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International Journal of Modern Physics B

A THEORY OF FERROMAGNETISM IN THE HUBBARD MODEL WITH INFINITE COULOMB INTERACTION

Yu. A. Izyumov, B. M. Letfulov, E. V. Shipitsyn, K. A. Chao

Published Nov 10, 1992• 18 citations
Publisher

Abstract

On the basis of a regular perturbation theory in the form of diagrammatic technique with X-operators, a ferromagnetic state in the Hubbard model with infinite on-site Coulomb interaction U is studied for a wide range of electron concentration n. Treating the kinetic energy of electrons as perturbation, we get the exact graphic representations for three Green’s functions [Formula: see text] and [Formula: see text] for electrons with spin ↑ and ↓ (spontaneous magnetic moment is in ↑ direction) and [Formula: see text] for spin waves. All of them are expressed through the unique system of four-point and three-point vertex parts for effective electron-magnon and electron-electron interactions. These vertex parts are calculated in two approximations: a low-density approximation for n≪1 or nh=1–n≫1, and the generalized random-phase approximation (GRPA), which was suggested earlier by us for the description of paramagnetic phase in this model. An important result in both cases is the understanding of essential difference of electron states with spin ↑ and ↓. The state for ↑ has coherent character in all region of electron concentration (n>nc) where a ferromagnetic state exists, while the state for spin ↓ has mixed characters including both coherent and incoherent contribution. For the saturated ferromagnetism (n>ns) when nh≪1, [Formula: see text] is an entirely incoherent (nonquasiparticle) state. Appearance of incoherent state is probably a general property of strongly correlated systems distinguishing its behavior from the Fermi liquid. We show also that the Hubbard-I approximation has no region of applicability for the electron with spin ↓ in a ferromagnet and we found a new factor describing the correlation-narrowing of ↓ spin electron band. Green’s function [Formula: see text] calculated in two different limits n≪1 and nh≪1, are jointed together to allow us to calculate ns which lies in the intermediate concentration regime. For simple cubic lattice we found ns=0.87. In the limit nh≪1 our results reduce to the earlier known ones, including Nagaoka’s result for the spin wave energy. We point out the way to construct a rigorous theory for ferromagnetic state in the intermediate range of electron concentration. In the conclusion several important problems are discussed in connection to the continuation of the present work.

DOI
10.1142/s0217979292001535

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On the basis of a regular perturbation theory in the form of diagrammatic technique with X-operators, a ferromagnetic state in the Hubbard model with infinite on-site Coulomb interaction U is studied for a wide range of electron concentration n. Treating the kinetic energy of electrons as perturbation, we get the exact graphic representations for three Green’s functions [Formula: see text] and [Formula: see text] for electrons with spin ↑ and ↓ (spontaneous magnetic moment is in ↑ direction) and [Formula: see text] for spin waves. All of them are expressed through the unique system of four-point and three-point vertex parts for effective electron-magnon and electron-electron interactions. These vertex parts are calculated in two approximations: a low-density approximation for n≪1 or nh=1–n≫1, and the generalized random-phase approximation (GRPA), which was suggested earlier by us for the description of paramagnetic phase in this model. An important result in both cases is the understanding of essential difference of electron states with spin ↑ and ↓. The state for ↑ has coherent character in all region of electron concentration (n>nc) where a ferromagnetic state exists, while the state for spin ↓ has mixed characters including both coherent and incoherent contribution. For the saturated ferromagnetism (n>ns) when nh≪1, [Formula: see text] is an entirely incoherent (nonquasiparticle) state. Appearance of incoherent state is probably a general property of strongly correlated systems distinguishing its behavior from the Fermi liquid. We show also that the Hubbard-I approximation has no region of applicability for the electron with spin ↓ in a ferromagnet and we found a new factor describing the correlation-narrowing of ↓ spin electron band. Green’s function [Formula: see text] calculated in two different limits n≪1 and nh≪1, are jointed together to allow us to calculate ns which lies in the intermediate concentration regime. For simple cubic lattice we found ns=0.87. In the limit nh≪1 our results reduce to the earlier known ones, including Nagaoka’s result for the spin wave energy. We point out the way to construct a rigorous theory for ferromagnetic state in the intermediate range of electron concentration. In the conclusion several important problems are discussed in connection to the continuation of the present work.
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.
We discuss, in connection with the problem of the ground state in the Hubbard model with U =∞, the normal (nonmagnetic) N -state of a system over the entire range of electron concentrations n ≤1. It is found that in a one-particle approximation, e.g., in the generalized Hartree-Fock approximation, the energy ε 0 ( n ) of the N -state is lower than the energy ε FM ( n ) of a saturated ferromagnetic state for all values of n . Using the random phase approximation we calculate the dynamical magnetic susceptibility and show that the N -state is stable for all values of n . A formally exact representation is derived for the mass operator of the one-particle electron Green’s function, and its expression in the self-consistent Born approximation is obtained. We discuss the first Born approximation and show that when correlations are taken into account, the attenuation vanishes on the Fermi surface and the electron distribution function at T =0 acquires a Migdal discontinuity, whose magnitude depends on n . The energy of the N -state in this approximation is still lower than ε FM ( n ) for n <1. We show that the spin correlation functions are isotropic, which is a characteristic feature of the singlet states of the system. We calculate the spin correlation function for the nearest neighbors in the zeroth approximation as a function of n . Finally, we conclude that the singlet state of the system in the thermodynamic limit is the ground state.
A Hubbard model with infinite Coulomb repulsion is studied in a many-electron operator representation. A picture of the density of states is constructed using expressions for the one-particle Green’s functions of first order in 1/ z . Its behavior is studied near the Fermi level, especially Condon effects. The stability of saturated ferromagnetism is examined. The corresponding critical current carrier densities are found for semielliptical and rectangular seed densities of state, and for square and cubic lattices. These results are compared with earlier work.
The Hubbard model in the limit of strong Coulomb interaction (the (t-J) model) is studied by the diagram technique for Hubbard operators. The generalized random-phase approximation (GRPA) is formulated as an approximation taking into account electron loop-type diagrams. Within the framework of this approximation, both the dynamic magnetic and dielectric susceptibilities are calculated for a wide interval of electron concentrations, 0<n<1. The magnetic susceptibility shows a crossover in the magnetic behaviour of the system from pure itinerant magnetism to magnetism with localized magnetic moments. The crossover occurs at a critical point, nc, when the chemical potential for electrons of the lower Hubbard band changes sign. Magnetic phase diagrams are constructed on the (t/U, n) plane at T=0 for different types of crystal lattices. The behaviour of magnetic phases with temperature is also studied. The GRPA leads to a too drastic crossover, especially at low temperatures, because of insufficient allowance for the charge and spin fluctuations on a site. Summation of special diagram series shows that the Gaussian fluctuations of the effective electric and magnetic fields lead to a smoother picture of the crossover.
A strongly correlated electronic system is studied within the framework of the t-J model with the help of diagram technique for the Hub- bard X-operators.The summing of some series of diagrams is suggested, corresponding to the following three types of approximations: the generalized random phase approximation, the mean field approximation, and the low density approximation.In generalized random phase approximation the dynamic magnetic susceptibility of paramagnetic phase is calculated, and boundaries of stability of this phase are found on the plane of Hamiltonian parameters and electron concentration n.It is shown that at some critical concentration nc the system undergoes a crossover from the itinerant magnetism (n < n c ) to the magnetism with localized magnetic moments (n > n c ). Simultaneously, the system transfers from the Fermi-liquid beh a v i o u r t o t h e r e g i m e o f s t r o n g e l e c t r o n c o r r e l a t i o n s w i t h n o n q u a s i p a r t i c l e states.In mean field approximation equations for the order parameters were derived and phase transitions temperatures in a ferromagnetic and antiferromagnetic phases were obtained.In low density approximation a state of the saturated ferromagnetism was investigated and a critical concentration n $ was calculated when the nonsaturated ferromagnetism follows the saturated ferromagnetism.
It is believed that strong ferromagnetic interactions in some solids are generated by subtle interplay between quantum many-body effects and spin-independent Coulomb interactions between electrons. It is a challenging problem to verify this scenario in the Hubbard model, which is an idealized model for strongly interacting electrons in a solid. Nagaoka's ferromagnetism is a well-known rigorous example of ferromagnetism in the Hubbard model. It deals with the limiting situation in which there is one fewer electron than in the half-filling and the on-site Coulomb interaction is infinitely large. There are relatively new rigorous examples of ferromagnetism in Hubbard models called flat-band ferromagnetism. Flat-band ferromagnetism takes place in carefully prepared models in which the lowest bands (in the single-electron spectra) are “flat.” Usually, these two approaches are regarded as two complimentary routes to ferromagnetism in the Hubbard model. In the present paper we describe Nagaoka's ferromagnetism and flat-band ferromagnetism in detail, giving all the necessary background as well as complete (but elementary) mathematical proofs. By studying an intermediate model called the long-range hopping model, we also demonstrate that there is indeed a deep relation between these two seemingly different approaches to ferromagnetism. We further discuss some attempts to go beyond these approaches. We briefly discuss recent rigorous example of ferromagnetism in the Hubbard model which has neither infinitely large parameters nor completely flat bands. We give preliminary discussion regarding possible experimental realizations of the (nearly-)flat-band ferromagnetism. Finally, we focus on some theoretical attempts to understand metallic ferromagnetism. We discuss three artificial one-dimensional models in which the existence of metallic ferromagnetism can be easily proved. We have tried to make the present paper as self-contained as possible, keeping in mind readers who are new to the field. Although the present paper is written as a review, it contains some material which appears for the first time.