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We study itinerant ferromagnetism in a $t_{2g}$ multi-orbital Hubbard systemin the cubic lattice, which consists of three planar oriented orbital bands of$d_{xy}$, $d_{yz}$, and $d_{zx}$. Electrons in each orbital band can only movewithin a two-dimensional plane in the three-dimensional lattice parallel to thecorresponding orbital orientation. Electrons of different orbitals interactthrough the on-site multi-orbital interactions including Hund's coupling. Thestrong coupling limit is considered in which there are no doubly occupiedorbitals but multiple on-site occupations are allowed. We show that, in thecase in which there is one and only one hole for each orbital band in eachlayer parallel to the orbital orientation, the ground state is a fullyspin-polarized itinerant ferromagnetic state, which is unique apart from thetrivial spin degeneracy. When the lattice is reduced into a singletwo-dimensional layer, the $d_{zx}$ and $d_{yz}$ bands becomequasi-one-dimensional while the $d_{xy}$ band remains two-dimensional. Theground state ferromagnetism also appears in the strong-coupling limit as ageneralization of the double exchange mechanism. Possible applications to thesystems of SrRuO$_3$ and LaAlO$_3$/SrTiO$_3$ interface are discussed.

Exact results for itinerant ferromagnetism in a $t_{2g}$ orbital system on cubic and square lattices