The Origin of Ferromagnetism

Abstract

In the present chapter we focus on ferromagnetism (or, more precisely, saturated ferromagnetism) in the ground states of the Hubbard model. Recalling that both the non-interacting models and the non-hopping models exhibit paramagnetism, we see that ferromagnetism can be generated only through nontrivial “competition” between wave-like nature and particle-like nature of electrons. After discussing basic properties of saturated ferromagnetism in the Hubbard model in Sect. 11.1, we present some rigorous results which establish that the ground states of certain versions of the Hubbard models exhibit ferromagnetism. They include Nagaoka’s ferromagnetism for systems with infinitely large U (Sect. 11.2), flat-band ferromagnetism by Mielke and by Tasaki (Sect. 11.3), and ferromagnetism in nearly-flat-band Hubbard model due to Tasaki (Sect. 11.4). The final result is of particular importance since it deals with a situation where ferromagnetism is intrinsically a non-perturbative phenomenon, and above mentioned competition plays an essential role. We end the chapter by briefly discussing the fascinating but extremely difficult problem of metallic ferromagnetism in Sect. 11.5.