Performance of Complex Langevin Simulation in 0+1 dimensional massive Thirring model at finite density

TL;DR

This paper investigates the effectiveness of the complex Langevin method in simulating the 0+1 dimensional Thirring model at finite density, addressing issues with drift singularities and proposing reweighting and model deformation techniques to improve accuracy.

Contribution

It introduces and evaluates reweighting and model deformation approaches to improve complex Langevin simulations of the Thirring model at finite density.

Findings

CL reproduces crossover behavior but deviates near transition

Reweighting and model deformation methods can recover correct behavior

Methods require exponential scaling evaluation of reweighting factors

Abstract

Statistical sampling with the complex Langevin (CL) equation is applied to (0+1)-dimensional Thirring model, and its uniform-field variant, at finite fermion chemical potential $\mu$. The CL simulation reproduces a crossover behavior which is similar to but actually deviating from the exact solution in the transition region, where we confirm that the CL simulation becomes susceptible to the drift singularities, i.e., zeros of the fermion determinant. In order to simulate the transition region with the CL method correctly, we examine two approaches, a reweighting method and a model deformation, in both of which a single thimble with an attractive fixed point practically covers the integration domain and the CL sampling avoids the determinant zeros. It turns out that these methods can reproduce the correct crossover behavior of the original model with using reference ensembles in the complexified space. However, they need evaluation of the reweighting factor, which scales with the system size exponentially. We discuss feasibility of applying these methods to the Thirring model and to more realistic theories.