Complex saddle points and the sign problem in complex Langevin simulation

TL;DR

This paper investigates why complex Langevin simulations sometimes give incorrect results, linking it to phase differences among saddle points, and proposes a reweighting method to address the sign problem.

Contribution

It introduces a phase-aware reweighting approach to improve the accuracy of complex Langevin simulations in the presence of multiple saddle points.

Findings

Complex Langevin can converge to wrong results due to phase differences.

Reweighting with phase information can correct convergence issues.

The method may reintroduce the sign problem in quantum many-body systems.

Abstract

We show that complex Langevin simulation converges to a wrong result, by relating it to the Lefschetz-thimble path integral, when the path-integral weight has different phases among dominant complex saddle points. Equilibrium solution of the complex Langevin equation forms local distributions around complex saddle points. Its ensemble average approximately becomes a direct sum of the average in each local distribution, where relative phases among them are dropped. We propose that by taking these phases into account through reweighting, we can solve the wrong convergence problem. However, this prescription may lead to a recurrence of the sign problem in the complex Langevin method for quantum many-body systems.