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

TL;DR

This review introduces ferromagnetism in the Hubbard model, covering Nagaoka's and flat-band ferromagnetism, their relation, recent advances, and potential experimental realizations, aimed at readers new to the field.

Contribution

It provides a comprehensive, accessible overview with complete proofs of key ferromagnetism mechanisms and explores recent developments beyond traditional models.

Findings

Deep relation between Nagaoka's and flat-band ferromagnetism

Existence of ferromagnetism without infinite parameters or flat bands

Theoretical models demonstrating metallic ferromagnetism

Abstract

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.