## Abstract

Quantum spin liquid is a disordered but highly entangled magnetic state with fractional spin excitations^{1}. The ground state of an exactly solved Kitaev honeycomb model is perhaps its clearest example^{2}. Under a magnetic field, a spin flip in this model fractionalizes into two types of anyon, a quasiparticle with more complex exchange statistics than standard fermions or bosons: a pair of gauge fluxes and a Majorana fermion^{2,3}. Here, we demonstrate this kind of fractionalization in the Kitaev paramagnetic state of the honeycomb magnet **α**-RuCl_{3}. The spin excitation gap determined by nuclear magnetic resonance consists of the predicted Majorana fermion contribution following the cube of the applied magnetic field^{2,4,5}, and a finite zero-field contribution matching the predicted size of the gauge flux gap^{2,6}. The observed fractionalization into gapped anyons survives in a broad range of temperatures and magnetic fields, which establishes **α**-RuCl_{3} as a unique platform for future investigations of anyons.

## Main

In many-body systems dominated by strong fluctuations, an excitation with a well-defined quantum number can break up into exotic quasiparticles with fractional quantum numbers. Well-known examples include fractionally charged quasiparticles in fractional quantum Hall effect^{7}, spin–charge separation in one-dimensional conductors^{8} and magnetic monopoles in spin ice^{9}. A major hunting ground for novel fractional quasiparticles is disordered magnetic states of interacting spin-1/2 systems governed by strong quantum fluctuations, called quantum spin liquids (QSLs). Most of their models predict that a spin-flip excitation fractionalizes into a pair of spinons, each carrying spin 1/2 (ref. ^{1}). Even more interesting in this respect is the Kitaev model^{2} of *S* = 1/2 spins on a two-dimensional (2D) honeycomb lattice with nearest neighbours interacting through an Ising exchange, whose axis depends on the bond direction, as shown in Fig. 1a. This is one of a few exactly solved 2D models supporting a QSL ground state. According to the solution, a spin flip fractionalizes into a pair of gauge fluxes and a Majorana fermion^{2,3}. As both types of quasiparticle behave as anyons under the magnetic field, they could potentially be used for decoherence-free topological quantum computation^{2}. The experimental detection of such anyons is thus the primary goal of current QSL research.

As fractional quasiparticles are always created in groups, their common signature is a continuous spin excitation spectrum, observed in recent QSL candidates on the kagome and triangular lattices^{10,11}, instead of sharp magnon modes found in ordered magnets. A Kitaev QSL also exhibits this feature^{6,12}, as well as additional, specific signatures, all related to the fact that fractionalization in this case leads to two types of quasiparticle. First, the fractionalization proceeds in two steps, with both types releasing their entropy at different temperatures^{13}. Second, although Majorana fermions themselves are gapless in zero magnetic field, the response of the QSL to a spin flip is gapped due to the inevitable simultaneous creation of a pair of gapped gauge fluxes^{6}. Third, in the presence of an external magnetic field, the Majorana fermions also acquire a gap, which is predicted to grow with the characteristic third power of the field in the low-field region^{2,4,5}. Currently, α-RuCl_{3} is the most promising candidate for the realization of the Kitaev QSL^{12,14,15,16,17}. Among the listed signatures, a spin excitation continuum was observed by Raman spectroscopy^{12,14} and inelastic neutron scattering^{15,16,17}, and the two-step thermal fractionalization was confirmed by specific-heat measurements^{17}, all in zero field. However, an application of a finite field, which should affect the gaps of both types of quasiparticle in different ways, is crucial to identify them. Using nuclear magnetic resonance (NMR), we determine the field dependence of the spin excitation gap *Δ* shown in Fig. 1c, which indeed exhibits a finite zero-field value predicted for gauge fluxes and the cubic growth predicted for Majorana fermions. This result clearly demonstrates the fractionalization of a spin flip into two types of anyon in α-RuCl_{3}.

α-RuCl_{3} is structurally related to the other two Kitaev QSL candidates, Na_{2}IrO_{3} (ref. ^{18}) and α-Li_{2}IrO_{3} (ref. ^{19}). All three are layered Mott insulators based on the edge-sharing octahedral units, RuCl_{6} and IrO_{6} (Fig. 1a), respectively, and driven by strong spin–orbit coupling^{20}, which together lead to a dominant Kitaev exchange coupling between the effective *S* = 1/2 spins of Ru^{3+} and Ir^{4+} ions, respectively^{21}. A monoclinic distortion of the IrO_{6} octahedra in both iridate compounds results in non-Kitaev exchange interactions between the spins, which lead to the low-temperature magnetic ordering and thus prevent the realization of the QSL ground state. Judging by the lower transition temperature with respect to the Kitaev exchange coupling *J*_{K}, these interactions are smaller in α-RuCl_{3} (refs ^{22,23,24}). Signatures of fractional quasiparticles should be sought in a region of the phase diagram outside the magnetically ordered phase, at temperatures low enough that the Kitaev physics is not yet destroyed by thermal fluctuations. This is the Kitaev paramagnetic phase (Fig. 1b) extending to a relatively high temperature of around 100 K, roughly half of *J*_{K} = 190 K (ref. ^{17}).

The boundary of the magnetically ordered phase measured in a large α-RuCl_{3} single crystal (see Methods) using ^{35}Cl NMR is displayed in Fig. 1b. The magnetic response of α-RuCl_{3} is known to be highly anisotropic^{23,25}, mainly due to the anisotropic Ru^{3+}*g*-tensor (Fig. 1a) with the components *g*_{ab} = 2.5 in the *a*–*b* plane and \({g}_{{c}^{*}}=1.1\) along the *c** axis (ref. ^{26}). Namely, the effect of an applied magnetic field **B** is described by the Zeeman term, which is proportional to \(\mathop{g}\limits_{}{\bf{B}}\) with magnitude *g*(*ϑ*)*B*, where \(\mathop{g}\limits_{}\) is the *g*-tensor and \(g(\vartheta )=\sqrt{{g}_{ab}^{2}{{\rm{\cos }}}^{2}\vartheta +{g}_{{c}^{* }}^{2}{{\rm{\sin }}}^{2}\vartheta }\) is the direction-dependent *g*-factor. Therefore, if **B** is applied at an angle *ϑ* from the *a*–*b* plane, it produces the same effect as the field with magnitude *B*_{ab} = *g*(*ϑ*)*B*/*g*_{ab} applied in the *a*–*b* plane. We exploit this to scan efficiently the phase diagram as a function of *B*_{ab} by varying both the direction and the magnitude of an applied field, instead of varying only the magnitude of a field applied in the *a*–*b* plane, as is usually done^{25}. This approach is valid if the *g*-tensor is the only source of anisotropy, a condition to be verified at the end (Fig. 1c). As shown in the inset of Fig. 1b, we determine the transition temperature *T*_{N2} as the onset of NMR line broadening (see Methods) monitored on the dominant NMR peak (inset of Fig. 2d). The obtained phase boundary extending up to the critical field *B*_{c} ≈ 8 T matches the result of a recent reference study^{25}. The observed transition temperature *T*_{N2} of around 14 K near zero field is consistent with a considerable presence of the two-layer *AB* stacking in the monoclinic *C*2/*m* crystal structure (Fig. 1a), in addition to the three-layer *ABC* stacking, which is characterized by a lower transition temperature *T*_{N1} of around 7 K in zero field^{15,24}. As our study is focused on the Kitaev paramagnetic region (Fig. 1b) governed by the physics of individual layers, it is not affected by the particular stacking type.

To detect and monitor the spin excitation gap as a function of the magnetic field, we use the NMR spin–lattice relaxation rate \({T}_{1}^{-1}\), which directly probes the low-energy limit of the local spin–spin correlation function and thus offers a direct access to the spin excitation gap. Figure 2a shows an exact theoretical \({T}_{1}^{-1}(T)\) dependence numerically calculated for the ferromagnetic Kitaev model in zero field^{27,28} adapted to the case of α-RuCl_{3} (see Methods). It is dominated by a broad maximum, which is a sign of thermally excited pairs of gauge fluxes over the two-flux gap^{27}, whose exact value amounts to *Δ*_{0} = 0.065*J*_{K} (refs ^{2,6}). The Kitaev paramagnetic phase is located between the lower temperature*T*_{L} = 0.012*J*_{K}, where gapped gauge fluxes start to be excited, and the higher temperature *T*_{H} = 0.375*J*_{K}, where thermal fluctuations already destroy the short-range spin correlations^{13,27}. As shown in Fig. 2a, in this broad temperature range, the theoretical zero-field \({T}_{1}^{-1}(T)\) dataset can be nicely reproduced with the empirical expression

where *Δ* is the field-dependent spin excitation gap, and the fitting procedure gives *n* = 0.67 when setting *Δ* = *Δ*_{0} for zero field. In the following, we use equation (1) with this value of *n* to extract *Δ* from the experimental \({T}_{1}^{-1}(T)\) datasets. Due to the unusual prefactor *T*^{−1} on the right-hand side of equation (1), *Δ* is encoded in the specific concave shape of the curve with the maximum at a temperature *nΔ*. As shown in the second inset of Fig. 2a, the gap is also directly accessible from the negative slope −*nΔ* of the linear dependence of \({\rm{ln}}\left({T}_{1}^{-1}T\right)\) on *T*^{−1}.

A broad maximum characteristic of the Kitaev spin excitations indeed appears in the ^{35}Cl \({T}_{1}^{-1}(T)\) dataset recorded in 9.4 T for the magnetic field orientation *ϑ* = 0° (*B*_{ab} = 9.4 T), as shown in Fig. 2c. The dataset is excellently reproduced with equation (1) using *Δ* = 46.8 K in the temperature range up to *T*_{H} = 0.375*J*_{K} ≈ 70 K for *J*_{K} = 190 K (ref. ^{17}). In the \({T}_{1}^{-1}(T)\) dataset for *ϑ* = 90° (*B*_{ab} = 4.1 T), a maximum would apparently develop at a lower temperature, if the dataset was not disrupted by a magnetic ordering transition at *T*_{N2} = 12 K (Fig. 2c). Between *T*^{*} ≈ 17 K, where the critical fluctuations preceding the magnetic ordering vanish (see Methods and Supplementary Information), and *T*_{H}, the dataset is perfectly reproduced with equation (1) using *Δ* = 13.8 K. A large difference between the two determined gaps in Fig. 2c points to a considerable *Δ*(*B*_{ab}) variation in the Kitaev paramagnetic phase. Below *T*_{N2}, two \({T}_{1}^{-1}\) components develop, both exhibiting a steep drop, one below *T*_{N2} and the other one below *T*_{N1} = 8 K. These two phase transitions were observed before and ascribed to the presence of *AB* and *ABC* stackings, respectively^{15,24}. Fitting the data below *T*_{N2} and *T*_{N1} with the expression \({T}_{1}^{-1}\propto {T}^{2}{\rm{\exp }}\left(-{\it{\Delta} }_{{\rm{m}}}{\rm{/}}T\right)\) valid for gapped magnon excitations in a three-dimensional magnetically ordered state (see Methods) gives comparable values of the magnon gap *Δ*_{m} = 32 K and 35 K, respectively, implying the same low-energy physics in both cases. The obtained values are consistent with the gap of 29 K determined by inelastic neutron scattering^{15,29}. Finally, the temperature-independent part of both \({T}_{1}^{-1}(T)\) datasets above 120 K (≈2*T*_{H}) indicates a crossover into a classical paramagnetic state (see Methods), in accordance with the theoretical dataset in Fig. 2a and with the experimental result of ref. ^{17}.

Although equation (1) is empirical, its functional form already bears signs of the involved fractional spin excitations. To show this, we first note that similar expressions are obtained for more conventional gapped magnon excitations in magnetic insulators at low temperatures \(T\ll {\it{\Delta} }_{{\rm{m}}}\) (see Methods). In that case, the prefactor *T*^{−1} is replaced by a more general *T*^{p} originating from the magnon density of states *g*(*E*) (where *E* is the energy of magnons), which depends on the dimensionality *D*, while *n* is generally the number of magnons involved in the process. For *n* = 1 (single-magnon scattering) and a quadratic dispersion relation for magnons, one obtains *p* = *D* − 1 ≥ 0, while higher *n* (multi-magnon scattering) lead to even higher powers *p* (see Methods). At higher temperatures *T* ≈ *Δ*_{m}, the effective *p* changes, but always remains positive. Therefore, a very unusual, theoretically confirmed *p* = −1 in equation (1) valid over a broad temperature range cannot be obtained for magnons. This, together with a fractional number *n* = 0.67 of the involved spin-flip excitations obtained in Fig. 2a, indicate that equation (1) is actually specific to the presence of fractional spin excitations.

In the ferromagnetic Kitaev model, the magnetic susceptibility *χ* exhibits a moderate, almost monotonic temperature dependence (Fig. 2b), in contrast to the non-monotonic \({T}_{1}^{-1}(T)\) behaviour (Fig. 2a)^{27,28}. As shown in Fig. 2c,d, we indeed observe contrasting temperature dependences of \({T}_{1}^{-1}\) and the local susceptibility monitored by the ^{35}Cl NMR shift in 9.4 T for *ϑ* = 0°. While both observables should exhibit a similar gapped behaviour in the presence of gapped magnon excitations, such a contrast between them can arise only in the presence of at least two types of fractional quasiparticle that enter the two observables in different ways^{27}.

To obtain *Δ* as a function of *B*_{ab} in Fig. 1c, the \({T}_{1}^{-1}(T)\) datasets in Fig. 3 taken in magnetic fields of different directions and magnitudes are fitted to equation (1) in the temperature range of the Kitaev paramagnetic phase. As the curve \({T}_{1}^{-1}\propto {T}^{-1}\) defined by *Δ* = 0 is steeper than any dataset in this range or, equivalently, as the datasets in the insets of Fig. 3 all exhibit a negative slope in this range, the obtained excitation gaps are obviously all finite. The inset of Fig. 1c showing the symmetric *Δ*(*ϑ*) dependence around 90° in 9.4 T, where *ϑ* traverses non-equivalent directions with respect to the Ising axes on both sides (inset of Fig. 1a), demonstrates that the *g*-tensor is indeed the only source of anisotropy as assumed when introducing *B*_{ab}. The obtained *Δ*(*B*_{ab}) in Fig. 1c can be perfectly reproduced as a sum of two terms: the two-flux gap *Δ*_{0} = 0.065*J*_{K} (refs ^{2,6}) and the gap acquired by Majorana fermions in a weak magnetic field, theoretically predicted to be proportional to the cube of the field^{2,4,5} (see Methods)

where \(\widetilde{B}={g}_{ab}{\mu }_{{\rm{B}}}{B}_{ab}{\rm{/}}{k}_{{\rm{B}}}\) is the field in kelvin units, *k*_{B} is the Boltzmann constant, *μ*_{B} is the Bohr magneton and *α* accounts for the sum over the excited states in the third-order perturbation theory, which is the origin of the \({\widetilde{B}}^{3}\) term^{2}. The fitting procedure leads to *α* = 1.2 and *J*_{K} = 183 ± 10 K, in perfect agreement with the value of 190 K determined by inelastic neutron scattering^{17}. This result demonstrates that a spin-flip excitation in α-RuCl_{3} indeed fractionalizes into a gauge flux pair and a Majorana fermion.

Focusing on the low-field Kitaev paramagnetic region in the phase diagram of α-RuCl_{3} in Fig. 1b is essential for our identification of two types of anyon. Instead, other recent experimental studies focused on the low-temperature region above *B*_{c}, observing the spin excitation continuum^{30} with either a gapless behaviour^{31} or the gap opening linearly^{32,33,34,35} or sublinearly^{36} with *B* − *B*_{c}, but without definite conclusions about the identity of the involved quasiparticles. These contradictory conclusions probably originate from the presence of additional, non-Kitaev interactions between the spins^{15,26,29,37}, whose role should be pronounced particularly at low temperatures. Our result shows that spin fractionalization into two types of anyon is robust against these interactions in a broad range of temperatures and magnetic fields. This is the main practical advantage of α-RuCl_{3} with respect to all other anyon realizations such as the fractional quantum Hall effect in 2D heterostructures^{7} or hybrid nanowire devices^{38} where anyons are observed only at extremely low temperatures and for certain field values. Our discovery thus establishes α-RuCl_{3} as a unique platform for future investigations of anyons.

## Methods

### Crystal growth

Crystals of α-RuCl_{3} were synthesized from anhydrous RuCl_{3} (Strem Chemicals). The starting material was heated in vacuum to 200 °C for one day to remove volatile impurities. In the next step, the powder was sealed in a silica ampoule under vacuum and heated to 650 °C in a tubular furnace. The tip of the ampoule was kept at lower temperature and the material sublimed to the colder end during one week. Phase pure α-RuCl_{3} (with a high-temperature phase of *C*2/*m* crystal structure) was obtained as thin crystalline plates. The residual in the hot part of the ampoule was black RuO_{2} powder. The purified α-RuCl_{3} was sublimed for the second time to obtain bigger crystal plates. The phase and purity of the compounds were verified by powder X-ray diffraction. All handling of the material was done under strictly anhydrous and oxygen-free conditions in glove boxes or sealed ampoules. Special care has to be taken when the material is heated in sealed-off ampoules. If gas evolves from the material, this may result in the explosion of the ampoule.

### NMR

The ^{35}Cl NMR experiments were performed on a foil-like α-RuCl_{3} single crystal of approximate dimensions 5 × 5 × 0.1 mm^{3} in a continuous-flow cryostat, allowing us to reach temperatures down to 4.2 K. When handling the sample, we took extreme care to minimize its exposure to air. A thin NMR coil tightly fitting the sample was made from a thin copper wire with 20–40 turns, depending on the required tuning frequency determined by the external magnetic field. The coil was covered with a mixture of epoxy and ZrO_{2} powder, which was allowed to harden to ensure the rigidity of the coil. The coil was then mounted on a Teflon holder attached to a rotator, which allowed us to vary the orientation of the sample with respect to the external magnetic field. To reduce the noise of an already weak ^{35}Cl NMR signal, a consequence of the extremely broad ^{35}Cl NMR spectrum, we used a bottom-tuning scheme. With the output radio frequency power of around 20 W, a typical π/2 pulse duration was 2 μs. The NMR signals were recorded using the standard spin-echo π/2 − *τ*_{d} − π pulse sequence with a typical delay of *τ*_{d} = 70 μs (much shorter than the spin–spin relaxation time *T*_{2}) between the π/2 and π pulses.

###
*T*
_{1} relaxation

The spin–lattice relaxation (*T*_{1}) experiment was carried out using an inversion recovery pulse sequence, *φ*_{i} − *τ* − π/2 − *τ*_{d} − π, with an inversion pulse *φ*_{i} < π (suitable for broad NMR lines) and a variable delay *τ* before the read-out spin-echo sequence. The spin–lattice relaxation datasets were typically taken at 20 increasing values of *τ*. The datasets were analysed using the model of magnetic relaxation for nuclear spin *I* = 3/2, appropriate for ^{35}Cl, monitored on the central −1/2 ↔ 1/2 transition:

where *T*_{1} is the spin–lattice relaxation time and *s* is the inversion factor. In the region of the phase diagram outside the magnetically ordered phase (Fig. 1b), this expression reproduces the experimental relaxation curves perfectly. In the magnetically ordered phase, two *T*_{1} components appear, and the relaxation curves are reproduced as a sum of two terms of the form given by equation (3). For instance, the temperature dependence of the corresponding two *T*_{1} values for *B* = 9.4 T with *ϑ* = 90° is given in Fig. 2c. In cases where only a narrow temperature region below the transition was covered, the two components in the relaxation curves were hard to separate, and we used equation (3) with a stretched exponent instead.

### Relation between orientation and field dependence of *T*
_{1}

Using the direction of the applied magnetic field (described by the angle *ϑ* from the crystal *a*–*b* plane) in addition to its magnitude to change the value of *B*_{ab} allows us to cover low *B*_{ab} values while keeping the applied magnetic field *B* high. This is beneficial for two reasons related to the strong quadrupole broadening of the ^{35}Cl NMR spectrum (inset of Fig. 2d and Supplementary Fig. 2): to minimize an already large NMR line width, and to keep the Larmor frequency well above the quadrupole splitting, which is of the order of 10 MHz as concluded in the following. The validity of this approach is supported by the fact that *Δ*(*B*_{ab}) data points for various angles *ϑ* and field values 2.35 T, 4.7 T and 9.4 T in Fig. 1c all collapse onto a smooth experimental curve. The *Δ*(*B*_{ab}) data points taken in lower fields obviously exhibit much larger error bars. Namely, the corresponding \({T}_{1}^{-1}(T)\) datasets in Fig. 3b are more scattered than the datasets taken in 9.4 T despite a much longer averaging for noise reduction.

### The boundary of the magnetically ordered phase

We measured the temperature dependence of the dominant ^{35}Cl NMR peak in magnetic fields of various magnitudes and various directions with respect to the crystal *a*–*b* plane of the sample, thus covering various *B*_{ab} values. For *B*_{ab} < 8 T, the frequency width of the peak exhibits a clear kink as a function of temperature (inset of Fig. 1b and Supplementary Fig. 3), which indicates the onset of NMR line broadening at the phase transition into the magnetically ordered state. Plotting the temperature of the kink as a function of *B*_{ab} in Fig. 1b (and in the inset of Supplementary Fig. 3), we obtain the boundary of the magnetically ordered phase, which perfectly matches the result of the reference study^{25}.

### Contributions to the NMR shift

The temperature dependence of the NMR frequency shift of the dominant NMR peak measured in 9.4 T with the field in the *a*–*b* plane (*ϑ* = 0°) is plotted in Fig. 2d and reproduced in Supplementary Fig. 4. To separate the magnetic contribution to the NMR shift from the temperature-independent quadrupole contribution, we plot the relative NMR shift (the NMR shift divided by the ^{35}Cl Larmor frequency *ν*_{L} = 39.18 MHz) against the rescaled magnetic susceptibility *χ*_{ab} in the inset of Supplementary Fig. 4. In ref. ^{25}, an experimental ratio between the susceptibility *χ* of the powdered sample and the susceptibility *χ*_{ab} of the single crystal with a field applied in the *a*–*b* plane is obtained as (2 + *r*)/3 with *r* = 0.157, leading to *χ*_{ab} = 3*χ*/(2 + *r*). We use this empirical relation to evaluate *χ*_{ab}(*T*) from our field-cooled *χ*(*T*) dataset shown in Supplementary Fig. 1. As we did not measure susceptibility in high magnetic fields, we rely on the dataset taken in 1.0 T. This is valid in a broad temperature range, except at low temperatures where this dataset starts to deviate from the high-field susceptibility^{22}. The relative shift is found to depend linearly on *χ*_{ab} up to 20 × 10^{−3} e.m.u. mol^{−1} (inset of Supplementary Fig. 4), that is, the NMR shift follows *χ*_{ab} down to 35 K (Supplementary Fig. 4). The proportionality constant 1.95 mol e.m.u.^{−1} between the relative shift and *χ*_{ab} translates to 1.09 T/*μ*_{B}. When further multiplied by the relevant *g*-factor, *g*_{ab} = 2.5, and divided by 2 (as each ^{35}Cl is coupled equally to two Ru^{3+}*S* = 1/2 spins), we obtain the component *A* = 1.35 T (in the *a*–*b* plane) of the hyperfine coupling tensor between ^{35}Cl and the Ru^{3+}*S* = 1/2 spin. In addition, a zero-temperature relative shift −0.039, when multiplied by *ν*_{L}, gives the quadrupole shift Δ*ν*_{Q} = −1.53 MHz.

From the obtained quadrupole shift, we can estimate the quadrupole splitting *ν*_{Q} between the successive ^{35}Cl NMR transitions. For the case of an axially symmetric electric field gradient (EFG) tensor and the field applied at an angle *ϑ*′ from the principal EFG axis *v*_{ZZ} with the largest EFG eigenvalue, the second-order quadrupole shift is given by \({\rm{\Delta }}{\nu }_{{\rm{Q}}}=-3{\nu }_{{\rm{Q}}}^{2}\)(1 − cos*ϑ*′^{2})(9cos*ϑ*′^{2} − 1)/(16*ν*_{L}) for the *I* = 3/2 nucleus^{39}. As the axes of the EFG tensor are not known, we assume a 45° typical tilt of *v*_{ZZ} from *c*^{*}, so that *ϑ*′ ≈ 45°. From the previously evaluated Δ*ν*_{Q}, we then obtain *ν*_{Q} ≈ 14.2 MHz. This is an estimate of the quadrupole splitting between the central ^{35}Cl NMR transition and the satellite transitions. We can thus conclude that the NMR peaks in the covered frequency range (inset of Fig. 2d and Supplementary Fig. 2) all belong to the central transition.

### Justification for the onset temperature *T*
^{*}

A simple estimate of the onset temperature *T*^{*} for the critical spin fluctuations preceding the magnetic ordering at *T*_{N2} can be obtained using a random phase approximation to express the dynamic susceptibility *χ* of α-RuCl_{3}. As the pure Kitaev magnet does not magnetically order, the magnetic ordering of α-RuCl_{3} is a consequence of additional non-Kitaev interactions. If we denote the effective coupling of all these additional interactions by *J*′, the random phase approximation reads *χ* = *χ*_{1}/(1 − *J*′*χ*_{1}), where *χ*_{1} is the dynamic susceptibility of a single Kitaev plane. As \({T}_{1}^{-1}(T)\) can be reasonably well approximated by a negative power law above *T*^{*} (as evident from the *ϑ* = 90° dataset in Fig. 2c, which is linear in a log–log scale), the same holds for *χ*_{1}, that is, *χ*_{1} = *cT*^{−r} with *r* > 0, where *c* is the proportionality constant. A condition *J*′*χ*_{1}(*T*_{N}) = 1 for the phase transition at *T*_{N} can then be written as \(cJ^{\prime} {T}_{{\rm{N}}}^{-r}=1\) This allows us to express *J*′ with *T*_{N}, so that the deviation of the dynamic susceptibility from the pure Kitaev case can finally be written in the form

The ratio *χ*/*χ*_{1} diverges when approaching *T*_{N} from above and gradually drops to 1 with increasing temperature. The temperature scale for this drop is apparently determined by *T*_{N}, which leads to the characteristic estimate *T*^{*} ≈ *T*_{N} + *T*_{N} = 2*T*_{N} consistent with our experimental determination (see Supplementary Information). Precisely such values of *T*^{*} with respect to *T*_{N} are also experimentally found in systems of coupled spin chains or ladders, whose *χ*_{1} values exactly obey power-law temperature dependences^{40,41}.

### Theoretical \({T}_{1}^{-1}(T)\) for Kitaev spin excitations

The theoretical temperature dependence of \({T}_{1}^{-1}\) is numerically calculated for the Kitaev model in zero field in ref. ^{27}. \({T}_{1}^{-1}\) contains two contributions, one coming from a single fluctuating spin (on-site) and the other one coming from fluctuating nearest-neighbouring spins in the Kitaev honeycomb lattice. As the ^{35}Cl nucleus in α-RuCl_{3} is located at equal distances from the closest two Ru^{3+}*S* = 1/2 spins (Fig. 1a), \({T}_{1}^{-1}\) generally contains both contributions. We evaluate their relative weights for the case of a magnetic field applied along *c*^{*} (*ϑ* = 90°) and assuming an isotropic hyperfine coupling *A*. A general expression for \({T}_{1}^{-1}\)(ref. ^{42}) can then be written as

where *γ* is the nuclear gyromagnetic ratio (26.2 MHz T^{–1} for ^{35}Cl), *t* is time, while *S*_{1} and *S*_{2} are the two involved Ru^{3+}*S* = 1/2 spins with relevant components perpendicular to the field direction, that is, along *a* and *b* (Fig. 1a). In the orthogonal system defined by the Ising axes *x*, *y* and *z* (Fig. 1a), unit vectors along *a*, *b* and *c*^{*} are written as

This allows us to switch from the spin components along *a* and *b* in equation (5) to the spin components along *x*, *y* and *z*, which are appropriate for the Kitaev model. Taking into account also the isotropy of the Kitaev model, that is, \(\left\langle {S}_{1}^{x}(t){S}_{1}^{x}\right\rangle\) = \(\left\langle {S}_{1}^{y}(t){S}_{1}^{y}\right\rangle\) = \(\left\langle {S}_{1}^{z}(t){S}_{1}^{z}\right\rangle\) and similar for the correlation functions \(\left\langle {S}_{1}^{x}(t){S}_{2}^{x}\right\rangle\) and \(\left\langle {S}_{2}^{x}(t){S}_{2}^{x}\right\rangle\), equation (5) simplifies to

Two terms under the integral represent precisely the on-site and nearest-neighbouring-sites contributions with the weights 4 and 4/3, respectively. A similar calculation for the magnetic field applied along *a* or *b* gives the same result. The theoretical curve plotted in Fig. 2a is based on these weights. As the curves for the on-site and nearest-neighbouring-sites contributions are almost identical up to *T*_{H} (ref. ^{27}), the values of the weights do not affect the form of equation (1).

### \({T}_{1}^{-1}\) in the classical paramagnetic state

In this state, nearest-neighbouring spins are not correlated, so that only the on-site spin correlations contribute to \({T}_{1}^{-1}\). A contribution of the correlation function \(\left\langle {S}_{1}^{x}(t){S}_{1}^{x}\right\rangle\) amounts to \({T}_{1x}^{-1}\) = \(\sqrt{{\rm{\pi }}}{\gamma }^{2}{A}^{2}\hslash {\rm{/}}\left(4J\sqrt{z}\right)\) (ref. ^{43}), where *ħ* is Planck’s constant divided by 2π, *J* is the exchange coupling (in this case, *J*_{K} = 183 K, as determined in Fig. 1c) and *z* is the number of neighbouring spins with the relevant component of the coupling (*x* in this case, *z* = 1 for the Kitaev honeycomb lattice). Using equation (7), we then obtain \({T}_{1}^{-1}\) = \(4{T}_{1x}^{-1}\) = \(\sqrt{{\rm{\pi }}}{\gamma }^{2}{A}^{2}\hslash {\rm{/}}{J}_{{\rm{K}}}\), which evaluates to 92 s^{−1} using *A* = 1.35 T determined above from Supplementary Fig. 4. This value is consistent with the observed values of 121 s^{−1} and 146 s^{−1} above 120 K (Fig. 2c) for *ϑ* = 0° and *ϑ* = 90°, respectively. In addition, these temperature-independent values are reached at around 2*T*_{H} (*T*_{H} = 0.375*J*_{K} ≈ 70 K), precisely as in the theoretical dataset in Fig. 2a. Both these conclusions demonstrate that the classical paramagnetic state is indeed reached.

###
*T*
_{1} relaxation due to gapped magnons

When spin fluctuations in the magnetic lattice are due to excited magnons, the corresponding spin–lattice relaxation rate for a single-magnon process is given by^{44}

where *f*(*E*) = [exp(*βE*) − 1]^{−1} is the Bose–Einstein distribution function and *β* = 1/(*k*_{B}*T*). Denoting the magnon gap by *Δ*_{m} (in kelvin units), we define *ε* = *E* − *k*_{B}*Δ*_{m} as the energy measured from the bottom of the magnon band. The power-law dispersion relation *ε* ∝ *k*^{s} in *D* dimensions, which includes the standard parabolic dispersion (*s* = 2) and the Dirac dispersion (*s* = 1) as special cases, leads to *g*(*E*) ∝ *ε*^{D/s − 1}. For low temperatures \(T\ll {\Delta }_{{\rm{m}}}\), *f*(*E*) can be approximated with the Boltzmann distribution, *f*(*E*) ≈ exp(−*βE*) = exp(−*Δ*_{m}/*T*)exp(−*βε*). Plugging these expressions for *g*(*E*) and *f*(*E*) into equation (8), we obtain

The integral on the right-hand side of equation (9) converges if *s* < 2*D* and evaluates to *Γ*(2*D*/*s* − 1) where *Γ* is the gamma function. We can thus rewrite equation (9) as

with the power of the prefactor *p* = 2*D*/*s* − 1. In the case of *D* = 2, which is relevant for the Kitaev honeycomb magnet, we obtain *p* = 1 for *s* = 2, and *p* = 3 for *s* = 1, so that the power *p* cannot be negative. Even in the case of *D* = 1, *p* can only approach the lowest value of 0 precisely for *s* = 2 (although care should be taken in this case, as the integral in equation (9) then formally diverges). If more than a single magnon is involved in the *T*_{1} process, the power *p* is also positive and becomes even higher^{44}. Gapped magnons thus cannot lead to the *T*_{1} relaxation described by equation (10) with *p* < 0, which is the case in equation (1).

Instead, we can use equation (9) in the three-dimensional magnetically ordered state, when the elementary excitations are indeed magnons with a gap *Δ*_{m}. In this case, *D* = 3 and *s* = 2, and this leads to \({T}_{1}^{-1}\propto {T}^{2}{\rm{\exp }}\left(-{\it{\Delta} }_{{\rm{m}}}{\rm{/}}T\right)\). We use this expression to analyse the \({T}_{1}^{-1}(T)\) data (Fig. 2c) in the low-temperature ordered state of α-RuCl_{3}.

All these examples show that a frequently used simple gapped model \({T}_{1}^{-1}\propto {\rm{\exp }}{(-{\it{\Delta} }_{{\rm{s}}}/T)}_{}^{}\) with the gap *Δ*_{s}, which was used before to analyse the \({T}_{1}^{-1}{(T)}_{}^{}\) datasets in α-RuCl_{3} (ref. ^{32}), is actually not justified in any region of the phase diagram of α-RuCl_{3}.

### Majorana fermion gap

In the presence of an external magnetic field, Majorana fermions in the Kitaev model acquire a gap^{2}. This is shown for a field applied perpendicularly to the honeycomb plane, that is, in the (111) direction in the coordinate system defined by the Kitaev axes *x*, *y* and *z*. In this case, the Zeeman term reads \({{\mathscr{H}}}_{{\rm{Z}}}=-h{\sum }_{j}\left({S}_{j}^{x}+{S}_{j}^{y}+{S}_{j}^{z}\right)\), where \(h=g{\mu }_{{\rm{B}}}B{\rm{/}}\sqrt{3}\) is a single component of the magnetic field *B* in energy units and *g* is the *g*-factor. When treated as a perturbation to the Kitaev Hamiltonian, the Zeeman term contributes to the Majorana fermion gap only at third order^{2}. The corresponding effective Hamiltonian is thus proportional to *h*^{3} and can be written as^{2,4,5}

where *Δ*_{0} is the two-flux gap (in kelvin units), while *α* (of the order of unity) accounts for the sum over the excited states, and its exact value is not known. The Kitaev model extended with such a three-spin exchange term \(-\kappa {\sum }_{jkl}{S}_{j}^{x}{S}_{k}^{y}{S}_{l}^{z}\) with \(\kappa =3\alpha {h}^{3}{\rm{/}}\left({k}_{{\rm{B}}}^{2}{\it{\Delta} }_{0}^{2}\right)\) is still exactly solvable, and the dispersion relation of the Majorana fermions is calculated as^{4}

where *i* is the square root of −1, **k** is the Majorana fermion momentum, and **a**_{1} and **a**_{2} are the base vectors of the honeycomb lattice. The dispersion relation given by equation (12) is gapped for *κ* ≠ 0, and the corresponding Majorana fermion gap *Δ*_{f} can be calculated numerically as a function of *κ* and thus as a function of the magnetic field. For small magnetic fields, that is, for \(\kappa \ll {k}_{{\rm{B}}}{J}_{{\rm{K}}}\), the Majorana fermion gap (in kelvin units) simplifies to

while for high magnetic fields, it saturates to *Δ*_{f} = 2*J*_{K}. The total spin excitation gap *Δ* is obtained by adding *Δ*_{f} to the two-flux gap *Δ*_{0} as in equation (2). The whole field dependence of *Δ* is shown in Supplementary Fig. 8 for *J*_{K} = 183 K as determined in Fig. 1c, *g* = *g*_{ab} and *α* = 1.2 (leading to the best fit of our *Δ*(*B*_{ab}) data points). The cubic approximation given by equation (13), which is plotted in Fig. 1c and, for comparison, in Supplementary Fig. 8, is apparently valid up to 15 T, well beyond the field range covered in this work.

### Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

M.K. acknowledges discussions with M. Horvatić and C. Berthier. The work was partly supported by the Slovenian ARRS program No. P1-0125 and project No. PR-07587. A.B. and Ch.R. acknowledge financial support by the Marie Curie FP7 COFUND PSI Fellowship programme, the Swiss National Science Foundation (Sinergia Network Mott Physics Beyond the Heisenberg Model), and the ERC Grant Hyper Quantum Criticality (HyperQC).

## Author information

### Affiliations

#### Jožef Stefan Institute, Ljubljana, Slovenia

- Nejc Janša
- , Andrej Zorko
- , Matjaž Gomilšek
- , Matej Pregelj
- & Martin Klanjšek

#### Department of Chemistry and Biochemistry, University of Bern, Bern, Switzerland

- Karl W. Krämer
- & Daniel Biner

#### Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, Villigen, Switzerland

- Alun Biffin
- & Christian Rüegg

#### Department of Quantum Matter Physics, University of Geneva, Geneva, Switzerland

- Christian Rüegg

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### Contributions

M.K. conceived, designed and led the project. N.J. and M.K. performed the NMR experiments and analysed the data. K.W.K. and D.B. grew the samples. A.B. performed the magnetic susceptibility measurements. All the authors discussed the results. M.K. wrote the paper with feedback from all the authors.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Martin Klanjšek.

## Supplementary information

### Supplementary Information

Supplementary Figures 1–9, Supplementary References

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