The continuum limit of the lattice Gribov problem, and a solution based on Hodge decomposition

TL;DR

This paper investigates the lattice Gribov problem in 2D U(1) gauge theory, revealing divergence of Gribov copies in the continuum limit and proposing a Hodge decomposition-based global gauge fixing method that ensures uniqueness and accuracy.

Contribution

It introduces a novel global gauge fixing approach using Hodge decomposition, overcoming lattice artifacts and ensuring a unique smooth gauge field in Landau gauge.

Findings

Number of Gribov copies diverges in the continuum limit on spherical topology.

Lattice artifacts can bias gauge-invariant correlation length measurements.

Hodge decomposition method yields a unique smooth gauge field, reducing artifacts.

Abstract

We study gauge fixing via the standard local extremization algorithm for 2-dimensional $U(1)$. On a lattice with spherical topology $S^2$ where all copies are lattice artifacts, we find that the number of these 'Gribov' copies diverges in the continuum limit. On a torus, we show that lattice artifacts can lead to the wrong evaluation of the gauge-invariant correlation length, when measured via a gauge-fixed procedure; this bias does not disappear in the continuum limit. We then present a new global approach, based on Hodge decomposition of the gauge field, which produces a unique smooth field in Landau gauge, and is economically powered by the FFT. We also discuss the use of this method for examining topological objects, and its extensions to non-abelian gauge fields.