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Journal of the Physical Society of Japan

On the Monte Carlo Method for Fermions in Multi-Dimensional Systems

Masatoshi Imada

Published Sep 15, 1984• 14 citations
Publisher

Abstract

The negative sign problem is a serious difficulty with most existing Monte Carlo methods of evaluating fermion systems of greater than one-dimension because it makes the computation of physical variables practically impossible. A simple and general averaging method is proposed for many-fermion systems. This method is completely free of the negative sign problem. The efficiency and convergence are verified in the Feynman's path integral calculations of a few simple models.

DOI
10.1143/jpsj.53.2861

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The negative sign problem is a serious difficulty with most existing Monte Carlo methods of evaluating fermion systems of greater than one-dimension because it makes the computation of physical variables practically impossible. A simple and general averaging method is proposed for many-fermion systems. This method is completely free of the negative sign problem. The efficiency and convergence are verified in the Feynman's path integral calculations of a few simple models.
The fermion sign problem, the biggest obstacle in quantum Monte Carlo calculations, is completely solved in this paper. Here, we find a strategy, in which the contribution from those negative-weighted paths is thoroughly cancelled or replaced by some positive-weighted paths. The crucial point lies on the Feynman path integral formula proposed in our group, which allows us to deeply analyze the Boltzmann weight of each path. Through mathematical proof, we demonstrate that physical quantities can be exactly calculated within a specific kind of paths, which have positive the Boltzmann weight. With this finding, a new Monte Carlo method is proposed, in which the fermion sign problem is absent. As an example, the current method is applied to the two-dimensional Hubbard model, and the results do manifest the correctness.
Abstract The fermion sign problem, the biggest obstacle in quantum Monte Carlo calculations, is completely solved in this paper. Here, we find a strategy, in which the contribution from those negative-weighted paths is thoroughly cancelled or replaced by some positive-weighted paths. The crucial point lies on the Feynman path integral formula proposed in our group, which allows us to deeply analyze the Boltzmann weight of each path. Through mathematical proof, we demonstrate that physical quantities can be exactly calculated within a specific wavefunction space, in which all paths have positive Boltzmann weight. With this finding, a new Monte Carlo method is proposed, in which the fermion sign problem is absent. As an example, the current method is applied to the two-dimensional Hubbard model, and the results do manifest the correctness. *Corresponding author: dysun@phy.ecnu.edu.cn
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.