Lattice-mediated interaction of ${\mathrm{Cu}}^{2+}$ Jahn-Teller ions in insulating cuprates

The interaction of the copper d orbitals with the lattice is investigated with the goal of finding the lattice and orbital ground state in ${\mathrm{La}}_2$${\mathrm{CuO}}_4$. Due to the considerable tetragonality of the crystal, the Kanamori interaction is strongly anisotropic. This results in a tendency to a lattice instability with rhombic distortion of the oxygen squares around the Cu ions. In the present approach, the experimentally observed partial occupation of ${\mathit{d}}_{3\mathit{z}}^2$-${\mathit{r}}^2$ orbitals can be accounted for by the Jahn-Teller effect. The renormalization of the phonon spectrum due to virtual ${\mathrm{Cu}}^{2+}$ d-d orbital transitions is calculated using appropriate electronic and phononic model parameters. A qualitative agreement with some features in the experimental phonon spectra is obtained.