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Journal of Physics C Solid State Physics

Exact and Monte Carlo studies of Gutzwiller's state for the localised-electron limit in one dimension

Peter Horsch, T. A. Kaplan

Published Dec 20, 1983• 69 citations
Publisher

Abstract

Exact calculations of the spin-spin correlation function q j in the localised-electron limit of Gutzwiller's (1963, 1965) variational approximation to the Hubbard-model group state, for rings with numbers of electrons N=4(2)18, strongly suggest that q j varies as (-) j /j at large separation j between the spins. Monte Carlo calculations for N up to 100 reproduce this behaviour.

DOI
10.1088/0022-3719/16/35/002

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Exact calculations of the spin-spin correlation function q j in the localised-electron limit of Gutzwiller's (1963, 1965) variational approximation to the Hubbard-model group state, for rings with numbers of electrons N=4(2)18, strongly suggest that q j varies as (-) j /j at large separation j between the spins. Monte Carlo calculations for N up to 100 reproduce this behaviour.
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.
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'').
What is commonly called the Gutzwiller wave function $\ensuremath{\psi}$, an approximate ground state of the single-band Hubbard Hamiltonian, is considered here for NNN-site NNN-electron rings, when the number DDD of doubly occupied sites is small (N=6,10,14,18N=6, 10, 14, 18N=6,10,14,18). For $D\ensuremath{\rightarrow}0$ spin correlations in $\ensuremath{\psi}$ surprisingly close to the exact values at zero bandwidth are found (i.e., those of the antiferromagnetic Heisenberg-model ground state). But the energy is grossly in error. A simple modification of $\ensuremath{\psi}$ reproduces the exact energy with remarkable accuracy.
Generalized Gutzwiller-type trial ground states for the Hubbard model are considered, which include extra variational parameters to control the nearest-neighbour charge-charge, charge-spin, and spin-spin correlations. The correlation between doubly occupied and empty sites is found to be particularly important. The results for the large- U regime show qualitative improvement when compared to the predictions of the simple Gutzwiller Ansatz: We get the correct order of magnitude for the energy density, and the band state occupation number is found to be a monotonically decreasing function, with the leading term corresponding to that of the t/U -expansion. We find that to get the small- U limit right, the pair-correlated Ansatz has to be used in conjunction with a cluster Gutzwiller approximation, and introduce a four-site cluster approximation which gives promising results. We argue that small-cluster Gutzwiller approximations get accurate in the limit of large lattice dimensionality.
The authors consider the variational Monte Carlo method of studying fermion systems in condensed matter physics. This method is a powerful tool for studying the properties of interacting electrons which are responsible for phenomena like the new high-Tc superconductors. Using a parallel algorithm for updating the determinants of large complex matrices on transputer arrays they study the energy and spin correlations for a 12*12 and a 22*22 lattice. Using the Hubbard model and the Gutzwiller projected wavefunction they have evaluated these quantities as functions of band filling and coupling constant. They find that although the Gutzwiller state is stable for densities near half filling it is not at low densities. For the stable states they find antiferromagnetic correlations. Statistically the thermodynamic limit is already attained with a 12*12 lattice.
As a continuation of a previous paper [J. Phys. Soc. Jpn. 56 (1987) 1490], the variational Monte-Carlo method is extended to include the antiferromagnetic long-range order. The theory is based on the Gutzwiller-type correlation factor and its effect is exactly taken into account by the Monte-Carlo procedure. An application is made to the half-filled-band case of one-dimensional lattice (50 sites), two-dimensional square lattice (up to 20×20 sites) and three-dimensional simple cubic lattice (6×6×6 sites). The result is qualitatively different from previous studies relying on the random-phase-type "Gutzwiller approximation." The variational energy for two and three dimensions is favorably compared with Hirsch's quantum Monte-Carlo data.