Ferromagnetism in correlated electron systems: Generalization of Nagaoka's theorem

Nagaoka's theorem on ferromagnetism in the Hubbard model with one electron fewer than half filling is generalized to the case where all possible nearest-neighbor Coulomb interactions (the density-density interaction $V$, bond-charge interaction $X$, exchange interaction $F$, and hopping of double occupancies $F^{\prime }$) are included. It is shown that for ferromagnetic exchange coupling ($F>\mathrm{}0\mathrm{}$) ground states with maximum spin are stable already at finite Hubbard interaction $U>\mathrm{}{U}_c$. For nonbipartite lattices this requires a hopping amplitude $t<~0$. For vanishing $F$ one obtains $U_c\to \infty$ as in Nagaoka's theorem. This shows that the exchange interaction $F$ is important for stabilizing ferromagnetism at finite $U$. Only in the special case $X=t$ is the ferromagnetic state stable even for $F=0\mathrm{}$, provided the lattice allows the hole to move around loops.