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Physics Physique Fizika

Band theory of magnetism in metals in context of exactly soluble model

Daniel Charles Mattis

Published Nov 1, 1964• 29 citations
Publisher

Abstract

The role of spatial forces, and of Hund's rule (exchange) forces, and the validity of the Hartree-Fock approximation in the band theory of magnetism of metals is reviewed. Some reasonable conjectures which have been made elsewhere by the author and others are examined in the light of an exactly soluble model of interacting electrons in one dimension. It is shown that within the framework of validity of this model, spatial i.e., direct (Coulomb) forces do not influence the spin susceptibility, and conversely, Hund's rule exchange forces do not affect the dielectric constant or plasmon spectrum. It is recalled that the time-independent Hartree-Fock approximation yields an incorrect spectrum of elementary excitations, and it is also shown explicitly that the magnetic susceptibility calculated in this approximation is incorrect. A paradox is noted, concerning whether ``correlations'' can in fact correct the errors in the H-F approximation. Finally, it is shown that when there are only space forces, such spin density waves as might be introduced in to the H-F ground state of the model are in fact spurious, because they are not representative of the correlations which exist in the true ground state.

DOI
10.1103/physicsphysiquefizika.1.183

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The role of spatial forces, and of Hund's rule (exchange) forces, and the validity of the Hartree-Fock approximation in the band theory of magnetism of metals is reviewed. Some reasonable conjectures which have been made elsewhere by the author and others are examined in the light of an exactly soluble model of interacting electrons in one dimension. It is shown that within the framework of validity of this model, spatial i.e., direct (Coulomb) forces do not influence the spin susceptibility, and conversely, Hund's rule exchange forces do not affect the dielectric constant or plasmon spectrum. It is recalled that the time-independent Hartree-Fock approximation yields an incorrect spectrum of elementary excitations, and it is also shown explicitly that the magnetic susceptibility calculated in this approximation is incorrect. A paradox is noted, concerning whether ``correlations'' can in fact correct the errors in the H-F approximation. Finally, it is shown that when there are only space forces, such spin density waves as might be introduced in to the H-F ground state of the model are in fact spurious, because they are not representative of the correlations which exist in the true ground state.
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 problem of the ground state of the electronic system in the Hubbard model for U=∞ is discussed. The author investigates the normal (singlet or nonmagnetic) N state of the electronic system over the entire range of electron densities n⩽1. It is shown that the energy of the N state ɛ 0 (1) (n) in a one-particle approximation, such as (e.g.) the extended Hartree-Fock approximation, is lower than the energy of the saturated ferromagnetic FM state ɛ FM(n) for all n. The dynamic magnetic susceptibility is calculated in the random phase approximation, and it is shown that the N state is stable over the entire range of electron densities: The static susceptibility (ω=0) does not have a band singularity in the zero-wave vector limit q→0. A formally exact representation is obtained for the mass operator of the one-particle Green’s function, and an approximation of this operator is proposed: M k(E)⋍λF(E), where λ=n(1−n)/(1−n/2)z is the kinematic interaction parameter, z is the number of nearest neighbors, and F(E) is the total single-site Green’s function. For an elliptical density of states the integral equation for F(E) is solved exactly, ad it is shown that the spectral intensity rigorously satisfies the sum rule. The calculated energy of the strongly correlated N state ɛ 0(n)<ɛ FM(n) for all n, and in light of this relationship the author discusses the hypothesis that the ground state of the system is the normal (singlet) state in the thermodynamic limit. The electron distribution function at T=0 differs significantly from the Fermi step; it is “smeared” along the entire energy spectrum, and discontinuities do not occur in the region of the chemical potential m.
A detailed investigation has been made of the application of the Hartree-Fock approximation to the stability and thermodynamic behaviour of spin-density waves in a model with a single electronic energy band and a delta-function-type interaction between electrons. It is shown that for a half-filled band the ground state at zero temperature is always a spin-density wave, provided only that the width of the band is sufficiently small. For other bands the conditions for the relative stability of the spin-density wave, ferromagnetic and paramagnetic states are more complicated, and are studied in detail for a special band form. For a half-filled band the energy per atom and specific heat have been calculated as functions of temperature for the spin-density wave, ferromagnetic and paramagnetic states. The temperature dependence of the amplitude for the spin-density wave and ferromagnetic states, and of the entropy for the spin-density wave and paramagnetic states have also been calculated. The formulation of the problem has been made using a Green function method in order to allow the development in a later paper of a higher-order approximation to the correlation problem.
A linear soluble Fermi-gas model is used to test the validity of perturbation theory and the criteria of instability obtained by applying transformations of the Bogoliubov type. The model Hamiltonian describing the interaction of the electrons in a simple band is H(ρ)=−Σ(cosk+ρ)ck+ck+ρΣcos q ck+q+cu−q+cuck. For a half-filled band, the energies of the low-lying states are calculated exactly, by using previous results concerning a linear chain of spins with interaction. It is shown that perturbation theory must be valid for −1 < ρ < 1, and, for values of ρ belonging to this range, the slope of the specific heat for T = 0 is calculated by direct application of Landau's theory. On the other hand, anomalous Hartree-Fock states can be built for all values of ρ, by using transformations of the Bogoliubov type, and these states are more stable than the normal Hartree-Fock state. Thus, it appears that the Bogoliubov method may lead to completely erroneous interpretations, when the coupling constants are small.
We discuss the spin-triplet pairing mechanism induced by the Hund rule ferromagnetic exchange.We include explicitly the eect of interband hybridization and treat the problem by starting from an extended Hubbard model for a doubly degenerate band, making the simplest HartreeFock approximation for the part involving the pairing and the Hubbard interaction.The conditions of stability of various phases are determined as a function of both band lling and microscopic parameters.The phase diagram contains regions of stability of the spin-triplet superconducting phase coexisting with either saturated or non-saturated ferromagnetism.Phase diagrams for the cases of constant density of states and that of square lattice have been provided.The inuence of hybridization on the stability of considered phases, as well as the temperature dependences of magnetic moment and the superconducting gap are also discussed.
We derive the random-phase approximation for spin excitations in general multi-band Hubbard models, starting from a collinear ferromagnetic Hartree-Fock ground state. The results are compared with those from a recently introduced variational many-body approach to spin waves in itinerant ferromagnets. As we exemplify for Hubbard models with one and two bands, the two approaches lead to qualitatively different results. The discrepancies can be traced back to the fact that the Hartree-Fock theory fails to describe properly the local moments which naturally arise in a correlated-electron theory.