$K$ dependent exchange interaction of the $1S$ ortho exciton in Cu$_2$O

Abstract: When treating the exchange interaction of Wannier excitons, usually only the leading terms of the analytic and the nonanalytic exchange interaction are considered. However, higher order terms can lead to a splitting of exciton states, for which reason a splitting of the $1S$~exciton in cuprous oxide $\left(\mathrm{Cu_{2}O}\right)$ depending on its total momentum $\hbar K$ has been attributed to a $K$ dependent analytic exchange interaction by Dasbach et al [Phys. Rev. Lett. 91, 107401 (2003)]. Going beyond the common treatment of the exchange interaction, we derive the correct expressions for these $K$ dependent higher order terms using $\boldsymbol{k}\cdot\boldsymbol{p}$ perturbation theory. We prove that the appearance of a $K$ dependent exchange interaction is inseparably connected with a $K$ independent exchange interaction of $P$ and $D$ excitons. We estimate the magnitude of these terms for $\mathrm{Cu_{2}O}$ from microscopic calculations and show that they are far too small to explain the observed $K$ dependent splitting. Instead, this splitting has to be treated in terms of the dispersion of the excitons. Furthermore, we prove the occurence of a coupling between longitudinal and transverse excitons in $\mathrm{Cu_{2}O}$ due to the $K$ dependent nonanalytic exchange interaction.