In this talk, I plan to discuss phases of matter with reflection symmetry (parity symmetry) with
interactions. While a systematic analysis is possible for non-interacting fermions, an important
challenge is to understand the effects of strong electron correlations. To get some insight into
this problem, I will discuss an example where by the effects of interactions the non-interacting
classification breaks down. I will also propose a generalization of Laughlin’
s thought experiment,
a theoretical method which is powerful enough to diagnose topological phases with U(1)
symmetry but no other symmetries, to the cases of various symmetry protected topological phases.
For the case of parity symmetry, the proposed generalization consists of putting they boundary
theories of a SPT phase on an unoriented surfaces, and hence is related to the so-called
quantum field theories.