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

TL;DR

This work develops a topological response framework for two-dimensional SPT phases with finite Abelian symmetry, recasting their bulk properties in an Abelian Chern-Simons-Higgs theory. By gauging the symmetry and analyzing the resulting flux statistics, the authors derive a compact classification of bosonic SPTs as ${\prod_i}{\mathbb Z}_{m_i}\times{\prod_{i<j}}{\mathbb Z}_{(m_i,m_j)}$, while highlighting that this captures only a subset of possible topological orders and may not exhaust non-Abelian flux phenomena. They show a nonperturbative relation between response and intrinsic topological order through ${\mathrm K}$-matrix equivalence under ${\mathrm GL}(N,\mathbb{Z})$, illustrating collapses in classification for concrete cases such as $G=\mathbb{Z}_n$ and $G=\mathbb{Z}_2\times\mathbb{Z}_2$. Extending to fermionic SPTs, the paper reveals an even-odd effect in the $\mathbb{Z}_m$ case: odd $m$ yields ${\mathbb Z}_m$ classes (equivalent to bosonic SPTs with trivial fermions), while even $m$ gives ${\mathbb Z}_{2m}$ with intrinsically fermionic classes; general Abelian groups acquire a mixed bosonic/fermionic structure with $m_i^*=m_i$ (odd) or $2m_i$ (even). Overall, the work provides a nonperturbative bulk invariant framework for Abelian SPTs and clarifies the relationship between bosonic and fermionic SPT classifications in two dimensions.

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.