Free energy and quark potential in Ising lattice gauge theory via cluster expansion

TL;DR

This paper analyzes the low-temperature behavior of Ising lattice gauge theory on b^m, deriving expansions for free energy and Wilson loop expectations, and establishing the static quark potential and perimeter law with rigorous estimates.

Contribution

It provides a rigorous cluster expansion approach to compute free energy and Wilson loop expectations in Ising lattice gauge theory at low temperature, including the static quark potential.

Findings

Existence and analyticity of the free energy in the infinite volume limit.

Explicit low-temperature expansion of free energy and Wilson loop expectations.

Verification of the perimeter law for the static quark potential.

Abstract

We revisit the cluster expansion for Ising lattice gauge theory on $\mathbb{Z}^m, \, m \ge 3,$ with Wilson action, at a fixed inverse temperature \( \beta\) in the low-temperature regime. We prove existence and analyticity of the infinite volume limit of the free energy and compute the first few terms in its expansion in powers of $e^{-\beta}$. We further analyze Wilson loop expectations and derive an estimate that shows how the lattice scale geometry of a loop is reflected in the large $\beta$ asymptotic expansion. Specializing to axis parallel rectangular loops $\gamma_{T,R}$ with side-lengths $T$ and $R$, we consider the limiting function $$ V_\beta(R) := \lim_{T \to \infty} - \frac{1}{T} \log \, \langle W_{\gamma_{T,R}} \rangle_\beta, $$ known as the static quark potential in the physics literature. We verify existence of the limit (with an estimate on the convergence rate) and compute the first few terms in the expansion in powers of $e^{-\beta}$. As a consequence, a strong version of the perimeter law follows. We also treat $- \log \, \langle W_{\gamma_{T,R}} \rangle_\beta / (T+R)$ as $T, R$ tend to infinity simultaneously and give analogous estimates.