• DOI:10.1103/PhysRevX.9.021005
• Corpus ID: 118964051

Classification of 3+1D Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions

@article{Lan2019ClassificationO,
  title={Classification of 
3+1D
 Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions},
  author={Tian Lan and Xiao-Gang Wen},
  journal={Physical Review X},
  year={2019}
}

• T. Lan, X. Wen
• Published 25 January 2018
• Mathematics
• Physical Review X

We call a topological order of 3+1-dimensional bosonic systems an all-boson (AB) topological order if all emergent point-like excitations are bosons. It was shown that AB topological orders, $\mathcal C^4_{AB}$, are classified by unitary pointed fusion 2-categories with only trivial 1-morphisms. In fact, AB topological orders can all be realized by Dijkgraaf-Witten gauge theories. In this paper, we consider emergent-fermion (EF) topological orders for 3+1D bosonic systems where some emergent point-like excitations are fermions. We show that all the EF topological orders, $\mathcal C^4_{EF}$, are classified by 2+1D anomalous topological orders $\mathcal A_b^3$ on the canonical boundary of the 3+1D topological order. The anomalous topological orders $\mathcal A_b^3$ is described by a unitary pointed 2-fusion category with only one invertible fermionic 1-morphism for each object. $\mathcal A_b^3$ sometimes also has quantum-dimension-$\sqrt 2$ 1-morphisms (which correspond to Majorana-zero-modes). By studying the mathematical structures of $\mathcal A^3_b$, we show that the EF topological orders $\mathcal C^4_{EF}$ are classified by 3+1D fermionic symmetry protected topological (SPT) orders with a finite unitary symmetry group after gauging all the symmetry group. We further propose that the general classification of 3+1D topological orders with symmetries can be obtained by gauging or partially gauging the finite symmetry group of 3+1D SPT phases of bosonic and fermionic systems.