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 $\alpha$ around another loop $\beta$, while both are linked to a third loop $\gamma$. 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

References in corpus (2)

• Twisted Quantum Double Model of Topological Phases in Two--Dimension
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