Complex Langevin method applied to the 2D $SU(2)$ Yang-Mills theory

TL;DR

This paper applies the complex Langevin method with gauge cooling to 2D SU(2) Yang-Mills theory, showing convergence issues at large gauge coupling phases despite analytical solvability.

Contribution

It demonstrates the effectiveness and limitations of the complex Langevin method with gauge cooling in a solvable gauge theory.

Findings

Convergence to incorrect values at large gauge coupling phases

Numerical evidence of convergence behavior

Insights into the method's limitations in certain regimes

Abstract

The complex Langevin method in conjunction with the gauge cooling is applied to the two-dimensional lattice $SU(2)$ Yang-Mills theory that is analytically solvable. We obtain strong numerical evidence that at large Langevin time the expectation value of the plaquette variable converges, but to a wrong value when the complex phase of the gauge coupling is large.