On the Segregation Phenomenon in Complex Langevin Simulation

TL;DR

This paper analyzes why complex Langevin simulations sometimes fail, identifying the treatment of singularities as the key issue, and proposes an effective algorithm to accurately simulate certain models in the small parameter limit.

Contribution

It clarifies the cause of complex Langevin failure and introduces a new algorithm to handle singularities, improving simulation accuracy.

Findings

Failure is due to singularity treatment, not the method itself.

Proposed algorithm accurately reproduces small parameter behavior.

Analysis based on a toy model with one degree of freedom.

Abstract

In the numerical simulation of certain field theoretical models, the complex Langevin simulation has been believed to fail due to the violation of ergodicity. We give a detailed analysis of this problem based on a toy model with one degree of freedom ($S=-\beta\cos\theta$). We find that the failure is not due to the defect of complex Langevin simulation itself, but rather to the way how one treats the singularity appearing in the drift force. An effective algorithm is proposed by which one can simulate the ${1/\beta}$ behaviour of the expectation value $<\cos\theta>$ in the small $\beta$ limit.