We apply the topological theory of symmetry indicators to interaction-induced exciton band structures in centrosymmetric semiconductors. Crucially, we distinguish between the topological invariants inherited from the underlying electron and hole bands and those that are intrinsic to the exciton wave function itself. Focusing on the latter, we show that there exists a class of exciton bands for which the maximally localized exciton Wannier states are shifted with respect to the electronic Wannier states by a quantized amount; we call these excitons shift excitons. Our analysis explains how the exciton spectrum can be topologically nontrivial and sustain exciton edge states in open boundary conditions even when the underlying noninteracting bands have a trivial atomic limit. We demonstrate the presence of shift excitons as the lowest energy neutral excitations of the Su-Schrieffer-Heeger model in its trivial phase when supplemented by local two-body interactions.
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