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Physical Review Letters

Efficient Monte Carlo Procedure for Systems with Fermions

J. E. Hirsch, Douglas J. Scalapino, Robert L. Sugar, R. Blankenbecler

Published Nov 30, 1981• 195 citations
Publisher

Abstract

A new Monte Carlo approach applicable to systems with fermion and boson degrees of freedom is proposed. The main advantage of the method is its speed: For systems in one space and one time dimension (1 + 1) the calculation reduces to doing Monte Carlo computations on a two-dimensional classical system with local interactions. Here the potential of the method is illustrated by application to various (1 + 1)-dimensional systems.

DOI
10.1103/physrevlett.47.1628

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A new Monte Carlo approach applicable to systems with fermion and boson degrees of freedom is proposed. The main advantage of the method is its speed: For systems in one space and one time dimension (1 + 1) the calculation reduces to doing Monte Carlo computations on a two-dimensional classical system with local interactions. Here the potential of the method is illustrated by application to various (1 + 1)-dimensional systems.
A new Monte Carlo method is proposed for fermion systems interacting with classical degrees of freedom. To obtain a weight for each Monte Carlo sample with a fixed configuration of classical variables, the moment expansion of the density of states by Chebyshev polynomials is applied instead of the direct diagonalization of the fermion Hamiltonian. This reduces a cpu time to scale as O(Ndim2log⁡Ndim)O(N_{\rm dim}^{2} \log N_{\rm dim})O(Ndim2​logNdim​) compared to O(Ndim3)O(N_{\rm dim}^{3})O(Ndim3​) for the diagonalization in the conventional technique; NdimN_{\rm dim}Ndim​ is the dimension of the Hamiltonian. Another advantage of this method is that parallel computation with high efficiency is possible. These significantly save total cpu times of Monte Carlo calculations because the calculation of a Monte Carlo weight is the bottleneck part. The method is applied to the double-exchange model as an example. The benchmark results show that it is possible to make a systematic investigation using a system-size scaling even in three dimensions within a realistic cpu timescale.
We have recently developed a formalism for carrying out Monte Carlo calculations on systems with coupled boson-fermion degrees of freedom. Here we present the results obtained using these techniques on some specific systems. First the quantum-mechanics problem of a single boson coordinate linearly coupled to a fermion is studied. The fermion Green's function and the boson coordinate displacement are calculated and compared with exact results. Extending this analysis to a one-dimensional array, a field theory in one space and one imaginary time dimension is investigated. The fermion and boson Green's functions are calculated and compared with analytic results obtained in the limit where the bare boson frequency is much larger or much smaller than the characteristic fermion energies.
We discuss a new method to perform numerical simulations of one-dimensional systems with fermion and boson degrees of freedom. The method is based on a direct-space, imaginary-time representation of the fermion field. It is fast so that systems having up to 100 sites can easily be simulated. In addition, the method provides an intuitive physical "picture" of the ground state of a one-dimensional many-body system. We discuss in detail how to implement the method and how to compute various physical quantities. In particular, we show how to extend the method to study averages of off-diagonal quantities in an occupation-number representation. To assess the accuracy of our procedure, we apply it to free fermions in one dimension and compare with exact results. We then study a model of spinless interacting fermions and obtain the expected phase structure and behavior of correlation functions. We also consider the extended Hubbard model at various points in its phase diagram and study the behavior of spin-density, charge-density, and pairing correlation functions. We then study the Gross-Neveu model and show how the behavior depends on the number of fermion flavors. Finally, we consider an electron-phonon model and study its behavior both in the one-particle polaron sector and in the half-filled-band case. Along the way we show pictures of the ground-state configurations that give physical insight into the properties of the systems, like charge-density-wave, spin-density-wave, and superconducting states, "fractional charges," and solitons. We conclude by comparing our method with other methods and discuss the possibility of extending it to higher dimensions.
A Monte Carlo method applicable to a class of noncommuting spin variables is developed. The technique relies on a stochastic procedure which simulates the dominant vacuum-to-vacuum graphs of quantum many-body theory. Computational speed is only weakly dependent on the dimensionality of the system and is fast enough to handle large lattices. This method is tested on the S=12S=\frac{1}{2}S=21​ ferromagnetic Heisenberg model in one, two, and three dimensions and on the S=12S=\frac{1}{2}S=21​ XY\mathrm{XY}XY model in one dimension where exact results are known.
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