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arXiv:2009.03580 · Journal of Statistical Physics

Ferromagnetism in d-Dimensional SU(n) Hubbard Models with Nearly Flat Bands

Kensuke Tamura, Hosho Katsura

Published Jan 1, 2021• 20 citations
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Abstract

Abstract We present rigorous results for the SU(n) Fermi–Hubbard models with finite-range hopping in d ( $$\ge 2$$ ≥ 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.

DOI
10.1007/s10955-020-02687-w

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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.
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.
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.
Whether spin-independent Coulomb interaction can be the origin of a realistic ferromagnetism in an itinerant electron system has been an open problem for a long time. Here we study a class of Hubbard models on decorated lattices, which have a special property that the corresponding single-electron Schrödinger equation has NdN_{\rm d}Nd​-fold degenerate ground states. The degeneracy NdN_{\rm d}Nd​ is proportional to the total number of sites $\absΛ$. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to $ρ_0=N_{\rm d}/(2\absΛ)$, and paramagnetism when the filling factor is sufficiently small. An important feature of the present work is that it provides examples of three dimensional itinerant electron systems which are proved to exhibit ferromagnetism in a finite range of the electron filling factor.
Recently certain Hubbard models with flat lowest bands were proved to exhibit ferromagnetism. Here we study perturbed models with nearly flat bands. We prove that the ferromagnetic state is stable against a single-spin flip for sufficiently large Coulomb interaction UUU, but is unstable for small U>0U>0U>0. This is the first time that the (local) stability of ferromagnetism is proved in nonsingular Hubbard models, in which we must overcome competition between the kinetic energy and the Coulomb interaction.