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In one dimensional chains of trapped Rydberg excitons in cuprous oxidesemiconductor the topological spin phase has been recently predicted [Phys.Rev. Lett. 123, 126801 (2019); arXiv:1903.11951]. This phase is characterizedby the diluted antiferromagnetic order of $p$-shell exciton angular momenta-$1$and the edge states behaving akin spin-$1/2$ fermions. Here we study theproperties of the ground state in the finite chains and its fine structureresulting from the effective interaction of the edge spins. We demonstrate thatthese edge states can detected optically via the enhancement of the circularpolarization of the edge emission as compared with the emission from the bulk.We calculate the distribution of the exciton angular momentum vs. trap numberin the chain numerically and analytically based on the variational ansatz.

Polarized edge state emission from topological spin phases of trapped Rydberg excitons in Cu$_2$O