Dipolar order across Bravais lattice space: classification, spin waves, and a four-attractor phase diagram

Abstract

Every crystal is a Bravais lattice decorated by a basis, so the dipolar ordering of the fourteen Bravais lattices is the natural starting point for any systematic theory of dipolar magnetism in three dimensions. We determine it here in a single common framework: the interaction tensor is Ewald-summed, the classical ground state is obtained by minimising the lowest Luttinger--Tisza band over the entire Brillouin zone, and the linear spin-wave spectrum with its zero-point corrections is computed for every lattice whose order is collinear. Three results emerge that are not properties of individual lattices but of the landscape. First, two structural principles --- the exact tracelessness of the dipolar tensor in three dimensions, and its identical vanishing at $\mathbf{k}=0$ for every cubic-symmetric lattice --- explain the ordering type, the absence of first-order cubic anisotropy, and the systematics of the zero-point moment reduction. Second, exactly one lattice defeats the Luttinger--Tisza construction: for face-centred orthorhombic the optimal eigenvector is not circular, the single-$\mathbf{k}$ state is a spin-density wave of non-constant length, and the tabulated energy is a strict lower bound; direct supercell minimisation gives the true ground state. Third, optimising each family over its free metric parameters collapses the whole of Bravais space onto only \emph{four} attractors, with body-centred tetragonal the global optimum and a single interior optimum at rhombohedral $\alpha =62.42^{\circ }$ lying below face-centred cubic. Three independent published benchmarks are reproduced.