A ${\mathbb Z}_2$-index of symmetry protected topological phases with reflection symmetry for quantum spin chains

Understand

For the classification of SPT phases, defining an index is a central problem.

• In the famous paper [PTBO1], Pollmann, Tuner, Berg, and Oshikawa introduced ${\mathbb Z}_2$-indices for injective matrix products states (MPS) which have either ${\mathbb Z}_2\times {\mathbb Z}_2$ dihedral group (of $\pi$-rotations about $x$, $y$, and $z$-axes) symmetry, time-reversal symmetry, or reflection symmetry.
• The first two are on-site symmetries.
• In [O4], an index for on-site symmetries, which generalizes the index in [PTBO1], was introduced for general unique gapped ground state phases in quantum spin chains.