Detection of gapped phases of a one-dimensional spin chain with on-site and spatial symmetries

We investigate the phase diagram of a quantum spin-1 chain whose Hamiltonian is invariant under a global on-site A4, translation, and lattice inversion symmetries. We detect different gapped phases characterized by a symmetry protected topological (SPT) order and symmetry breaking using matrix product state order parameters. We observe a rich variety of phases of matter characterized by a combination of symmetry breaking and symmetry fractionalization and also the interplay between the on-site and spatial symmetries. Examples of continuous phase transitions directly between topologically nontrivial SPT phases are also observed.