of springs and masses or optical devices which have an underlying topological band structure. Their edges can be seen as robust waveguides which have potential engineering applications, such as delay lines for light and sound [@Hafezi2011]. Notwithstanding, it is difficult to view such phenomena as a distinct phase of matter, since the topological nature of the band structure does not induce any sharp measurable features in equilibrium. Further, at present the effect of non-linearities on these systems is unclear. (See however [@Chen09092014].) Both these issues can be seen as a classical reflection of the aforementioned difficulty of finding topological equilibrium phases of non-interacting bosons. As in the quantum case, an alternative route may thus be to consider strongly interacting systems.

One approach to obtain such models is to start from known quantum SPT models and attempt to write their discretized Euclidean time partition function in a sign-problem free and local manner. When possible, the resulting partition function can then be viewed as a classical statistical mechanical system. Nonetheless, the models thus obtained have several drawbacks. First, the notion of symmetry protection does not generally carry through to the classical problem, in the following sense: We define classical symmetries as those 