Bosonic symmetry-protected topological phases with reflection symmetry.

We study two-dimensional bosonic symmetry-protected topological (SPT) phases that are protected by reflection symmetry and local symmetry [ZN XI R, ZN × R, U(1)XIR, or U(l)&#×R], in the search for twodimensional bosonic analogs of topological crystalline insulators in integer-S spin systems with reflection and spin-rotation symmetries. To classify them, we employ a Chern-Simons approach and examine the stability of edge states against perturbations that preserve the assumed symmetries. We find that SPT phases protected by ZN XI R symmetry are classified as ℤ2 × ℤ2 for even N and 0 (no SPT phase) for odd V, while those protected by U(1)XI R symmetry are ℤ2. We point out that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of S = 2 spins on the square lattice is a ℤ2 SPT phase protected by reflection and it -rotation symmetries. [ABSTRACT FROM AUTHOR]