Magnetic field induced evolution of intertwined orders in the Kitaev magnet $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$

Recent scattering experiments in the 3D Kitaev magnet $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ have shown that a relatively weak magnetic field along the crystallographic $\mathbf{b}$ axis drives the system from its incommensurate counter-rotating order to a correlated magnet, with a significant uniform ‘zigzag’ component superimposing the magnetization along the field. Here it is shown that the zigzag order is not emerging from its linear coupling to the field (via a staggered, off-diagonal element of the $\mathbf{g}$ tensor) but from its intertwining with the incommensurate order and the longitudinal magnetization. The emerging picture explains all qualitative experimental findings at zero and finite fields, including the rapid decline of the incommensurate order with field and the so-called intensity sum rule. The latter are shown to be independent signatures of the smallness of the Heisenberg exchange $J$, compared to the Kitaev coupling $K$ and the off-diagonal anisotropy $\mathrm{\Gamma }$. Remarkably, in the regime of interest, the field $H^{*}$ at which the incommensurate component vanishes, depends essentially only on $J$, which allows us to extract an estimate of $J\simeq 4$ K from reported measurements of $H^{*}$. We also comment on recent experiments in pressurized $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ and conclude that $J$ decreases with pressure.

Figure 1

(a) The relevant parameter space in $(r,\varphi )$ [see Eq. (4)], along with the $K$ and $\mathrm{\Gamma }$ regions proposed in Ref. [60]. The points $A$ and $B$, referred to in subsequent figures, correspond to $(r,\varphi )=(0.32\pi ,1.52\pi )$ and $(0.32\pi ,1.625\pi )$. (b),(c) The structure of the $K$ state at low $\left(H\le H^{*}\right)$ and high $\left(H\ge H^{*}\right)$ fields. The solid (dashed) green and red bonds depict the $xy\left(x^{\prime }{y}^{\prime }\right)$ chains running along $\widehat{\mathbf{a}}+\widehat{\mathbf{b}}\left(\widehat{\mathbf{a}}\text{−}\widehat{\mathbf{b}}\right)$. The blue vertical segments point along the $\mathbf{c}$ axis and depict the $z$ bonds. The Cartesian components of the spins are shown in the side panels. Both states respect the combined operation $\mathrm{\Theta }{C}_{2\mathbf{c}}$, where $\mathrm{\Theta }$ is time reversal and $C_{2\mathbf{c}}$ is the twofold rotation around $\mathbf{c}$. The high-field $K$ state is qualitatively similar to the state ‘FM-${\mathrm{SZ}}_{\text{FM}}$’ of Ref. [61], which is stabilized in a finite region with $\varphi <3\pi /2$.

Figure 2

Evolution of the various Fourier components of the static structure factor of the $K$ state with a magnetic field $H$ along $\mathbf{b}$, for the points $A$ (panel a) and $B$ (panel b) in Fig. 1.

Figure 3

Evolution of the coefficients $x_1,y_1$, etc. with a magnetic field $H$ along $\mathbf{b}$, for the points $A$ (panel a) and $B$ (panel b) in Fig. 1.

Figure 4

Evolution of the combinations $I_I,I_V$, and $I_{\text{tot}}$ [see Eq. (10)] with a magnetic field $H$ along $\mathbf{b}$, for the points $A$ (panel a) and $B$ (panel b) of Fig. 1. In (a), $\alpha \simeq 0.503$ [very close to the expected value of $1/2$ when $\varphi \to {(3\pi /2)}^{+}$, see text], while in (b), $\alpha \simeq 0.535$.

Figure 5

Same as in Fig. 3 but now we show the behavior up to fields much higher than $H^{*}$. Panels (a) and (b) correspond again to the parameter points $A$ and $B$, respectively, of Fig. 1.

Figure 6

Same as in Fig. 2 but now we show the behavior up to fields much higher than $H^{*}$. Panels (a) and (b) correspond again to the parameter points $A$ and $B$, respectively, of Fig. 1.

Figure 7

Effect of $g_{ab}$ on the evolution of the various components of the structure factor. Data are taken at the point $A$ of Fig. 1, with $g_{bb}=2$ and $g_{ab}=0.1$ [dotted lines, same as in Fig. 2], 0 (solid) and $-0.1$ (long dashed).

Figure 8

(a) Evolution of $H^{*}$ as a function of $K$ for fixed $\mathrm{\Gamma }=-1$ and various values of $J$. (b) Evolution of $H^{*}$ as a function of $\mathrm{\Gamma }$ for fixed $K=-1$ and various values of $J$. (c) Collected data shown in panels (a) and (b), with $J$ on the horizontal axis. The line shown is a guide to the eye. For all cases we have taken $g_{ab}=0.1$ and $g_{bb}=2$.