Interacting fermionic symmetry-protected topological phases in two dimensions

Abstract

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