Analytical studies of the complex Langevin equation with a Gaussian Ansatz and multiple solutions in the unstable region

TL;DR

This paper combines numerical and analytical methods to study the complex Langevin equation, revealing multiple solutions and proposing ways to improve convergence in unstable regions.

Contribution

It demonstrates that the Gaussian Ansatz effectively captures key features of CLE results and identifies an alternative solution that improves convergence.

Findings

Gaussian Ansatz reproduces CLE features well

Multiple solutions exist along the Lefschetz thimble

A new prescription enhances CLE convergence

Abstract

We investigate a simple model using the numerical simulation in the complex Langevin equation (CLE) and the analytical approximation with the Gaussian Ansatz. We find that the Gaussian Ansatz captures the essential and even quantitative features of the CLE results quite well including unwanted behavior in the unstable region where the CLE converges to a wrong answer. The Gaussian Ansatz is therefore useful for looking into this convergence problem and we find that the exact answer in the unstable region is nicely reproduced by another solution that is naively excluded from the stability condition. We consider the Gaussian probability distributions corresponding to multiple solutions along the Lefschetz thimble to discuss the stability and the locality. Our results suggest a prescription to improve the convergence of the CLE simulation to the exact answer.