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arXiv:cond-mat/9712240 · Physical review. B, Condensed matter

Multiband Gutzwiller wave functions for general on-site interactions

Jörg Bünemann, W. Weber, Florian Gebhard

Published Mar 15, 1998• 245 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.57.6896

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arXiv:cond-mat/0308513•Aug 26, 2003•0 citations
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.
This work covers three lectures on Gutzwiller wave functions for correlated electron systems. In the first lecture, some basic ideas on model Hamiltonians and their motivation are outlined, Gutzwiller's variational approach to Landau Fermi-liquid theory is sketched, and, as a first application, the theory is applied to the one-band Hubbard model where the Brinkman–Rice metal-to-insulator transition is found. The second lecture is devoted to the perturbative calculation of expectation values. In this rather technical part the variational analog of Wick's theorem, Feynman diagrams, and the linked-cluster theorem are introduced. The important differences between the variational approach and the standard Green function technique are the absence of a time-dependence and the flexibility to select the expansion parameter of the variational perturbation theory. When this flexibility is used appropriately, not a single self-energy diagram must be calculated in the limit of large dimensions, i.e. in the limit of large lattice coordination number the expectation values can be evaluated exactly for general Hubbard-type models for all interaction strengths and band fillings. The third lecture covers the application of the theory to ferromagnetism of nickel. After a discussion of the nickel problem, multi-band Hubbard models and Gutzwiller wave functions are introduced and parameterized. The favourable comparison between the results from Gutzwiller theory and a large number of experiments shows that correlations play an important role in itinerant ferromagnets.
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.
The variational scheme of Gutzwiller has been reexamined so as to apply it to the case of degenerate narrow bands. The ground-state wave function for electrons is investigated for an arbitrary density of electrons and an arbitrary strength of interaction. The typical features of a metal, particularly the Fermi surface, are preserved in this approach as opposed to some other methods. The Coulomb repulsion among electrons scatters the electrons from below to above the Fermi surface of uncorrelated bands. Under the assumption that only the intra-atomic Coulomb interaction is important, the probability of electrons being scattered is larger for the paramagnetic state than for the ferromagnetic state. Therefore the ferromagnetic ground state is favored if the density of states is large at the band edges. The degenerate-band-model Hamiltonian is spin dependent, hence the results obtained from it do not have to obey the theorem of Lieb and Mattis.
We develop an extension of the Gutzwiller approximation (GA) formalism that includes the effects of Coulomb interactions of arbitrary range (including density density, exchange, pair hopping, and Coulomb-assisted hopping terms). This formalism reduces to the ordinary GA formalism for the multiband Hubbard models in the presence of only local interactions. This is accomplished by combining the 1/z1/z1/z expansion---where zzz is the coordination number, and only the leading-order terms contribute in the limit of infinite dimensions---with a ${P}_{R}^{\ifmmode\dagger\else\textdagger\fi{}}{P}_{R}\ensuremath{-}I$ expansion, where PR{P}_{R}PR​ is the Gutzwiller projector on the site RRR. The method is conveniently formulated in terms of a Gutzwiller Lagrange function. We apply our theory to the extended single-band Hubbard model. Similarly to the usual Brinkman-Rice mechanism, we find a Mott transition. A valence skipping transition is observed, where the occupation of the empty and doubly occupied states for the Gutzwiller wave function is enhanced with respect to the uncorrelated Slater determinant wave function.
We make use of a recently developed diagrammatic theory to calculate correlation functions for interacting fermions of Hubbard-type models in terms of the Gutzwiller wave function. Of the eleven nontrivial correlation functions involving the spin, density, empty and doubly occupied sites, and local Cooper pairs, four are shown to be independent. They are expressed as a power series in a suitably chosen correlation parameter, whose terms are represented diagrammatically. In one dimension these terms may be evaluated to arbitrary order by employing symmetry relations. This allows for an analytic, approximation-free calculation of the correlation functions for arbitrary momentum, particle density, and interaction strength. In the atomic limit the momentum-dependent spin-correlation function shows an antiferromagnetic divergence at half filling in all dimensions. In one dimension the behavior is in very good agreement with all exact analytic and numerical results for the antiferromagnetic Heisenberg chain. Hole-hole correlations also compare very well with exact results. However, correlations between holes and doubly occupied sites appear insufficient. Superconducting correlations involving on-site, singlet Cooper pairs are suppressed. The results allow for an analytic evaluation of the ground-state energy of a large class of extended Hubbard models in terms of the Gutzwiller wave function. Thus they provide exact upper bounds for their ground-state energies.
Laughlin's construction of exact gossamer ground states is applied to normal metals. We show that for each variational parameter 0<=g<=1, the paramagnetic or ferromagnetic Gutzwiller wave function is the exact ground state of an extended Hubbard model with correlated hopping, with arbitrary particle density, non-interacting dispersion, and lattice dimensionality. The susceptibility and magnetization curves are obtained, showing that the Pauli susceptibility is enhanced by correlations. The elementary quasiparticle excitations are gapless, except for a half-filled band at g=0, where a Mott transition from metal to insulator occurs.