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Journal of the Physical Society of Japan

Reweighting Method for Quantum Monte Carlo Simulations with the Negative-Sign Problem

Tota Nakamura, Naomichi Hatano, Hidetoshi Nishimori

Published Oct 15, 1992• 24 citations
Publisher

Abstract

We propose a new method for quantum Monte Carlo simulations with the negative-sign problem. In the conventional method, one simulates a virtual system whose Boltzmann factors are the absolute values of the original one, and analyzes the data so that the correct expression for physical quantities may be recovered. We simulate a different virtual system with properties closer to the original system and yet without the negative-sign problem. Comparison between the present method and the conventional one is made for the antiferromagnetic Heisenberg model on the 4×h4 square lattice with frustrations ( J 1 - J 2 model). We conclude that the present method is useful in the low-temperature region where the negative-sign problem becomes serious. It is also shown that the method is considerably efficient on the point that quantities for different parameters can be obtained in a single run.

DOI
10.1143/jpsj.61.3494

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We propose a new method for quantum Monte Carlo simulations with the negative-sign problem. In the conventional method, one simulates a virtual system whose Boltzmann factors are the absolute values of the original one, and analyzes the data so that the correct expression for physical quantities may be recovered. We simulate a different virtual system with properties closer to the original system and yet without the negative-sign problem. Comparison between the present method and the conventional one is made for the antiferromagnetic Heisenberg model on the 4×h4 square lattice with frustrations ( J 1 - J 2 model). We conclude that the present method is useful in the low-temperature region where the negative-sign problem becomes serious. It is also shown that the method is considerably efficient on the point that quantities for different parameters can be obtained in a single run.
When one tries to simulate quantum spin systems by the Monte Carlo method, often the ``minus-sign problem'' is encountered. In such a case, an application of probabilistic methods is not possible. In this paper, the method has been proposed which allows to avoid the minus-sign problem for certain class of frustrated Heisenberg models. The systems where this method is applicable are, for instance, the pyrochlore lattice and the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. The method works in singlet sector. It relies on the expression of wave functions in dimer (pseudo)basis and writing down the Hamiltonian as a sum over plaquettes. In such a formulation, matrix elements of the exponent of Hamiltonian are positive.
Building on a recent investigation of the Shastry-Sutherland model [S. Wessel et al., Phys. Rev. B 98, 174432 (2018)], we develop a general strategy to eliminate the Monte Carlo sign problem near the zero-temperature limit in frustrated quantum spin models. If the Hamiltonian of interest and the sign-problem-free Hamiltonian, obtained by making all off-diagonal elements negative in a given basis, have the same ground state and this state is a member of the computational basis, then the average sign returns to one as the temperature goes to zero. We illustrate this technique by studying the triangular and kagome lattice Heisenberg antiferrromagnet in a magnetic field above saturation, as well as the Heisenberg antiferromagnet on a modified Husimi cactus in the dimer basis. We also provide detailed Appendices on using linear programming techniques to automatically generate efficient directed loop updates in quantum Monte Carlo simulations.
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.
We propose a sampling method to calculate the ground state of interacting quantum systems. This approximate method, which we call the adaptive sampling quantum Monte Carlo (ASQMC) method, utilizes information from the short projection time density matrix derived from the Monte Carlo steps. The ASQMC method circumvents the negative-sign problem by introducing a conditional probability distribution function and guiding walkers to avoid a set of canceling fields. With the ASQMC method, the negative-sign ratio is greatly reduced. The ratio becomes zero in the limit $\ensuremath{\Delta}\ensuremath{\tau}$ goes to zero without prior knowledge of the exact ground state wave function. Comparisons with numerical results obtained by using other methods are made, and we observe that the ASQMC method gives accurate results with less than 0.50.5%0.5 error for the energy over wide regions of physical parameter values.
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