Lefschetz-thimble analysis of the sign problem in one-site fermion model

TL;DR

This paper applies the Lefschetz-thimble method to a one-site Hubbard model to address the sign problem, revealing the importance of multiple saddle point interference for accurate results and stability.

Contribution

It demonstrates the necessity of multiple Lefschetz thimbles and proposes a criterion to determine their number, advancing understanding of the sign problem in quantum models.

Findings

Interference among multiple Lefschetz thimbles is crucial for accurate non-analytic behavior.

Insufficient thimbles lead to discrepancies and thermodynamic instabilities.

A semiclassical criterion for the necessary number of thimbles is proposed.

Abstract

The Lefschetz-thimble approach to path integrals is applied to a one-site model of electrons, i.e., the one-site Hubbard model. Since the one-site Hubbard model shows a non-analytic behavior at the zero temperature and its path integral expression has the sign problem, this toy model is a good testing ground for an idea or a technique to attack the sign problem. Semiclassical analysis using complex saddle points unveils the significance of interference among multiple Lefschetz thimbles to reproduce the non-analytic behavior by using the path integral. If the number of Lefschetz thimbles is insufficient, we found not only large discrepancies from the exact result, but also thermodynamic instabilities. Analyzing such singular behaviors semiclassically, we propose a criterion to identify the necessary number of Lefschetz thimbles. We argue that this interference of multiple saddle points is a key issue to understand the sign problem of the finite-density quantum chromodynamics.