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arXiv:2303.15820 · Journal of Physics A Mathematical and Theoretical

Flat-band ferromagnetism in the SU(N) Hubbard and Kondo lattice models

Kensuke Tamura, Hosho Katsura

Published Aug 17, 2023• 3 citations
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Abstract

Abstract We develop a general theory of flat-band ferromagnetism in the SU(N) Fermi–Hubbard model, which describes the behavior of N-component fermions with SU(N) symmetric interactions. We focus on the case where the single-particle spectrum has a flat band at the bottom and establish a necessary and sufficient condition for the SU(N) Hubbard model to exhibit ferromagnetism when the number of particles is the same as the degeneracy. We show that the occurrence of ferromagnetism is equivalent to the irreducibility of the projection matrix onto the space of single-particle ground states. We also demonstrate that this result can be exploited to establish a rigorous result for the ferromagnetic SU(N) Kondo lattice model with a flat band. Specifically, we prove that when the SU(N) Hubbard model is ferromagnetic, the ferromagnetic SU(N) Kondo lattice model with the same hopping matrix also exhibits SU(N) ferromagnetism.

DOI
10.1088/1751-8121/acf100

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Abstract We develop a general theory of flat-band ferromagnetism in the SU(N) Fermi–Hubbard model, which describes the behavior of N-component fermions with SU(N) symmetric interactions. We focus on the case where the single-particle spectrum has a flat band at the bottom and establish a necessary and sufficient condition for the SU(N) Hubbard model to exhibit ferromagnetism when the number of particles is the same as the degeneracy. We show that the occurrence of ferromagnetism is equivalent to the irreducibility of the projection matrix onto the space of single-particle ground states. We also demonstrate that this result can be exploited to establish a rigorous result for the ferromagnetic SU(N) Kondo lattice model with a flat band. Specifically, we prove that when the SU(N) Hubbard model is ferromagnetic, the ferromagnetic SU(N) Kondo lattice model with the same hopping matrix also exhibits SU(N) ferromagnetism.
Abstract We present rigorous results for the SU(n) Fermi–Hubbard models with finite-range hopping in d ( ≥2\ge 2≥2 ≥ 2 ) dimensions. The models are defined on a class of decorated lattices. We first study the models with flat bands at the bottom of the single-particle spectrum and prove that the ground states exhibit SU(n) ferromagnetism when the number of particles is equal to the number of unit cells. We then perturb the models by adding particular hopping terms and make the bottom bands dispersive. Under the same filling condition, it is proved that the ground states remain SU(n) ferromagnetic when the bottom bands are sufficiently flat and the Coulomb repulsion is sufficiently large.
We present rigorous results for the SU(n)\mathrm{SU}(n)SU(n) Fermi-Hubbard model on the railroad-trestle lattice. We first study the model with a flat band at the bottom of the single-particle spectrum and prove that the ground states exhibit SU(n)\mathrm{SU}(n)SU(n) ferromagnetism when the total fermion number is the same as the number of unit cells. We then perturb the model by adding extra hopping terms and make the flat band dispersive. Under the same filling condition, it is proved that the ground states of the perturbed model remain SU(n)\mathrm{SU}(n)SU(n) ferromagnetic when the bottom band is nearly flat. This is the first rigorous example of the ferromagnetism in nonsingular SU(n)\mathrm{SU}(n)SU(n) Hubbard models in which both the single-particle density of states and the on-site repulsive interaction are finite.
We study the ground-state properties of the SU(NNN) generalization of the Kondo lattice model in one dimension when the Kondo coupling JK{J}_{\text{K}}JK​ (both ferromagnetic and antiferromagnetic) is sufficiently strong. Both cases can be realized using alkaline-earth-metal-like cold gases in optical lattices. Specifically, we first carry out the strong-coupling expansion and identify two insulating phases [one of which is the SU(NNN) analog of the well-known gapped Kondo singlet phase]. We then rigorously establish that the ground state in the low-density (for JK<0{J}_{\text{K}}<0JK​<0) or the high-density (for JK>0{J}_{\text{K}}>0JK​>0) region is ferromagnetic. The results are accounted for by generalizing the double-exchange mechanism to SU(NNN) ``spins.'' Possible realizations of Bose-Fermi supersymmetry SU(N∣1N|1N∣1) in the (generalized) SU(NNN) Kondo lattice model are discussed as well.
The Hubbard model on the kagome lattice has highly degenerate ground states (the flat lowest band) in the corresponding single-electron problem and exhibits the so-called flat-band ferromagnetism in the many-electron ground states as was found by Mielke [J. Phys. A 24, L73 (1991)]]. Here we study the model obtained by adding extra hopping terms to the above model. The lowest single-electron band becomes dispersive, and there is no band gap between the lowest band and the other band. We prove that, at half filling of the lowest band, the ground states of this perturbed model remain saturated ferromagnetic if the lowest band is nearly flat.
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 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 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.
Exact SU(NNN) symmetry resolution of bosonic Hubbard models: general boson-fermion mappings and flat-band singlet-cluster states
We show how to perform exact diagonalization of SU(NNN)-symmetric Bose-Hubbard models separately in each irreducible representation of SU(NNN), by exploiting the Howe duality between U(LLL) and U(NNN). Three structural consequences follow for any lattice and any Hamiltonian built from the U(LLL) generators: (i) the spectrum in a given orbital sector is independent of NNN and of the particle statistics, (ii) the canonical partition functions for bosons and fermions obey the exact identity Zb(−N)=(−1)NbZf(N)Z_{\mathrm{b}}(-N)=(-1)^{N_\mathrm{b}}Z_{\mathrm{f}}(N)Zb​(−N)=(−1)Nb​Zf​(N), (iii) with unfrustrated hopping the groundstate is always a color ferromagnet, so that the color-resolved method we introduce is useful to study excited sectors, frustrated lattices and finite temperatures. As applications, we first derive the finite-temperature phase diagram of attractive SU(NNN) bosons on few-site rings at large NNN for arbitrary filling. We then show the emergence of singlet-cluster states from frustrated hopping, the counterpart of fermionic ferromagnetism, with the example of the sawtooth and kagome lattice.