Improved spin-wave estimate for Wilson loops in $U(1)$ lattice gauge theory

TL;DR

This paper derives bounds on Wilson loop expectations in 4D $U(1)$ lattice gauge theory, highlighting the effects of topological defects and improving previous estimates for the free-energy correction term.

Contribution

It extends non-perturbative techniques to provide new quantitative bounds on topological defect fluctuations in continuous Abelian gauge theories.

Findings

Bound on Wilson loop expectation decays exponentially with loop size.

Improved correction term from $e^{-2 ext{pi}^2eta}$ to $e^{- ext{pi}^2eta}$ in free-energy estimate.

Provides lower bounds matching renormalization group predictions.

Abstract

In this paper, we obtain bounds on the Wilson loop expectations in 4D $U(1)$ lattice gauge theory which quantify the effect of topological defects. In the case of a Villain interaction, by extending the non-perturbative technique introduced in [GS20a], we obtain the following estimate for a large loop $\gamma$ at low temperatures: \[ |\langle W_\gamma\rangle_{\beta}| \leq \exp \left(-\frac{C_{GFF}} {2\beta}(1+C \beta e^{- 2\pi^2 \beta} )(|\gamma|+o(|\gamma|)) \right)\,. \] Our result is in the line of recent works [Cha20, Cao20, FLV20, For21] which analyze the case where the gauge group is discrete. In the present case where the gauge group is continuous and Abelian, the fluctuations of the gauge field decouple into a Gaussian part, related to the so-called {\em free electromagnetic wave} [Gro83, Dri87], and a gas of {\em topological defects}. As such, our work gives new quantitative bounds on the fluctuations of the latter which complement the works by Guth and Fr\"ohlich-Spencer [Gut80, FS82]. Finally, we improve, also in a non-perturbative way, the correction term from $e^{-2\pi^2\beta}$ to $e^{-\pi^2\beta}$ in the case of the free-energy of the system. This provides a matching lower-bound with the prediction of Guth [Gut80] based on renormalization group techniques.