Low-temperature orbital ordering and dynamical frustration of spins in KCuF$_{3 }$: Theoretical model

A prototypical orbital ordering material, the charge transfer insulator KCuF$_{3}$ is believed to undergo orbital ordering at a temperature $T_{OO}$~$\sim $ 800~K. Recent Raman and X-ray measurements reveal a low-temperature structural transition just preceding the onset of three-dimensional magnetic ordering at $T_{N}$~$\sim $ 40~K. We present a model of how orbital, structural and magnetic fluctuations are coupled at temperatures between $T_{OO}$ and $T_{N}$, leading to the dynamical frustration of in-plane spin order. The low-temperature structural transition quenches the orbital fluctuations, stabilising A-type N\'{e}el spin order. A striking implication of our results is that the ground state of KCuF$_{3}$ lies near a quantum critical point associated with an orbital/spin liquid phase that is obscured by emergent N\'{e}el ordering of the spins. This exotic liquid phase might be accessible with the application of pressure.