Spin order of the classical Kagome antiferromagnet: via effective Hamiltonians

The classical Heisenberg Kagom\'e-lattice antiferromagnet (KAF) is only known to have a coplanar ``spin nematic'' (or octupole~\footnote{ M. E. Zhitomirsky, Phys. Rev. B 78, 094423 (2008).}) order, so that low-energy states are labeled by colorings. Contrary to accepted phenomenology,$^1$ I propose that these colorings develop {\it long-range order}.~\footnote{ C. L. Henley, arxiv:0811.0026.} First, from the spin-wave Hamiltonian up to 4th order, most modes are integrated out, leaving an effective quartic Hamiltonian $Q$ for just the ``soft'' (zero at harmonic order) modes. Writing it $Q=Q_0+Q'$, where only $Q'$ depends on the discrete coplanar state, $Q'$ is treated as a perturbation, and its expectation in the $Q_0$ ensemble becomes an effective Hamiltonian $\Phi$ for the colorings. The couplings in $\Phi$ are estimated using ``Coulomb phase'' coarse-grainings.$^2$ Following Huse \& Rutenberg,~\footnote{ D. A. Huse and A. D. Rutenberg, Phys. Rev. B 45, 7536 (1992).} I observe the unweighted coloring model sits at a roughening transition, hence $\Phi$ drives the KAF to long-range order of the $\sqrt{3}\times\sqrt{3}$ type (modulo the inevitable gradual orientation fluctuations of the spin plane). A similar effective Hamiltonian exists for related $d=3$ lattices,~\footnote{ C. L. Henley, arxiv:0809.0079.} but cannot produce order.