Stable Wave-Function Zeros Indicate Exciton Topology

Excitons are bound states of electrons and holes whose band topology arises from an interplay between the topology of the underlying electronic bands and the structure of the electron-hole interaction. In crystalline solids, symmetry representations and topological invariants of the conduction and valence bands constrain the structure of the exciton envelope wave function. In particular, we show that crystalline symmetry can enforce stable zeros in the exciton wave function. These occur at high-symmetry momenta, including the optically accessible total momentum p=0. We work out how the stable zeros constrain both the relative exciton-band topology (the difference of exciton and non-interacting topological invariants) and the relative band topology (the difference of valence and conduction band invariants), all without requiring detailed knowledge of the band structure or interactions. We establish these results for two-band excitons in inversion- and rotation-symmetric systems in one and two dimensions, where the relevant topological invariants are the Berry phase in one dimension and the Chern number (modulo the rotation order) in two dimensions. In two dimensions, the exciton Chern number itself can also be constrained by zero patterns.