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Physical review. B, Condensed matter

Flat-band ferromagnetism in extended Δ-chain Hubbard models

M. Ichimura, Koichi Kusakabe, Shinji Watanabe, T. Onogi

Published Oct 15, 1998• 17 citations
Publisher

Abstract

We examined the magnetic property of newly extended \ensuremath{\Delta}-chain Hubbard models which may be relevant to quantum atomic wires formed on solid surfaces. We rigorously show that the extended models exhibit so-called flat-band ferromagnetism, based on the Mielke-Tasaki theorem. Moreover, we numerically find that the ferromagnetic state is stable against some deviations from completely flat band, by using the exact diagonalization method. The present result will be also useful as a guiding principle to design magnetic materials that possibly display the flat-band ferromagnetism.

DOI
10.1103/physrevb.58.9595

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We examined the magnetic property of newly extended \ensuremath{\Delta}-chain Hubbard models which may be relevant to quantum atomic wires formed on solid surfaces. We rigorously show that the extended models exhibit so-called flat-band ferromagnetism, based on the Mielke-Tasaki theorem. Moreover, we numerically find that the ferromagnetic state is stable against some deviations from completely flat band, by using the exact diagonalization method. The present result will be also useful as a guiding principle to design magnetic materials that possibly display the flat-band ferromagnetism.
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We have studied the magnetic properties of a certain class of Hubbard models proposed as models for quantum atomic wires with flat single-particle bands. These Hubbard models belong to a class of models to which Mielke-Tasaki theorem for flat-band ferromagnetism can be applied. We also found that the Mielke-Tasaki theorem is applicable in networks of quantum atomic wires. The present findings will be useful for producing flat-band ferromagnetism in quantum atomic wires and two dimensional networks.
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.
The Hubbard model on the kagome lattice has highly degenerate ground states (the flat lowest band) in the corresponding single-electron problem and exhibits the so-called flat-band ferromagnetism in the many-electron ground states as was found by Mielke [J. Phys. A 24, L73 (1991)]]. Here we study the model obtained by adding extra hopping terms to the above model. The lowest single-electron band becomes dispersive, and there is no band gap between the lowest band and the other band. We prove that, at half filling of the lowest band, the ground states of this perturbed model remain saturated ferromagnetic if the lowest band is nearly flat.
We consider the standard repulsive Hubbard model with a flat lowest-energy band for two one-dimensional lattices (diamond chain and ladder) as well as for a two-dimensional lattice (bilayer) at half filling of the flat band. The considered models do not fall in the class of Mielke-Tasaki flat-band ferromagnets, since they do not obey the connectivity conditions. However, the ground-state ferromagnetism can emerge, if the flat band becomes dispersive. To study this kinetic-energy-driven ferromagnetism we use perturbation theory and exact diagonalization of finite lattices. We find as a typical scenario that small and moderate dispersion may lead to a ferromagnetic ground state for sufficiently large on-site Hubbard repulsion U>UcU>{U}_{c}U>Uc​, where Uc{U}_{c}Uc​ increases monotonically with the acquired bandwidth. However, we also observe for some specific parameter cases, that (i) ferromagnetism appears at already very small Uc{U}_{c}Uc​, (ii) ferromagnetism does not show up at all, (iii) the critical on-site repulsion Uc{U}_{c}Uc​ is a nonmonotonic function of the bandwidth, or that (iv) a critical bandwidth is needed to open the window for ground-state ferromagnetism.
We model conducting pentagon chains with a multi-orbital Hubbard model and prove that well below half filling, exact ferromagnetic ground states appear. The rigorous method we use is based on the transformation of original hamiltonian into positive semidefinite form. This technique is independent of the spatial dimension and does not require integrability of the model. The obtained ferromagnetism is connected to dispersionless bands but in a much broader sense than flat-band ferromagnetism requires, where on every site a Hubbard term is present. In our case, only a small percentage of, even randomly distributed, sites are only interacting.