The 10 phases of spin chains with two Ising symmetries

We explore the topological properties of quantum spin-1/2 chains with two Ising symmetries. This class of models does not possess any of the symmetries that are required to protect the Haldane phase. Nevertheless, we show that there are 4 symmetry-protected topological phases, in addition to 6 phases that spontaneously break one or both Ising symmetries. By mapping the model to one-dimensional interacting fermions with particle-hole and time-reversal symmetry, we obtain integrable parent Hamiltonians for the conventional and topological phases of the spin model. We use these Hamiltonians to characterize the physical properties of all 10 phases, identify their local and nonlocal order parameters, and understand the effects of weak perturbations that respect the Ising symmetries. Our study provides the first explicit example of a class of spin chains with several topologically non-trivial phases, and binds together the topological classifications of interacting bosons and fermions.