$q$-state Potts ice

Classical and quantum spin ice arguably provide the simplest route towards spin liquids and their emergent gauge fields. $q-$state Potts ice models have been constructed that generalize spin ice, hosting multiple emergent $\text{U}(1)$ gauge fields and excitations charged under non-trivial combinations of these fields. We present a general treatment of classical $q-$state Potts ices relating their properties to the $\mathfrak{su}(q)$ Lie algebras, and demonstrate how the properties of charged excitations in the classical model can be related to this symmetry group. We also introduce quantum generalizations of the Potts ice models, and demonstrate how charge flavor changing interactions unique to $q>2$ models dominate their low energy physics. We further show how symmetries inherited from the $\mathfrak{su}(q)$ can lead to flux vacuum frustration, greatly modifying the dynamical properties of charged excitations.