# Critical Slowing-Down in $SU(2)$ Landau Gauge-Fixing Algorithms

## TL;DR

This paper investigates the critical slowing-down phenomenon in various gauge-fixing algorithms for $SU(2)$ lattice gauge theory, demonstrating that Fourier acceleration effectively eliminates it, while other methods exhibit different critical exponents.

## Contribution

The study compares five gauge-fixing algorithms on 2D lattices, quantifies their critical slowing-down behavior, and shows Fourier acceleration removes this issue entirely.

## Key findings

- Fourier acceleration eliminates critical slowing-down.
- Different algorithms have distinct critical exponents.
- All observables relax at the same rate within each algorithm.

## Abstract

We study the problem of critical slowing-down for gauge-fixing algorithms (Landau gauge) in $SU(2)$ lattice gauge theory on a $2$-dimensional lattice. We consider five such algorithms, and lattice sizes ranging from $8^{2}$ to $36^{2}$ (up to $64^2$ in the case of Fourier acceleration). We measure four different observables and we find that for each given algorithm they all have the same relaxation time within error bars. We obtain that: the so-called {\em Los Alamos} method has dynamic critical exponent $z \approx 2$, the {\em overrelaxation} method and the {\em stochastic overrelaxation} method have $z \approx 1$, the so-called {\em Cornell} method has $z$ slightly smaller than $1$ and the {\em Fourier acceleration} method completely eliminates critical slowing-down. A detailed discussion and analysis of the tuning of these algorithms is also presented.