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Physical review. B, Condensed matter

Sum rule and symmetry-controlled expansion for generalized Gutzwiller wave functions

Y. M. Li, N. d’Ambrumenil

Published Dec 1, 1992• 2 citations
Publisher

Abstract

A systematic expansion scheme that goes beyond the Gutzwiller-type approximation is developed for generalized Gutzwiller wave functions. The expansion satisfies sum rules and symmetries of the trial wave function in a self-consistent way. We apply the scheme to an antiferromagnetic generalization of the Gutzwiller wave function for the one-dimensional Hubbard model. We find good agreement with known results from variational Monte Carlo calculations. We also demonstrate that the expansion scheme converges rapidly.

DOI
10.1103/physrevb.46.13928

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Physical review. B, Condensed matter•May 1, 1990•24 citations
A systematic expansion scheme that goes beyond the Gutzwiller-type approximation is developed for generalized Gutzwiller wave functions. The expansion satisfies sum rules and symmetries of the trial wave function in a self-consistent way. We apply the scheme to an antiferromagnetic generalization of the Gutzwiller wave function for the one-dimensional Hubbard model. We find good agreement with known results from variational Monte Carlo calculations. We also demonstrate that the expansion scheme converges rapidly.
A systematic expansion scheme that goes beyond the Gutzwiller-type approximation is developed for generalized Gutzwiller wave functions. The expansion satisfies sum rules and symmetries of the trial wave function in a self-consistent way. We apply the scheme to an antiferromagnetic generalization of the Gutzwiller wave function for the one-dimensional Hubbard model. We find good agreement with known results from variational Monte Carlo calculations. We also demonstrate that the expansion scheme converges rapidly.
We present a systematic formalism for the variational evaluation of ground-state properties of Hubbard-type models in finite dimensions d. The formalism starts from generalized Gutzwiller correlated wave functions, which are then studied in a systematic (1/d) expansion around the limit of high dimensions (d=\ensuremath{\infty}). The limit of d\ensuremath{\rightarrow}\ensuremath{\infty} has recently been introduced by Metzner and Vollhardt (MV) for itinerant lattice fermion systems. The approach, presented in this paper, is particularly efficient since results in d=\ensuremath{\infty} are obtained without having to calculate a single graph. Our results confirm the finding of MV that counting approximations in the spirit of the Gutzwiller approximation become exact in d=\ensuremath{\infty} for translationally invariant wave functions. This type of approximation is no longer exact for more complicated (e.g., antiferromagnetic) wave functions. In addition, we completely reproduce the results of the Kotliar-Ruckenstein path-integral approach to the Hubbard model. Performing a (1/d) expansion for the Gutzwiller wave function, we show that the lowest orders in (1/d) are sufficient to reproduce all numerical findings in d=2,3 quantitatively. We therefore conclude that the limit of d=\ensuremath{\infty} is a very fruitful starting point for the study of finite-dimensional systems. On the basis of our study we propose new variational wave functions for the numerical investigation of antiferromagnetism in the Hubbard, the t-J, and the spin-1/2 Heisenberg model. For the first two models we calculate in d=\ensuremath{\infty} only, for the Heisenberg model we also derive corrections up to order (1/d). To this order we obtain complete agreement with linear spin-wave theory. Since our trial state is based on a fermionic description of the Heisenberg model, we interpret this analytically determined wave function as Fermi sea of spin-1/2 quasiparticles (``spinons'').
A variational Monte-Carlo method, which is a combination of the variational theory and the Monte-Carlo technique is proposed to study the Hubbard model with the Gutzwiller's wave function. An application is made to one-, two-, and three-dimensional models having sites up to 216 (6×6×6) and arbitrary values of U / t (on-site repulsion U and hopping integral t ). Contrary to resorting to the Gutzwiller approximation, it provides a virtually exact way to carry out the Gutzwiller-type variational theory and opens up a possibility of further development.
Antiferromagnetic states of the half-filled Hubbard model on the d=1, 2, 3, and \ensuremath{\infty} dimensional lattices are systematically studied by using the Gutzwiller-type analytical approximation based on the slave-boson functional-integral approach. In all the cases investigated, antiferromagnetic states have lower energies than paramagnetic states even for infinitesimally small interactions, U. Calculated results of the ground-state energy E and the sublattice moment m for d=2 and 3 are in very good agreement with those of the Monte Carlo simulations. Even for d=1 the antiferromagnetic state in our Gutzwiller-type approximation has energy much closer to the exact one than the paramagnetic states in Gutzwiller's approximation and in Metzner-Vollhardt's approximation-free method. On the basis of the 1/d plot of E and m, a possibility of the 1/d expansion starting from the limit of d=\ensuremath{\infty} is discussed. It is also shown that the Brinkman-Rice metal-insulator transition does not take place at finite U for all d.