Topological phases protected by reflection symmetry and cross-cap states

Twisting symmetries provides an efficient method to diagnose symmetry-protected topological (SPT) phases. In this paper, edge theories of (2+1)-dimensional topological phases protected by reflection as well as other symmetries are studied by twisting reflection symmetry, which effectively puts the edge theories on an unoriented space-time, such as the Klein bottle. A key technical step taken in this paper is the use of the so-called cross-cap states, which encode entirely the unoriented nature of space-time, and can be obtained by rearranging the space-time geometry and exchanging the role of space and time coordinates. When the system is in a nontrivial SPT phase, we find that the corresponding cross-cap state is noninvariant under the action of the symmetries of the SPT phase, but acquires an anomalous phase. This anomalous phase, with a proper definition of a reference state, on which symmetry acts trivially, reproduces the known classification of (2+1)-dimensional bosonic and fermionic SPT phases protected by reflection symmetry, including in particular the Z8 classification of topological crystalline superconductors protected by reflection and time-reversal symmetries.