We classify and construct models for two-dimensional (2D) interacting fermionic symmetry-protected topological (FSPT) phases with general finite Abelian unitary symmetry Gf. To obtain the classification, we couple the FSPT system to a dynamical discrete gauge field with gauge group Gf 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 Z4f×Z4×Z4 symmetry. In addition, we show that all 2D FSPT phases with a finite Abelian symmetry of the form Z2f×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.
Bibliographical noteFunding Information:
This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. C.-H.L. acknowledges the funding from the Canada Research Chair (CRC) program and the University of Alberta. Z.-C.G. acknowledges start-up support, Direct Grants No. 4053163 and No. 3132745 from The Chinese University of Hong Kong and the funding from RGC/ECS (Grant No. 2191110) and GRF (Grant No. 14306714).
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