Topology Obstructs Nodeless Excitons

Lowest-energy excitons are usually expected to have nodeless $1s$-like envelope wavefunctions. We show that band topology can obstruct this expectation. The exciton envelope wavefunction is a section of the valence-conduction transition bundle, whose topology can enforce momentum-space zeros, with their parity or signed total index fixed by the corresponding characteristic class. We exemplify this behavior in a minimal $PT$-symmetric three-band model with a rank-two transition bundle characterized by Euler number ${\nu }^e=2$. By directly solving the Bethe-Salpeter equation with both contact and Rytova-Keldysh interactions, we find that the lowest exciton contains two zeros, each with index $+1$, and is less strongly bound than its topologically trivial counterpart. We further classify topology-enforced exciton nodal structures for real and complex transition bundles in dimensions $d\le 3$. Our results establish the topology of the transition bundle as a direct constraint on exciton wavefunctions and spectra.