Symmetry indicators and stable zeros in exciton wavefunctions

The full quantum state of an exciton can be decomposed into an exciton envelope wavefunction that binds the free electron and hole wavefunctions into a localized state. In the presence of crystalline symmetries, the symmetry indicators and topological invariants of the free bands enter the exciton state directly via their respective wavefunctions, and can thereby make it topologically nontrivial. On the other hand, the exciton envelope function is the home of intrinsic exciton topology that may be nontrivial even when all free topological invariants themselves are trivial. We systematically derive the constraints that the crystalline symmetries and topological invariants of the free bands impose on the exciton envelope function. In particular, we show that the combination of free and exciton symmetry indicators can imply stable zeros of the envelope wavefunction. Among other high-symmetry momenta, these zeros persist at the optically accessible total momentum p=0. We show that knowledge of the stable zeros can be used to guarantee intrinsic exciton topology, without any additional assumptions about the band structure or interactions. In addition, the mere existence of a gapped exciton state with stable zeros at p=0 imposes nontrivial constraints on the topological invariants of the free band structure. Finally, the stable zeros inform the construction of simple microscopic models with intrinsic exciton topology. Our results pave the way for a full topological quantum chemistry description of excitons.