Susceptibility anisotropy and its disorder evolution in models for Kitaev materials

Mott insulators with strong spin-orbit coupling display a strongly anisotropic response to applied magnetic fields. This applies in particular to Kitaev materials, with $\alpha -{\mathrm{RuCl}}_3$ and ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ representing two important examples. Both show a magnetically ordered zigzag state at low temperatures, and considerable effort has been devoted to properly modeling these systems in order to identify routes towards realizing a quantum spin liquid. Here, we investigate the relevant Heisenberg-Kitaev-$\mathrm{\Gamma }$ model primarily at elevated temperatures, focusing on the characteristic anisotropy between the in-plane and out-of-plane uniform susceptibility. For $\alpha -{\mathrm{RuCl}}_3$, we find that the experimentally observed anisotropy, including its temperature dependence, can be reproduced by combining a large off-diagonal ${\mathrm{\Gamma }}_1$ coupling with a moderate $g$-factor anisotropy. Moreover, we study in detail the effect of magnetic dilution and provide predictions for the doping evolution of the temperature-dependent susceptibilities.

Figure 1

(a)–(c) Specific heat $c_v$ and (d)–(f) zigzag correlation length $\xi$ as a function of temperature $T$ for (a),(d) model 1, (b),(e) model 2, and (c),(f) model 3, and different linear lattice sizes $L$. The vertical dashed line marks the position of the ordering temperature $T_{\mathrm{N}}$.

Figure 2

Out-of-plane susceptibility ${\chi }_{\perp }$ (black) and in-plane susceptibilities ${\chi }_{\parallel }$ (red) for (a) model 1, (b) model 2, and (c) model 3, as a function of temperature $T$. Susceptibility anisotropy ratio ${\chi }_{\parallel }/{\chi }_{\perp }$ in units of ${(g_{\parallel }/g_{\perp })}^2$ for (d) model 1, (e) model 2, and (f) model 3 as a function of temperature $T$. (a)–(c) MC results on an $L=24$ lattice in units of $g_{\parallel }^2$ and $g_{\perp }^2$, displaying the $g$-factor independent intrinsic anisotropy arising from the ${\mathrm{\Gamma }}_1$ interaction in (a) and (b). Inset: Comparison between MC results (full lines) and high-temperature series expansion (dashed lines) in a log-log scale, using $c=S^2$. The vertical dashed line marks the position of the ordering temperature $T_{\mathrm{N}}$. (d)–(f) Susceptibility anisotropy ratio ${\chi }_{\parallel }/{\chi }_{\perp }$ for two values of the ratio ${(g_{\parallel }/g_{\perp })}^2$ from the data in (a)–(c). We considered (d),(e) $g_{\parallel }=2.3$ and $g_{\perp }=1.3$ and (f) $g_{\parallel }=1.9$ and $g_{\perp }=2.7$.

Figure 3

Uniform susceptibility $\chi$, in units of $g^2$, as a function of temperature $T$ in the diluted Heisenberg model for different vacancy concentrations $x$, as obtained from classical MC simulations for $L=24$ and uniform $g$ factors. The black dot shows the analytical result for $x=0$ in the low-temperature limit: $\chi (x=0)=g^2/\left(9J_1\right)$. Inset: Zoom into the low-temperature region with the dashed lines showing the fit given by Eq. (10).

Figure 4

Uniform susceptibilities ${\chi }_{\nu }, \nu =\perp ,\parallel$, as a function of temperature $T$ in units of $g_{\perp }^2$ and $g_{\parallel }^2$, respectively, for (a) model 1, (b) model 2, and (c) model 3, from classical MC simulations on an $L=24$ lattice. Insets: Susceptibilities for $x=0.20$ in a log-log scale. The dashed lines correspond to the high-temperature results, given by Eq. (11).

Figure 5

Susceptibility anisotropy ratio ${\chi }_{\parallel }/{\chi }_{\perp }$ in units of ${(g_{\parallel }/g_{\perp })}^2$ as a function of doping level $x$ for different temperatures for (a) model 1 and (b) model 2, from classical MC simulations on an $L=24$ lattice. For small doping level $x$, the low-temperature anisotropy increases with increasing $x$, while its high-temperature counterpart decreases.

Figure 6

In-plane, $\nu =\parallel$, and out-of-plane, $\nu =\perp$, uniform magnetic susceptibilities ${\chi }_{\nu }$ in units of $g_{\parallel }^2$ and $g_{\perp }^2$, respectively, as a function of temperature for different values of ${\mathrm{\Gamma }}_1^{\prime }$ (in units of meV). These results were obtained from classical MC simulations on an $L=24$ lattice with $J_1, K_1, {\mathrm{\Gamma }}_1$, and $J_3$ chosen as in model 1. Inset: Susceptibilities for ${\mathrm{\Gamma }}_1^{\prime }=0.25$ meV in a log-log scale. The dashed lines show the results from the high-temperature expansion.