Optical conductivity in doped manganites with planar x$^2$-y$^2$ orbital order

We investigate a planar model for the ferromagnetic (FM) phase of manganites, which develops orbital order of $e_g$ electrons with x$^2$-y$^2$-symmetry at low temperature. The dynamic structure factor of orbital excitations and the optical conductivity $\sigma(\omega)$ are studied with help of a finite-temperature diagonalization method. Our calculations provide a theoretical prediction for $\sigma(\omega)$ for the 2D FM state and are of possible relevance for the recently found A-type phase of manganites at high doping which consists of FM layers coupled antiferromagnetically. In the x$^2$-y$^2$ ordered regime $\sigma(\omega)$ shows both a Drude peak and a gapped incoherent absorption due to a gap in the orbital excitations.