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We present rigorous results for the SU($n$) Fermi-Hubbard models withfinite-range hopping in $d$ ($\ge 2$) dimensions. The models are defined on aclass of decorated lattices. We first study the models with flat bands at thebottom of the single-particle spectrum and prove that the ground states exhibitSU($n$) ferromagnetism when the number of particles is equal to the number ofunit cells. We then perturb the models by adding particular hopping terms andmake the bottom bands dispersive. Under the same filling condition, it isproved that the ground states remain SU($n$) ferromagnetic when the bottombands are sufficiently flat and the Coulomb repulsion is sufficiently large.

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