STABILITY OF FERROMAGNETISM IN HUBBARD MODELS WITH NEARLY FLAT BANDS

Whether spin-independent Coulomb interaction in an electron system can be the origin of ferromagnetism has been an open problem for a long t ime. Recently, a ''constructive'' approach to this problem has been de veloped, and the existence of ferromagnetism in the ground states of c ertain Hubbard models was established rigorously. A special feature of these Hubbard models is that their lowest bands (in the corresponding single-electron problems) are completely flat. Here we study models o btained by adding small but arbitrary translation-invariant perturbati on to the hopping Hamiltonian of these Bat-band models. The resulting models have nearly flat lowest bands. We prove that the ferromagnetic state is stable against a single-spin flip provided that Coulomb inter action U is sufficiently large. (It is easily found that the same stat e is unstable against a single-spin flip if U is small enough.) We als o prove upper and lower bounds for the dispersion relation of the lowe st energy eigenstate with a single flipped spin, which bounds establis h that the model has ''healthy'' spin-wave excitation. It is notable t hat the (local) stability of ferromagnetism is proved in nonsingular H ubbard models, in which we must overcome competi tion between the kine tic energy and the Coulomb interaction. We also note that this is one of the very few rigorous and robust results which deal with truly nonp erturbative phenomena in many-electron systems. The local stability st rongly suggests that the Hubbard models with nearly fat bands have fer romagnetic ground states. We believe that the present models can be st udied as paradigm models for (insulating) ferromagnetism in itinerant electron systems.