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

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 repulsive/attractive pairwise interaction shifts the sign problem to larger/smaller imaginary time, but does not make it more severe than the non-interacting case. More surprisingly, for closed-shell number of fermions, the sign problem goes away at large imaginary time. 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.