Understanding excitons using spherical geometry

Using 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 the exact results for a wide range of ε values.