  We discuss the nearest neighbour spin ice model in the presence of a magnetic
field placed along the cubic [100] direction. As recently shown in Phys. Rev.
Lett. 100, 067207, 2008, the symmetry sustaining ordering transition observed
at low temperature is a three dimensional Kasteleyn transition. We confirm this
with numerical data using a non-local algorithm that conserves the topological
constraints at low temperature and from analytic calculations from a Bethe
lattice of corner sharing tetrahedra . We present a thermodynamic description
of the Kasteleyn transition and discuss the relevance of our results to recent
neutron scattering experiments on spin ice materials.
