Metallic ferromagnetism supported by a single band in a multi-band Hubbard model

We construct a multi-band Hubbard model on the lattice obtained by "decorating" a closely packed $d$-dimensional lattice $\mathcal{M}$ (such as the triangular lattice) where $d\ge2$. We take the limits in which the Coulomb interaction and the band gap become infinitely large. Then there remains only a single band with finite energy, on which electrons are supported. Let the electron number be $N_\mathrm{e}=|\mathcal{M}|-N_\mathrm{h}$, where $|\mathcal{M}|$ corresponds to the electron number which makes the lowest (finite energy) band half-filled, and $N_\mathrm{h}$ is the number of "holes". It is expected that the model exhibits metallic ferromagnetism if $N_\mathrm{h}/|\mathcal{M}|$ is nonvanishing but sufficiently small. We prove that the ground states exhibit saturated ferromagnetism if $N_\mathrm{h}\le(\text{const.})|\mathcal{M}|^{2/(d+2)}$, and exhibit (not necessarily saturated) ferromagnetism if $N_\mathrm{h}\le(\mathrm{const.})|\mathcal{M}|^{(d+1)/(d+2)}$. This may be regarded as a rigorous example of metallic ferromagnetism provided that the system size $|\mathcal{M}|$ is not too large.