Controlling Complex Langevin simulations of lattice models by boundary term analysis

TL;DR

This paper investigates the causes of failure in the Complex Langevin method for lattice models, focusing on boundary terms at infinity, and demonstrates how to estimate systematic errors through boundary term analysis.

Contribution

It extends boundary term analysis from simple models to complex lattice models like HDQCD and XY, enabling systematic error estimation in CL simulations.

Findings

Boundary terms at infinity cause CL failures.

Systematic error estimation via boundary terms is feasible.

Analysis applied to models including HDQCD and 3D XY.

Abstract

One reason for the well known fact that the Complex Langevin (CL) method sometimes fails to converge or converges to the wrong limit has been identified long ago: it is 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, in a previous paper we have studied the emergence of such boundary terms thoroughly in a simple model, where analytic results can be compared with numerics. Here we continue this type of analysis for more physically interesting models, focusing on the boundaries at infinity. We start with abelian and non-abelian one-plaquette models, then we proceed to a Polyakov chain model and finally to high density QCD (HDQCD) and the 3D XY model. We show that the direct estimation of the systematic error of the CL method using boundary terms is in principle possible.