Intermediate Field Spin(on) Dynamics in $α$-RuCl$_3$

TL;DR

This study addresses the field-induced quantum spin liquid state in $α$-RuCl$_3$ by mapping spin excitations with inelastic neutron scattering across two in-plane orientations. The INS data reveal a gapped spin-excitation continuum that remains broad and becomes more two-dimensional as the field increases, consistent with fractionalized spinons rather than conventional magnons; sharp bound-state features can emerge within the continuum. A spinon-based theory using a two-dimensional KJΓ Hamiltonian and random-phase approximation reproduces the field dependence of the gap and the emergence of relatively sharp modes bound by the continuum, supporting a KQSL interpretation. These findings provide strong evidence for fractionalization in $α$-RuCl$_3$ under field and impose stringent constraints on the microscopic Hamiltonian.

Abstract

We present comprehensive inelastic neutron spectroscopic maps of the magnetic field-induced disordered phase of the Kitaev quantum spin liquid candidate material $α$-RuCl$_3$. For fields along both in-plane high-symmetry directions we observe that the spin excitation spectrum at and above a magnetic field of 8~T is gapped. Excitation modes then sharpen for increasing field but are consistently broader than experimental resolution even at 13.5~T. The out-of-plane dispersion diminishes in the 7-10~T regime, signifying enhanced two-dimensional behavior as the in-plane liquid correlations are established. In this regime, excitations are very broad and largely flat for all accessible energy-momenta, which is kinematically at odds with a magnon-decay picture. By contrast, a continuum of fractionalized excitations naturally yields a broad continuum response, which crucially may be accompanied by sharper modes of bound states of fractionalized excitations. Their damping by the continuum accounts for the observed spectral broadening and field dependence. Our results provide strong evidence for the existence of fractionalized excitations in $α$-RuCl$_3$ in a magnetic field.

Figures (5)

• Figure 1: Evolution of continuum with $B_\perp$: (a) Honeycomb layer structure of $\alpha$-RuCl$_3$ showing the bond-dependent Kitaev interactions $K^{x}$, $K^{y}$, and $K^{z}$. We have defined the two applied magnetic field orientations as follows: $B_\perp$ was defined to be along the crystallographic $b$-axis (corresponding to ($\bar{1}~2~0$) in reciprocal space). $B_{||}$ was defined to be along the ($\bar{1}~1~0$) reciprocal space direction. (b) Schematic magnetic field vs temperature phase diagram for $\alpha$-RuCl$_3$ with field applied within honeycomb planes. For zero field at temperatures above the ordering transition the system is a correlated paramagnet (CPM), while below $T_{N}$ it orders into a 3-layer zigzag AFM order (ZZ1). Under an applied magnetic field, a 6-layer zigzag AFM phase (ZZ2) emerges at $B_{\textrm{C1}}$ before entering the quantum disordered (QD) phase at $B_{\textrm{C2}}$. For high fields, the system tends toward a partially field-polarised limit (PFP). (Inset) Picture of 2.0 g sample and aluminum mount. (c) The first (blue honeycomb) and the second (red honeycomb) 2D Brillouin Zones corresponding to panel a with some high symmetry point marked. Under field (directions marked with black arrows) the three zero-field magnetic domains become nonequivalent, as explained in Fig S1. (d-o) INS data taken throughout the intermediate and high-field regime with $B_\perp$. All data were acquired at $T$ = 0.25 K with E$_i$ = 5.5 meV throughout the intermediate and high-field limit (12 T data taken with 6.5 meV, see Methods). (d-i) Out-of-plane momentum (0 0 L) vs energy transfer pseudo-color plots. (j-o) The corresponding in-plane momentum, (H 0 1.5), vs energy transfer pseudo-color plots centered at the local minima (0 0 1.5). Integration ranges for (H 0 0), (-H 2H 0), and (0 0 L) were [-0.076:0.076], [-0.053:0.053], and [1.25:1.75] r.l.u. respectively. Some remnants of the magnet background persist after background subtraction (SI section 1).
• Figure 2: Evolution of continuum for $B_\perp$: (a) Momentum integrated cuts of data presented in Fig. 1d-o centred on (0 0 1.5) showing a flat continuum at 7-8 T. Intensities are offset linearly in field. Black arrows show onset definitions used in panel e. (b-c) In-plane momentum transfer vs energy transfer pseudocolor plots near (0 0 1.5) for 8 T and 7.3 T respectively, after subtracting the 13.5 T data as a low-energy background. (d) INS data acquired on CTAX for 7.3 T and 8 T. A clear onset is observed in the 8 T data. (e) Evolution of the spin gap at (0 0 1.5) as a function of field applied along $B_\perp$. Model-independent estimates from the data are presented for onset beyond experimental background. The visible first onset (purple dots) of the excitation reflect a more conservative estimate of the spin gap compared to previous INS results (black dots)balz2019finite. Fitted estimates using three methods (model 1-3) (red, yellow and green dots) are also provided, particularly for lower field values. Integration ranges in (a-c) for (H 0 0), (-H 2H 0), and (0 0 L) were [-0.076:0.076], [-0.053:0.053] and [1.25:1.75] r.l.u. respectively. For model-specific fits, all errorbars represent a 95% confidence interval of fitted parameters.
• Figure 3: Field evolution of spectra for $B_{||}$ between 7 T and 13.5 T: INS data taken at $T$ = 1.5 K with $E_i$ = 15 meV. (a-f) Out-of-plane momentum (0 0 L) vs energy transfer ($\Delta$E) pseudo-color plots. (g-l) The corresponding in-plane momentum vs energy transfer pseudo-color plots presented at a minimum in the out-of-plane dispersion, (H H 1.5). Data shows the two sharper modes (M) progressively broaden into continuum-like (C) excitations with decreasing field from 13.5 T to 7 T. Two further features - a sharp feature corresponding to an acoustic phonon mode sharply dispersing from (0 0 6), as well as another spurious feature near (0 0 5) appear to be field independent and hence unlikely to be related to magnetism. (m-r) Momentum integrated cuts centered at (0 0 1.5) as a function of field. For L = 1.5, we observe the smallest observed gap and considerable magnetic intensity. The results for L = 3 is presented in supplementary information with greatly reduced intensity due to the Ru$^{3+}$ form factor sarkis2024experimental. The quasielastic (QE) regime (red) was fit at 13.5 T, where the data was robustly gapped, and held fixed for the rest of the fields. A Gaussian assumption misses the low-energy spin gap (Fig. 4), so the fits are meant to be guides to the eye). As field is decreased, the system undergoes a seemingly smooth collapse of the two sharp modes into a broader continuum. At and below 8 T (near $B_{\textrm{C2}}$=7.8 T), the continuum seems to co-exist with additional sharper modes, likely connected to ZZ2 and ZZ1. Integration ranges for (-H H 0), (H H 0), and (0 0 L) were [-0.097:0.097], [-0.043:0.043], and [1.25:1.75] r.l.u. respectively. Errorbars represent 1$\sigma$.
• Figure 4: Low energy INS data for $B_{||}$: (a) Out of plane (0 0 L) Momentum vs energy transfer pseudocolor plots of INS data acquired at $E_{i}$ = 6 meV and $T$ = 1.5 K for $B_{||}$. Using the robustly gapped 13.5 T as a subtraction for the 8 T, the low energy portion of the spectra becomes cleanly gapped. (b) Momentum integrated cuts taken at (0 0 1.5) showing the clarity of the gapped onset in the 8 T data. Integration ranges for (-H H 0), (H H 0), and (0 0 L) were [-0.097:0.097], [-0.043:0.043], and [1.25:1.75] r.l.u. respectively. (c) The Spin gap evolution derived from the onset in Fig. 4b plotted as a function of field for $B_{||}$ for local both the minimum (0 0 1.5) (red circles) and maximum (0 0 3) 3D quasi-$\Gamma$ points (blue circles) (reliably extracted from the data for $B_{||}$$>$ 8 T, while for 7.5 T and 7 T, only upper limits can be estimated.) To compare with ESR measurements reported by Ponomaryov, $et~al.$ponomaryov2020nature, plots of the nearest ESR modes denoted modes "B", "C", and "F" are presented alongside our data.
• Figure 5: Spinon band structure and susceptibility: (a) Spinon bands at $|B_{||}|$ = 0.1$|K|$ (solid lines) and $|B_{||}|$ = 0.2$|K|$ (dashed lines). (b) Calculated INS intensity at zero momentum for $|B_{||}|$ = 0.1$|K|$ (dashed lines) and $|B_{||}|$ = 0.2$|K|$ (solid lines). As magnetic field increases, the broadened collective mode becomes sharper and shifts to higher energies. (c-d) INS intensity as a function of in-plane momentum and frequency at $|B_{||}|$ = 0.1$|K|$ (panel c) and $|B_{||}|$ = 0.2$|K|$ (panel d). The broad scattering arises from the two-spinon continuum. The two-spinon bound states appear as sharp collective modes, which in turn are broadened by the overlapping continuum.