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arXiv:hep-lat/9605026

Suppression of the negative sign problem in quantum Monte Carlo

Galli, A.

Published May 22, 1996• 2 citations
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Abstract

We present a new Monte Carlo algorithm for simulating quantum spin systems which is able to suppress the negative sign problem. This algorithm has only a linear complexity in the lattice size used for the simulation. A general description and a rigorous proof of its correctness is given. Its efficiency is tested on a simple 2-dimensional fermionic model. For this model we show that our algorithm eliminates the sign problem.

DOI
10.48550/arxiv.hep-lat/9605026

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We present a new Monte Carlo algorithm for simulating quantum spin systems which is able to suppress the negative sign problem. This algorithm has only a linear complexity in the lattice size used for the simulation. A general description and a rigorous proof of its correctness is given. Its efficiency is tested on a simple 2-dimensional fermionic model. For this model we show that our algorithm eliminates the sign problem.
All known Quantum-Monte-Carlo algorithms for fermions suffer from the so-called “minus-sign-problem” which is detrimental to the application of these simulation methods to fermion systems at very low temperatures and/or of very many lattice sites. We identify the origin of this fundamental problem, demonstrate that it is a very general feature, not necessarily related to the presence of fermionic degrees of freedom. We describe an novel algorithm which does not suffer from the minus-sign problem. Illustrative results for the two-dimensional Hubbard model are presented
Quantum Monte Carlo (QMC) simulations involving fermions have a notorious sign problem. Some well-known exceptions to the auxiliary field QMC algorithm rely on the factorizibility of the fermion determinant. Recently, a fermionic QMC algorithm [C. Wu, J. Hu, and S. Zhang, Phys. Rev. Lett. 91, 186402 (2003)] has been found in which the fermion determinant may not necessarily be factorizable, but can instead be expressed as a product of complex conjugate pairs of eigenvalues, thus eliminating the sign problem for a much wider class of models. In this paper, we present the general conditions for the applicability of this algorithm and point out that it is deeply related to the time-reversal symmetry of the fermion matrix. We apply this method to various models of strongly correlated systems at all doping levels and lattice geometries, and show that many phases can be simulated without the sign problem.
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.
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.