An $H^{3}(G,{\mathbb T})$-valued index of symmetry-protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systems

Abstract

Abstract We consider symmetry-protected topological phases with on-site finite group G symmetry $\beta $ for two-dimensional quantum spin systems. We show that they have $H^{3}(G,{\mathbb T})$-valued invariant.