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Physical review. B, Condensed matter

Stable monte carlo algorithm for fermion lattice systems at low temperatures

J. E. Hirsch

Published Dec 1, 1988• 76 citations
Publisher

Abstract

We discuss Monte Carlo algorithms for fermion lattice systems based on exact updating of the fermion Green's function. A space and a space-time algorithm are shown to be limiting cases of a formulation that smoothly interpolates between them. This new formulation can to a large extent preserve the desirable features of both limits and suppress the undesirable ones.

DOI
10.1103/physrevb.38.12023

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We discuss Monte Carlo algorithms for fermion lattice systems based on exact updating of the fermion Green's function. A space and a space-time algorithm are shown to be limiting cases of a formulation that smoothly interpolates between them. This new formulation can to a large extent preserve the desirable features of both limits and suppress the undesirable ones.
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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.