Topological Origin of the Fermion Sign Problem

TL;DR

This paper reveals that the fermion sign problem in quantum Monte Carlo simulations stems from topological properties of configurations, linking geometric phases to the negative signs encountered.

Contribution

It demonstrates that the sign problem arises from topological phases like Berry and Aharonov-Anandan phases in auxiliary field approaches, clarifying its origin.

Findings

Negative signs are geometric phases in auxiliary field QMC.

Sign problem linked to topological properties of configurations.

Provides a topological classification of the sign problem.

Abstract

Monte Carlo simulations are a powerful tool for elucidating the properties of complex systems across many disciplines. Not requiring any a priori knowledge, they are particularly well suited for exploring new phenomena. However, when applied to fermionic quantum systems, quantum Monte Carlo (QMC) algorithms suffer from the so-called "negative sign problem", which causes the computational effort to grow exponentially with problem size. Here we demonstrate that the fermion sign problem originates in topological properties of the configurations. In particular, we show that in the widely used auxiliary field approaches the negative sign of a configuration is a geometric phase that is the imaginary time counterpart of the Aharonov-Anandan phase, and reduces to a Berry phase in the adiabatic limit. This provides an intriguing connection between QMC simulations and classification of topological states. Our results shed clarify the controversially debated origin of the sign problem in fermionic lattice models.