Bloch Waves in Minimal Landau Gauge and the Infinite-Volume Limit of Lattice Gauge Theory

TL;DR

This paper introduces a novel method using Bloch waves to effectively simulate infinite-volume gauge configurations in lattice gauge theory, enabling more accurate calculations of the gluon propagator from smaller lattice data.

Contribution

The paper applies a Bloch wave-based approach to approximate infinite-volume Landau gauge configurations, extending the reach of numerical simulations in lattice gauge theory.

Findings

Successfully evaluated the gluon propagator on larger lattices

Demonstrated the method's effectiveness in 2D and 3D SU(2) gauge theories

Provided insights into the infinite-volume limit in lattice gauge simulations

Abstract

By exploiting the similarity between Bloch's theorem for electrons in crystalline solids and the problem of Landau gauge-fixing in Yang-Mills theory on a "replicated" lattice, one is able to obtain essentially infinite-volume results from numerical simulations performed on a relatively small lattice. This approach, proposed by D. Zwanziger in \cite{Zwanziger:1993dh}, corresponds to taking the infinite-volume limit for Landau-gauge field configurations in two steps: firstly for the gauge transformation alone, while keeping the lattice volume finite, and secondly for the gauge-field configuration itself. The solutions to the gauge-fixing condition are then given in terms of Bloch waves. Applying the method to data from Monte Carlo simulations of pure SU(2) gauge theory in two and three space-time dimensions, we are able to evaluate the Landau-gauge gluon propagator for lattices of linear extent up to sixteen times larger than that of the simulated lattice. The approach is reminiscent of Fisher and Ruelle's construction of the thermodynamic limit in classical statistical mechanics.