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Ferromagnetism in Hubbard Models

Published Jan 1, 1999• 0 citations
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Abstract

We present the first rigorous examples of non-singular Hubbard models which exhibit ferromagnetism at zero temperature.The models are defined in arbitrary dimensions, and are characterized by finite-ranged hoppings, dispersive bands, and finite on-site Coulomb interaction U .The picture, which goes back to Heisenberg, that sufficiently large Coulomb interaction can revert Pauli paramagnetism into ferromagnetism has finally been confirmed in concrete examples.

DOI
10.1142/9789812386274_0008

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We present the first rigorous examples of non-singular Hubbard models which exhibit ferromagnetism at zero temperature.The models are defined in arbitrary dimensions, and are characterized by finite-ranged hoppings, dispersive bands, and finite on-site Coulomb interaction U .The picture, which goes back to Heisenberg, that sufficiently large Coulomb interaction can revert Pauli paramagnetism into ferromagnetism has finally been confirmed in concrete examples.
We present the first rigorous examples of nonsingular Hubbard models which exhibit ferromagnetism at zero temperature. The models are defined in arbitrary dimensions, and are characterized by finite-range hoppings, dispersive bands, and finite on-site Coulomb interaction UUU. The picture, which goes back to Heisenberg, that sufficiently large Coulomb interaction can revert Pauli paramagnetism into ferromagnetism has been confirmed in concrete examples.
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.
Abstract We present rigorous results for the SU(n) Fermi–Hubbard models with finite-range hopping in d ( ≥2\ge 2≥2 ≥ 2 ) dimensions. The models are defined on a class of decorated lattices. We first study the models with flat bands at the bottom of the single-particle spectrum and prove that the ground states exhibit SU(n) ferromagnetism when the number of particles is equal to the number of unit cells. We then perturb the models by adding particular hopping terms and make the bottom bands dispersive. Under the same filling condition, it is proved that the ground states remain SU(n) ferromagnetic when the bottom bands are sufficiently flat and the Coulomb repulsion is sufficiently large.
Recently certain Hubbard models with flat lowest bands were proved to exhibit ferromagnetism. Here we study perturbed models with nearly flat bands. We prove that the ferromagnetic state is stable against a single-spin flip for sufficiently large Coulomb interaction UUU, but is unstable for small U>0U>0U>0. This is the first time that the (local) stability of ferromagnetism is proved in nonsingular Hubbard models, in which we must overcome competition between the kinetic energy and the Coulomb interaction.