On the validity of complex Langevin method for path integral computations

TL;DR

This paper investigates the complex Langevin method's reliability for path integral computations, revealing conditions for its success and limitations, especially regarding probability density localization and the effects of gauge cooling.

Contribution

It provides a rigorous analysis of the CL method's bias mechanisms, formulates it on general groups, and evaluates the impact of gauge cooling on its stability and accuracy.

Findings

The CL method converges to biased results when the probability density is not localized.

Localized probability densities are absent in lattice field theories for general compact groups.

Gauge cooling can improve localization but has limitations, especially for Abelian groups.

Abstract

The complex Langevin (CL) method is a classical numerical strategy to alleviate the numerical sign problem in the computation of lattice field theories. Mathematically, it is a simple numerical tool to compute a wide class of high-dimensional and oscillatory integrals. However, it is often observed that the CL method converges but the limiting result is incorrect. The literature has several unclear or even conflicting statements, making the method look mysterious. By an in-depth analysis of a model problem, we reveal the mechanism of how the CL result turns biased as the parameter changes, and it is demonstrated that such a transition is difficult to capture. Our analysis also shows that the method works for any observables only if the probability density function generated by the CL process is localized. To generalize such observations to lattice field theories, we formulate the CL method on general groups using rigorous mathematical languages for the first time, and we demonstrate that such localized probability density function does not exist in the simulation of lattice field theories for general compact groups, which explains the unstable behavior of the CL method. Fortunately, we also find that the gauge cooling technique creates additional velocity that helps confine the samples, so that we can still see localized probability density functions in certain cases, as significantly broadens the application of the CL method. The limitations of gauge cooling are also discussed. In particular, we prove that gauge cooling has no effect for Abelian groups, and we provide an example showing that biased results still exist when gauge cooling is insufficient to confine the probability density function.