Simple proof that there is no sign problem in Path Integral Monte Carlo simulations of fermions in one dimension

TL;DR

This paper provides a direct proof that Path Integral Monte Carlo simulations of one-dimensional fermions do not suffer from the sign problem, by analyzing the sign of the fermionic propagator and its relation to particle displacements.

Contribution

It offers the first direct proof that the sign problem is absent in 1D PIMC fermion simulations, based on the properties of the fermionic propagator and particle displacements.

Findings

The sign of the N-fermion propagator is determined by pairwise particle displacements.

Closed-loop products of propagators are positive in one dimension.

Permutation sampling still exhibits a sign problem even in one dimension.

Abstract

It is widely known that there is no sign problem in Path Integral Monte Carlo (PIMC) simulations of fermions in one dimension. Yet, as far as the author is aware, there is no direct proof of this in the literature. This work shows that the $sign$ of the $N$-fermion anti-symmetric free propagator is given by the product of all possible pairs of particle separations, or relative displacements. For a non-vanishing closed-loop product of such propagators, as required by PIMC, all relative displacements from adjacent propagators are paired into perfect squares, and therefore the loop product must be positive, but only in one dimension. By comparison, permutation sampling, which does not evaluate the determinant of the anti-symmetric propagator exactly, remains plagued by a low-level sign problem, even in one dimension.