An integrable 3D lattice model with positive Boltzmann weights

TL;DR

This paper introduces the first known 3D integrable lattice model with positive Boltzmann weights, featuring infinite discrete states and satisfying the tetrahedron equation, advancing the understanding of 3D solvable models.

Contribution

It constructs a novel 3D solvable lattice model with non-negative weights and continuous parameters, satisfying the tetrahedron equation, and forming a commutative family of transfer matrices.

Findings

First example of a 3D solvable lattice model with positive weights

Boltzmann weights depend on parameter q and continuous fields

Model satisfies the tetrahedron equation

Abstract

In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run over an infinite number of discrete states. The Boltzmann weights satisfy the tetrahedron equation, which is a 3D generalisation of the Yang-Baxter equation. The weights depend on a free parameter 0<q<1 and three continuous field variables. The layer-to-layer transfer matrices of the model form a two-parameter commutative family. This is the first example of a solvable 3D lattice model with non-negative Boltzmann weights.