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Physical Review Letters

Close Relation between Localized-Electron Magnetism and the Paramagnetic Wave Function of Completely Itinerant Electrons

T. A. Kaplan, Peter Horsch, Peter Fulde

Published Sep 20, 1982• 214 citations
Publisher

Abstract

What is commonly called the Gutzwiller wave function $\ensuremath{\psi}$, an approximate ground state of the single-band Hubbard Hamiltonian, is considered here for $N$-site $N$-electron rings, when the number $D$ of doubly occupied sites is small ($N=6, 10, 14, 18$). 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.

DOI
10.1103/physrevlett.49.889

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What is commonly called the Gutzwiller wave function \ensuremathψ\ensuremath{\psi}\ensuremathψ, 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→0D\ensuremath{\rightarrow}0D\ensuremath→0 spin correlations in \ensuremathψ\ensuremath{\psi}\ensuremathψ 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 \ensuremathψ\ensuremath{\psi}\ensuremathψ reproduces the exact energy with remarkable accuracy.
We introduce Gutzwiller wave functions for multiband models with general on-site Coulomb interactions. As these wave functions employ correlators for the exact atomic eigenstates, they are exact both in the noninteracting and atomic limits. We evaluate them in infinite lattice dimensions for all interaction strengths without any restrictions on the structure of the Hamiltonian or the symmetry of the ground state. The results for the ground-state energy allow us to derive an effective one-electron Hamiltonian for Landau quasiparticles, applicable for finite temperatures and frequencies within the Fermi-liquid regime. As applications for a two-band model we study the Brinkman-Rice metal-to-insulator transition at half-band-filling, and the transition to itinerant ferromagnetism for two specific fillings, at and close to a peak in the density of states of the noninteracting system. Our results significantly differ from those for earlier Gutzwiller wave functions where only density-type interactions were included. When the correct spin symmetries for the two-electron states are taken into account, the importance of the Hund's-rule exchange interaction is even more pronounced, and leads to paramagnetic metallic ground states with large local magnetic moments. Ferromagnetism requires fairly large interaction strengths, and the resulting ferromagnetic state is a strongly correlated metal.
The well-known Gutzwiller wave function is generalized by including new variational parameters to control nearest-neighbour charge-charge, charge-spin, and spin-spin correlations. The non-magnetic state of the one-dimensional, half-filled Hubbard model is studied. Within the Gutzwiller approximation, the expression for the ground state energy can be worked out analytically. The correlation between empty and doubly occupied sites is found to play the most essential role. Minimization in the large-U limit shows that the Brinkman-Rice transition has been pushed to U → ∞, and the leading term of the ground state energy density is of order −t2/ U . In contrast to results obtained with the Gutzwiller wave function, we find that the band occupation number nk is monotonically decreasing both above and below kF. The dominant k–dependence is given by ~(t/U) cos k, in agreement with t/U–expansion results. nk has also a weak step-function component, with the discontinuity at kF vanishing as (t/U)2 in the limit U/t ≫ 1.
We introduce a new lattice version of the correlated-basis-function approach for the study of strongly interacting electrons. As a first application we have chosen the one-dimensional Hubbard model. For this system, we have implemented different types of correlations: namely, (a) Gutzwiller; (b) Jastrow; (c) nearest-neighbor spin-spin; (d) excitonic (e-d); and (f) pairing (BCS). The resulting energies, momentum distribution, and other quantities are compared with exact results (when available) and are discussed, both from the methodological and physical point of view. The most important correlations that are needed to obtain good ground-state energies are found to be Gutzwiller (reduced double occupancy) plus e-d correlations (between empty and doubly occupied sites), in agreement with other studies.
A new analytic approach to the evaluation of ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function is presented. It is based on the observation that expectation values in terms of this wave function may be expressed by sums over different lattice sites. This makes the application of Wick's theorem and the resulting contractions extremely simple, since the latter involve only anticommuting numbers as in a Grassmann algebra. Expressions for the momentum distribution ${n}_{\mathrm{k}\ensuremath{\sigma}}$ and the Hubbard interaction in terms of a power series in a particular correlation parameter are derived which are valid for all dimensions. An explicit diagrammatic evaluation of the coefficients is described. In one dimension these coefficients may be determined to all orders which yields an approximation-free calculation of nk{n}_{\mathrm{k}}nk​ and ground-state energy E for arbitrary density n and interaction strength U. In the case of a half-filled band and large U, hitherto unexpected nonanalyticities are found. The results allow for the first approximation-free assessment of the properties of the Gutzwiller wave function. It is shown that the well-known Gutzwiller approximation may be derived diagrammatically, too. The approximation is seen to yield the exact results for expectation values in terms of this wave function in the limit of infinite dimensionality.
Whether spin-independent Coulomb interaction can be the origin of a realistic ferromagnetism in an itinerant electron system has been an open problem for a long time. Here we study a class of Hubbard models on decorated lattices, which have a special property that the corresponding single-electron Schrödinger equation has NdN_{\rm d}Nd​-fold degenerate ground states. The degeneracy NdN_{\rm d}Nd​ is proportional to the total number of sites $\absΛ$. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to $ρ_0=N_{\rm d}/(2\absΛ)$, and paramagnetism when the filling factor is sufficiently small. An important feature of the present work is that it provides examples of three dimensional itinerant electron systems which are proved to exhibit ferromagnetism in a finite range of the electron filling factor.