Low-temperature ordered states of dysprosium, terbium, and holmium aluminum garnets

The low-temperature ordered states of dysprosium, terbium, and holmium aluminum garnets have been investigated by the Luttinger-Tisza method, taking rigorously into account the dipole-dipole interaction. It is found that for each of these cases, in the absence of an external magnetic field, the lowest-energy configuration corresponds to antiferromagnetic ordering with three pairs of sublattices, in agreement with neutron scattering experiments. In the presence of external field along the [001] direction, the lowest-energy configuration turns out to be that of ferromagnetic chains in the $x$ and $y$ directions, with the nearest ferromagnetic chains pointing in the opposite directions, the same as that proposed by Landau and Keen. When the external magnetic field is along the [110] direction, there are two configurations which correspond to lowest energy; the individual chains are ferromagnetic but not all ferromagnetic chains are parallel, there being chains which point in opposite directions.