Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions

TL;DR

This paper develops a topological response theory to classify bosonic and fermionic symmetry-protected topological phases with finite Abelian symmetries in two dimensions, revealing new phases beyond existing frameworks.

Contribution

It introduces a systematic scheme for classifying SPT phases using topological response theory, including phases beyond Chern-Simons descriptions and fermionic phases with parity-dependent classifications.

Findings

Classifies bosonic SPT phases with any finite Abelian symmetry group.

Identifies fermionic SPT phases with $ ext{Z}_m$ symmetry, showing classification depends on parity of m.

Reveals existence of bosonic SPT phases beyond current Chern-Simons framework.

Abstract

It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with $\mathbb{Z}_m$ symmetry and find the classification of SPT phases depends on the parity of $m$: for even $m$ there are $2m$ classes, $m$ out of which is intrinsically fermionic SPT phases and can not be realized in any bosonic system. Finally we propose a classification scheme of fermionic SPT phases for any finite, Abelian symmetry.