Kitaev-Heisenberg model in a magnetic field: Order-by-disorder and commensurate-incommensurate transitions

We present a theoretical study of field-induced magnetic phases in the honeycomb Kitaev-Heisenberg model, which is believed to describe the essential physics of Mott insulators with strong spin-orbit coupling such as $A_2{\mathrm{IrO}}_3$ and $\alpha \text{−}{\mathrm{RuCl}}_3$. We obtain a rich finite temperature phase diagram in which the competition between the Zeeman coupling and thermal fluctuations gives rise to both collinear zigzag phases and noncoplanar magnetic orders. Our large-scale classical Monte Carlo simulations also unveil intriguing commensurate-incommensurate transitions and multiple-$\mathbf{Q}$ incommensurate phases at high field. Experimental implications are also discussed.

Figure 1

The field-temperature $\left(H\text{−}T\right)$ phase diagram of the KH model with parameter $\phi =0.7\pi$. Dashed and solid lines denote first- and second-order phase transitions, respectively. There are five ordered phases at low temperatures. Other than the $\sqrt{3}\times \sqrt{3}$ order at high field, the phase diagram is dominated by four distinct zigzag phases: single-$\mathbf{Q}$ canted zigzag (I), triple-$\mathbf{Q}$ commensurate zigzag (II), triple-$\mathbf{Q}$ partial incommensurate zigzag (III), and fully incommensurate $3\mathbf{Q}$ zigzag (IV). The corresponding structure factors and spin snapshots are shown in Fig. 2. $T$ and $H$ are measured in units of overall exchange energy scale $A$.

Figure 2

Magnetic phases in the honeycomb KH model. Top row shows the spin structure factors obtained from simulations at $T=0.005$; corresponding snapshots of the spin configurations are shown in the bottom row. The three spin components are shown here with red, green, and blue colors. The five phases shown here are (a) single-$\mathbf{Q}$ collinear zigzag order $\left(H=0.016\right)$, (b) commensurate triple-$\mathbf{Q}$ noncoplanar zigzag $\left(H=0.48\right)$, (c) coexistence of commensurate and incommensurate triple-$\mathbf{Q}$ zigzag phase $\left(H=1.02\right)$, (d) incommensurate triple-$\mathbf{Q}$ zigzag phase $\left(H=1.34\right)$, and (e) $\sqrt{3}\times \sqrt{3}$ order $\left(H=1.4\right)$. $T$ and $H$ are measured in units of $A$.

Figure 3

Monte Carlo simulations of KH model subject to a magnetic field along the $\left[111\right]$ symmetric direction. (a) Magnetization projected onto the field direction as a function of $H$ for varying temperatures. (b) Amplitude of zigzag order parameter $\varphi =\left|\mathbf{\varphi }\right|$ (left axis) and the $\sqrt{3}\times \sqrt{3}$ order parameter $\psi$ (right axis) versus field strength. (c) Field dependence of the order parameter $\zeta =\left|\mathbf{\zeta }\right|$ characterizing the disparity of the three zigzags. Both temperature $T$ and field strength $H$ are in units of the exchange energy scale $A$. The simulations are performed on the KH model with parameter $\phi =0.7\pi$, where single-$\mathbf{Q}$ collinear zigzag order is the ground state. The number of spins is $N_s=2\times {60}^2$.

Figure 4

The honeycomb lattice with three types of nearest neighbor bonds. Here ${\mathit{t}}^x=(\frac{1}{2},\frac{-\sqrt{3}}{2}),{\mathit{t}}^y=(\frac{1}{2},\frac{\sqrt{3}}{2})$ are two primitive translations. Extended magnetic unit cells used in our variational calculation of the KH model. The quadrupled unit cell (yellow shaded sites) corresponding to the general ordering composed of three wave vectors ${\mathbf{Q}}_1=(-\pi ,-\pi /\sqrt{3}),{\mathbf{Q}}_2=(0,2\pi /\sqrt{3})$, and ${\mathbf{Q}}_3=(+\pi ,-\pi /\sqrt{3})$. The tripled unit cell (green shaded sites), on the other hand, describes the $\sqrt{3}\times \sqrt{3}$ type ordering with a wave vector $\mathbf{K}=(4\pi /3,0)$.

Figure 5

Contour plot of the minimum eigenvalue of $\mathbb{H}\left(\mathbf{k}\right)$, showing minimum at the $K$ points ${\mathbf{Q}}_K=(\frac{4\pi }{3},0)$.

Figure 6

Temperature dependence of order parameters $\varphi =\left|\mathbf{\varphi }\right|$ and $\zeta =\left|\mathbf{\zeta }\right|$ from annealing and heating simulations. Panels (a) and (b) are obtained with $H=0.2$, while (c) and (d) are obtained with $H=0.96$.

Figure 7

Variational ground-state calculation of KH model at $\phi =0.7\pi$ in the magnetic field along the $\left[111\right]$ direction: (a) Magnetization given by the ferromagnetic order parameter $m$ as a function of field strength. Also shown for comparison is the magnetization curve obtained from Monte Carlo simulations at a temperature $T=0.005$. (b) The amplitude of the various order parameters defined in Eqs. (A2) and (A5) versus $H$. $T$ and $H$ are measured in units of $A$.

Figure 8

Field dependence of the uniaxial nematic order parameter ${\lambda }_Q$ at various temperatures. The arrows indicate the small jumps of ${\lambda }_Q$ at the commensurate-incommensurate phase transitions. $T$ and $H$ are measured in units of $A$.