Interacting fermionic symmetry-protected topological phases in two dimensions

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. In addition, 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), (ii), or a simple stack of them. Finally, we study the stability of BSPT phases when they are embedded into fermionic systems.

Figure 1

Trajectories of ${\xi }_{\mu }$ in the braiding processes associated with the topological invariants ${\mathrm{\Theta }}_{\mu \nu }$ (a) and ${\mathrm{\Theta }}_{\mu \nu \lambda }$ (b).

Figure 2

Diagrammatic calculations of the Berry phases ${\mathrm{\Theta }}_{\mu \mu \nu }$ (a) and $m{\mathrm{\Theta }}_{0\mu \nu }$ (b).

Figure 3

Diagrammatic calculations of Berry phases ${\eta }_0$ (a) and ${\eta }_1$ (b).