Improved Spin-Wave Estimate for Wilson Loops in U(1) Lattice Gauge Theory

Abstract

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 [24], we obtain the following estimate for a large loop $\gamma $ at low temperatures: $ |\langle W_\gamma \rangle _{\beta }|\leq \exp \Big (-\frac {C_{GFF}} {2\beta }(1+C \beta e^{- 2\pi ^2 \beta } )(|\gamma |+o(|\gamma |)) \Big )\,.$ Our result is in line with recent works [4, 9, 13, 15] 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 free electromagnetic wave [11, 23], and a gas of topological defects. As such, our work gives new quantitative bounds on the fluctuations of the latter which complement the works by Guth and Frölich-Spencer [17, 27]. 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 [27] based on renormalization group techniques.