Title: The 10 phases of spin chains with two Ising symmetries,
Abstract: 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.