Honeycomb-Lattice Heisenberg-Kitaev Model in a Magnetic Field: Spin Canting, Metamagnetism, and Vortex Crystals

The Heisenberg-Kitaev model is a paradigmatic model to describe the magnetism in honeycomb-lattice Mott insulators with strong spin-orbit coupling, such as $A_2{\mathrm{IrO}}_3$ ($A=\mathrm{Na}$, $\mathrm{Li}$) and $\alpha \text{−}{\mathrm{RuCl}}_3$. Here, we study in detail the physics of the Heisenberg-Kitaev model in an external magnetic field. Using a combination of Monte Carlo simulations and spin-wave theory, we map out the classical phase diagram for different directions of the magnetic field. Broken SU(2) spin symmetry renders the magnetization process rather complex, with sequences of phases and metamagnetic transitions. In particular, we find various large-unit-cell and multi-$Q$ phases including a vortex-crystal phase for a field in the $[111]$ direction. We also discuss quantum corrections in the high-field phase.

Figure 1

(a) Phase diagram of the classical HK model for $T\to 0$, with $J=A\cos \phi$, $K=A\sin \phi$, and magnetic field $\overrightarrow{h}$ in the $[001]$ direction. Solid (dash-dotted) lines show first-order (second-order) phase transitions. For $\overrightarrow{h}$ along $[110]$, the phase diagram is identical. The dash-dotted line bounding the polarized phase represents $h_{\mathrm{c}0}(\phi )$. (b) Same as (a), but for $\overrightarrow{h}\parallel [111]$, showing a multitude of novel phases. Multi-$Q$ phases are asterisked; see text for details. (c) Phase diagram for $h=0$ for comparison.

Figure 2

Low-$T$ MC static structure factor for (a)–(c) $\phi =0.62\pi$, (d), (e) $\phi =1.687\pi$, and (f) $\phi =1.813\pi$, respectively, and different strengths $\overrightarrow{h}\parallel [111]$, showing magnetic Bragg peaks for various phases. Inner white dashed hexagon indicates the first Brillouin zone of the honeycomb lattice; high-symmetry points are labeled $\mathrm{\Gamma }$, ${\mathbf{M}}_{1,2,3}$, and $\mathbf{K},{\mathbf{K}}^{\prime }$ in standard notation. Here, $N=2·18^2$ and $T/|J{S}^2|=0.005$.

Figure 3

Spin configurations of selected phases in three-dimensional spin space for $\overrightarrow{h}\parallel [111]$ (chosen vertical, see inset). The dashed lines denote the respective magnetic unit cell. $\phi$ and $h$ are chosen as in Figs. 2, 2, 2, and 2, respectively.

Figure 4

Total magnetization of the HK model, $\overrightarrow{m}$, in field direction $\widehat{h}\parallel [111]$ as function of field strength $h$ for different scans above the (a), (b) zigzag, and (c) stripy states. The curves show the classical magnetization obtained from cooling a low-$T$ MC configuration and analytical spin-pattern parametrization; in addition, the spin-wave result (SW) for $S=1/2$ is shown in the high-field phase, for details see text. Color shading refers to phases in Fig. 1.