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

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