Orbital order in La$_{0.5}$Sr$_{1.5}$MnO$_{4}$: a failure of the local Jahn-Teller physics

Orbital order (OO) occurs quite often in orbitally degenerate correlated transition-metal compounds. It has been generally accepted that there is one-to-one correspondence between a specific orbital order and a local Jahn-Teller distortion. Here we demonstrate that this is not always true, by demonstrating a failure of the local Jahn-Teller physics in the layered perovskite La$_{0.5}$Sr$_{1.5}$MnO$_{4}$ which is one of prototype OO materials. We studied both the site- and bond-centered charge orderings, crystal field levels, orbital states and their dependence on the varying local lattice distortions, through detailed $ab$ $initio$ electronic structure calculations. We conclude that this material has the site-centered charge ordering, and that the local Jahn-Teller physics fails and the type of occupied orbitals (3x$^{2}$-r$^{2}$/3y$^{2}$-r$^{2}$ ones) contradicts the local compression of Mn$^{3+}$O$_{6}$ octahedra which could require x$^{2}$-z$^{2}$/y$^{2}$-z$^{2}$ occupation. We explain this by the contribution of the long-range crystal-field in this anisotropic layered material and by the maximization of kinetic energy. Our theoretical results are confirmed by x-ray absorption linear dichroism.