  We investigate the recently reported analogies between pinned vortices in
nano-structured superconductors or colloids in optical traps, and spin ice
materials. The frustration of colloids and vortices differs essentially from
spin ice. However, their effective energetics is made identical by the
contribution of an emergent field associated to the topological charge, thus
leading to a (quasi) ice manifold for lattices of even (odd) coordination. The
equivalence extends to the local low-energy dynamics of the ice manifold, where
the effect of geometric hard constraints can be subsumed into the spatial
modulation of the emergent field, which mediates an entropic interaction
between topological charges. There, as in spin ice materials, genuine ice
manifolds enter a Coulomb phase, whereas quasi-ice manifolds posses a well
defined screening length, provided by a plasma of embedded topological charges.
The equivalence between the two systems breaks down in lattices of mixed
coordination because of topological charge transfer between sub-latices. We
discuss extensions to social and economical networks.

Statistical Mechanics