Stability of the Nagaoka-type Ferromagnetic State in a $t_{2g}$ Orbital System on a Cubic Lattice

TL;DR

This paper extends the understanding of Nagaoka-type ferromagnetic states to multi-hole, three-dimensional $t_{2g}$-orbital systems on cubic lattices, proving their stability under certain conditions in the strong coupling limit.

Contribution

It generalizes previous results to multiple holes and demonstrates the degeneracy and stability of Nagaoka ferromagnetic states in a $t_{2g}$-orbital system with strong coupling.

Findings

Degeneracy of ferromagnetic states with the ground state in the thermodynamic limit.

Stability of the Nagaoka-type state at finite electron densities against single spin-flip.

Valid for arbitrary ferromagnetic Hund's coupling and inter-orbital repulsion.

Abstract

We generalize the previous exact results of the Nagaoka-type itinerant ferromagnetic states in a three dimensional $t_{2g}$-orbital system to allow for multiple holes. The system is a simple cubic lattice with each site possessing $d_{xy}$, $d_{yz}$, and $d_{xz}$ orbitals, which allow two-dimensional hopping within each orbital plane. In the strong coupling limit of $U\to \infty$, the orbital-generalized Nagaoka ferromagnetic states are proved degenerate with the ground state in the thermodynamic limit when the hole number per orbital layer scales slower than $L^{\frac{1}{2}}$. This result is valid for arbitrary values of the ferromagnetic Hund's coupling $J>0$ and inter-orbital repulsion $V\ge 0$. The stability of the Nagaoka-type state at finite electron densities with respect to a single spin-flip is investigated. These results provide helpful guidance for studying the mechanism of itinerant ferromagnetism for the $t_{2g}$-orbital materials.