Orbital ordering in the layered perovskite material CsVF$_4$

In strongly correlated electronic systems, several novel physical properties are induced by the orbital degree of freedom. In particular, orbital degeneracy near the Fermi level leads to spontaneous symmetry breaking, such as the nematic state in FeSe and the orbital ordering in several perovskite systems. Here, the novel layered perovskite material CsVF$_4$, with a $3d^2$ electronic configuration, was systematically studied using density functional theory and a multiorbital Hubbard model within the Hatree-Fock approximation. Our results show that CsVF$_4$ should be magnetic, with a G-type antiferromagnetic arrangement in the $ab$ plane and weak antiferromagnetic exchange along the $c$-axis, in agreement with experimental results. Driven by the Jahn-Teller distortion in the VF$_6$ octahedra that shorten the $c$-axis, the system displays an interesting electron occupancy $d_{xy}^1(d_{xz}d_{yz})^1$ corresponding to the lower nondegenerate $d_{xy}$ orbital being half-filled and the other two degenerate $d_{yz}$ and $d_{xz}$ orbitals sharing one electron per site. We show that this degeneracy is broken and a novel $d_{yz}$/$d_{xz}$ staggered orbital pattern is here predicted by both the first-principles and Hubbard model calculations. This orbital ordering is driven by the electronic instability associated with degeneracy removal to lower the energy.