  Topological phases of matter are one of the hallmarks of quantum condensed
matter physics. One of their striking features is a bulk-boundary
correspondence wherein the topological nature of the bulk manifests itself on
boundaries via exotic massless phases. In classical wave phenomena analogous
effects may arise; however, these cannot be viewed as equilibrium phases of
matter. Here we identify a set of rules under which robust equilibrium
classical topological phenomena exist. We write down simple and analytically
tractable classical lattice models of spins and rotors in two and three
dimensions which, at suitable parameter ranges, are paramagnetic in the bulk
but nonetheless exhibit some unusual long-range or critical order on their
boundaries. We point out the role of simplicial cohomology as a means of
classifying, writing-down, and analyzing such models. This opens a new
experimental route for studying strongly interacting topological phases of
spins.
