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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Simple numerical approach to quantum lattice models

Eduardo R. Gagliano, Silvia B. Bacci

Published Jul 15, 1987• 15 citations
Publisher

Abstract

The purpose of this paper is to present some simple variants of a recently developed iterative algorithm for improvement of any given trial wave function. These modifications improve the convergence rate of the iterative process and allow us to obtain a good variational state for larger lattices. We apply the method to the one-dimensional Gross-Neveu model with one flavor in a finite-size lattice. The ground-state energy, the density-density correlation function, and the single-particle Green's function are calculated. They are in good agreement with existing results.

DOI
10.1103/physrevd.36.546

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The purpose of this paper is to present some simple variants of a recently developed iterative algorithm for improvement of any given trial wave function. These modifications improve the convergence rate of the iterative process and allow us to obtain a good variational state for larger lattices. We apply the method to the one-dimensional Gross-Neveu model with one flavor in a finite-size lattice. The ground-state energy, the density-density correlation function, and the single-particle Green's function are calculated. They are in good agreement with existing results.
In the context of the Hamiltonian variational approach we propose a systematic method for the improvement of any given trial wave function. The approach has many similarities with the Lanczos scheme and it may be used for quantum-mechanical problems as well as field-theory and statistical-mechanics systems. We apply the method to the Mathieu equation and to the Ising one-dimensional model in a finite lattice. The agreement between our results and the exact ones is excellent in the whole range of parameters. We also briefly discuss the application of these ideas to lattice gauge theories and the similarities with other recently proposed methods.
We develop a general numerical method to study the zero-temperature properties of strongly correlated electron models on large lattices. The technique, which resembles Green's-function Monte Carlo, projects the ground-state component from a trial wave function with no approximations. We use this method to determine the phase diagram of the two-dimensional $t\ensuremath{-}J$ model, using the Maxwell construction to investigate electronic phase separation. The shell effects of fermions on finite-sized periodic lattices are minimized by keeping the number of electrons fixed at a closed-shell configuration and varying the size of the lattice. Results obtained for various electron numbers corresponding to different closed shells indicate that the finite-size effects in our calculation are small. For any value of interaction strength, we find that there is always a value of the electron density above which the system can lower its energy by forming a two-component phase separated state. Our results are compared with other calculations on the $t\ensuremath{-}J$ model. We find that the most accurate results are consistent with phase separation at all interaction strengths.
The initial trial wave function used in a simple ground-state projection method, the power method, is systematically improved by using Lanczos algorithm. Much faster convergence to the ground state achieved by using these wave functions significantly reduces the effect of the fermion sign problem. The results for the ground state of the two-dimensional t-J model are presented. The density correlation function for the t-J model at small J shows a surprisingly good agreement with that of a system of noninteracting hard-core bosons.