  Artificial spin ices are ideal frustrated model systems in which to explore
or design emergent phenomena with unprecedented characterization of the
constituent degrees of freedom. In square spin ice, violations of the ice rule
are topological excitations essential to the kinetics of the system, providing
an ideal testbed for studying the dynamics of such defects under varied quench
rates. In this work we describe the first test of the Kibble-Zurek mechanism
and critical coarsening in colloidal square and colloidal hexagonal ice under
quenches from a weakly interacting liquid state into a strongly interacting
regime. As expected, for infinitely slow quenches, the system is defect free,
while for increasing quench rate, an increasing number of defects remain in the
sample. For square ice, we find regimes in which the defect population
decreases as a power law with decreasing quench rate. A detailed scaling
analysis shows that for a wide range of parameters, including quench rates that
are accessible by experiments, the behavior is described by critical coarsening
rather than by the Kibble-Zurek mechanism, since the defect-defect interactions
are long ranged. For quenches closer to the critical point, however, there can
be a competition between the two mechanisms.
