Edge theories of 2D fermionic symmetry protected topological phases protected by unitary Abelian symmetries

TL;DR

This paper uses Abelian Chern-Simons theory to analyze fermionic symmetry protected topological phases, revealing new edge theories and features distinct from bosonic cases, including gapless Majorana modes and intrinsic phases.

Contribution

It introduces a framework for deriving edge theories of fermionic SPT phases with Abelian symmetries, highlighting novel features like Majorana modes and intrinsic phases.

Findings

Constructed Luttinger liquid edge theories with central charge n-1 for certain bosonic SPT phases.

Derived edge theories for intrinsically interacting fermionic SPT phases.

Identified new features such as gapless Majorana modes and trivialization of some bosonic SPT phases.

Abstract

Abelian Chern-Simons theory, characterized by the so-called $K$ matrix, has been quite successful in characterizing and classifying Abelian fractional quantum hall effect (FQHE) as well as symmetry protected topological (SPT) phases, especially for bosonic SPT phases. However, there are still some puzzles in dealing with fermionic SPT(fSPT) phases. In this paper, we utilize the Abelian Chern-Simons theory to study the fSPT phases protected by arbitrary Abelian total symmetry $G_f$. Comparing to the bosonic SPT phases, fSPT phases with Abelian total symmetry $G_f$ has three new features: (1) it may support gapless majorana fermion edge modes, (2) some nontrivial bosonic SPT phases may be trivialized if $G_f$ is a nontrivial extention of bosonic symmetry $G_b$ over $\mathbb{Z}_2^f$, (3) certain intrinsic fSPT phases can only be realized in interacting fermionic system. We obtain edge theories for various fSPT phases, which can also be regarded as conformal field theories (CFT) with proper symmetry anomaly. In particular, we discover the construction of Luttinger liquid edge theories with central charge $n-1$ for Type-III bosonic SPT phases protected by $(\mathbb{Z}_n)^3$ symmetry and the Luttinger liquid edge theories for intrinsically interacting fSPT protected by unitary Abelian symmetry. The ideas and methods used here might be generalized to derive the edge theories of fSPT phases with arbitrary unitary finite Abelian total symmetry $G_f$.