Classical dipolar Heisenberg models on the Archimedean and Laves lattices

Abstract

We study classical Heisenberg spins interacting through the dipole--dipole interaction on the eleven Archimedean and eight Laves planar lattices at canonical geometry, which reduce to fifteen distinct vertex sets. For each lattice we compute the Fourier-space interaction matrix $\Am(\bk)$, extract the Luttinger--Tisza ordering wave vector $\bk_{0}$, determine the classical ground state by unconstrained minimization on the commensurate magnetic cell, and evaluate the linearized spin-wave dispersion $\varepsilon_{i}(\bk)$ along the high-symmetry path of the (magnetic) Brillouin zone together with the Holstein--Primakoff moment reduction. The ground states fall into three classes: five are collinear, two are non-collinear at angles commensurate with the lattice symmetry, and eight cant at transcendental angles, which we determine to thirty digits. Across the family, failure of the Luttinger--Tisza strong condition coincides exactly with the appearance of incommensurate canting, with no exceptions in either direction. Truncating the interaction range shows that whether a lattice cants is fixed by the local coordination geometry, whereas the value of the canting angle is set by the long-range tail. The quantum corrections are governed by the magnon spectrum through the size of the magnetic basis and are uncorrelated with frustration, so that the frustration and fluctuation classifications are independent; the corner-sharing lattices host the narrowest low-energy branches. The results constitute a spectral atlas of dipolar Heisenberg systems on the full family of $1$-uniform planar tilings and their duals, and provide predictions for inelastic neutron scattering in materials whose magnetic ions occupy Archimedean or Laves lattices, and for artificial arrays of dipolar-coupled nanomagnets.