Why spin ice obeys the ice rules

The low-temperature entropy of the spin ice compounds, such as Ho2Ti2O7 and Dy2Ti2O7, is well described by the nearest-neighbor antiferromagnetic Ising model on the pyrochlore lattice, i.e., by the ice rules. This is surprising since the dominant coupling between the spins is their long ranged dipole interaction. We show that this phenomenon can be understood rather elegantly: one can construct a model dipole interaction, by adding terms of shorter range, which yields precisely the same ground states, and hence T = 0 entropy, as the nearest-neighbor interaction. A treatment of the small difference between the model and true dipole interactions reproduces the numerical work by Gingras et al. in detail. We are also led to a more general concept of projective equivalence between interactions.