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Physical Review Letters

Positive-projection Monte Carlo simulation: A new variational approach to strongly interacting fermion systems

Stephen Fahy, Donald R. Hamann

Published Dec 31, 1990• 41 citations
Publisher

Abstract

We show that very accurate ground-state energies and correlation functions can be obtained from an approximate, variational form of the widely used auxiliary-field simulation method. It is derived by recasting this method as a diffusion problem, and does not exhibit the poor statistical behavior due to vanishing normalization or ``sign'' common to exact fermion simulations. We give illustrative results for two-dimensional Hubbard models.

DOI
10.1103/physrevlett.65.3437

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We show that very accurate ground-state energies and correlation functions can be obtained from an approximate, variational form of the widely used auxiliary-field simulation method. It is derived by recasting this method as a diffusion problem, and does not exhibit the poor statistical behavior due to vanishing normalization or ``sign'' common to exact fermion simulations. We give illustrative results for two-dimensional Hubbard models.
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Quantum Monte Carlo simulations provide a powerful tool to compute static properties of quantum many-particles models and realistic systems. A vast array of ground-state energies and properties defined by equal-time correlation functions can be computed with high accuracy. On the other hand, the extensions needed to study time-displaced correlation functions, which provide access to excited states, have been challenging, especially for fermionic strongly correlated systems. The authors present a methodology that allows a high-accuracy calculation of the dynamical Green functions of many-fermions systems in the framework of the auxiliary-field quantum Monte Carlo method. They extract the the charge gap of the repulsive two-dimensional Hubbard model at half-filling as a function of the interaction strength and propose strategies for applying their methodology to more realistic systems.
We propose a new quantum Monte Carlo algorithm to compute fermion ground-state properties. The ground state is projected from an initial wave function by a branching random walk in an over-complete basis space of Slater determinants. By constraining the determinants according to a trial wave function $|{\ensuremath{\Psi}}_{T}〉$, we remove the exponential decay of signal-to-noise ratio characteristic of the sign problem. The method is variational and is exact if $|{\ensuremath{\Psi}}_{T}〉$ is exact. We report results on the two-dimensional Hubbard model up to size $16\ifmmode\times\else\texttimes\fi{}16$, for various electron fillings and interaction strengths.
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The ground-state projection version of the Hubbard-Stratonovich auxiliary-field simulation method has been extended to Hubbard-style models with arbitrary range interactions between pairs of sites. For repulsive interactions, a transformation based on three auxiliary fields per site performs comparably to the usual single-field approach for local interactions. These fields have unit-variance Gaussian distributions, and couple to the number operators on each site through several matrices calculated from the original pairwise interaction matrix. Tests of this approach have been performed on small clusters of sites representing highly simplified models of atoms and molecules. Our recently introduced ``positive-projection'' approximation remains effective in controlling the ``sign problem.'' Comparisons to exact diagonalization demonstrate that accurate energies and correlation functions are obtained using this approach.
We solve the zero-temperature unitary Fermi gas problem by incorporating a BCS importance function into the auxiliary-field quantum Monte Carlo method. We demonstrate that this method does not suffer from a sign problem and that it increases the efficiency of standard techniques by many orders of magnitude for strongly paired fermions. We calculate the ground-state energies exactly for unpolarized systems with up to 66 particles on lattices of up to 273{27}^{3}273 sites, obtaining an accurate result for the universal parameter $\ensuremath{\xi}$. We also obtain results for interactions with different effective ranges and find that the energy is consistent with a universal linear dependence on the product of the Fermi momentum and the effective range. This method will have many applications in superfluid cold atom systems and in both electronic and nuclear structures where pairing is important.