Ferromagnetism in single-band Hubbard models with a partially flat band

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. PACS-numbers: 75.10.Lp, 71.10.Fd Ferromagnetism of itinerant electrons is an old and still unsolved problem in theoretical physics. One of the motivations to introduce the Hubbard model [1] more than 30 years ago has been to understand this problem [2]. Conceptually, the Hubbard model is very simple. It describes electrons on a lattice interacting via a repulsive, purely local Coulomb interaction. Originally, only a single energy band was considered. But it is relatively easy to generalize the model including more energy bands or further interactions. Despite the simplicity of the model, there are only few rigorous results concerning the existence of ferromagnetism. The first rigorous result is the so called Nagaoka theorem [3]. It states that on certain lattices, for an infinitely large Coulomb repulsion, and close to half filling, the ground state is ferromagnetic. But the problem is that this ferromagnetic state disappears if the repulsion has realistic values or if the number of electrons is varied [4, 5], at least for usual cubic lattices in d dimensions.[6] Ferromagnetic ground states also occur, if one of several bands of the Hubbard model is dispersionless (Lieb’s ferrimagnetism [7] and the flat-band ferromagnetism [8]). Tasaki [9] was able to show the local stability of ferromagnetic ground states in related models with a nearly flat band. But these models (as well as the flat-band ferromagnetism) are already extensions of the single band Hubbard model since they contain more than one band. Furthermore, these models show characteristic features of an insulator: The (nearly) flat band is half filled and there is a gap in the single-particle spectrum. Other extensions of the Hubbard model have been proposed as well, for instance an additional ferromagnetic interaction between the electrons [10] or degenerate bands with Hund’s coupling between the bands [11]. The question, if and under which conditions a simple, single band Hubbard model can show ferromagnetism, is still open. Ferromagnetism is a strong coupling phenomenon. To obtain ferromagnetism ∗E–mail: [email protected]