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International Journal of Modern Physics C

A GEOMETRICAL VIEW OF THE MINUS-SIGN PROBLEM

A. Muramatsu, Gil Zumbach, X. Zotos

Published Feb 1, 1992• 9 citations
Publisher

Abstract

It is shown that the sign of the fermionic determinant in the projector Monte Carlo method is directly related to a topological invariant. A key ingredient to obtain this result is the identification of the appropriate manifolds to describe the evolution of a fermionic trial wavefunction. They allow for a purely geometrical consideration of the minus-sign problem.

DOI
10.1142/s0129183192000154

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It is shown that the sign of the fermionic determinant in the projector Monte Carlo method is directly related to a topological invariant. A key ingredient to obtain this result is the identification of the appropriate manifolds to describe the evolution of a fermionic trial wavefunction. They allow for a purely geometrical consideration of the minus-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
Monte Carlo simulations are a powerful tool for elucidating the properties of complex systems, but when applied to fermionic quantum systems, quantum Monte Carlo (QMC) algorithms suffer from the so-called ``negative sign problem,'' causing the computational effort to grow exponentially with problem size. Here we demonstrate that the fermion sign problem originates in topological properties of the configurations. Using the widely used determinantal approaches, that remove the trivial sign problem due to particle exchange, we prove that the negative sign of a configuration is a topological invariant---an imaginary time counterpart of the Aharonov-Anandan phase---and reduces to a Berry phase in the adiabatic limit. This provides an intriguing connection between QMC simulations and classification of topological states.
The quantum Monte Carlo method for the Hubbard model using the Hubbard-Stratonovich transformation is reexamined. It is observed that the value of the ground state energy for a 3×2 and 4×4 system is estimated correctly by ignoring the minus-sign of the negative sampling weight appearing in the simulation using the projector method. It is also observed that the average of sign is positive and the correlation between the energy and the sign is weak for these systems. The reasons why the ground state energy is estimated correctly is given on a phenomenological base.
We tutorially review the determinantal Quantum Monte Carlo method for fermionic systems, using the Hubbard model as a case study. Starting with the basic ingredients of Monte Carlo simulations for classical systems, we introduce aspects such as importance sampling, sources of errors, and finite-size scaling analyses. We then set up the preliminary steps to prepare for the simulations, showing that they are actually carried out by sampling discrete Hubbard-Stratonovich auxiliary fields. In this process the Green's function emerges as a fundamental tool, since it is used in the updating process, and, at the same time, it is directly related to the quantities probing magnetic, charge, metallic, and superconducting behaviours. We also discuss the as yet unresolved "minus-sign problem", and two ways to stabilize the algorithm at low temperatures.