Complex Langevin and boundary terms

TL;DR

This paper investigates why the Complex Langevin method sometimes fails by analyzing boundary terms that affect convergence, using a simple model to compare analytic and numerical results, and proposing modifications to improve stability and correctness.

Contribution

It provides a detailed analysis of boundary terms causing CL failures, demonstrates how to stabilize the process, and suggests a criterion for correctness in lattice simulations.

Findings

Boundary terms can cause CL convergence failures.

Modifications can stabilize CL and improve accuracy.

A correctness criterion for CL is proposed.

Abstract

As is well known the Complex Langevin (CL) method sometimes fails to converge or converges to the wrong limit. We identified one reason for this long ago: insufficient decay of the probability density either near infinity or near poles of the drift, leading to boundary terms that spoil the formal argument for correctness. To gain a deeper understanding of this phenomenon, we analyze the emergence of such boundary terms thoroughly in a simple model, where analytic results can be compared with numerics. We also show how some simple modification stabilizes the CL process in such a way that it can produce results agreeing with direct integration. Besides explicitly demonstrating the connection between boundary terms and correct convergence our analysis also suggests a correctness criterion which could be applied in realistic lattice simulations.