Integrability in $[d+1]$ dimensions: combined local equations and commutativity of the transfer matrices

TL;DR

This paper introduces new local integrability equations for high-dimensional vertex models within the Algebraic Bethe Ansatz framework, demonstrating their effectiveness and constructing novel 3D solutions with potential quantum computing applications.

Contribution

It proposes combined local integrability equations for general dimensions, constructs 3D solutions with three-state R-matrices, and introduces a 3D analogue of star-triangle equations.

Findings

Demonstrates efficiency of transfer matrix commutation in low dimensions

Constructs simple 3D solutions with three-state R-matrices

Defines new 3D integrability equations and a star-triangle analogue

Abstract

We propose new inhomogeneous local integrability equations - combined equations, for statistical vertex models of general dimensions in the framework of the Algebraic Bethe Ansatz (ABA). For the low dimensional cases the efficiency of the step by step consideration of the transfer matrices' commutation is demonstrated. We construct some simple 3D solutions with the three-state $R$-matrices of certain 20-vertex structure; the connection with the quantum three-qubit gates is discussed. New, restricted versions of 3D local integrability equations with four-state $R$-matrices are defined, too. Then we construct a new 3D analogue of the two-dimensional star-triangle equations.