Some remarks on Lefschetz thimbles and complex Langevin dynamics

TL;DR

This paper investigates the relationship between Lefschetz thimbles and complex Langevin dynamics in models with sign problems, revealing insights into their interplay, stability issues, and the impact of singular drifts.

Contribution

It provides new findings on the connection between classical runaways, stable thimbles, and degenerate fixed points in complex Langevin dynamics for specific models.

Findings

Evidence linking classical runaways to stable thimbles

Identification of degenerate fixed points in the models

Sampled distributions relate to thimbles with caveats due to unstable fixed points

Abstract

Lefschetz thimbles and complex Langevin dynamics both provide a means to tackle the numerical sign problem prevalent in theories with a complex weight in the partition function, e.g. due to nonzero chemical potential. Here we collect some findings for the quartic model, and for U(1) and SU(2) models in the presence of a determinant, which have some features not discussed before, due to a singular drift. We find evidence for a relation between classical runaways and stable thimbles, and give an example of a degenerate fixed point. We typically find that the distributions sampled in complex Langevin dynamics are related to the thimble(s), but with some important caveats, for instance due to the presence of unstable fixed points in the Langevin dynamics.