Interacting fermionic symmetry-protected topological phases in two dimensions

TL;DR

This paper classifies and constructs models for two-dimensional interacting fermionic symmetry-protected topological phases with finite Abelian symmetry, revealing almost all phases can be realized through known models except one intrinsically fermionic class.

Contribution

It provides a comprehensive classification of 2D FSPT phases with Abelian symmetry and constructs models that realize nearly all these phases, identifying a unique intrinsically interacting class.

Findings

Almost all FSPT phases are realizable by stacking free-fermion and bosonic SPT models.

Identifies a unique intrinsically interacting and fermionic FSPT phase class.

Shows stability of bosonic SPT phases when embedded into fermionic systems.

Abstract

We classify and construct models for two-dimensional (2D) interacting fermionic symmetry-protected topological (FSPT) phases with general finite Abelian unitary symmetry $G_f$. To obtain the classification, we couple the FSPT system to a dynamical discrete gauge field with gauge group $G_f$ and study braiding statistics in the resulting gauge theory. Under reasonable assumptions, the braiding statistics data allows us to infer a potentially complete classification of 2D FSPT phases with Abelian symmetry. The FSPT models that we construct are simple stacks of the following two kinds of existing models: (i) free-fermion models and (ii) models obtained through embedding of bosonic symmetry-protected topological (BSPT) phases. Interestingly, using these two kinds of models, we are able to realize almost all FSPT phases in our classification, except for one class. We argue that this exceptional class of FSPT phases can never be realized through models (i) and (ii), and therefore can be thought of as intrinsically interacting and intrinsically fermionic. The simplest example of this class is associated with $\mathbb Z_4^f\times\mathbb Z_4\times\mathbb Z_4$ symmetry. We show that all 2D FSPT phases with a finite Abelian symmetry of the form $ \mathbb Z_2^f\times G$ can be realized through the above models (i), or (ii), or a simple stack of them. Finally, we study the stability of BSPT phases when they are embedded into fermionic systems.