Classical topological order of the Rys F-model and its breakdown in realistic spin ice: Topological sectors of Faraday loops

Both the Rys F-model and antiferromagnetic square ice posses the same ordered, antiferromagnetic ground state, but the ordering transition is of second order in the latter, and of infinite order in the former. To tie this difference to topological properties and their breakdown, we introduce a Faraday line representation where loops carry the energy and magnetization of the system. Because of the absence of monopoles in the F-model, its loops have distinct topological properties, absent in square ice, and which allow for a natural partition of its phase space into topological sectors. Then the N\'eel temperature corresponds to a transition from trivial to non-trivial topological sectors. Moreover, its zero susceptibility below a critical field is explained by the homotopy invariance of its magnetization. In spin ices, instead, monopoles destroy the homotopy invariance of the magnetization, and thus erase this rich topological structure. Consequently, even trivial loops can be magnetized, and their susceptibility is never zero.