Braiding statistics of loop excitations in three dimensions

Abstract

While it is well known that three dimensional quantum many-body systems can support non-trivial braiding statistics between particle-like and loop-like excitations, or between two loop-like excitations, we argue that a more fundamental quantity is the statistical phase associated with braiding one loop $\hat{I}\pm$ around another loop ${\hat{I}}^2$, while both are linked to a third loop ${\hat{I}}^3$. We study this three-loop braiding in the context of $({\mathbb{Z}}_N{)}^K$ gauge theories which are obtained by gauging a gapped, short-range entangled lattice boson model with $({\mathbb{Z}}_N{)}^K$ symmetry. We find that different short-range entangled bosonic states with the same $({\mathbb{Z}}_N{)}^K$ symmetry (i.e. different symmetry-protected topological phases) can be distinguished by their three-loop braiding statistics.

5+5 pages, 10 figures. Minor changes made to improve clarity, published version