Lefschetz-thimble approach to the Silver Blaze problem of one-site fermion model

TL;DR

This paper investigates the sign problem in finite-density QCD using a one-site fermion model and the Lefschetz-thimble method, revealing how the integration cycle decomposes and affects phase transitions.

Contribution

It demonstrates the decomposition of the integration cycle into multiple Lefschetz thimbles and the importance of phase interference in understanding the sign problem.

Findings

Decomposition of the integration cycle at a critical chemical potential.

Fictitious phase transition occurs on each Lefschetz thimble.

Complex phases interfere, affecting the system's description.

Abstract

The sign problem of finite-density QCD at the zero temperature becomes very severe if the quark chemical potential exceeds half of the pion mass. In order to understand its property, we consider the sign problem of the one-site fermion model appearing in its path-integral expression by using the Lefschetz-thimble method. We show that the original integration cycle becomes decomposed into multiple Lefschetz thimbles at a certain value of the fermion chemical potential, which would correspond to half of the pion mass of finite-density QCD. This triggers a fictitious phase transition on each Lefschetz thimble, and the interference of complex phases among them plays an important role for the correct description of the system. We also show that the complex Langevin method does not work in this situation.