Citation: Wang, Chenjie, and Michael Levin. "Braiding statistics of loop excitations in three dimensions." Physical review letters 113.8 (2014): 080403.
Web: https://arxiv.org/abs/1403.7437
Tags: SPT/SETs, Abelian-anyons, Higher-dimensional
This paper argues that an important physical quantity for understanding topological order in (3+1) dimensions is three-loop braiding. That is, the braiding of two loop charges which simultaneously hooked on a third loop charge. They show that these braiding phases can be used to distinguish different SPT phases. The follow-up paper
> Lin, Chien-Hung, and Michael Levin. "Loop braiding statistics in exactly soluble 3D lattice models." arXiv preprint arXiv:1503.00142 (2015).
shows an explicit example of two phases whose pointlike and looplike excitations have the same statistics, but who differ in their three-loop braiding.
This three-loop invariant can be understood nicely in the context of braided fusion 2-categories:
> Else, Dominic V., and Chetan Nayak. "Cheshire charge in (3+ 1)-dimensional topological phases." Physical Review B 96.4 (2017): 045136.
The authors say that the results of their paper " support the hypothesis that 2-categories are the correct mathematical framework for (3 + 1)-dimensional topological phases". This also gives a partial explanation for why 2-categories need to have so much more data than 1-categories. There are so many more ways for objects to interact in three dimensions!