Symmetry protected topological phases in spin-1 ladders and their phase transitions

We study two-legged spin-1 ladder systems with $D_2\times \sigma$ symmetry group, where $D_2$ is discrete spin rotational symmetry and $\sigma$ means interchain reflection symmetry. The system has one trivial phase and seven nontrivial symmetry protected topological (SPT) phases. We construct Hamiltonians to realize all of these SPT phases and study the phase transitions between them. Our numerical results indicate that there is no direct continuous transition between any two SPT phases we studied. We interpret our results via topological nonlinear sigma model effective field theory, and further conjecture that generally there is no direct continuous transition between two SPT phases in one dimension if the symmetry group is discrete at all length scales.