Spiral antiferromagnets beyond the spin-wave approximation: Frustrated $XY$ and Heisenberg models on the honeycomb lattice

We examine the stability of classical states with a generic incommensurate spiral order against quantum fluctuations. Specifically, we focus on the frustrated spin-1/2 $XY$ and Heisenberg models on the honeycomb lattice with nearest-neighbor $J_1$ and next-nearest-neighbor $J_2$ antiferromagnetic couplings. Our variational approach is based on the Jastrow wave functions, which include quantum correlations on top of classical spin waves. We perform a systematic optimization of wave vectors and Jastrow pseudopotentials within this class of variational states and find that quantum fluctuations favor collinear states over generic coplanar spirals. The Néel state with $\mathbf{Q}=(0,0)$ extends its stability well beyond the classical value $J_2/J_1=1/6$. Most importantly, the collinear states with $\mathbf{Q}=(0,2\pi /\sqrt{3})$ (and the two symmetry-related states) are found to be stable in a large regime with intermediate frustration, while at the classical level they are limited to the point $J_2/J_1=0.5$. For large frustration, the ${120}^{\circ }$ state is stabilized for finite values of $J_2/J_1$ in both models.

Figure 1

The manifold of classically degenerate spiral wave vectors for a few values of $J_2/J_1$. For $J_2/J_1<1/6$, the lowest-energy state has ${\mathbf{Q}}^{*}=(0,0)$, while for $J_2/J_1>1/6$ the ground state is degenerate: the ${\mathbf{Q}}^{*}$ wave vectors form closed contours around the $\mathit{\Gamma }$ point for $1/6<J_2/J_1<1/2$ and around the $\mathbf{K}$ (or ${\mathbf{K}}^{\prime }$) point for $J_2/J_1>1/2$. For $J_2/J_1\to \infty$, ${\mathbf{Q}}^{*}=\mathbf{K}$ or ${\mathbf{K}}^{\prime }$.

Figure 2

Real-space illustration of some of the ordered states considered in this work. (a) Néel antiferromagnet, for which $\mathbf{Q}=\mathit{\Gamma }=(0,0)$ and $\eta =0$. (b) $(\pi ,0)$ spiral, for which $\mathbf{Q}=(\pi ,0)$ and $\eta =0$. (c) Collinear state, for which $\mathbf{Q}=\mathbf{M}=(0,2\pi /\sqrt{3})$ and $\eta =\pi$. (d) ${120}^{\circ }$ ordered phase, for $\mathbf{Q}=\mathbf{K}$ or ${\mathbf{K}}^{\prime }$ and $\eta$ arbitrary.

Figure 3

The locus of wave vectors for different values of $J_2/J_1$ giving the minimal energy of the Jastrow variational wave functions. The results are shown for the $XY$ model with $L=18$ (i.e., 648 sites).

Figure 4

Variational energies for the $XY$ model for $0\le J_2/J_1\le 1$ (a) and $1\le J_2/J_1\le 5$ (b) as a function of $J_2/J_1$ for $L=18$. Statistical errors are smaller than the symbols and solid lines are provided as a guide to the eye.

Figure 5

Energy versus wave vector for classical (a), $XY$ (b), and Heisenberg (c) models for $J_2/J_1=0.3$ and $L=18$.

Figure 6

Finite-size scaling of the energy of the spin-1/2 $XY$ model for different values of $J_2/J_1$ in the Néel and collinear phases. Lines depict the results of fits according to Eq. (13). Red circles (blue squares) indicate the Néel (collinear) phase.

Figure 7

The same as Fig. 3, but for the Heisenberg model.

Figure 8

The same as Fig. 4, but for the Heisenberg model.

Figure 9

The same as Fig. 6 but for the Heisenberg model. Red circles (blue squares) indicate the Néel (collinear) phase. The black triangles indicate the $(\pi ,0)$ spiral.

Figure 10

Comparison between our variational approach for magnetically ordered states and other numerical approaches that have been considered recently.