Relief and generation of frustration in pyrochlore magnets by single-ion anisotropy

We show that a strongly anisotropic classical Heisenberg magnet on the pyrochlore lattice can be mapped onto an Ising model with an exchange constant of the opposite sign. This mapping can be applied to the recently discovered ``spin ice'' Ho₂Ti₂O₇ [Harris et al., Phys. Rev. Lett. 79, 2554 (1997)], a pyrochlore ferromagnet which---counterintuitively---is frustrated. We describe the implications of this mapping for the ordering properties of the magnets at low temperatures. Finally, we extend this discussion to the ordering properties of Heisenberg pyrochlore antiferromagnets of any anisotropy strength.