Two-stage orbital order and dynamical spin frustration in KCuF$_3$

Results from our x-ray and Raman scattering studies on KCuF$_3$, a model orbital order system, strongly link a low-temperature orbital order transition to a previously unidentified structural phase transition at 50 K. Raman scattering shows softening of phonon modes linked to F---ions that ceases upon a splitting of a degenerate E$_g$ mode at 50 K. This, along with the emergence of diffuse scattering around an orbital order Bragg peak at low temperature, suggests an onset of GdFeO$_3$-like octahedral tilting, which serves to stabilize the Neel spin order at 39 K. To explain these effects, we have added to the Kugel-Khomskii model a term for direct orbital exchange driven by electron-electron interactions and ligand distortions. This term creates a near degeneracy, which dynamically frustrates the spin order at high temperature, that is lifted by orbital---lattice interactions at low temperature.