Linear-response description of superexchange-driven orbital ordering in K2CuF4

We study the nature of orbital and magnetic order in the layered perovskite ${\mathrm{K}}_{2}{\mathrm{CuF}}_{4}$, and compare to the case of the infinite-layer system ${\mathrm{KCuF}}_{3}$. To this end, we augment the local-density approximation $+$ dynamical mean-field theory technique with linear-response functions. We explain orbital and magnetic order, and their evolution with increasing pressure. We show that both the tetragonal (${\ensuremath{\varepsilon}}_{\mathrm{T}}$) and the Jahn-Teller (${\ensuremath{\varepsilon}}_{\mathrm{JT}}$) crystal-field splitting play a key role. We find that surprisingly, unlike in ${\mathrm{KCuF}}_{3},\phantom{\rule{0.16em}{0ex}}{\ensuremath{\varepsilon}}_{\mathrm{T}}$ is comparable to, or even larger than, ${\ensuremath{\varepsilon}}_{\mathrm{JT}}$; in addition, ${\ensuremath{\varepsilon}}_{\mathrm{T}}$ is mostly determined by the layered structure itself and by the compression of the K cage, rather than by the deformations of the ${\mathrm{CuF}}_{6}$ octahedra. Next, we study the nature of orbital order. We calculate the superexchange transition temperature, finding ${T}_{\mathrm{KK}}\ensuremath{\sim}300\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, a value close to the one for ${\mathrm{KCuF}}_{3}$. Thus, in ${\mathrm{K}}_{2}{\mathrm{CuF}}_{4}$ as in ${\mathrm{KCuF}}_{3},\phantom{\rule{0.16em}{0ex}}{T}_{\mathrm{KK}}$ is too small to explain the existence of orbital order up to the melting temperature. We show, however, that in the case of the layered perovskite, an additional superexchange mechanism is at work. It is an orbital Zeeman term, ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{h}}_{\mathrm{KK}}$, and it is active also above ${T}_{\mathrm{KK}}$. We show that due to ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{h}}_{\mathrm{KK}}$, phases with different types of ordering can coexist at temperatures below ${T}_{\mathrm{KK}}$. Similar effects are likely to play a role in other layered correlated systems.