Stability of complex Langevin dynamics in effective models

TL;DR

This paper investigates the stability of complex Langevin dynamics in effective models, explaining why it succeeds in some cases and fails in others, with a focus on the role of the Haar measure in stabilizing the process.

Contribution

It analyzes the difference in complex Langevin stability between models, highlighting the stabilizing effect of the Haar measure in the SU(3) spin model.

Findings

Complex Langevin fails in the disordered phase of the XY model.

It succeeds across the entire phase diagram of the SU(3) spin model.

The Haar measure plays a key role in stabilizing the dynamics.

Abstract

The sign problem at nonzero chemical potential prohibits the use of importance sampling in lattice simulations. Since complex Langevin dynamics does not rely on importance sampling, it provides a potential solution. Recently it was shown that complex Langevin dynamics fails in the disordered phase in the case of the three-dimensional XY model, while it appears to work in the entire phase diagram in the case of the three-dimensional SU(3) spin model. Here we analyse this difference and argue that it is due to the presence of the nontrivial Haar measure in the SU(3) case, which has a stabilizing effect on the complexified dynamics. The freedom to modify and stabilize the complex Langevin process is discussed in some detail.