Correlation inequalities for the uniform 8-vertex model and the toric code model

TL;DR

This paper establishes correlation inequalities for the toric code and uniform eight-vertex models by connecting them to classical models and utilizing GKS-inequalities, advancing understanding of their probabilistic properties.

Contribution

It introduces new correlation inequalities for the toric code and eight-vertex models through novel connections with classical statistical models.

Findings

Correlation inequalities for the toric code model.

Correlation inequalities for the uniform eight-vertex model.

Connections between quantum and classical models.

Abstract

We elucidate connections between four models in statistical physics and probability theory: (1) the toric code model of Kitaev, (2) the uniform eight-vertex model, (3) random walk on a hypercube, and (4) a classical Ising model with four-body interaction. As a consequence of our analysis (and of the GKS-inequalities for the Ising model) we obtain correlation inequalities for the toric code model and the uniform eight-vertex model.