$\mathbb{Z}_3$ Topological Order in Face Centered Cubic Quantum Plaquette Model

We examine the topological order in the resonating singlet valence plaquette (RSVP) phase of the hard-core quantum plaquette model (QPM) on the face centered cubic (FCC) lattice. To do this, we construct a Rohksar-Kivelson type Hamiltonian of local plaquette resonances. This model is shown to exhibit a $\mathbb{Z}_3$ topological order, which we show by identifying a $\mathbb{Z}_3$ topological constant (which leads to a $3^3$-fold topological ground state degeneracy on the $3$-torus) and topological point-like charge and loop-like magnetic excitations which obey $\mathbb{Z}_3$ statistics. We also consider an exactly solvable generalization of this model, which makes the geometrical origin of the $\mathbb{Z}_3$ order explicitly clear. For other models and lattices, such generalizations produce a wide variety of topological phases, some of which are novel fracton phases.