Dipolar Order by Disorder in the Classical Heisenberg Antiferromagnet on the Kagome Lattice

Ever since the experiments which founded the field of highly frustrated magnetism, the kagome Heisenberg antiferromagnet has been the archetypical setting for the study of fluctuation induced exotic ordering. To this day the nature of its classical low-temperature state has remained a mystery: the nonlinear nature of the fluctuations around the exponentially numerous harmonically degenerate ground states has not permitted a controlled theory, while its complex energy landscape has precluded numerical simulations at low temperature, $T$. Here we present an efficient Monte Carlo algorithm which removes the latter obstacle. Our simulations detect a low-temperature regime in which correlations asymptote to a remarkably small value as $T\to 0$. Feeding these results into an effective model and analyzing the results in the framework of an appropriate field theory implies the presence of long-range dipolar spin order with a tripled unit cell.

Figure 1

Top: Candidate phases of the kagome antiferromagnet (from left to right: $q=0$; Kosterlitz-Thouless phase; $q=\sqrt{3}\times \sqrt{3}$) in the phase diagram of an extended Potts model as defined in the text. $J_2=0$ corresponds to the pure Potts model on the kagome lattice, the unweighted ensemble of coplanar Heisenberg ground states. Bottom: Cartoon of the effective entropy landscape for ground states—soft fluctuations yield minima at coplanar states. Our algorithm surmounts the entropic free energy barriers, exploring many coplanar states. The resulting ensemble average leads to a small but nonzero spin order. Inset: Transitions between harmonically equivalent configurations—spins on loops of alternating color are rotated such that the angle a spin makes with its local Potts axis is not changed as the dark green and blue spins are exchanged for the light ones.

Figure 2

Spin correlation functions $C_s(r)=⟨\mathbf{S}(r)·\mathbf{S}(0)⟩$ along the nearest-neighbor directions at various temperatures. Also shown for comparison is the case of Potts model. The simulations were done on a lattice with $N\approx 9\times {15}^2$ spins with open boundary conditions. Inset: The correlations acquire a stable low-temperature value below $T={10}^{-3}J$.

Figure 3

(a) Spin correlations of the Heisenberg model at $T={10}^{-5}$ and the extended Potts model with $J_2=0.019$ on a lattice with 396 spins. (b) Order parameter $m_{\sqrt{3}}$ as a function of linear system size $L$ for different values of $J_2$. The solid red points for small systems are from the Heisenberg simulations. Inset: Scaling collapse of data for $L\ge 60$.

Figure 4

(a) The function $\Theta (q)\equiv ⟨|{\mathbf{h}}_{\mathrm{Potts}}(q){|}^2⟩/⟨|{\mathbf{h}}_{\mathrm{Heisenberg}}(q){|}^2⟩$ denotes the ratio of the averaged height fields. A ratio $\Theta (q)>1$ for $q^2\to 0$ reflects the increased stiffness of the Heisenberg model compared to the Potts model, and the resultant ordering at low temperature. (b) Relative frequency of the averaged height vector for Heisenberg spins at $T={10}^{-4}J$ for a lattice with 729 spins and open boundary conditions. The maxima correspond to $\sqrt{3}\times \sqrt{3}$ order.