Origin of Spin Ice Behavior in Ising Pyrochlore Magnets with Long Range Dipole Interactions: an Insight from Mean-Field Theory

Recent experiments suggest that the Ising pyrochlore magnets ${\rm Ho_{2}Ti_{2}O_{7}}$ and ${\rm Dy_{2}Ti_{2}O_{7}}$ display qualitative properties of the ferromagnetic nearest neighbor spin ice model proposed by Harris {\it et al.}, Phys. Rev. Lett. {\bf 79}, 2554 (1997). The manifestation of spin ice behavior in these systems {\it despite} the energetic constraints introduced by the strength and the long range nature of dipole-dipole interactions, remains difficult to understand. We report here results from a mean field analysis that shed some light on the origin of spin ice behavior in (111) Ising pyrochlores. Specifically, we find that there exist a large frustrating effect of the dipolar interactions beyond the nearest neighbor, and that the degeneracy established by effective ferromagnetic nearest neighbor interactions is only very weakly lifted by the long range interactions. Such behavior only appears beyond a cut-off distance corresponding to $O(10^2)$ nearest neighbor. Our mean field analysis shows that truncation of dipolar interactions leads to spurious ordering phenomena that change with the truncation cut-off distance.