The range of non-Kitaev terms and fractional particles in RuCl$_3$

Significant efforts have focused on the magnetic excitations of relativistic Mott insulators, predicted to realize the Kitaev quantum spin liquid (QSL). This exactly solvable model involves a highly entangled state resulting from bond-dependent Ising interactions that produce excitations which are non-local in terms of spin flips. A key challenge in real materials is identifying the relative size of the non-Kitaev terms and their role in the emergence or suppression of fractional excitations. Here, we identify the energy and temperature boundaries of non-Kitaev interactions by direct comparison of the Raman susceptibility of RuCl3 with quantum Monte Carlo (QMC) results for the Kitaev QSLs. Moreover, we further confirm the fractional nature of the magnetic excitations, which is given by creating a pair of fermionic quasiparticles. Interestingly, this fermionic response remains valid in the non-Kitaev range. Our results and focus on the use of the Raman susceptibility provide a stringent new test for future theoretical and experimental studies of QSLs.

In these candidate materials, as with other QSL candidates, the presence of additional symmetry allowed terms (Heisenberg and bond-dependent off-diagonal interactions in this case), produces long range magnetic order. 10,10,29,37 Despite extensive studies and evidence for fractional particles 11,34,38,39 , the relative size of the non-Kitaev terms and the range over which they are relevant remains controversial 37,40 . In α-RuCl 3 , these non-Kitaev terms lead to a magnetically ordered phase below 7 K, which could be destroyed by an in-plane magnetic field [41][42][43][44][45] . The exact nature of the field induced QSL state remains unclear 27, 38, 42 as the zero field Hamiltonian is still unresolved.
In particular, non-Kitaev interactions dominant energy and temperature ranges have not yet been experimentally established. Additionally, there is a need to determine if excitations in these ranges maintain their fractional nature.
Raman scattering is a powerful probe of magnetic materials, revealing the presence of long range order, symmetry and statistics of the excitations, as well as the strength and nature of the exchange, even in single 2D atomic layers 3, 46-53 . Indeed, Raman scattering was the first to reveal the continuum from magnetic excitations in α-RuCl 3 11 . However, a careful study of the Kitaev term's temperature and energy dependence is still a challenge, as one requires a very high temperature and energy resolution to show the spectral change and directly compare the spectra with theoretical calculations 54 . Previously, Raman efforts relied on spectral integration over a certain energy range which averaged out the energy dependence of the excitations, and, the low scattering intensity made it difficult to directly compare the spectra with theoretical calculations from the exact Kitaev model. As such the role of the non-Kitaev terms, and their size, could be identified in previous efforts. Furthermore, demonstration of the fractional nature relied on the integrated Raman intensity and thus required subtraction of a bosonic background, without justification. This approach also meant fitting the data with an average energy in the fermi function, further limiting the ability to uncover if the non-Kitaev terms affected the statistical response of the excitations. 26, 34 .
Here, we overcome all these previous limitations with new Raman spectra with dramatically improved signal levels, high temperature and energy resolution. Firstly, having improved the optics, our Raman measurements now obtain a signal level 18 times larger than before 11 55 We also checked that the susceptibility integration is governed by a Fermi function with half energy, which further confirms each fractional particle holds one half of the scattering energy in both Kitaev and non-Kitaev dominant regimes. Interestingly this is revealed without the need to subtract the bosonic background.
In inelastic light scattering, the measured intensity is determined by symmetry, Fermi's golden rule, and from the fluctuation-dissipation theorem, is proportional to the Raman susceptibility (Im(χ[ω, T ])) times a Bose function 56,57 . In magnets this can produce peaks from single magnons, broad features reflecting the two-magnon joint density of states (JDos), or QES from thermal fluctuations 11,13 . For the Kitaev QSL, Raman predominantly excites pairs of fractional particles in the energy range considered here (≈ 0.5J K < ω <≈ 2J K ), leading to the energy loss (I S [ω, T ]) and gain (I aS [ω, T ]) intensities 34,58 : As shown in Fig. 1a, we collected both the Stokes and anti-Stokes spectra of bulk α-RuCl 3 from 10 K to 300 K. Our Rayleigh scattering half width is 2.3 meV, enabling measurement down into the low energy regime. The temperature dependent spectra show a clear magnetic excitation continuum (2.3meV ∼ 10 meV) below the first phonon, which mostly results from the Kitaev interaction and is consistent with previous predictions and measurements 11,26,34,39 Fig. 1b shows the comparison of the QMC results for the pure Kitaev limit and the Raman susceptibility at 10 and 40 K. While excellent agreement is seen at 40 K, the data at 10 K only matches the model between 6 to 10 meV. Noting that this temperature is still above the magnetic ordering temperature of 7 K, this additional susceptibility results from non-Kitaev terms, as recently suggested by exact diagonalization (ED) calculations 40 .
To further investigate the temperature and energy dependence of the non-Kitaev interactions, we consider the energy and temperature dependent colormap in Fig. 1c. Here χ δ = χ measured − χ QM C is the difference between the measured Raman susceptibility and that of the pure Kitaev model (determined by the QMC calculation). The green color indicates the measured susceptibility is higher than the QMC results and the blue color indicates regions of very good overlap between the measurement and the calculation. The black dots suggest the temperature and energy boundaries where the system perfectly resembles the pure Kitaev QSL. Specifically, there is a large χ δ in the quarter circle area below 6 meV and 40 K, which can be explained as the region where non-Kitaev interactions become dominant in the response. The deviation above 150 K and below 8 meV results from the QES induced by thermal fluctuations in the system, which is well known in frustrated magnets 7,11,26,48,59 . The high energy deviation (>12meV) is from the low energy tail of the phonon. Nonetheless, the low energy and temperature deviation from the pure Kitaev model is consistent with the calculated intensities of recent ED results for a model with only Γ and Kitaev terms in the system (K-Γ model) 40 with −J K /Γ = 5. Furthermore, the ED results suggest enhanced response over that expected for the pure Kitaev limit for ω Γ ∼ 2.5Γ. As shown in our colormap, when the temperature is low, the disagreement occurs for ω Γ < 6 meV. based on K-Γ model, this suggests Γ ≈ 2.4 meV, where the Heisenberg interaction and terms beyond nearest neighbors are neglected. We note that regardless of the specific non-Kitaev terms, this can be interpreted as an upper bound on the ratio of Kitaev to non-Kitaev terms in this system. Furthermore, we find the best agreement for the pure Kitaev limit with J K = 10 meV (see Supplemental Sup_Fig. 4), consistent with our observed bandwidth of the continuum (Fig. 1a) of 30 meV. 11,58 The Γ ≈ 2.4 meV we obtained here is also consistent with the results obtained from neutron scattering (2.5 meV) 29 , from thermal Hall measurements (2.5 meV) 60 , and from THz measurements (2.4 meV) 61 .
Having established the size and extent of the non-Kitaev terms, we examine the statistics of the excitations in α-RuCl 3 to see if they are truly fractional. As the statistics depends on both temperature and energy, one should make sure the system is in detailed balance 56 and that laser heating is negligible, which was not quantitatively shown before. As discussed in the supplemental, the fermionic response written above is consistent with the fluctuation-dissipation theorem with the presence of time-reversal symmetry, requiring ously, the discrepancy between the prediction of the Bose factor and the measured intensity at low temperatures was attributed to fractional statistics 26, 34 . However, these works did not exclude the possibility that laser induced heating kept the measured area at a fixed temperature, while the bulk was cooled. This is not unlikely, given the small specific heat and thermal conductivity of RuCl 3 at low temperatures 55, 62-65 . Furthermore, as described in the supplemental, previous uses of the anti-Stokes responses were unreliable due to the low signal levels 11 . Most importantly, unless the temperature is well known, it is difficult to directly compare with the theoretical prediction for fractional statistics. In our current work we have made substantial improvements to the thermal anchoring and collection efficiency to allow for much higher temperature resolution and lower Raman frequency. In this way, we can observe the spectra change between different temperatures and directly compare it with QMC. Most importantly, due to enhanced signals and lower probing frequencies, we have been able to collect anti-Stokes response at lower temperatures to ensure that laser induced heating is not an issue. Returning to the actual sample temperature, in Fig. 2d, we compare the anti-Stokes intensity and Stokes intensity times a Boltzmann factor with the measured temperature. The excellent agreement between them reveals that there is nearly no heating in the laser spot and thus we can use the measured crystal temperature. Unlike previous studies 11,26 , our new quantitative comparison between Stokes and anti-Stokes limits the possibility of laser heating to explain the low temperature upturn and confirms the sample is in detailed balance.
We explore the possibility that the Raman susceptibility results from purely fermionic excitions in Fig. 2a. If the excitation is fractional, one expects Im[χ(ω, To cancel the constant term and focus only on the fermionic part, we show The utility of such an analysis is quite clear: the energy and temperature extent of the continuum can be directly observed -without contributions from high temperature QES fluctuations or phonons. To test the predicted fermionic response from fractional particles, we plot For Raman scattering, it is a particle creation/absorption process, so the temperature and energy dependent of fractional particles is determined by occupation, which is described by Fermi function. Therefore, the good agreement between the data and Fermi functions with half of the scattering energy provides strong evidence for the presence of pairs of fractional particles. We note this is done without any artificial subtraction of a bosonic background. This approach relies on a nearly temperature and energy independent JDos[ω, T ], which is expected from numerical calculations for the Kitaev system at temperatures above the flux gap 55 . This assumption appears to generally hold in our data, whose temperature an energy evolution are generally described by a Fermi function. Nonetheless at the lowest tempera-tures, there is some deviation of the data for energies above 6 meV. The origin of this discrepncy is not clear, but likely results from a the temperature and energy dependence of the JDos[ω, T ]. Additionally, we find poor agreement if the full scattering energy (n F [ω, T ]) is used (not shown).
We additionally performed the same analysis on another honeycomb system Cr 2 Ge 2 Te 6 ( Fig. 2b), which was grown by established methods and which is ferromagnetic below 60 K with a similar Curie-Weiss temperature as α-RuCl As discussed later, the response at higher temperatures is consistent with quasi-elastic scattering.
We note that the exact temperature at which the response will set in, depends on the energy scale at which it is measured. As such the integrated response investigated in Fig. 3C, appears to have a higher onset temperature for the QES due to the inclusion of higher energy scales. Specifically, a Lorentzian at zero energy results from thermal fluctuations of the magnetism that confirm the magnetic specific heat is consistent with a standard paramagnet at high temperatures. Lastly, this analysis also provides new insights into the phonons overlapping the continuum. Specifically, consistent with previous works we also find the phonons have a low energy tail due to their coupling with the continuum (see Fig. 1c and supplemental Fig. 4b). Thus with just three parameters, one fixed by the lowest temperature, we fully explain the SW for all energy ranges, temperatures, and polarizations. To further confirm this, we tried the same analysis on our new Cr 2 Ge 2 Te 6 data. As shown in Fig. 3e&  Raman spectroscopy experiments. Since Raman scattering involves a photon in and photon out, it allows one to measure both the symmetry and energy change of an excitation. Furthermore, one can choose an energy and/or symmetry channel to separate the magnetic, electronic and lattice responses 11,13,26,34,52,56,59 . The majority of the Raman experiments were performed with a custom built, low temperature microscopy setup 72 . A 532 nm excitation laser, whose spot has a diameter of 2 µm, was used with the power limited to 30 µW to minimize sample heating while allowing for a strong enough signal. The sample was mounted by thermal epoxy onto a copper xyz stage.
At both room and base temperature the reported spectra were averaged from three spectra in the same environment to ensure reproducibility. The spectrometer had a 2400 g/mm grating, with an Andor CCD, providing a resolution of ≈ 1 cm −1 . Dark counts are removed by subtracting data collected with the same integration time with the laser off. To minimize the effects of hysteresis from the crystal structural transition, data was taken by first cooling the crystal to base temperature, and once cooled to base temperature, spectra were acquired either every 5 or 10 K by directly heating to that temperature. The absence of hysteresis effects was confirmed by taking numerous spectra at the same temperature after different thermal cycles (100 K in the middle of the hysteresis region). In addition, recent studies of the Raman spectra of RuCl 3 suggest an effect of the surface structure upon exposure to air 49,69 . To minimize this, crystals were freshly cleaved and immediately placed in vacuum within three minutes. Lastly, a recently developed wavelet based approach was employed to remove cosmic rays 72,73 .
Quantum Monte Carlo Calculations. The Hamiltonian of the Kitaev model on the honeycomb lattice is given by where S j represents an S = 1/2 spin on site j, and J K γ stands for a nearest-neighbor (NN) γ(= x, y, z) bond shown in Fig. 1a. In the calculation for the spectrum of the Raman scattering we adopt the Loudon-Fleury (LF) approach. The LF operator for the Kitaev model is given by where in and out are the polarization vectors of the incoming and outgoing photons and d α is the vector connecting a NN α bond 47,58 . Using this LF operator, the Raman spectrum is calculated as where R(t) = e iHt Re −iHt is the Heisenberg representation. The temperature dependence of I(ω) is numerically evaluated using the Monte Carlo simulation in the Majorana fermion representation without any approximation 16 . In the following we show the details of the calculation procedure 34 .
Using the Jordan-Wigner transformation, the Hamiltonian is mapped onto the Majorana fermion model as where (jj ) γ is the NN pair satisfying j < j on the γ bond, and η r is a Z 2 conserved quantity defined on the z bond (r is the label for the bond), which takes ±1. This Hamiltonian is simply written as using the Hermitian matrix A jk ({η r }) depending on the configuration of {η r }. The LF operator shown in Eq. (2) is also given by the bilinear form of the Majorana fermion: where B({η r }) is a Hermitian matrix. To evaluate Eq. (3), we separate the sum over the states into {c j } and {η r } parts: withĪ Wick's theorem to Eq. (8), we calculate the Raman spectrum at ω( = 0) for a given configuration  The temperature and energy dependent map of χ δ (χ δ = χ measured − χ QM C ). χ δ at low temperature and low energy range shows the temperature and energy boundary of non-Kitaev (NK) interactions in the system. χ δ at the high temperature and low energy range indicates the quasi-elastic scattering (QES) in the system.