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

We study itinerant ferromagnetism in a $t_{2g}$ multiorbital Hubbard system in 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 move within a two-dimensional plane in the three-dimensional lattice parallel to the corresponding orbital orientation. Electrons of different orbitals interact through the on-site multiorbital interactions including Hund's coupling. The strong-coupling limit is considered in which there are no doubly occupied orbitals but multiple on-site occupations are allowed. We show that in the case in which there is one and only one hole for each orbital band in each layer parallel to the orbital orientation, the ground state is a fully spin-polarized itinerant ferromagnetic state, which is unique apart from the trivial spin degeneracy. When the lattice is reduced into a single two-dimensional layer, the $d_{zx}$ and $d_{yz}$ bands become quasi-one-dimensional while the $d_{xy}$ band remains two-dimensional. The ground-state ferromagnetism also appears in the strong-coupling limit as a generalization of the double-exchange mechanism. Possible applications to the systems of ${\mathrm{SrRuO}}_3$ and ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface are discussed.

Figure 1

The Wannier orbital wave functions of $t_{2g}$ orbitals: $d_{xy},d_{yz}$, and $d_{zx}$. For electrons in the $d_a$ orbitals $(a=xy,yz,zx)$, they can only move along the $xy,yz$, or $zx$ plane, respectively, but not perpendicular to the orbital orientation plane.

Figure 2

Representative orbital configurations along a bond $\langle ij\rangle$ with two orbitals at sites $i$ and $j$ initially occupied by spins $\uparrow$ and $\downarrow$, respectively. Case (a): $[d_{zx,\uparrow }\left(i\right);d_{zx,\downarrow }\left(j\right)]$, case (b): $[d_{zx,\uparrow }\left(i\right);d_{yz,\downarrow }\left(j\right)]$, case (c): $[d_{zx,\uparrow }\left(i\right);d_{xy,\downarrow }\left(j\right)]$, case (d): $[d_{xy,\uparrow }\left(i\right);d_{xy,\downarrow }\left(j\right)]$. Any two cases of (a)–(d) are nonequivalent under the lattice symmetry transformation.

Figure 3

The 12 configurations with three electrons (2 spin-$\uparrow$ and 1 spin-$\downarrow$) and a single hole in a square plaquette. The hole's motion builds up transitivity among all these states. From Ref. [3].