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arXiv:2407.01618 · Physical review. E

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

Siu A. Chin

Published Jun 25, 2024• 3 citations
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Abstract

It is widely known that there is no sign problem in path integral Monte Carlo (PIMC) simulations of fermions in one dimension. 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 antisymmetric free propagator is given by the product of all possible pairs of particle separations, or relative displacements. For a nonvanishing 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 antisymmetric propagator exactly, remains plagued by a low-level sign problem, even in one dimension.

DOI
10.1103/physreve.109.065312

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It is widely known that there is no sign problem in path integral Monte Carlo (PIMC) simulations of fermions in one dimension. 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 antisymmetric free propagator is given by the product of all possible pairs of particle separations, or relative displacements. For a nonvanishing 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 antisymmetric propagator exactly, remains plagued by a low-level sign problem, even in one dimension.
In the Path Integral Monte Carlo (PIMC) simulation of fermions in a harmonic trap, with and without pairwise harmonic interactions, the partition functions for any discrete number of imaginary time slices (or beads) and for any choice of the short-time propagator can be analytically obtained from the contracted determinant form of the propagator. This work shows that the resulting recursion relation can be reformulated in the λλλ-ring language, yielding a closed-form finite-bead partition function in two dimensions in terms of permutation statistics. This closed-form partition function can be evaluated by a special algorithm in O(n2)O(n^2)O(n2) time, providing an exact and numerically stable scheme for reproducing the energies of the original (undoable) fermion PIMC simulation for n=104n=10^4n=104 or more fermions. This result provides a concrete framework in which the numerical instability of the sign problem is completely bypassed, and serves as a counterexample to the prevailing view that all truly fermionic PIMC sign problems are NP-hard.
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 DDD dimension, with n=D+1n=D+1n=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.
This work shows that the recently discovered operator contraction identity for solving the discrete 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 interactions can shift the sign problem to larger/smaller imaginary time but do 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 fermions, 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 dimensions. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and are compared to results obtained by modern neural networks.
We present a method for performing path integral molecular dynamics (PIMD) simulations for fermions and address its sign problem. PIMD simulations are widely used for studying many-body quantum systems at thermal equilibrium. However, they assume that the particles are distinguishable and neglect bosonic and fermionic exchange effects. Interacting fermions play a key role in many chemical and physical systems, such as electrons in quantum dots and ultracold trapped atoms. A direct sampling of the fermionic partition function is impossible using PIMD since its integrand is not positive definite. We show that PIMD simulations for fermions are feasible by employing our recently developed method for bosonic PIMD and reweighting the results to obtain fermionic expectation values. The approach is tested against path integral Monte Carlo (PIMC) simulations for up to seven electrons in a two-dimensional quantum dot for a range of interaction strengths. However, like PIMC, the method suffers from the sign problem at low temperatures. We propose a simple approach for alleviating it by simulating an auxiliary system with a larger average sign and obtaining an upper bound to the energy of the original system using the Bogoliubov inequality. This allows fermions to be studied at temperatures lower than would otherwise have been feasible using PIMD, as demonstrated in the case of a three-electron quantum dot. Our results extend the boundaries of PIMD simulations of fermions and will hopefully stimulate the development of new approaches for tackling the sign problem.