  The two dimensional F-model is an ice-rule obeying model, with a low
temperature antiferroelectric state and high temperature critical Coulomb
phase. Polarization in the system is associated with topological defects in the
form of system-spanning windings which makes it an ideal system on which to
observe topological sector fluctuations, as have been discussed in the context
of spin ice and Berezinskii-Kosterlitz-Thouless (BKT) systems. Here we develop
Lieb and Baxter's historic solutions of the F-model to exactly calculate
relevant properties, several apparently for the first time. We further
calculate properties not amenable to exact solution by an approximate cavity
method and by referring to established scaling results. Of particular relevance
to topological sector fluctuations are the exact results for the applied field
polarization and the "energetic susceptibility". The latter is a both a measure
of topological sector fluctuations and, surprisingly, in this case, a measure
of the order parameter correlation exponent. In the high temperature phase, the
temperature tunes the density of topological defects and algebraic
correlations, with the energetic susceptibility undergoing a jump to zero at
the antiferroelectric ordering temperature, analogous to the "universal jump"
in BKT systems. We discuss how these results are relevant to experimental
systems, including to spin ice thin films and three-dimensional dipolar spin
ice and water ice, where we find that an analogous "universal jump" has
previously been established in numerical studies. This unexpected result
suggests a universal limit on the stability of perturbed Coulomb phases that is
independent of dimension and of the order of the transition. Experimental
results on water ice Ih are not inconsistent with this proposition. We complete
the paper by relating our results to experimental studies of artificial spin
ice arrays.
