Ferromagnetic ordering in Hubbard models

Abstract

One of the long-standing and only partially solved problems of theoretical condensed matter physics and mathematical physics is to demonstrate that ground states of some of the versions of the Hubbard model can exhibit a ferromagnetic ordering. It has long been speculated that the opportunity crucial for the occurrence of ferromagnetism is the structure of the lattice on which the Hubbard model is formulated \cite{TasakiMB}. As a consequence, while on simple cubic lattices no ferromagnetic ordering seems to be possible, it can naturally arise, even for low densities of magnetic moment carriers, on so-called frustrated lattices. We investigate the problem of ground state ferromagnetic ordering with the use of the formula for ground-state energy of interacting fermions as the first term of `density expansion', proven rigorously by Lieb, Seiringer and Solovej \cite{fermi exact} in continuum and by Giuliani \cite{hub exact} for the simple cubic lattice. Assuming that analogous expansion holds also for certain another lattices we apply this formula to five frustrated lattices -- among them to the face-centered cubic one. The hypothesis is confirmed: most of examined models formulated on frustrated lattices do indeed have ferromagnetic ground states already for densities being moderate or even low. Although the approach adopted cannot be treated as a rigorous proof that the ground state is ferromagnetic, the results obtained here strongly indicate that it can be the case. Moreover, as in some cases FM occurs at low densities, one can hope that it would be possible to prove convergence of the density expansion and prove rigorously the occurrence of `wealthy ferromagnetism' in these cases.