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 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-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 α=62.42∘ lying below face-centred cubic. Three independent published benchmarks are reproduced.
Related
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BibTeX
@misc{batle2026dipolar,
title = {{Dipolar order across Bravais lattice space: classification, spin waves, and a four-attractor phase diagram}},
author = {Josep Batle},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02936},
eprint = {2609.02936},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02936},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dipolar order across Bravais lattice space: classification, spin waves, and a four-attractor phase diagram
AU - Josep Batle
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.02936
UR - https://arxiv.org/abs/2609.02936
AB - 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 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-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 α=62.42^∘ lying below face-centred cubic. Three independent published benchmarks are reproduced.
ER -