Loop braiding statistics in exactly soluble 3D lattice models

Understand

We construct two exactly soluble lattice spin models that demonstrate the importance of three-loop braiding statistics for the classification of 3D gapped quantum phases.

• The two models are superficially similar: both are gapped and both support particle-like and loop-like excitations similar to that of charges and vortex lines in a $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge theory.
• Furthermore, in both models the particle excitations are bosons, and in both models the particle and loop excitations have the same mutual braiding statistics.
• The difference between the two models is only apparent when one considers the recently proposed three-loop braiding process in which one loop is braided around another while both are linked to a third loop.