Analytical studies of the complex Langevin equation with a Gaussian Ansatz and multiple solutions in the unstable region

Abstract

We investigate a simple model using the numerical simulation in the complex Langevin equation (CLE) and the analytical approximation with the Gaussian Ansatz. We find that the Gaussian Ansatz captures the essential and even quantitative features of the CLE results quite well including unwanted behavior in the unstable region where the CLE converges to a wrong answer. The Gaussian Ansatz is therefore useful for looking into this convergence problem and we find that the exact answer in the unstable region is nicely reproduced by another solution that is naively excluded from the stability condition. We consider the Gaussian probability distributions corresponding to multiple solutions along the Lefschetz thimble to discuss the stability and the locality. Our results suggest a prescription to improve the convergence of the CLE simulation to the exact answer.

arXiv comment10 pages, 10 figures

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• Why is the mission impossible? -- Decoupling the mirror Ginsparg-Wilson fermions in the lattice models for two-dimensional abelian chiral gauge theories
• Stochastic quantization and diffusion models
• Lefschetz thimble-inspired weight regularizations for complex Langevin simulations
• Lefschetz Thimbles and Quantum Phases in Zero-Dimensional Bosonic Models