Topological order of the Rys F-model and its breakdown in realistic square spin ice: Topological sectors of Faraday loops
Both the Rys F-model and antiferromagnetic square ice possess 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 lines representation where loops carry the energy and magnetization of the system. Because the F-model does not admit monopoles, its Faraday 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 Néel temperature corresponds to a transition from topologically trivial to non-trivial Faraday loops. Because magnetization is a homotopy invariant of the Faraday loops, and it is zero for topologically trivial ones, the susceptibility is zero below a critical field. In square spin ice, instead, monopoles destroy the homotopy invariance and the parity distinction among loops, thus erasing this rich topological structure. Consequently, even trivial loops can be magnetized in square ice, and their susceptibility is never zero.