Abstract
Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group $G$ and its group 4-cocycle $\omega_4 \in \mathcal H^4[G;U(1)]$ up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.
Highlights
Large classes of matter states have only been completely classified a few times
Since 3 þ 1D AB topological orders can be obtained by gauging [17] the symmetry of 3 þ 1D bosonic symmetry-protected topological (SPT) states, and since Dijkgraaf-Witten models only correspond to gauging the pure SPT states, we see that the classification results in this paper imply the following: In 3 þ 1D, there is no mixed bosonic SPT order for unitary finite symmetry group G
We note that 3 þ 1D topological orders contain both pointlike and stringlike excitations
Summary
Large classes of matter states have only been completely classified a few times. The pointlike excitations of a 3 þ 1D topological order can always be viewed as carrying irreducible representations of the group [32] They behave exactly like the quasiparticle excitations above a product state with G symmetry. This is quite an amazing result: A 3 þ 1D topological order whose quasiparticles are all bosonic is always related to a finite group G [33]. We want to show that there is nothing more general: All 3 þ 1D topological orders, whose pointlike excitations are all bosons, are classified by a finite group G and its group 4-cocycle ω4 ∈ H41⁄2G; Uð1Þ, up to group automorphisms. We show the main result of the paper following the above four steps
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