Understanding the sign problem from an exact Path Integral Monte Carlo model of interacting harmonic fermions

TL;DR

This paper presents an exact Path Integral Monte Carlo model for interacting harmonic fermions, revealing insights into the sign problem and demonstrating applications to quantum dots with up to 110 electrons.

Contribution

It introduces an exactly solvable model for the sign problem using an operator contraction identity applicable in any dimension.

Findings

The sign problem is mainly due to the free fermion propagator.

Interaction shifts the severity of the sign problem but does not eliminate it.

Closed-shell states in any dimension have no sign problem at large imaginary time.

Abstract

This work shows that the recently discovered operator contraction identity for solving the discreet Path Integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the Path Integral Monte Carlo energy at any time step for any number of fermions is known analytically, or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interaction can shift the sign problem to larger/smaller imaginary time but does not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in $D$ dimension, with $n=D+1$ fermion, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three-dimension. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and compared to results obtained by modern neural networks.