Ferromagnetism in Hubbard models.

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. Introductions: The origin of ferromagnetism has been a mystery in physical science for quite a long time [1]. It was Heisenberg [2] who first realized that ferromagnetism is intrinsically a quantum many-body effect, and proposed a scenario that spin-independent Coulomb interaction and the Pauli exclusion principle generate “exchange interaction” between electronic spins. One of the motivations to study the so-called Hubbard model has been to establish and understand the generation of ferromagnetism in simplified situations [3, 4]. Unfortunately, rigorous examples of ferromagnetism (or ferrimagnetism) in the Hubbard models have been limited to singular models which have infinitely large Coulomb interaction (Nagaoka-Thouless ferromagnetism [5]), or in which magnetization is supported by a dispersionless band (Lieb’s ferrimagnetism [6], and flat-band ferromagnetism due to Mielke [7] and the present author [8]). In [9, 10], local stability of ferromagnetism in a generic family of Hubbard models with nearly-flat bands was proved. In the present Letter, we treat a class of Hubbard models in arbitrary dimensions, which are non-singular in the sense that they have finite ranged hoppings, dispersive (singleelectron) bands, and finite Coulomb interaction U . We prove that the models exhibit ferromagnetism in their ground states provided that U is sufficiently large. We recall that Hubbard models with dispersive bands (like ours) exhibit Pauli paramagnetism when U = 0, and remain non-ferromagnetic for sufficiently small U . The appearance of ferromagnetism is a purely non-perturbative phenomenon. As far as we know, this is the first time that the existence of ferromagnetism is established in non-singular itinerant electron systems. We stress that our examples finally provide the definite affirmative answer to the long standing fundamental problem; whether spinindependent Coulomb interaction can be the origin of ferromagnetism in itinerant electron systems [11]. See [8, 10, 12] for further discussions on ferromagnetism in the Hubbard models. 1 Phys. Rev. Lett. 75, 4678–4681, 1995. Archived as cond-mat/9509063.