Correctness criteria for complex Langevin

TL;DR

This paper systematically compares various correctness criteria for the complex Langevin method, assessing their applicability and predictive power across simple models to improve reliability in complex systems.

Contribution

It provides a comprehensive comparison of prominent correctness criteria for complex Langevin, highlighting their strengths and limitations in simple models.

Findings

Different criteria vary in predictive power and ease of use.

Some criteria reliably identify convergence to correct results.

The conclusions are expected to extend to more realistic theories.

Abstract

The complex Langevin approach is a promising method for the numerical treatment of systems with a sign problem, for which conventional lattice field theory techniques based on importance sampling cannot be applied. However, complex Langevin dynamics may fail to converge in some cases and converge to a wrong limit in others, motivating the development of various diagnostic tools over the years to assess the correctness of given simulation results. This work aims at providing a systematic comparison between the most prominent such correctness criteria. In particular, the main goal is to contrast their applicability, ease of use, and - most importantly - their predictive power. To this end, four simple but nontrivial models are considered and the criteria applied to each of them. The obtained conclusions are expected to carry over to more realistic theories as well.