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arXiv:cond-mat/9606115

A note on ferromagnetism in the Hubbard model on the complete graph

Andreas Mielke, Hal Tasaki

Published Jun 17, 1996• 0 citations
arXivHTMLPDF

Abstract

Recently there have appeared some papers which discuss the existence of ferromagnetism in the Hubbard model defined on the complete graph. At least for the special electron number N_e=N-1, where N denotes the number of sites in the lattice, the existence of ferromagnetism in this model was established rigorously some time ago, as special (and indeed the simplest) cases of more general classes of models. Here we explain these implications to clarify the situation, although we believe the implications are straightforward. We are posting this note to the preprint archive to make it public, but we are not planning to publish it in other forms. This is because we do not think the problem warrants any extra publications, and we believe that the validity of our explanation is evident to the readers.

DOI
10.48550/arxiv.cond-mat/9606115

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Recently there have appeared some papers which discuss the existence of ferromagnetism in the Hubbard model defined on the complete graph. At least for the special electron number N_e=N-1, where N denotes the number of sites in the lattice, the existence of ferromagnetism in this model was established rigorously some time ago, as special (and indeed the simplest) cases of more general classes of models. Here we explain these implications to clarify the situation, although we believe the implications are straightforward. We are posting this note to the preprint archive to make it public, but we are not planning to publish it in other forms. This is because we do not think the problem warrants any extra publications, and we believe that the validity of our explanation is evident to the readers.
In this work, the ground states of the Hubbard model on complete graph are studied, for a finite lattice size LLL and arbitrary on-site energy UUU. We construct explicitly the ground states of the system when the number of the electrons Ne≥L+1N_e \ge L+1Ne​≥L+1. In particular, for Ne=L+1N_e=L+1Ne​=L+1, the ground state is ferromagnetic with total spin sg=(Ne−2)/2s_g=(N_e-2)/2sg​=(Ne​−2)/2.
Let L(G) be the line graph of a graph G=(V,E). The Hubbard model on L(G) has ferromagnetic ground states with a saturated spin if the interaction is repulsive (U>0) and if the number of electrons N satisfies N>or=M. M= mod E mod + mod V mod -1 if G is bipartite and M= mod E mod + mod V mod otherwise. The author shows that the ferromagnetic ground state is unique if N=M. Further he gives a sufficient condition for the existence of other ground states if N>M. The results are valid also for a multi-band Hubbard model on a bipartite graph. In the case of a periodic lattice, the results are related to the existence of a flat energy band.
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.
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.
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.