High-statistics finite size scaling analysis of U(1) lattice gauge theory with a Wilson action

Abstract

We describe the results of a systematic high-statistics Monte Carlo study of finite-size effects at the phase transition of compact U(1) lattice gauge theory with a Wilson action on a hypercubic lattice with periodic boundary conditions. We find unambiguously that the critical exponent $\ensuremath{\nu}$ is lattice-size dependent for volumes ranging from ${4}^{4}$ to ${12}^{4}.$ Asymptotic scaling formulas yield values decreasing from $\ensuremath{\nu}(L>~4)\ensuremath{\approx}0.33$ to $\ensuremath{\nu}(L>~9)\ensuremath{\approx}0.29.$ Our statistics are sufficient to allow the study of different phenomenological scenarios for the corrections to asymptotic scaling. We find evidence that corrections to a first-order transition with $\ensuremath{\nu}=0.25$ provide the most accurate description of the data. However the corrections do not always follow the expected first-order pattern of a series expansion in the inverse lattice volume ${V}^{\ensuremath{-}1}.$ Reaching the asymptotic regime will require lattice sizes greater than $L=12.$ Our conclusions are supported by the study of many cumulants which all yield consistent results after proper interpretation.