Kernels and integration cycles in complex Langevin simulations

TL;DR

This paper investigates the complex Langevin method for simulating systems with complex actions, highlighting issues with convergence and proposing kernel modifications to improve correctness by controlling integration cycles.

Contribution

It demonstrates that boundary terms alone are insufficient for correctness and introduces kernel techniques to manage integration cycles, enhancing the reliability of complex Langevin simulations.

Findings

Boundary terms are not enough to ensure correct convergence.

Kernel modifications can control sampling of integration cycles.

Proper kernel choice restores correct equilibrium distributions.

Abstract

The method of complex Langevin simulations is a tool that can be used to tackle the complex-action problem encountered, for instance, in finite-density lattice quantum chromodynamics or real-time lattice field theories. The method is based on a stochastic evolution of the dynamical degrees of freedom via (complex) Langevin equations, which, however, sometimes converge to the wrong equilibrium distributions. While the convergence properties of the evolution can to some extent be assessed by studying so-called boundary terms, we demonstrate in this contribution that boundary terms on their own are not sufficient as a correctness criterion. Indeed, in their absence complex Langevin simulation results might still be spoiled by unwanted so-called integration cycles. In particular, we elaborate on how the introduction of a kernel into the complex Langevin equation can - in principle - be used to control which integration cycles are sampled in a simulation such that correct convergence is restored.