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We construct two exactly soluble lattice spin models that demonstrate theimportance of three-loop braiding statistics for the classification of 3Dgapped quantum phases. The two models are superficially similar: both aregapped and both support particle-like and loop-like excitations similar to thatof charges and vortex lines in a $\mathbb{Z}_2 \times \mathbb{Z}_2$ gaugetheory. Furthermore, in both models the particle excitations are bosons, and inboth models the particle and loop excitations have the same mutual braidingstatistics. The difference between the two models is only apparent when oneconsiders the recently proposed three-loop braiding process in which one loopis braided around another while both are linked to a third loop. We find thatthe statistical phase associated with this process is different in the twomodels, thus proving that they belong to two distinct phases. An importantfeature of this work is that we derive our results using a concrete approach:we construct string and membrane operators that create and move the particleand loop excitations and then we extract the braiding statistics from thecommutation algebra of these operators.

Loop braiding statistics in exactly soluble 3D lattice models