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Ferromagnetism in $d$-dimensional SU($n$) Hubbard models with nearly flat bands

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