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International Journal of Modern Physics B

Determinant Monte Carlo for the Hubbard Model with Arbitrarily Gauged Auxiliary Fields

Liang Chen, A.–M. S. Tremblay

Published Mar 1, 1992• 20 citations
Publisher

Abstract

Monte Carlo methods for the Hubbard model rely on a Hubbard-Stratonovich (HS) decomposition (auxiliary field method) to perform importance sampling on classical variables. Freedom in the choice of the local HS fields can be formally seen as a gauge choice. While the choice of gauge does not influence observable quantities, it may influence intermediate quantities in the calculation, such as the famous “fermion sign”, and it may also influence the efficiency with which the algorithm explores phase space. The effect of arbitrary gauge choices on both aspects of the algorithm are investigated. It is found that in the single spin-flip determinantal approach, certain gauges lead to a better exploration of phase space. This improvement is demonstrated, in the intermediate coupling regime, by histograms which for the first time show the behavior expected from grand canonical simulations. It is also found that the improved phase space exploration can in practice offset the apparent disadvantage of a smaller fermion sign.

DOI
10.1142/s0217979292000323

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Monte Carlo methods for the Hubbard model rely on a Hubbard-Stratonovich (HS) decomposition (auxiliary field method) to perform importance sampling on classical variables. Freedom in the choice of the local HS fields can be formally seen as a gauge choice. While the choice of gauge does not influence observable quantities, it may influence intermediate quantities in the calculation, such as the famous “fermion sign”, and it may also influence the efficiency with which the algorithm explores phase space. The effect of arbitrary gauge choices on both aspects of the algorithm are investigated. It is found that in the single spin-flip determinantal approach, certain gauges lead to a better exploration of phase space. This improvement is demonstrated, in the intermediate coupling regime, by histograms which for the first time show the behavior expected from grand canonical simulations. It is also found that the improved phase space exploration can in practice offset the apparent disadvantage of a smaller fermion sign.
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.
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.
Within the ground-state auxiliary-field quantum Monte Carlo technique, we introduce discrete Hubbard-Stratonovich transformations (HSTs) that are also suitable for spatially inhomogeneous trial functions. The discrete auxiliary fields introduced here are coupled to local spin or charge operators fluctuating around their Hartree-Fock values. The formalism can be considered a generalization of the discrete HSTs by J. E. Hirsch [Phys. Rev. B 28, 4059 (1983)] or a compactification of the shifted-contour auxiliary-field Monte Carlo formalism by N. Rom et al. [Chem. Phys. Lett. 270, 382 (1997)]. An improvement of the acceptance ratio is found for a real auxiliary field, while an improvement of the average sign is found for a purely imaginary auxiliary field. Efficiencies of the different HSTs are tested in the single-band Hubbard model at and away from half filling by studying the staggered magnetization and energy expectation values, respectively.
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 attractive SU(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.
$A\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ computations of finite-temperature properties are important in fermion systems such as correlated models, ultracold Fermi atomic gases, and optical lattices, and realistic materials. Here, the authors present a highly accurate approach that eliminates the sign problem in standard determinantal quantum Monte Carlo (DQMC) using a gauge constraint in auxiliary-field space. The constraint can be systematically improved by a self-consistent procedure coupling the many-body computation with an independent-particle calculation. A benchmark is performed in the Hubbard model across the entire temperature range.
Continuous-time determinantal algorithm is proposed for the quantum Monte Carlo simulation of the interacting fermions. The scheme does not invoke Hubbard-Stratonovich transformation. The fermionic action is divided into two parts. One of them contains the interaction and certain additional terms; another one is purely Gaussian. The first part is considered as a perturbation. Terms of the series expansion for the partition function are generated in a random walk process. The sign problem and the complexity of the algorithm are analyzed. We argue that the scheme should be useful particularly for the systems with non-local interaction.