Magnetization processes of zigzag states on the honeycomb lattice: Identifying spin models for $\alpha \text{−}{\mathrm{RuCl}}_3$ and ${\mathrm{Na}}_2{\mathrm{IrO}}_3$

We study the field-induced magnetization processes of extended Heisenberg-Kitaev models on the honeycomb lattice, taking into account off-diagonal and longer-range exchange interactions, using a combination of Monte Carlo simulations, classical energy minimization, and spin-wave theory. We consider a number of different parameter sets, previously proposed to describe the magnetic behavior of $\alpha \text{−}{\mathrm{RuCl}}_3$ and ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ with their antiferromagnetic zigzag ground states. By classifying these parameter sets, we reveal the existence of three distinct mechanisms to stabilize zigzag states, which differ in the sign of the nearest-neighbor Kitaev interaction, the role of longer-range interactions, and the magnitude of the off-diagonal ${\mathrm{\Gamma }}_1$ interaction. While experimentally hardly distinguishable at zero field, we find that the three different scenarios lead to significantly different magnetization processes in applied magnetic fields. In particular, we show that a sizable off-diagonal interaction ${\mathrm{\Gamma }}_1>0$ naturally explains the strongly anisotropic field responses observed in $\alpha \text{−}{\mathrm{RuCl}}_3$ without the need for a strong anisotropy in the effective $g$ tensor. Moreover, for a generic field direction, it leads to a high-field state with a finite transversal magnetization, which should be observable in $\alpha \text{−}{\mathrm{RuCl}}_3$.

Figure 1

(a) Ferromagnetic zigzag chains along $x$ and $y$ bonds with antiferromagnetically ordered $z$ bonds (“$z$ zigzag”), together with the definition of the staggered magnetization ${\vec{m}}_{\text{stagg}}$. Dotted rectangle, magnetic unit cell. (b) Hexagonal structure of ${\mathrm{Ru}}^{3+}$ ions with edge-shared octahedra of ${\mathrm{Cl}}^{-}$ ions. The cubic axes ${\vec{e}}_x$, ${\vec{e}}_y$, and ${\vec{e}}_z$ are along Ru-Cl bonds, while the $\left[\overline{1}10\right]$ direction is along Ru-Ru $z$ bonds of the honeycomb plane. In the conventions of Ref. [29] the latter corresponds to the crystallographic $b$ axis, while the $\left[11\overline{2}\right]$ direction in our cubic basis then coincides with the $a$ axis, with $a\perp b$. The $\left[111\right]$ direction is perpendicular to the honeycomb plane and is sometimes referred to as the $c^{*}$ axis [8]. (c) Classical energy gain on each bond in Scenario 1 (antiferromagnetic $K_1$, ferromagnetic $J_1$). In the $z$-zigzag state, ${\vec{m}}_{\text{stagg}}$ is parallel or antiparallel to the $z$ axis (“cubic-axis $z$ zigzag”). (d) In $\alpha \text{−}{\mathrm{RuCl}}_3$, the spins point along Ru-Cl bonds. (e) Same as (c), but in Scenario 2 (ferromagnetic $K_1$, antiferromagnetic $J_3$). In the $z$-zigzag state, ${\vec{m}}_{\text{stagg}}$ can (classically) point everywhere in the $xy$ plane (“cubic-plane $z$ zigzag”). (f) In $\alpha \text{−}{\mathrm{RuCl}}_3$, this corresponds to the planes perpendicular to the Ru-Cl bonds. (g) Same as (e), but in Scenario 3 (strong ${\mathrm{\Gamma }}_1>0$). Here, we display the limit ${\mathrm{\Gamma }}_1\gg \left|K_1\right|$ with ${\mathrm{\Gamma }}_1/(-J_1)>0$ finite. In the $z$-zigzag state, ${\vec{m}}_{\text{stagg}}$ lies in the $\left[11\overline{1}\right]$ direction. (h) In $\alpha \text{−}{\mathrm{RuCl}}_3$, the spins point towards the centers of opposite faces of ${\mathrm{Cl}}^{-}$ octahedra surrounding each ${\mathrm{Ru}}^{3+}$ ion (“face-center $z$ zigzag”).

Figure 2

Magnetization in the field direction for the $J_1\text{−}{K}_1$ model with large $K_1>0$ and small $J_1<0$, using parameter set 1, from MC simulations (red crosses) and semianalytical variational calculation [blue (dark) curves]. For comparison: spin-wave magnetization for $S=1/2$ [red (light) curves, high-field phases only]. The small octahedra are given as a reminder of the spin alignment at zero field; cf. Fig. 1. For (b) $\vec{h}\parallel \left[001\right]$ and (c) $\vec{h}\parallel \left[\overline{1}10\right]$, the classical magnetization curves are linear in the intermediate-field regime and exhibit a continuous transition towards the uniform high-field phase at $h_0$, indicating a uniform canting mechanism with low-field spin alignment ${\vec{S}}_i\perp \vec{h}$ for $h\searrow 0$. For (a) $\vec{h}\parallel \left[111\right]$ and (d) $\vec{h}\parallel \left[11\overline{2}\right]$, the magnetization curve becomes nonlinear and a direct continuous transition towards the high-field phase becomes impossible.

Figure 3

Magnetization in the field direction for the $K_1\text{−}{J}_3$ model with $K_1<0$ and $J_3>0$, using parameter set 2. At zero field, the spins lie in the $xy$ plane ($z$ zigzag), $yz$ plane ($x$ zigzag), or $xz$ plane ($y$ zigzag) and can always align perpendicular to $\vec{h}$ upon switching on a small magnetic field in an arbitrary direction. Therefore, the classical magnetization curve is independent of the field direction in this model with ${\mathrm{\Gamma }}_1=0$. The quantum corrections [red (light) curves, high-field phases only], however, do depend on the field axis as a consequence of the order-from-disorder mechanism.

Figure 4

Magnetization in the field direction for the $J_1\text{−}{K}_1\text{−}{\mathrm{\Gamma }}_1\text{−}{J}_2\text{−}{K}_2\text{−}{J}_3$ model with $K_1<0$, $J_3>0$, and small $0<{\mathrm{\Gamma }}_1\ll \left|K_1\right|$, using parameter set $2+\mathrm{\Gamma }$. At zero field, the spins align along the $\left[11\overline{\epsilon }\right]$, $\left[1\overline{\epsilon }1\right]$, or $\left[\overline{\epsilon }11\right]$ direction, with $0<\epsilon \ll 1$, and then cant nonuniformly towards the field for $0<h<h_{\mathrm{c}1}$ and (a) $\vec{h}\parallel \left[111\right]$ or (b) $\vec{h}\parallel \left[001\right]$. At $h_{\mathrm{c}1}$, there is a discontinuous transition towards a phase in which the spins cant uniformly, leading to a linear magnetization curve as in Fig. 3. For $\vec{h}\parallel \left[001\right]$ this intermediate phase is tiny: see expanded view in (b). In this case, the transition is characterized by a change of the static structure factor from $z$ zigzag to $x/y$ zigzag. There are no intermediate phases for (c) $\vec{h}\parallel \left[\overline{1}10\right]$. The curve for (d) $\vec{h}\parallel \left[11\overline{2}\right]$ can be understood as a (very weak) crossover. For all field directions, the transition towards the uniform high-field phase at $h_0$ is continuous.

Figure 5

Magnetization in the field direction for the $J_1\text{−}{K}_1\text{−}{\mathrm{\Gamma }}_1\text{−}{J}_3$ model with $K_1<0$, $J_3>0$, and large ${\mathrm{\Gamma }}_1>0$, using parameter set 2/3. In contrast to Fig. 4, there are no intermediate phases as the would-be transition at $h_{\mathrm{c}1}$ is preempted by a first-order transition at $h_0$ towards the uniform high-field phase. For $\vec{h}\parallel \left[001\right]$, the high-field phase exhibits a finite transversal magnetization, leading to a nonsaturated classical magnetization $\vec{m}·\widehat{h}/S<1$ for any finite $h$; see Sec. 7.

Figure 6

Magnetization in the field direction for the $J_1\text{−}{K}_1\text{−}{\mathrm{\Gamma }}_1\text{−}{J}_3$ model with $J_1<0$, $K_1<0$, $J_3>0$, and large ${\mathrm{\Gamma }}_1>0$ using parameter set $3+J_3$. The critical field $h_0$ is strongly anisotropic, with the largest $h_0$ among the four shown here occurring for (a) the out-of-plane direction $\vec{h}\parallel \left[111\right]$. For this particular field direction, the leading-order quantum correction to the classical magnetization happens to vanish due to an accidental cancellation in linear spin-wave theory [red (light) curves, high-field phases only]. For (b) the intermediate field direction $\vec{h}\parallel \left[001\right]$, the uniform high-field phase is characterized by a finite transversal magnetization, $\langle \vec{S}\rangle \nparallel \vec{h}$, leading to a nonsaturated magnetization in the field direction already on the classical level. The transition at $h_0$ is typically discontinuous, except for (c) $\vec{h}\parallel \left[\overline{1}10\right]$, which corresponds to a field direction that is perpendicular to the zero-field moments.

Figure 7

(a) Susceptibility $\chi (\vartheta ,\phi )=d(\vec{m}·\widehat{h})/d(h/|K_1\left|\right)$ at zero temperature and in the classical limit as a function of magnetic field axis $\vec{h}/h$ for parameter set $3+J_3$ with sizable ${\mathrm{\Gamma }}_1>0$. $\vartheta$ and $\phi$ are the spherical angles in the $a,b,c^{*}$ basis, such that $\vartheta =0$ corresponds to the crystallographic $c^{*}$ axis perpendicular to the honeycomb layer, denoted by $\left[111\right]$. $\vartheta =\pi /2$ with $\phi =0$ ($\phi =\pi /2$) corresponds to the in-plane crystallographic $a$ ($b$) axis, denoted by $\left[11\overline{2}\right]$ ($\left[\overline{1}10\right]$). Red (lower) curve, out-of-plane susceptibility $\chi (\vartheta ,0)$ from $a$ to $c^{*}$ to $-a$; blue (upper) curve, in-plane susceptibility $\chi (0,\phi )$ from $a$ to $b$ to $-a$. (b) Same as (a), but now showing the critical field strength $h_0$ of the transition towards the high-field phase.

Figure 8

Low-$T$ MC static structure factor for $J_1\text{−}{K}_1\text{−}{\mathrm{\Gamma }}_1$ model with $(K_1,{\mathrm{\Gamma }}_1)=(-6.8,+9.5)$ meV (set 3) and different values of $J_1\le 0$ at zero external field. The color scale is logarithmic. Bright spots indicate Bragg peaks. The dotted inner hexagon denotes the first Brillouin zone. The Bragg-peak pattern reveals that the ground state for (a) $J_1=0$ is not a zigzag antiferromagnet, but a multi-$\vec{Q}$ state. This remains true upon inclusion of a small ferromagnetic Heisenberg interaction $J_1<0$, until there is, for some finite $J_1$, a direct transition towards (b) an incommensurate phase. Upon further increasing $-J_1$, the ground state becomes (c) the simple ferromagnet.

Figure 9

Low-$T$ MC static structure factor for $J_1\text{−}{\mathrm{\Gamma }}_1$ model. For $J_1=0$, the ground state is (a) a classical spin liquid [57], characterized by a structure factor that is finite everywhere except at a few isolated points in the Brillouin zone. By including a small ferromagnetic Heisenberg interaction $J_1<0$, the ground-state degeneracy is lifted in favor of (b) the zigzag antiferromagnet, evidenced by its characteristic Bragg peaks at the three $\mathbf{M}$ points. Upon increasing $-J_1$, we recover (c) the incommensurate phase, before the ground state is again the ferromagnet at large values of $-J_1$ (not shown).