  A class of three dimensional classical lattice systems with macroscopic
ground state degeneracies, most famously the spin ice system, are known to
exhibit "Coulomb" phases wherein long wavelength correlations within the ground
state manifold are described by an emergent Maxwell electrodynamics. We discuss
a new example of this phenomenon-the four state Potts model on the pyrochlore
lattice-where the long wavelength description now involves three independent
gauge fields as we confirm via simulation. The excitations above the ground
state manifold are bions, defects that are simultaneously charged under two of
the three gauge fields, and exhibit an entropic interaction dictated by these
charges. We also show that the distribution of flux loops shows a scaling with
loop length and system size previously identified as characteristic of Coulomb
phases.
