  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. We find that
the statistical phase associated with this process is different in the two
models, thus proving that they belong to two distinct phases. An important
feature of this work is that we derive our results using a concrete approach:
we construct string and membrane operators that create and move the particle
and loop excitations and then we extract the braiding statistics from the
commutation algebra of these operators.
