One-dimensional symmetric phases protected by frieze symmetries

We make a systematic study of symmetry-protected topological gapped phases of quantum spin chains in the presence of the frieze space groups in one dimension using matrix product states. Here, the spatial symmetries of the one-dimensional lattice are considered together with an additional vertical reflection, which we take to be an on-site Z2 symmetry. We identify seventeen distinct non-trivial phases, define canonical forms, and compare the topological indices obtained from the MPS analysis with the group cohomological predictions. We furthermore construct explicit renormalization group fixed-point wave functions for symmetry-protected topological phases with global on-site symmetries, possibly combined with time reversal and parity symmetry. En route, we demonstrate how group cohomology can be computed using the Smith normal form.

Automated summary

In this study, we systematically investigate the symmetry-protected topological gapped phases of quantum spin chains using matrix product states. We consider the spatial symmetries of the one-dimensional lattice together with an additional vertical reflection. We identify seventeen distinct non-trivial phases and compare the topological indices obtained from the MPS analysis with the group cohomological predictions.