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The Journal of Chemical Physics

Formally exact path integral Monte Carlo calculations using approximate projection operators

Randall W. Hall

Published Nov 1, 1992• 7 citations
Publisher

Abstract

A class of approximate projection operators is used to reduce the variance in path integral Monte Carlo calculations in a formally exact manner. Paths are classified according to the projection operators, allowing the identification of paths whose contribution to the variance is negligible. The approach is applied to two canonical systems: Two like-spin electrons in a three dimensional harmonic well and the two dimensional Hubbard model, representing localized and extended electronic states, respectively. Time savings of 15–900 over straightforward Monte Carlo calculations are observed.

DOI
10.1063/1.463709

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A class of approximate projection operators is used to reduce the variance in path integral Monte Carlo calculations in a formally exact manner. Paths are classified according to the projection operators, allowing the identification of paths whose contribution to the variance is negligible. The approach is applied to two canonical systems: Two like-spin electrons in a three dimensional harmonic well and the two dimensional Hubbard model, representing localized and extended electronic states, respectively. Time savings of 15–900 over straightforward Monte Carlo calculations are observed.
Feynman’s path integral formulation of quantum mechanics, supplemented by an approximate projection operator (exact in the case of noninteracting particles), is used to study the 2D Hubbard model. The projection operator is designed to study Hamiltonians defined on a finite basis set, but extensions to continuous basis sets are suggested. The projection operator is shown to reduce the variance by a significant amount relative to straightforward Monte Carlo integration. Approximate calculations are usually within one standard deviation of exact results and virtually always within two to three standard deviations. In addition, the algorithm scales with the number of discretization points P as either P or P2 (depending on the method of implementation), rather than the P3 of the Hubbard–Stratonovich transformation. Accuracy to about 5%–10% in energies and spin–spin correlation functions are found using moderate amounts of computer time.
A projection operator, similar to one previously used by us for problems with a finite set of basis functions, is suggested for use with continuous basis sets. This projection operator requires knowledge of the nodes of the density matrix at all temperatures. We show that a class of nodes, determined from the noninteracting density matrix and present at high temperatures in the interacting system are preserved to first order in the interaction at low temperatures. While we cannot show that the nodes are present at intermediate temperatures, we suspect they do exist and, as a test of this conjecture, we perform a calculation of two electrons confined in a harmonic well, using the projection operator. We find that accurate results are obtained at a range of temperatures, suggesting that our conjecture is indeed correct. We find that the error limits determined using the projection operator are 1–2 times smaller than those obtained with straightforward Monte Carlo integration (corresponding to a reduction in time of 1–4 in obtaining a desired level of accuracy).
Mean-field molecular dynamics based on path integrals is used to approximate canonical quantum observables for particle systems consisting of nuclei and electrons. A computational bottleneck is the Monte Carlo sampling from the Gibbs density of the electron operator, which due to the fermion sign problem has a computational complexity that scales exponentially with the number of electrons. In this work, we construct an algorithm that approximates the mean-field Hamiltonian by path integrals for fermions. The algorithm is based on the determinant of a matrix with components built on Brownian bridges connecting permuted electron coordinates. The computational work for n electrons is O ( n 3 ) , which reduces the computational complexity associated with the fermion sign problem. We analyze a bias resulting from this approximation and provide a rough computational error indicator. It remains to rigorously explain the surprisingly high accuracy for high temperatures. The method becomes infeasible at low temperatures due to a large sample variance.
A kink-based expression for the canonical partition function is developed using Feynman’s path integral formulation of quantum mechanics and a discrete basis set. The approach is exact for a complete set of states. The method is tested on the 3×3 Hubbard model and overcomes the sign problem seen in traditional path integral studies of fermion systems. Kinks correspond to transitions between different N-electron states, much in the same manner as occurs in configuration interaction calculations in standard ab initio methods. The different N-electron states are updated, based on which states occur frequently during a Monte Carlo simulation, giving better estimates of the true eigenstates of the Hamiltonian.
We present an exact formulation of the single-particle Green's functions for the two-dimensional three-band Hubbard model of the CuO2{\mathrm{CuO}}_{2}CuO2​ lattice based on a projection-operator formalism of Mori. The model is general and incorporates the various known local-density-approximation (LDA) interaction parameters. In this paper we present only the mean-field results. The inclusion of self-energy effects will be presented elsewhere. Already, in the mean-field approximation, we find that properties such as hole occupations and magnetic moments compare remarkably well with those from quantum Monte Carlo (MC) calculations. The densities of states are also calculated for various hole dopings, again using the parameters from the LDA calculations. Finally the detailed effect of hopping between different O p orbitals, are also presented.