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Physical review. B, Solid state

Magnetic Transitions in a Degenerate Band

K. A. Chao

Published Dec 1, 1971• 39 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.4.4034

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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.
The ground-state wave function for the electrons in a narrow sss band is investigated for arbitrary density of electrons and arbitrary strength of interaction. An approximation is proposed which limits all the calculations to counting certain types of configurations and attaching the proper weights. The expectation values of the one-particle and two-particle density matrix are computed for the ferromagnetic and for the non-ferromagnetic case. The ground-state energy is obtained under the assumption that only the intra-atomic Coulomb interaction is of importance. Ferromagnetism is found to occur if the density of states is large at the band edges rather than in the center, and if the intra-atomic Coulomb repulsion is sufficiently strong. The relation of this approximation to certain exact results for one-dimensional models is discussed.
The Hubbard hamiltonian is analysed by means of Gutzwiller's variational scheme for both the nondegenerate and the degenerate band models. The single-particle excitations in the correlated ground state, which is determined by a balance between the electron hopping energy and the intra-atomic interaction energy, are investigated for arbitrary electron density and arbitrary correlation strength. It is found that the single-particle excitation energies forming a quasi-energy band. The centre of gravity of the quasi band is shifted from that of the bare band, and the bulk part of the density of states function for the quasi particles is narrowed by the correlation effect. Both the shift and the narrowing are spin-dependent. In the limit of strong correlation, our results agree with those obtained by Harris and Lange (1967) and Roth (1969).
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.
The method of Gutzwiller in the problem of electron correlation in the Hubbard model is extended to the antiferromagnetic state. It is shown that the transition from the paramagnetic to the antiferromagnetic state occurs at a certain value of U (intraatomic Coulomb energy) for a fixed value of the electron concentration. In contrast to the result in the Hartree-Fock approximation, this value of U does not vanish even in the case when the magnetic zone boundary coincides with the Fermi surface. For extremely large values of U, the critical electron concentration is shown to approach to 1/2. Special care is taken in the case of the half-filled band, and our ground state energy is compared with that in the Hartree-Fock approximation. It is shown that for large value of U the Hartree-Fock energy is lower, but that for intermediate value of U our energy becomes lower for the special band structure.
We present analytic results for ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function with nonzero values of the magnetization m. In dimension D=1D=1D=1 approximation-free evaluations are made possible by appropriate canonical transformations and an analysis of umklapp processes. We calculate the double occupation and the momentum distribution, as well as its discontinuity at the Fermi surface, for arbitrary values of the interaction parameter g, density n, and magnetization m. These quantities determine the expectation value of the one-dimensional Hubbard Hamiltonian for any symmetric, monotonically increasing dispersion ${\ensuremath{\epsilon}}_{k}.$ In particular for nearest-neighbor hopping and densities away from half filling the Gutzwiller wave function is found to predict ferromagnetic behavior for sufficiently large interaction U.