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

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.

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)

References in corpus (1)

• Stability of Ferromagnetism in Hubbard Models with Nearly-Flat Bands

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• Identifying Instabilities with Quantum Geometry in Flat Band Systems
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• Boundary Flat Bands with Topological Spin Textures Protected by Sub-chiral Symmetry
• Ferromagnetic Pairing States on Two-Coupled Chains
• Low-temperature thermodynamics of one class of flat-band models
• Prerelaxation in quantum, classical, and quantum-classical two-impurity models
• Collective resonances near zero energy induced by a point defect in bilayer graphene
• Dynamical $t/U$ Expansion of the Doped Hubbard Model
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• Stability of Nagaoka phase, spin effective action and delocalized free holes
• Bad Metal Behavior and Lifshitz Transition of a Nagaoka Ferromagnet
• Ferromagnetism and doublon localization in a Wannier-Hubbard chain
• Quantum Magnetism in Wannier-Obstructed Mott Insulators
• Insulator-metal transition and quasi-flat-band of Shastry-Sutherland lattice
• Thermal stability of the Nagaoka-Thouless theorems
• Interplay of mass imbalance and frustration in correlated band insulators
• Geometric Frustration Assisted Kinetic Ferromagnetism in Doped Mott Insulators
• Complete spectrum of the infinite-$U$ Hubbard ring using group theory
• Non-ergodic one-magnon magnetization dynamics of the kagome lattice antiferromagnet
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• Symmetry-Based Real-Space Framework for Realizing Flat Bands and Unveiling Nodal-Line Touchings
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• Influence of flat bands on RKKY interaction: perspective of Fano defects
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• Invisible flat bands on a topological chiral edge
• Flat bands and band touching from real-space topology in hyperbolic lattices
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• Defect-induced edge ferromagnetism and fractional spin excitations of the SU(4) $\pi$-flux Hubbard model on honeycomb lattice
• Electric-field driven flat bands in the distorted sawtooth chain via the Katsura-Nagaosa-Balatsky mechanism
• RKKY couplings in the Lieb lattice: flat-band induced frustration
• Singular flat bands in three dimensions: Landau level spreading, quantum geometry, and Weyl reconstruction