Topological Spin Phases of Trapped Rydberg Excitons in Cu_{2}O

We theoretically study Rydberg excitons in one-dimensional chains of traps in Cu_{2}O coupled via the van der Waals interaction. The triplet of optically active p-shell states acts as an effective spin 1, and the interactions between the excitons are strongly spin dependent. We predict that the system has the topological Haldane phase with the diluted antiferromagnetic order, long-range string correlations, and finite excitation gap. We also analyze the effect of the trap geometry and interactions anisotropy on the Rydberg exciton spin states and demonstrate that a rich spin phase diagram can be realized showing high tunability of the Rydberg exciton platform.