From Nagaoka's ferromagnetism to flat-band ferromagnetism and beyond: An introduction to ferromagnetism in the Hubbard model

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  • 5 March 1998
Abstract
This is a self-contained review about ferromagnetism in the Hubbard model, which should be accessible to the readers with various background who are new to the field. We describe Nagaoka's ferromagnetism and flat-band ferromagnetism in detail, giving all the necessary background as well as complete (but elementary) mathematical proofs. By studying an intermediate model called long-range hopping model, we also demonstrate that there is indeed a deep relation between these two seemingly different approaches to ferromagnetism. We further discuss some attempts to go beyond these approaches. We briefly discuss recent rigorous example of ferromagnetism in the Hubbard model which has neither infinitely large parameters nor completely flat bands. We give preliminary discussions about possible experimental realizations of the (nearly-)flat-band ferromagnetism. Finally we focus on some theoretical attempts to understand metallic ferromagnetism. We discuss three artificial one-dimensional models in which the existence of metallic ferromagnetism can be easily proved.

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