Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral

TL;DR

This paper introduces a Lefschetz-thimble path integral approach to address the fermion sign problem in mean-field approximations, ensuring real physical quantities and improving computational accuracy.

Contribution

It develops a systematic scheme using Lefschetz-thimbles to overcome the sign problem in mean-field calculations, maintaining reflection symmetry and realness of physical quantities.

Findings

Lefschetz-thimble method respects reflection symmetry.

The approach yields real physical quantities at all approximation orders.

Demonstrated with Airy integral and applied to dense QCD models.

Abstract

The fermion sign problem appearing in the mean-field approximation is considered, and the systematic computational scheme of the free energy is devised by using the Lefschetz-thimble method. We show that the Lefschetz-thimble method respects the reflection symmetry, which makes physical quantities manifestly real at any order of approximations using complex saddle points. The formula is demonstrated through the Airy integral as an example, and its application to the Polyakov-loop effective model of dense QCD is discussed in detail.