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

TL;DR

This paper introduces an exact, solvable PIMC model for harmonic fermions across dimensions by using an operator contraction identity that preserves the nodal structure of the determinant propagator. It derives analytic expressions for discrete partition functions and energies at each time step, and analyzes how interactions and closed-shell degeneracy affect the fermion sign problem, revealing that closed-shell states avoid sign problems at large imaginary time. The authors demonstrate the practical utility of fourth-order propagators and a new class of Variable-Bead algorithms to compute ground-state energies for quantum dots containing up to about 110 electrons, achieving accuracy competitive with modern neural-network methods. The findings provide both fundamental insight into the sign problem in PIMC and practical computational strategies for large fermionic systems, with potential cross-fertilization into neural-network approaches.

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.

Figures (14)

• Figure 1: (color online) Left: The Hamiltonian energy of a 90-electron unpolarized quantum dot as a function of imaginary time $\tau$. Algorithm VB2's energy of 396.69(6) at $\tau=3$, is only 0.5% above Ref.nor23's SJ neural network energy of 394.621(4). Right: The Hamiltonian energy for a 110-electron unpolarized quantum dot. Algorithm VB2's energy at $\tau=2.5$ is 556.5(6). See Sect.\ref{['lqd']} for details.
• Figure 2: (color online) The PA propagator's non-interacting 2, 3, 4 and 6-fermion $N$-bead thermodynamic and Hamiltonian energies in a 2D harmonic oscillator, denoted by T$N$ and H$N$, are plotted as a function of the imaginary time $\tau=N\epsilon$. The data points are PIMC calculations and smooth curves are analytical results from (\ref{['eth']}) and (\ref{['eh']}).
• Figure 3: (color online) The Hamiltonian energies of the fourth-order Best 3-Bead (BB3) algorithm in solving for the ground state energy of 6 and 28 non-interacting fermions in a 2D harmonic oscillator. Notice that two energy minima are possible for $t_2=0.24$.
• Figure 4: (color online) The PA propagator's average sign $\langle {\rm sgn}\rangle$ for $n=2,3,4,6$ non-interacting harmonic fermions at various bead number $N$ as a function of $\tau$. The horizontal black lines in the two-fermion case are average sign values at large $\tau$ deduced in Appendix \ref{['ass']}.
• Figure 5: (color online) The average sign of a $N=4$ bead calculation in 2D and 3D for various $n$ non-interacting harmonic fermions. The average sign reverts back to one for closed-shell number of fermions in the large $\tau$ limit.