Semiclassical Routes to the $α$-RuCl$_3$ Scattering Continuum via Model Meta$-$Analysis

TL;DR

This study examines the origin of a broad, Γ-point-centered spin continuum in the proximate Kitaev magnet $\alpha$-RuCl$_3$ by combining semi-classical stochastic Landau-Lifshitz dynamics with quantum-corrections to compute the temperature evolution of the dynamical spin structure factor across correlated and conventional paramagnetic regimes. Through a meta-analysis of 38 proposed exchange-parameter sets and Bayesian optimization, the authors identify parameter families that reproduce the observed continuum, including a best-fit set that features a dominant off-diagonal $\Gamma'$ term, and demonstrate that spectral features collapse when temperatures are scaled by the Curie-Weiss scale $\bar{\theta}_{\mathrm{CW}}$, indicating the high-$T$ dynamics are governed primarily by correlated paramagnetism rather than Kitaev fractionalization. Complementary 24-site exact diagonalization clarifies finite-size effects and the proximity to zigzag/incommensurate order, yielding distinct low-temperature spectra that help discriminate between competing parameterizations. The resulting data-driven, reproducible framework enables inference of effective spin models in magnets with broad continua and offers a path to applying this approach to other candidate Kitaev materials beyond $\alpha$-RuCl$_3$.

Abstract

$α$-RuCl$_3$ is a leading material for proximate Kitaev magnetism. We address the origin of the broad, $Γ$-point centered excitation continuum observed by inelastic neutron scattering at elevated temperatures in this compound. Using stochastic Landau-Lifshitz dynamics augmented with quantum-equivalent corrections, we reproduce the temperature-dependent dynamical spin structure factor across both the correlated and conventional paramagnetic regimes. A meta-analysis of 38 published exchange parameter sets identifies those most consistent with the full temperature evolution. A Bayesian optimization procedure is used to derive parameters that capture the low-energy star-like momentum dependence and the overall bandwidth of the continuum. Rescaling temperatures by the Curie--Weiss scale produces a collapse of spectral measures, demonstrating that the high-$T$ dynamics are governed by correlated paramagnetism below $θ_{\mathrm{CW}}$ rather than by the Kitaev crossover to fractionalized excitations. Complementary 24-site exact diagonalization clarifies finite-size systematics at low temperature and the proximity to zigzag/incommensurate ordering. Beyond $α$-RuCl$_3$, our simulation pipeline provides a reproducible, data-driven framework to infer effective spin models in magnets that exhibit broad continua.

Figures (12)

• Figure 1: Crystal and magnetic structures, exchange interactions, and interpretation of temperature-dependent regimes in $\alpha$-RuCl$_3$. (a) Illustration of the crystal structure of $\alpha$-RuCl$_3$ where the top honeycomb shows the arrangement of RuCl$_6$ octahedra with magenta spheres denoting chloride ions and gray spheres denoting ruthenium ions. The bottom left honeycomb illustrates the zigzag long-range magnetic order with first- and third-nearest-neighbor intralayer couplings $J$ and $J_3$. The bottom right honeycomb depicts the ideal Kitaev model, where three orthogonal Ising axes are associated with bonds of different colors. The bottom matrices present the exchange-interaction tensor of the generalized Heisenberg–Kitaev model for each bond, defined in the cubic frame (with axes shown inside the bottom right honeycomb). (b) Schematic representation of the temperature-dependent regimes in the pure Kitaev model. The yellow hexagon and red circles indicate $W_p\!=\!1$ and $W_p\!=\!-1$ , black dots represent itinerant Majorana fermions, and the blue ellipsoid denotes localized Majorana fermions, and black arrows indicate the uncorrelated paramagnetic phase. (c) Schematic representation of the temperature-dependent regimes expected in a model of $\alpha$-RuCl$_3$ treating spins semi-classically. The color of the arrows is determined by the spin component along the $\bf a$ direction.
• Figure 2: Temperature evolution of the spin dynamics for several minimal models of $\alpha$-RuCl$_3$. (a) Temperature dependence of momentum- and energy- slices through the neutron scattering intensity calculated by LLD for $\mathcal{H}_{\rm min}$. The data in the first column were extracted from Ref. Do2017. The momentum transfer ${\bf Q}=(H,K,0)$ follows high-symmetry directions in the 2D BZ (see Fig.\ref{['fig:8']} for definition). Data and simulations were integrated over the out-of-plane momentum transfer $\Delta L$ = [-2.5, 2.5] r.l.u. Each column represents a different parameter set originating from a different determination method. Each row was simulated (or measured in the case of the experimental data) at the same ratio $T/\bar{\theta}_{\rm CW}$. (b) Temperature dependence of constant-energy cuts across the data from panel (a) with colors indicating the energy integration range $\Delta E$ in meV. Colored circles indicate the data [first column of (a)], and the colored area with solid lines indicates our best parameter set [second column of (a)].
• Figure 3: Temperature dependence of the momentum-dependence of low-energy excitations. (a) Constant-energy cut through the INS data of Ref. Do2017 (left column) compared to our optimized LLD simulations (right column). The effective temperature $T/\bar{\theta}_{\mathrm{CW}}$ for each row is written on the right side of the figure. White dashed lines represent the Brillouin zone boundary. The data and simulations were integrated over $\Delta E = [1.5, 3]$ meV. (b) Corresponding line cuts of the intensity along the ${\bf Q} = [1, 0, 0]$ direction (gold color) and ${\bf Q} = [1, 1, 0]$ direction (blue color). Colored circles indicate the INS data, and solid lines show the LLD simulations. Two colored dashed lines on each row indicate the high-symmetric M (or K) points for each direction. The grey area in the data was masked due to the contamination from the direct beam.
• Figure 4: Temperature dependence of the magnetic signal at the Brillouin zone center ${\bf Q} = \Gamma$. (a) Temperature-energy map of the INS intensity at the (repeated) Brillouin zone center extracted from Ref. Do2017. (b) Corresponding LLD simulations for our optimized parameters. The intensity was integrated by $\Delta H = \pm 0.12$ and $\Delta K = \pm 0.2$ r.l.u in the $(H,K,0)$-plane with similar $L$-integration as Figs. \ref{['fig:2']} and Figs. \ref{['fig:3']}. (c) Corresponding energy-dependent line cuts at selected values of $T/\bar{\theta}_{\mathrm{CW}}$. The black dashed lines are guides representing the vertical plotting offset for each temperature. (d) Line cuts of energy-integrated neutron scattering intensity as a function of temperature. Each color is labeled as a different energy integration range. Dots and diamond marks indicate the experimental data, and lines indicate the LLD simulations, respectively.
• Figure 5: Evaluation of goodness-of-fit ($\chi^2$) with previously reported models. The definition of the numbering of each model is given in the Supplementary information. (a) Summary of the $\chi^2$ values of previously suggested parameter sets in the chronological sequence. Each parameter sets were categorized by the absence of $J_3$ and $\Gamma'$ with different symbols and colors. (b) Presentation of $\chi^2$ for each parameter set. Color indicates the value of the $\chi^2$. Grey lines in (a-b) represent the minimum value of $\chi^2$.