Excitations in the field-induced quantum spin liquid state of alpha-RuCl3

The Kitaev model on a honeycomb lattice predicts a paradigmatic quantum spin liquid (QSL) exhibiting Majorana Fermion excitations. The insight that Kitaev physics might be realized in practice has stimulated investigations of candidate materials, recently including alpha-RuCl3. In all the systems studied to date, non-Kitaev interactions induce magnetic order at low temperature. However, in-plane magnetic fields of roughly 8 Tesla suppress the long-range magnetic order in alpha-RuCl3 raising the intriguing possibility of a field-induced QSL exhibiting non-Abelian quasiparticle excitations. Here we present inelastic neutron scattering in alpha-RuCl3 in an applied magnetic field. At a field of 8 Tesla, the spin waves characteristic of the ordered state vanish throughout the Brillouin zone. The remaining single dominant feature of the response is a broad continuum centered at the Gamma point, previously identified as a signature of fractionalized excitations. This provides compelling evidence that a field-induced QSL state has been achieved.

The Kitaev model on a honeycomb lattice [1] has been exactly solved to reveal a unique quantum spin liquid (QSL) exhibiting itinerant Majorana Fermion and gauge-flux excitations. The Kitaev candidate system α-RuCl3 is an insulating magnetic material comprised of van der Waals coupled honeycomb layers of 4d 5 Ru 3+ cations nearly centered in edge-sharing RuCl6 octahedra. A strong cubic crystal field combined with spin-orbit coupling leads to a Kramer's doublet, nearly perfect J = 1/2 ground state [2][3][4], thus satisfying the conditions necessary for producing Kitaev couplings in the low energy Hamiltonian [5]. Similar to the widely studied honeycomb [6] and hyper-honeycomb [7] Iridates, at low temperatures α-RuCl3 exhibits small-moment antiferromagnetic zigzag order [3,[8][9][10][11] with TN ≈ 7 K for crystals with minimal stacking faults. In the zigzag state the magnetic excitation spectrum shows well-defined low-energy spin waves with minima at the M points (See Supplementary Materials (SM) Fig. S1 for the Brillouin Zone (BZ) definition) as well as a broad continuum that extends to much higher energies centered at the Γ points [12,13]. Above TN the spin waves disappear but the continuum remains, essentially unchanged until high temperatures of the order of 100 K [3,12,13]. In analogy with the situation for coupled spin-½ antiferromagnetic Heisenberg chains [14], the high energy part of the continuum has been interpreted as a signature of fractionalized excitations [3,12,13]. The overall features of the inelastic neutron scattering (INS) response resemble those of the Kitaev QSL [15][16][17] and are consistent with an unusual response seen in Raman scattering [16,18,19], suggesting that the system is proximate to a QSL state exhibiting magnetic Majorana fermion excitations [3,12,13]. Magnetic field offers a clean quantum tuning parameter for Kitaev materials [7][8][9]20] and can be applied on large single crystals facilitating INS studies. It is known to suppress the magnetic order in α-RuCl3 [8,9,[21][22][23][24][25][26] raising the intriguing possibility of a field-induced QSL [20]. It is thus of great interest to investigate the nature of the excitations in the magnetic fieldinduced disordered state.
To investigate this phenomenon, high-quality single crystals of α-RuCl3 were grown using vapor-transport techniques [3,10]. Figure 1 shows bulk susceptibility and neutron diffraction measurements, demonstrating the suppression of the zigzag order (Fig. 1a) when the field is applied along the ζ = (-1, 2, 0) (trigonal notation, see e.g., ref. [10][11][12] or equivalent directions (Fig. 1b). The cusp in the susceptibility occurs at TN = 7.5 K at 0.1 T, close to the location of the zero-field heat capacity anomaly in single-phase RuCl3 crystals with ABC magnetic stacking [10]. With increasing field, TN shifts to lower temperatures and the transition is not observed beyond BC = 7.3 (3) T.
Magnetic Bragg peaks associated with the zigzag spin order appear below TN at (½, 0, L), (0, ½, L) and (½, -½, L) where L ≠3n [9][10][11][12]21]. Consistent with Sears et al. [21], magnetic fields of 2 T along {1 1 0}-equivalent directions completely suppress the intensity of magnetic Bragg peaks with Q ⊥ B as seen in Fig. 1c. Conversely, peaks with a significant projection of Q along B gain intensity at low fields with increasing fields. This intensity redistribution can signify a reorientation of the ordered moments to lie perpendicular to the field direction, or a depopulation of domains with moments aligned along the field direction [21]. The weighted average intensity of magnetic peaks with L = 1 is roughly flat up to B ≈ 3.5 T and then follows a downward trend. (See Figs. S2 and S3 for further details on the behavior of (½, 0, 1) and (½, 0, 2) orders with applied field to 2 T). direction (see Methods). The scattering at 2 K in zero field (Fig. 2a) shows low-energy, gapped spin waves with the expected minima at the (±1/2, 0, L) M points. The spin waves also show local minima at the Γ (H = 0) and Y (H = 1) points. The scattering near the Γ point, however, is dominated by a broad continuum extending to higher energies, consistent with previous measurements [3,12,13].
Figures 2b-e, show the evolution of the spectrum at 2 K in 2 T increments from B = 2 to 8 T. At B = 2 T the spin wave intensity is increased. This is consistent with expectations for the neutron scattering cross-section. As noted above, one likely explanation (Fig. 1c) is at 2 T the ordered moment direction becomes perpendicular to the applied field, and nearly parallel to (1/2, 0, 1). This naturally leads to a decreased intensity for (1/2, 0, 1) peaks and an enhanced scattering intensity for the concomitant M-point spin waves (Fig.   2b) since the later represent fluctuations of components perpendicular to the ordered moment. The spin waves persist as the field is increased to 4 T (Fig. 2c), and begin to lose intensity by 6 T (Fig. 2d). At 8 T, above BC, spin-wave scattering is totally suppressed and the intensity of the continuum is enhanced (Fig. 2e). The response is an intense, and apparently gapped, column of scattering at the Γ point. Over most of the energy range, the 8 T spectrum bears strong resemblance to the zero-field spectrum just above TN (Fig. 2f) [12,13].
Further details of the in-field spectra are presented in Fig. 3. Constant-Q cuts at the M point between 0 and 6 T in Fig. 3a show a spin wave at E = 2.25 ± 0.11 meV, consistent with previous zero-field experimental results [12,27]. The cuts verify that the spin-wave intensity is enhanced at 2 T, gradually reduced by 6 T, and completely absent at 8 T, consistent with the suppression of magnetic order above BC. Figure 3b shows constant-Q cuts at the zone center (Γ point, H = 0). The zero-field cut at 2 K exhibits a peak at E = 2.69 ± 0.11 meV, consistent with a recent observation of the zone-center magnon by THz spectroscopy [28]. The scattering is largely unchanged up to 4 T. Conversely, at 6 T the low energy scattering has "filled-in", suggesting that the energy gap at the Γ point has closed. On further increasing the field to 8 T, the scattering consists of a broad peak centered around 3.4 meV that merges into the higher energy continuum.
Extrapolation of the low-energy Γ-point spectrum suggests a re-opening of a new gap by 8 T. The appearance of a field-induced gap in the spectrum of spin excitations above BC has been observed in NMR [26] and inferred from thermal transport [24] measurements. Figure 3c plots the wave-vector dependence of the T = 2 K scattering at different fields, integrated over the energy interval [5 , 7] meV. As seen previously [12,13], the scattering profile is a peak centered at the Γ point. At low temperatures and low fields the peak width is broader, reflecting contributions associated with enhanced correlations related to the zigzag order [12]. A similar comparison of the T = 2 K, B = 8 T with the T = 15 K, zero-field scattering in Fig. 3d demonstrates near quantitative agreement between the two magnetically disordered states. This is true over the available data range for energy transfers above the maximum of the spin wave band.
It has been shown previously [12,13] that the energy continuum observed at the Γ point in zero field both above and below TN is similar to the T = 0 response function of a Kitaev QSL [15][16][17], and therefore may be a signature of fractionalized excitations associated with proximity to the QSL. Combined exact diagonalization and DMRG calculations [20] for extended Kitaev-Heisenberg Hamiltonians have provided indications for a magnetic field-induced transition from zigzag order to a gapped QSL state. The disappearance of spin waves at BC, combined with the appearance of a gap in the continuum excitation spectrum that is unconnected with spin waves provides significant evidence for such a field-induced QSL in α-RuCl3. This observation is consistent with the interpretation of NMR experiments by Baek et al. [26].
The removal of magnetic long-range order by the field enables the comparison of the measured energy-dependent scattering to a QSL-based theory over a large portion of the bandwidth of the magnetic excitations. The full effective magnetic Hamiltonian describing α-RuCl3 has not been definitively determined so that the concomitant dynamic response functions remain unknown. In view of this, as a starting point it is reasonable to compare the results to exact calculations for a Kitaev QSL. Previous such comparisons [3,12] of the zero-field measurements were restricted to high-energy features with relatively small spectral weight. The expected scattering is broad in energy for both ferromagnetic (FM) and antiferromagnetic (AFM) Kitaev QSLs, however in zero field the momentum dependence of the scattering was seen to be similar to the response calculated for an AFM Kitaev QSL at T = 0. With a phenomenological extension of the calculations to nonzero T, the full energy dependence of the intensity at the Γ point appears to be closer to the response calculated for a FM Kitaev model in an effective magnetic field (see supplementary materials, Figs. S5, S6). The latter exhibits the following features that are qualitatively consistent with experiment: For temperatures comparable to, or larger than, the flux gap, the signal near Q = 0 is a broad continuum, exhibiting only a moderately intense peak at the lower-energy threshold. Secondly, the high frequency part of the spectrum is resilient as a function of temperature (up to T ~ JK, the Kitaev constant) or magnetic field (both below and above Bc). Finally, with field, the low frequency response acquires a low-energy gap with an intensity enhanced at higher energies, similar to the Γ point continuum scattering seen in α-RuCl3 above Bc (Fig. 3b). This fits with the idea that To conclude, we have shown that with a magnetic field applied in the {110} direction, long-range magnetic order in α−RuCl3 disappears above a threshold field BC, and the high-frequency spectrum of magnetic excitations for B > BC resembles that expected for a Kitaev QSL. This QSL is expected to be topologically different from the zero-field Kitaev QSL, and hence the next challenge is to identify and explore experimentally its most compelling signatures. These may include topologically protected edge states and quasiparticle excitations with non-Abelian statistics [1], which have generated much enthusiasm about topological quantum computation [30][31][32].

Methods:
Synthesis and bulk characterization: Single crystals of α-RuCl3 were prepared using vaportransport techniques from pure α-RuCl3 powder as described previously [10,12]. Crystals grown by the same method have been extensively characterized via bulk and neutron scattering techniques [10,12]. All samples measured in the current work exhibit a single magnetic phase at low temperature with a transition temperature Tc ~ 7 K, indicating high crystal quality with minimal stacking faults [10]. DC magnetization and AC susceptibility in 12 mg and 17 mg single crystals, respectively, were collected in DC fields of up to 14 T in a Quantum Design Physical Property Measurement System (PPMS). The magnetic field was applied along the reciprocal {1 1 0} directions as identified by Laue diffraction as described in , as well as the out-of-plane peaks with L= ±2. The data were reduced using Mantid [34].
Diffraction measurements were also obtained on the same 740 mg crystal measured used in the inelastic study using the HB1A and HB3 triple axis instruments at the High Flux Isotope Reactor (HFIR). For both HFIR measurements the sample was aligned with the (H, 0, L) scattering plane horizontal and with an applied vertical field of up to 5 T. An incident energy Ei=14.7 meV was used. Detailed mesh scans and order parameter measurements confirmed that the magnetic phase transition remains sharp as the field is increased from zero (Fig. S3) and the in-plane magnetic peaks remain at the commensurate position.
Inelastic scattering experiments: Single-crystal inelastic neutron scattering measurements were carried out on a 740 mg α-RuCl3 crystal in an 8 T vertical-field cryomagnet using the HYSPEC instrument at SNS [35]. The sample was aligned in the horizontal (H, 0, L) scattering plane, with the magnetic field (B) parallel to the vertical ζ = (−1, 2, 0) direction (See Fig. 1b). Hz, which increases the intensity, but decreases the energy resolution, both by a factor of 2, as compared to 240 Hz. The 15 K data in Fig. 3(d) is scaled by the factor of 2 to allow a direct comparison of the intensity with the 2 K data taken at 240 Hz. These settings provide reasonable (Q, E) coverage and resolution with which to examine the spin-wave spectrum in the (H, 0, 0) direction, however using Ei = 17 meV, the higher-energy part of the sample spectrum (i.e., above roughly 6 meV) is limited by kinematic restrictions. Data were collected in 1° steps as the sample was rotated through 200° about the vertical axis for every temperature and field condition. Empty aluminum sample holder measurements were performed under identical conditions as the sample measurements and the resultant scattering has been subtracted from all data presented in the manuscript. Reduction of the raw data was carried out using standard data reduction routines and Python codes available within Mantid software [34].    The field dependence hints at a discontinuous jump close to 6 T (crossover region, shaded.) Lines are guides to the eye. (f) AC susceptibility (Re(χAC)) measured at T = 2 K with a frequency of 1 kHz shows two anomalies at BC1 = 6.1 ± 0.5 T and BC2 = 7.3 ± 0.3 T. In all panels  respectively. The Tc and the exponent β have only a feeble field dependence in this range, with average Tc = 6.91 ± 0.04 K, and average β = 0.126 ± 0.03. These values match zerofield results reported in a different sample before [11,12]. The lines in panel (c) are splines serving as guides to the eye. Errorbars represent 1 σ. Here we provide details of our theoretical calculation of the dynamical response of the Kitaev QSL. The Kitaev model stands out as an example of a non-trivial interacting spin system beyond one dimension for which even dynamical properties can be calculated exactly [15][16][17]. The actual Hamiltonian describing the material comprises additional interactions beyond the Kitaev exchange; these are not generally agreed upon but at any rate, for a generic model Hamiltonian the calculation of physical observables is typically restricted to purely numerical approaches [S2-S5] which instead suffer, e.g., from finite size effects. Hence, we follow our previous approach [2,12,13] of comparing INS measurements to reliable calculations of an arguably fine-tuned pure Kitaev Hamiltonian.

C. Energy profile of spin-waves at various Q.
In the following, we sketch a phenomenological extension beyond exact solubility of our calculations to include nonzero temperatures and the magnetic field.
The addition of a magnetic field spoils the solubility of the pure Kitaev model, but as was already noted by Kitaev [30], time reversal symmetry can instead be broken with a soluble effective three spin interaction heff, which may perturbatively be interpreted as an effective magnetic field. However, establishing a quantitative equivalence is not possible, not least because this also depends on the precise form of the full Hamiltonian, for example heff~h 3 for a pure Kitaev Hamiltonian but heff~h Γ 2 for a Kitaev model with a small spin-off diagonal Gamma (Γ) term [S5]. Here, we therefore concentrate on the main qualitative features of the effect of breaking TRS and use its strength as an undetermined parameter.
We have shown previously that good quantitative results of the dynamical structure factor at the Gamma point S(q=0,ω) are obtained by a scattering problem of Majorana fermions on a local pair of Z2 fluxes [15]. Here we employ an extension of this idea to finite temperature by averaging this 'adiabatic response' over a disordered flux background [S6].
Approximating fluxes as non-interacting yields a thermally excited background flux density Δ/ in terms of the flux gap ∆ in the form of the Fermi function 1/ 1 . In the notation of Ref. [16] the response is obtained from: Figure S5 shows the resulting dynamical structure factor at the Gamma point for a system of 800 spins sampled over 2000 appropriately sampled random flux configurations. As the effective field heff is increased at low temperature (T=∆=0.065JK), the response at low frequencies is suppressed, with weight being transferred into a peak which grows as it moves to higher frequencies. Although the 6T data may be slightly affected by the remnants of weak spin wave intensity, it is very interesting to note that the calculation is in good qualitative agreement with the trend of the experimental data as the field is increased.
As an aside, we also show in Fig. S6 a comparison between FM and AFM Kitaev coupling both at low/high T (solid/dashed lines for T= ∆/7.5∆) and zero/nonzero value of the timereversal symmetry-breaking term (black/red lines for heff = 0/0.1JK). For both FM and AFM models an applied field results in an upward renormalization of the main peak in the data.
Notably, the low-frequency portion of the FM response is most strongly temperature dependent. While considerable uncertainties with respect to the precise microscopic Hamiltonian remain, for the data presented here, the FM choice of JK , resulting in a peak in the response on the low energy side, resembles the data more closely. This would then be in keeping with the idea that the dominant interaction in α-RuCl3 is a FM Kitaev-type exchange [13, 20, S3, S4], which is consistent with the observed sensitivity of the magnetization to an applied magnetic field [26].
We finally emphasize that a hallmark of the Kitaev QSL in a magnetic field is provided by the appearance of Majorana flux bound states. While it might be difficult to separate a broadened peak in a small gap from the large continuum response, it would be desirable to measure at temperatures well below the flux gap and also for a number of magnetic fields above Bc. Combined observations of the characteristic scaling of the gap, position and intensity of the peak [S6] would be the strong signature of the non-Abelian Kitaev QSL.