Bosonic symmetry-protected topological phases with reflection symmetry

Abstract

We study two-dimensional bosonic symmetry protected topological (SPT) phases\nwhich are protected by reflection symmetry and local symmetry [$Z_N\\rtimes R$,\n$Z_N\\times R$, U(1)$\\rtimes R$, or U(1)$\\times R$], in the search for\ntwo-dimensional bosonic analogs of topological crystalline insulators in\ninteger-$S$ spin systems with reflection and spin-rotation symmetries. To\nclassify them, we employ a Chern-Simons approach and examine the stability of\nedge states against perturbations that preserve the assumed symmetries. We find\nthat SPT phases protected by $Z_N\\rtimes R$ symmetry are classified as\n$\\mathbb{Z}_2\\times\\mathbb{Z}_2$ for even $N$ and 0 (no SPT phase) for odd $N$\nwhile those protected by U(1)$\\rtimes R$ symmetry are $\\mathbb{Z}_2$. We point\nout that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of $S=2$ spins\non the square lattice is a $\\mathbb{Z}_2$ SPT phase protected by reflection and\n$\\pi$-rotation symmetries.\n