Refined Phase Diagram for a Spin-1 System Exhibiting a Haldane Phase

We provide the phase diagram of a 2-parameter spin-1 chain that has a symmetry-protected topological (SPT) Haldane phase using computational algorithms along with tensor-network tools. We improve previous results, showing the existence of a new phase and new triple points. New striking features are the triple end of the Haldane phase and the complexity of phases bordering the Haldane phase in proximity allowing moving to nearby non-SPT phases via small perturbations. These characteristics make the system, which appears in Rydberg excitons, e.g. in Cu${}_2$O, a prime candidate for applications.