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A Generic Solution of Fermion Sign Problem

D. Y. Sun, Jianpan Wang

Published Mar 17, 2022• 0 citations
Publisher

Abstract

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

DOI
10.21203/rs.3.rs-1456011/v1

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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
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.
The path-integral formulation of nonrelativistic quantum mechanics was introduced by Feynman in 1948. The use of Path Integral Monte Carlo can be put on a rigorous footing using conditional Wiener integrals. This thesis addresses the topics both of numerical error and of Monte Carlo error.\nA piecewise constant numerical method which is of second order of accuracy for computing conditional Wiener integrals for a rather general class of sufficiently smooth functional is proposed. The method is based on simulation of Brownian bridges via the corresponding stochastic differential equations (SDEs) and on ideas of the weak-sense numerical integration of SDEs. A convergence theorem is proved. Special attention is paid to integral-type functionals. Results of some numerical experiments are presented.\nIn a further part of the research, the goal is to develop Monte Carlo methods for fermion simulations that are resistant to the explosion of variance which happens due to the fermion sign problem. A novel approach is developed which represents a radical departure from the current approaches. This is based on the principle of using a geometrical interpretation of the problem in order to find ways to maximize the negative covariance between the countersigned functional contributions. The fundamental connection between quantum exchange and the fermion sign problem is exploited. It is shown that this leads to a mathematical proof of the well-known exact solution to the sign problem for 1-dimensional fermion systems, and also to a novel exact solution in the case of a pair of 2-dimensional fermions.
The minus sign problem, as a problem of the non-positive-definite measure of the path integrals, is examined in the Monte Carlo simulation of the Hubbard model. A new correction method in terms of the ratio of two measured quantities is proposed to calculate physical properties. The momentum distribution and the spin correlation as well as the energy in the ground state are tested against the exactly diagonalized results. The results show that this ratio correction method is a promising prescription to reduce the minus sign problem.
A general algorithm toward the solution of the fermion sign problem in finite-temperature quantum Monte Carlo simulations has been formulated for discretized fermion path integrals with nearest-neighbor interactions in the Trotter direction. This multilevel approach systematically implements a simple blocking strategy in a recursive manner to synthesize the sign cancellations among different fermionic paths throughout the whole configuration space. The practical usefulness of the method is demonstrated for interacting electrons in a quantum dot.
By precisely writing down the matrix element of the local Boltzmann operator (e(−τhe^(-{\tau}he(−τh), where h is the paired operator formed by two interacting terms conjugate to each other), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. Our formula presents remarkable outcomes such that: 1) the Hubbard-Stratonovich transformation is not necessary, and the auxiliary field is no longer needed, which can improve the computational efficiency. 2) The new formula improves the accuracy of the Suzuki-Trotter decomposition. Our method also provides a potential theoretical framework for analyzing and solving the fermion sign problem. As an example, we studied the one-dimensional half-filled Hubbard model at a finite temperature. The obtained results were in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be used in various studies.