Chiral symmetry on the edge of two-dimensional symmetry protected topological phases

Symmetry protected topological (SPT) states are short-range entangled states with symmetry. The boundary of a SPT phases has either gapless excitations or degenerate ground states, around a gapped bulk. Recently, we proposed a systematic construction of SPT phases in interacting bosonic systems, however it is not very clear what is the form of the low-energy excitations on the gapless edge. In this paper, we answer this question for two-dimensional (2D) bosonic SPT phases with Z₍ and U (1) symmetry. We find that while the low-energy modes of the gapless edges are nonchiral, symmetry acts on them in a ``chiral'' way, i. e. , acts on the right movers and the left movers differently. This special realization of symmetry protects the gaplessness of the otherwise unstable edge states by prohibiting a direct scattering between the left and right movers. Moreover, understanding of the low-energy effective theory leads to experimental predictions about the SPT phases. In particular, we find that all the 2D U (1) SPT phases have even integer quantized Hall conductance.