Bosonic symmetry protected topological phases with reflection symmetry

We study two-dimensional bosonic symmetry protected topological (SPT) phases which are protected by reflection symmetry and local symmetry [$Z_N⋊R$, $Z_N\times R$, U(1)$⋊R$, or U(1)$\times R$], in the search for two-dimensional 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 $Z_N⋊R$ symmetry are classified as ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ for even $N$ and 0 (no SPT phase) for odd $N$ while those protected by U(1)$⋊R$ symmetry are ${\mathbb{Z}}_2$. We point out that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of $S=2$ spins on the square lattice is a ${\mathbb{Z}}_2$ SPT phase protected by reflection and $\pi$-rotation symmetries.