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Journal of the Physical Society of Japan

A New Hybrid Algorithm for Fermion Simulations Applied to the Hubbard Model

Masatoshi Imada

Published Jan 15, 1988• 16 citations
Publisher

Abstract

A new hybrid-type algorithm is proposed for simulations of fermion lattices. The direct calculation of the matrix inversion operation is replaced with a stochastic iteration procedure. The computation time grows linearly with the increase of spatial system size. Applications to the Hubbard model on the 6 2 , 12 2 and 16 2 square lattices are illustrated with no indication of enhancement for superconducting correlation in the non-half-filled band.

DOI
10.1143/jpsj.57.42

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A new hybrid-type algorithm is proposed for simulations of fermion lattices. The direct calculation of the matrix inversion operation is replaced with a stochastic iteration procedure. The computation time grows linearly with the increase of spatial system size. Applications to the Hubbard model on the 6 2 , 12 2 and 16 2 square lattices are illustrated with no indication of enhancement for superconducting correlation in the non-half-filled band.
We propose a Monte Carlo method, which is a hybrid method of the quantum Monte Carlo method and variational Monte Carlo theory, to study the Hubbard model. The theory is based on the off-diagonal and the Gutzwiller type correlation factors which are taken into account by a Monte Carlo algorithm. In the 4x4 system our method is able to reproduce the exact results obtained by the diagonalization. An application is given to investigate the half-filled band case of two-dimensional square lattice. The energy is favorably compared with quantum Monte Carlo data.
We present an algorithm for the numerical simulation of many electron systems, which is based upon the hybrid stochastic approach presently being used to study lattice gauge theory. A method for preconditioning the fermion matrices is described which significantly reduces the computing time needed in the simulation. Results are presented for the three-dimensional Hubbard model on lattices ranging in size from 43{4}^{3}43 to 103{10}^{3}103.
We have studied the two-dimensional Hubbard model on a square lattice with nearest-neighbor hopping. We first discuss the properties of the model within the mean-field approximation: Because of the form of the band structure, some peculiar features are found. We then discuss the simulation algorithm used and compare simulation results with exact results for 6-site chains to test the reliability of the approach. We present results for thermodynamic properties and correlation functions for lattices up to 8\ifmmode\times\else\texttimes\fi{}8 in spatial size. The system is found to be an antiferromagnetic insulator for all values of the coupling constant at zero temperature in the half-filled-band case, but the long-range order is much smaller than predicted by mean-field theory. We perform a finite-size-scaling analysis to determine the character of the transition at zero coupling. For non-half-filled-band cases, our results suggest that the system is always paramagnetic, in contradiction with Hartree-Fock predictions. The system does not show tendency to ferromagnetism nor triplet superconductivity in the parameter range studied. We also discuss some properties of the attractive Hubbard model in the half-filled-band case.
A comparative study of the efficiency of different algorithms (Metropolis, Langevin, and Hybrid) for the simulation of a one-dimensional spinless fermion model with a half-filled band is presented. The Metropolis updating of the fields is performed with an algorithm introduced by Blankenbecler, Scalapino, and Sugar. The density-density correlation function, the order parameter, and the autocorrelation are calculated. A study of the dependence of the systematic errors on the temperature and the step size for the Langevin and hybrid algorithms is presented. The relation between the accuracy and the speed of the algorithms is studied.