Deconvolving the components of the sign problem

TL;DR

This paper investigates the underlying causes of the sign problem in auxiliary field Quantum Monte Carlo simulations, distinguishing between random fluctuations and biased sampling effects that lead to the exponential decay of the average sign.

Contribution

It provides a detailed analysis of whether the sign problem arises from random determinant signs or from the sampling process favoring negative determinants.

Findings

Identifies the dominant mechanism behind the sign problem.

Clarifies the role of sampling bias in the sign problem.

Offers insights for developing mitigation strategies.

Abstract

Auxiliary field Quantum Monte Carlo simulations of interacting fermions require sampling over a Hubbard-Stratonovich field $h$ introduced to decouple the interactions. The weight for a given configuration involves the products of the determinant of matrices $M_\sigma(h)$, where $\sigma$ labels the species, and hence is typically not positive definite. Indeed, the average sign $\langle {\cal S} \rangle$ of the determinants goes to zero exponentially with increasing spatial size and decreasing temperature for most Hamiltonians of interest. This statement, however, does not explicitly separate two possible origins for the vanishing of $\langle {\cal S} \rangle$. Does $\langle {\cal S} \rangle \rightarrow 0$ because {\it randomly} chosen field configurations have ${\rm det}\big(M(h)\big) < 0$, or does the `sign problem' arise because the specific subset of configurations chosen by the weighting function have a greater preponderance of negative values? In the latter case, the process of weighting the configurations with $|{\rm det}\big(M(h)\big)|$ might steer the simulation to a region of configuration space of $h$ where positive and negative determinants are equally likely, even though randomly chosen $h$ would preferentially have determinants with a single dominant sign. In this paper we address the relative importance of these two mechanisms for the vanishing of $\langle {\cal S} \rangle$ in quantum simulations.