Mixing of $t_{2g}$-$e_g$ orbitals in 4d and 5d transition metal oxides

Abstract

Using exact diagonalization, we study the spin-orbit coupling and interaction-induced mixing between $t_{2g}$ and $e_g$ $d$-orbital states in a cubic crystalline environment, as commonly occurs in transition metal oxides. We make a direct comparison with the widely used $t_{2g}$ only or $e_g$ only model, depending on electronic filling. We consider all electron fillings of the $d$-shell and compute the total magnetic moment, the spin, the occupancy of each orbital, and the effective spin-orbit coupling strength (renormalized through interaction effects) in terms of the bare interaction parameters, spin-orbit coupling, and crystal field splitting, focusing on the parameter ranges relevant to 4d and 5d transition metal oxides. In various limits we provide perturbative results consistent with our numerical calculations. We find that the $t_{2g}$-$e_g$ mixing can be large, with up to 20\% occupation of orbitals that are nominally "empty", which has experimental implications for the interpretation of the branching ratio in experiments, and can impact the effective local moment Hamiltonian used to study magnetic phases and magnetic excitations in transition metal oxides. Our results can aid the theoretical interpretation of experiments on these materials, which often fall in a regime of intermediate coupling with respect to electron-electron interactions.