  Spin ice in a magnetic field in the [111] direction displays two
magnetization plateaux, one at saturation and an intermediate one with finite
entropy. We study the crossovers between the different regimes from a point of
view of (entropically) interacting defects. We develop an analytical theory for
the nearest-neighbor spin ice model, which covers most of the magnetization
curve. We find that the entropy is non-monotonic, exhibiting a giant spike
between the two plateaux. This regime is described by a monomer-dimer model
with tunable fugacities. At low fields, we develop an RG treatment for the
extended string defects, and we compare our results to extensive Monte Carlo
simulations. We address the implications of our results for cooling by
adiabatic (de)magnetization.

Statistical Mechanics;Disordered Systems and Neural Networks;Strongly Correlated Electrons