On a modification method of Lefschetz thimbles

TL;DR

This paper proposes a modification of Lefschetz thimbles to reduce the sign problem in complexified theories like finite-density QCD, by restructuring thimbles so only one contributes, demonstrated in a toy model.

Contribution

The authors introduce a novel method to modify Lefschetz thimbles, ensuring only a single thimble influences the partition function, simplifying calculations and mitigating interference issues.

Findings

Method effectively reduces sign problem in toy model

Observable relations between original and modified theories established

Potential applicability to complex theories like QCD at finite density

Abstract

The QCD at finite density is not well understood yet, where standard Monte Carlo simulation suffers from the sign problem. In order to overcome the sign problem, the method of Lefschetz thimble has been explored. Basically, the original sign problem can be less severe in a complexified theory due to the constancy of the imaginary part of an action on each thimble. However, global phase factors assigned on each thimble still remain. Their interference is not negligible in a situation where a large number of thimbles contribute to the partition function, and this could also lead to a sign problem.In this study, we propose a method to resolve this problem by modifying the structure of Lefschetz thimbles such that only a single thimble is relevant to the partition function. It can be shown that observables measured in the original and modified theories are connected by a simple identity. We exemplify that our method works well in a toy model.