Ferromagnetism in the Hubbard models with degenerate single-electron ground states

We study a class of Hubbard models in which the corresponding single-electron ground states have bulk degeneracy. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to ${\mathrm{\ensuremath{\rho}}}_{0}$=\ensuremath{\Vert}V\ensuremath{\Vert}/2\ensuremath{\Vert}\ensuremath{\Lambda}\ensuremath{\Vert} (where \ensuremath{\Vert}V\ensuremath{\Vert} is the dimension of degeneracy and \ensuremath{\Vert}\ensuremath{\Lambda}\ensuremath{\Vert} is the number of sites), and exhibit paramagnetism when the filling factor is sufficiently small. This is the first time that a three-dimensional itinerant-electron system is proved to exhibit ferromagnetism in a finite range of the electron filling factor.