Understanding excitons using spherical geometry

Using the spherical geometry, we introduce a novel model to study excitons confined in a three-dimensional space, which offers unparalleled mathematical simplicity while retaining much of the key physics. This new model consists of an exciton trapped on the 3-sphere (i.e. the surface of a four-dimensional ball), and provides a unified treatment of Frenkel and Wannier-Mott excitons. Moreover, we show that one can determine, for particular values of the dielectric constant $ϵ$, the closed-form expression of the exact wave function. We use the exact wave function of the lowest bound state for $ϵ=2$ to introduce an intermediate regime which gives satisfactory agreement with \alert{the} exact results for a wide range of $ϵ$ values.