CHANG-TSE HSIEHSule O.M.Cho G.Y.Ryu S.Leigh R.G.2022-06-302022-06-30201410980121https://www.scopus.com/inward/record.uri?eid=2-s2.0-84908402403&doi=10.1103%2fPhysRevB.90.165134&partnerID=40&md5=d00589627a84c5a258c0a689d2f42931https://scholars.lib.ntu.edu.tw/handle/123456789/614634We generalize Laughlin's flux insertion argument, originally discussed in the context of the quantum Hall effect, to topological phases protected by non-on-site unitary symmetries, in particular by parity symmetry or parity symmetry combined with an on-site unitary symmetry. As a model, we discuss fermionic or bosonic systems in two spatial dimensions with CP symmetry, which are, by the CPT theorem, related to time-reversal symmetric topological insulators (e.g., the quantum spin Hall effect). In particular, we develop the stability/instability (or "gappability"/"ingappablity") criteria for nonchiral conformal field theories with parity symmetry that may emerge as an edge state of a symmetry-protected topological phase. A necessary ingredient, as it turns out, is to consider the edge conformal field theories on unoriented surfaces, such as the Klein bottle, which arises naturally from enforcing parity symmetry by a projection operation. © 2014 American Physical Society.journal article10.1103/PhysRevB.90.1651342-s2.0-84908402403