quantized dimer models, linking to known integrable systems.

Abstract

We study the integrability of the quantized six-vertex model with four parameters on a torus. It is a three-dimensional integrable lattice model in which a layer transfer matrix, depending on two spectral parameters associated with the homology cycles of the torus, can be defined not only on the square lattice but also on more general graphs. For a class of graphs that we call admissible, we establish the commutativity of the layer transfer matrices by introducing four types of tetrahedron equations and two types of inversion relations. Expanding in the spectral parameters yields a family of commuting quantum Hamiltonians. The quantized six-vertex model can also be reformulated in terms of (quantized) dimer models, and encompasses known integrable systems as special cases, including the free parafermion model and the relativistic Toda chain.