This is a self-contained review about ferromagnetism in the Hubbard model,
which should be accessible to readers with various backgrounds who are new to
the field.
We describe Nagaoka's ferromagnetism and flat-band ferromagnetism in detail,
giving all necessary backgrounds 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.Comment: LaTeX2e, 72 pages, 17 epsf figures. Many minor corrections made in
March 1998. This is the final version, which will appear in Prog. Theor.
Phys. 99 (invited paper
