A $${{\mathbb {Z}}}_2$$-index of Symmetry Protected Topological Phases with Reflection Symmetry for Quantum Spin Chains

For the classification of SPT phases, defining an index is a central problem. In famous paper (Pollmann et al. Phys Rev B 81:064439, 2010), Pollmann, Tuner, Berg, and Oshikawa introduced $${{\mathbb {Z}}}_2$$ -indices for injective matrix products states which have either {Z}}}_2\times {{\mathbb dihedral group (of $$\pi $$ -rotations about x, y, z-axes) symmetry, time-reversal or reflection symmetry. The first two are on-site symmetries. Ogata in A $${\mathbb {Z}}_2$$ -index symmetry protected topological phases with time reversal quantum spin chains. arXiv:1810.01045 , symmetries, generalizes Pollmann (2010), was general unique gapped ground state It proved that invariant $$C^1$$ -classification phases. not left as open question. this paper, we introduce symmetric complete generalization problem by We also show -classification.