Long Range Order at Low Temperature in Dipolar Spin Ice

Recently it has been suggested that long range magnetic dipolar interactions are responsible for spin ice behavior in the Ising pyrochlore magnets ${\rm Dy_{2}Ti_{2}O_{7}}$and${\rm Ho_{2}Ti_{2}O_{7}}$. We report here numerical results on the low temperature properties of the dipolar spin ice model, obtained via a new loop algorithm which greatly improves the dynamics at low temperature. We recover the previously reported missing entropy in this model, and find a first order transition to a long range ordered phase with zero total magnetization at very low temperature. We discuss the relevance of these results to $\mathrm{D}{\mathrm{y}}_2\mathrm{T}{\mathrm{i}}_2{\mathrm{O}}_7$ and $\mathrm{H}{\mathrm{o}}_2\mathrm{T}{\mathrm{i}}_2{\mathrm{O}}_7$.