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

Abstract

© 2019 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the «https://creativecommons.org/licenses/by/4.0/» Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Topological orders describe a new class of gapped quantum phases of matter at zero temperature, that has various patterns of many-body quantum entanglement. Previously, topological orders in one and two spatial dimensions have been systematically understood and classified. This paper [together with Phys. Rev. X 8, 021074 (2018)PRXHAE2160-330810.1103/PhysRevX.8.021074] develops a systematic and classifying understanding of topological orders in three-dimensional bosonic systems, where the number of topological types for the pointlike and stringlike excitations is assumed to be finite. Our systematic understanding comes from the unique canonical boundary for each 3+1D topological order. We find that the pointlike and stringlike excitations on the canonical boundary are described fully by a mathematical theory - the so-called fusion 2-categories. This theory allows us to classify 3+1D topological orders in bosonic systems in terms of a subset of fusion 2-categories. This systematic understanding further leads to a systematic understanding of 3+1D topological orders in bosonic and fermionic systems with arbitrary finite unitary symmetry.