Continuum field theory of 3D topological orders with emergent fermions and braiding statistics

• DOI: 10.48550/arxiv.2307.09983
• Publication Date: 2023-01-01
• Authors: Zhifeng Zhang; Qing-Rui Wang; Peng Ye
• Keywords: Topological algebra; Topological degeneracy

Universal topological data of topologically ordered phases can be captured by quantum field theory in continuous space time taking the limit low energies and long wavelengths. While previous continuum field-theoretical studies orders $3$D real focus on either self-statistics, braiding statistics, shrinking rules, fusion rules or dimensions, it is yet to systematically put all together a unified framework. Here, we construct $BF$ with twisted terms (e.g., $AAdA$ $AAB$) as well $K$-matrix $BB$ term, order simultaneously explore such reach anomaly-free orders. Following spirit famous Chern-Simons $2$D orders, present general formulas show how term confines excitations, self-statistics particles transmuted between bosonic one fermionic one. In explore, within framework, principle gauge invariance fundamentally influences possible realizations data. More concretely, actions (i) particle-loop braidings emergent fermions, (ii) multiloop (iii) Borromean-Rings calculate their universal Together efforts, our work paves way toward more systematic complete analysis exotic properties Several interesting future directions are also discussed.