Regularization of Complex Langevin Method

TL;DR

This paper introduces regularization techniques for the complex Langevin method to improve stability, along with two approaches, 2R and 3R, to recover unbiased results, validated through lattice field theory experiments.

Contribution

The paper proposes novel regularization methods, 2R and 3R, for the complex Langevin method, enabling bias correction and efficiency improvements in complex ensemble computations.

Findings

Regularization stabilizes the complex Langevin method.

2R and 3R methods effectively recover unbiased results.

Numerical experiments confirm the approaches' effectiveness.

Abstract

The complex Langevin method, a numerical method used to compute the ensemble average with a complex partition function, often suffers from runaway instability. We study the regularization of the complex Langevin method via augmenting the action with a stabilization term. Since the regularization introduces biases to the numerical result, two approaches, named 2R and 3R methods, are introduced to recover the unbiased result. The 2R method supplements the regularization with regression to estimate the unregularized ensemble average, and the 3R method reduces the computational cost by coupling the regularization with a reweighting strategy before regression. Both methods can be generalized to the SU(n) theory and are assessed from several perspectives. Several numerical experiments in the lattice field theory are carried out to show the effectiveness of our approaches.