Topological order and topological entropy in classical systems

We show that the concept of topological order, introduced to describe ordered quantum systems which cannot be classified by broken symmetries, also applies to classical systems. We discuss some of the fundamental properties of this type of classical order and propose how to expose it via a generalized topological entropy. Starting from a specific example, we show how to use (quantum) pure state density matrices to construct corresponding (classical) thermally mixed ones that retain precisely half of the original topological entropy, a result that we generalize to a whole class of systems