AUXILIARY FIELDS AND THE SIGN PROBLEM

J.H. Samson ()
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J.H. Samson: Department of Physics, Loughborough University of Technology, Loughborough, Leicestershire LE11 3TU, England

International Journal of Modern Physics C (IJMPC), 1995, vol. 06, issue 03, 427-465

Abstract: The auxiliary-field quantum Monte Carlo method is reviewed. The Hubbard-Stratonovich transformation converts an interacting Hamiltonian into a non-interacting Hamiltonian in a time-dependent stochastic field, allowing calculation of the resulting functional integral by Monte Carlo methods. The method is presented in a sufficiently general form to be applicable to any Hamiltonian with one- and two-body terms, with special reference to the Heisenberg model and one- and many-band Hubbard models. Many physical correlation functions can be related to correlation functions of the auxiliary field; general results are given here. Issues relating to the choice of auxiliary fields are addressed; operator product identities change the relative dimensionalities of the attractive and repulsive parts of the interaction. Frequently the integrand is not positive-definite, rendering numerical evaluation unstable. If the auxiliary field violates time-reversal invariance, the integrand is complex and this sign problem becomes a phase problem. The origin of this sign or phase is examined from a number of geometrical and other viewpoints and illustrated by simple examples: the phase problem by the spin (1/2) Heisenberg model, and the sign problem by the attractiveSU(N)Hubbard model on a triangular molecule with negative hopping integrals. In the latter case, widely studied in the Jahn-Teller literature, the sign is due neither to fermions nor spin, but to frustration. This system is used to illustrate a number of suggested interpretations of the sign problem.

Keywords: Quantum Monte Carlo; Auxiliary Fields; Functional Integration; Sign Problem (search for similar items in EconPapers)
Date: 1995
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DOI: 10.1142/S0129183195000290

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