From Nagaoka's Ferromagnetism to Flat-Band Ferromagnetism and Beyond: An Introduction to Ferromagnetism in the Hubbard Model

Abstract

It is believed that strong ferromagnetic interactions in some solids are generated by subtle interplay between quantum many-body effects and spin-independent Coulomb interactions between electrons. It is a challenging problem to verify this scenario in the Hubbard model, which is an idealized model for strongly interacting electrons in a solid.Nagaoka's ferromagnetism is a well-known rigorous example of ferromagnetism in the Hubbard model. It deals with the limiting situation in which there is one fewer electron than in the half-filling and the on-site Coulomb interaction is infinitely large. There are relatively new rigorous examples of ferromagnetism in Hubbard models called flat-band ferromagnetism. Flat-band ferromagnetism takes place in carefully prepared models in which the lowest bands (in the single-electron spectra) are "flat." Usually, these two approaches are regarded as two complimentary routes to ferromagnetism in the Hubbard model.In the present paper 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 the longrange 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 discussion regarding 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.We have tried to make the present paper as self-contained as possible, keeping in mind readers who are new to the field. Although the present paper is written as a review, it contains some material which appears for the first time.7 Possible experimental realizations of (nearly-)flat-band ferromagnetism 7.1 Ferromagnetism in La 4 Ba 2 Cu 2 O 10 .

Exotic electronic states in the world of flat bands: From theory to material

“…We limit our discussions to the most well-studied 2D FBs, and purposely avoid invoking abstract mathematical derivations in this article. With the intuitive pictures provided here, we encourage the readers to work on more specialized literature for the rigorous proof [24][25][26]. Also, we primarily focus on realizing electronic FBs in solid-state materials using the charge degree of freedom.…”

Geometric Origin of Superfluidity in the Lieb-Lattice Flat Band

“…Bands that are nearly flat and/or feature a nontrivial topological invariant, similar to Landau levels producing the quantum Hall effects [2][3][4], have been considered in recent theoretical works [5][6][7][8][9][10][11][12] and can be realized in ultracold gas experiments [13][14][15]. Flat-band ferromagnetism has been studied first by Lieb [16] and, subsequently, by Tasaki and Mielke [17][18][19][20]. More recently it has been shown that the high density of states of flat bands enhances the superconducting critical temperature [21,22].…”

“…Flat bands, or quasiflat bands, can be realized in bipartite lattices [16] and other models [6][7][8]20,24]. A simple bipartite lattice featuring a strictly flat band is the Lieb lattice [ Fig.…”

Ground state spin and excitation energies in half-filled Lieb lattices

“…[1,[24][25][26]), which is located at the middle of the spectrum. This flat band is one from the total of three bands of the Lieb lattice, consistent with the three atoms unit cell.…”