In this talk, I review the "Coulomb phase", an emergent state of various lattice models (particularly highly frustrated antiferromagnets) which have local constraints that can be mapped to a divergence-free "flux". The coarse-grained versions of this flux or polarization behave analogously to electric or magnetic fields. In consequence:
(1) correlation functions have the same functional form as a dipole-dipole interaction, meaning they are (surprisingly) long-ranged; correspondingly, in reciprocal space, the (all diffuse) scattering has characteristic "pinch-point" singularities
(2) topological defects (where the local constraint is violated) behave like effective charges with Coulomb interactions.
Time permitting, I will mention extensions and applications of the Coulomb phase idea to (a) dynamics, (b) quantum mechanics, (c) phase transitions out of such states, and/or (d) models with disorder.