Chiral symmetry on the edge of 2D symmetry protected topological phases

TL;DR

This paper investigates the low energy excitations on the edges of 2D bosonic SPT phases with Z_N and U(1) symmetry, revealing a chiral action of symmetry that protects gapless edge states and predicts quantized Hall conductance.

Contribution

It provides a detailed analysis of the edge excitations in 2D bosonic SPT phases, showing how symmetry acts chirally and protects gapless modes, with implications for experimental observations.

Findings

Edge modes are non-chiral but symmetry acts chirally on them.

Symmetry prevents scattering between left and right movers, protecting gaplessness.

All 2D U(1) SPT phases have even integer quantized Hall conductance.

Abstract

Symmetry protected topological (SPT) states are short-range entangled states with symmetry, which have symmetry protected gapless edge 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 bosonic SPT phases with Z_N and U(1) symmetry. We find that while the low energy modes of the gapless edges are non-chiral, 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.