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 Nel 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.

Rys F模型和反铁磁方冰都具有相同的有序反铁磁基态，但是有序跃迁在后者中是二阶的，在前者中是无穷大的。为了将此差异与拓扑属性及其分解联系起来，我们引入了法拉第线表示法，其中回路承载了系统的能量和磁化强度。由于F模型不允许单极子，因此它的法拉第环具有独特的拓扑特性，在方冰中不存在，并且可以将其相空间自然地划分为拓扑部分。然后，尼尔温度对应于从拓扑上平凡的法拉第环到非平凡的法拉第环的过渡。由于磁化是法拉第环的同态不变，而对于拓扑琐碎的磁化，它为零，在临界场以下的敏感性为零。相反，在正方形自旋冰中，单极子破坏了同态不变性和环之间的奇偶性区别，从而擦除了这种丰富的拓扑结构。因此，即使是微小的环也可以在方冰中磁化，其磁化率永远不会为零。