Complex Langevin: Etiology and Diagnostics of its Main Problem

TL;DR

This paper analyzes the complex Langevin method's main issues, providing a criterion for correctness, and tests its effectiveness on toy models to diagnose and understand convergence failures.

Contribution

It offers a formal justification, a practical truncated criterion for correctness, and empirical validation on toy models to diagnose convergence problems.

Findings

The truncated criterion effectively detects correctness in toy models.

Failures are linked to specific points in the formal justification.

The method shows promise with certain limitations in diagnosing convergence.

Abstract

The complex Langevin method is a leading candidate for solving the so-called sign problem occurring in various physical situations. Its most vexing problem is that in some cases it produces `convergence to the wrong limit'. In the first part of the paper we go through the formal justification of the method, identify points at which it may fail and identify a necessary and sufficient criterion for correctness. This criterion would, however, require checking infinitely many identities, and therefore is somewhat academic. We propose instead a truncation to the check of a few identities; this still gives a necessary criterion, but a priori it is not clear whether it remains sufficient. In the second part we carry out a detailed study of two toy models: first we identify the reasons why in some cases the method fails, second we test the efficiency of the truncated criterion and find that it works perfectly at least in the toy models studied.