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arXiv:cond-mat/9904258 · Physical Review Letters

Ferromagnetism in Single-Band Hubbard Models with a Partially Flat Band

Andreas Mielke

Published May 24, 1999• 75 citations
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Abstract

A Hubbard model with a single, partially flat band has ferromagnetic ground states. It is shown that local stability of ferromagnetism implies its global stability in such a model: The model has only ferromagnetic ground states if there are no single spin-flip ground states. Since a single-band Hubbard model away from half filling describes a metal, this result may open a route to metallic ferromagnetism in single-band Hubbard models.

DOI
10.1103/physrevlett.82.4312

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A Hubbard model with a single, partially flat band has ferromagnetic ground states. It is shown that local stability of ferromagnetism implies its global stability in such a model: The model has only ferromagnetic ground states if there are no single spin-flip ground states. Since a single-band Hubbard model away from half filling describes a metal, this result may open a route to metallic ferromagnetism in single-band Hubbard models.
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.
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.
A Hubbard model with a N d -fold degenerate single-particle ground state has ferromagnetic ground states if the number of electrons is less than or equal to N d . It is shown rigorously that the local stability of ferromagnetism in such a model implies global stability: the model has only ferromagnetic ground states, if there are no single spin-flip ground states. If the number of electrons is equal to N d , it is well known that the ferromagnetic ground state is unique if and only if the single-particle density matrix is irreducible. We present a simplified proof for this result.
Accepted for publication in J. Phys. A A Hubbard model with a Nd-fold degenerate single-particle ground state has ferromagnetic ground states if the number of electrons is less or equal to Nd. It is shown rigorously that the local stability of ferromagnetism in such a model implies global stability: The model has only ferromagnetic ground states, if there are no single spin-flip ground states. If the number of electrons is equal to Nd, it is well known that the ferromagnetic ground state is unique if and only if the single-particle density matrix is irreducible. We present a simplified proof for this result. PACS-numbers: 75.10.Lp, 71.10.Fd 1
It is well known that the Hubbard model on a line graph has a flat band and ferromagnetic ground states in a certain density range. We show that for a Hubbard model on a line graph of a planar bipartite graph the ferromagnetic ground state is stable if one adds a special contribution to the kinetic energy which lifts the degeneracy of the lowest single particle state. Stability holds for sufficiently strong repulsion U. The model has extended single particle eigenstates, no degeneracy, and no band gap. It is therefore a good candidate for metallic ferromagnetism.
We examined a one-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping with up to 32 sites by means of the density-matrix renormalization-group method. The model has two energy bands and includes the flat-band model studied by Tasaki as a special case. The model with finite band dispersion is found to show metallic (saturated) ferromagnetism for a sufficiently strong interaction. No partially polarized ground state is found. The interaction necessary to realize a ferromagnetic ground state becomes weak when the electron density is decreased from the quarter-filling.