Observation of two types of anyons in the Kitaev honeycomb magnet
N. Janša, A. Zorko, M. Gomilšek, M. Pregelj, K. W. Krämer, D. Biner, A. Biffin, Ch. Rüegg, M. Klanjšek
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Observation of two types of anyons in the Kitaev honeycomb magnet
N. Janˇsa, A. Zorko, M. Gomilˇsek, M. Pregelj, K. W. Kr¨amer, D. Biner, A. Biffin, Ch. R¨uegg,
3, 4 and M. Klanjˇsek ∗ Joˇzef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia Department of Chemistry and Biochemistry, University of Bern, CH-3012 Bern, Switzerland Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, CH-5232 Villigen, Switzerland Department of Quantum Matter Physics, University of Geneva, CH-1211 Geneva, Switzerland (Dated: August 22, 2017)Quantum spin liquid is a disordered magnetic state with fractional spin excitations. Its clearestexample is found in an exactly solved Kitaev honeycomb model where a spin flip fractionalizes intotwo types of anyons, quasiparticles that are neither fermions nor bosons: a pair of gauge fluxes anda Majorana fermion. Here we demonstrate this kind of fractionalization in the Kitaev paramagneticstate of the honeycomb magnet α -RuCl . The spin-excitation gap measured by nuclear magneticresonance consists of the predicted Majorana fermion contribution following the cube of the appliedmagnetic field, and a finite zero-field contribution matching the predicted size of the gauge-flux gap.The observed fractionalization into gapped anyons survives in a broad range of temperatures andmagnetic fields despite inevitable non-Kitaev interactions between the spins, which are predicted todrive the system towards a gapless ground state. The gapped character of both anyons is crucialfor their potential application in topological quantum computing. In many-body systems dominated by strong fluctua-tions, an excitation with an integer quantum number canbreak up into exotic quasiparticles with fractional quan-tum numbers. Well known examples include fraction-ally charged quasiparticles in fractional quantum Hall ef-fect [1], spin-charge separation in one-dimensional con-ductors [2], and magnetic monopoles in spin ice [3]. Amajor hunting ground for novel fractional quasiparticlesare disordered magnetic states of interacting spin-1 / / S = 1 / α -RuCl stands as themost promising candidate for the realization of the Ki-taev QSL [13, 18–21]. Among the listed signatures, aspin-excitation continuum was observed by Raman spec-troscopy [13, 18] and inelastic neutron scattering [19–21],and the two-step thermal fractionalization was confirmedby specific-heat measurements [21], all in zero field. How-ever, an application of a finite field, which should affectthe gaps of both types of quasiparticles differently, is cru-cial to identify them. Using nuclear magnetic resonance(NMR), we determine the field dependence of the spin-excitation gap ∆ shown in Fig. 1(c), which indeed ex-hibits a finite zero-field value predicted for gauge fluxesand the cubic growth predicted for Majorana fermions.This result clearly demonstrates the fractionalization ofa spin flip into two types of anyons in α -RuCl . α -RuCl is structurally related to the other two Ki-taev QSL candidates, Na IrO [22] and α -Li IrO [23].All three are layered Mott insulators based on the edge-sharing octahedral units, RuCl and IrO [Fig. 1(a)], re-spectively, and driven by strong spin-orbit coupling [24],which together lead to a dominant Kitaev exchange cou-pling between the effective S = 1 / andIr ions, respectively [25]. A monoclinic distortion ofthe IrO octahedra in both iridate compounds resultsin the presence of non-Kitaev exchange interactions be- FIG. 1:
Structure of α -RuCl and the key signature of anyons. (a) The structure of a single layer of α -RuCl in themonoclinic C /m (no. 12) setting with the monoclinic axis b ( c ∗ ⊥ a, b ). Spin-1 / ions (red spheres) at the centers of theedge-sharing RuCl octahedra (gray) form an almost perfect honeycomb lattice. Ising axes of the exchange interactions betweennearest-neighboring spins are perpendicular to the bond directions, pointing along x , y or z for blue, green and orange bonds,respectively. Red arrows show the employed magnetic field directions (described by the angle ϑ from the ab plane) with respectto the oblate Ru g -tensor (yellow ellipsoid) of axial symmetry around c ∗ . The field directions form a fan (red semicircle)perpendicular to the ab plane, at 15 ◦ from the b axis (inset). (b) Phase diagram of α -RuCl as a function of temperature T and the effective magnetic field B ab = g ( ϑ ) B/g ab (so that B ab = B for B ⊥ c ∗ ) selected by the direction-dependent g -factor.The boundary of the magnetically ordered phase extending up to B c ≈ Cl linewidth δν ( T ) (inset)and T − ( T ) (Fig. 3), matches the result of Ref. [34] (gray line). (c) The spin-excitation gap ∆ as a function of B ab (obtainedfrom the fits in Fig. 3) follows the theoretically predicted cubic dependence (blue line) with a finite initial value correspondingto the two-flux gap ∆ = 0 . J K [11] with J K = 190 K [21]. The inset shows ∆( ϑ ) for 9 . B ab ) (blue line). The only field direction outside the red fan in (a) is represented by the ϑ = 180 ◦ point. tween the spins, which lead to the low-temperature mag-netic ordering and thus prevent the realization of theQSL ground state. Judging by the lower transition tem-perature, these interactions are smaller in α -RuCl [28–31]. Signatures of fractional quasiparticles should thusbe sought in a region of the phase diagram outside themagnetically ordered phase, at temperatures where theKitaev physics is not yet destroyed by thermal fluctua-tions. This is the Kitaev paramagnetic region [Fig. 1(b)]extending to a relatively high temperature around 100 Kwhere the nearest-neighbor spin correlations vanish [21].The boundary of the magnetically ordered phase mea-sured in a large α -RuCl single crystal [33] using ClNMR is displayed in Fig. 1(b). Magnetic properties of α -RuCl are known to be highly anisotropic [30, 34], mainlybecause of the anisotropic Ru g -tensor [Fig. 1(a)] with g ab = 2 . g c ∗ = 1 . ab plane [34]. Namely, as theZeeman term contains the product gB , a magnetic field B applied at an angle ϑ from the ab plane is equiva-lent to the effective field B ab = g ( ϑ ) B/g ab applied in the ab plane, where g ( ϑ ) = q g ab cos ϑ + g c ∗ sin ϑ is thedirection-dependent g -factor. This is valid if the studiedunderlying physics is close to isotropic, a condition to beverified at the end. As shown in the inset of Fig. 1(b), we determine the transition temperature T N as the on-set of NMR line broadening [33] monitored on the domi-nant NMR peak [inset of Fig. 2(b)]. The obtained phaseboundary extending up to the critical field B c ≈ T N of around 14 K nearzero field is consistent with a considerable presence ofthe two-layer AB stacking in the monoclinic C /m crys-tal structure [Fig. 1(a)], in addition to the three-layer ABC stacking, which is characterized by a lower transi-tion temperature T N of around 7 K in zero field [19, 31].As our study is focused on the Kitaev paramagnetic re-gion [Fig. 1(b)] governed by the physics of individual lay-ers, it is not affected by the particular stacking type.To detect and monitor the spin-excitation gap as afunction of the magnetic field, we use the NMR spin-lattice relaxation rate T − , which directly probes thelow-energy limit of the local spin-spin correlation func-tion and thus offers a direct access to the spin-excitationgap. Fig. 2(a) shows the Cl T − ( T ) datasets taken onthe dominant NMR peak [inset of Fig. 2(b)] in 9 . T − ( T ) dataset for B ⊥ c ∗ (i.e., in the ab plane, B ab = 9 . T − ( T ) dataset for B k c ∗ , such a feature would appar-ently develop at a lower temperature, if the dataset wasnot disrupted by the phase transition at T N = 12 K [in a theory (a) T N2 B = 9.4 T m = K l n ( T - T ) T -1 (K -1 )50 K B || c*B ^ c* T N1 T N2 (b) m agne t i c s h i ft ( M H z ) T (K) -2 0 2 4 6 - L (MHz)T = 20 K K ) -101 N M R s h i ft ( M H z ) FIG. 2:
Evidence for the spin-excitation continuum. (a) Cl T − as a function of temperature T in 9 . B k c ∗ ( B ab = 4 . B ⊥ c ∗ ( B ab = 9 . J K = 190 K [21] is rescaled in vertical direction to matchthe B k c ∗ dataset between 17 K and 100 K. Dashed linesmark the transition temperatures T N = 12 K ( AB stacking)and T N = 8 K ( ABC stacking) into the magnetically or-dered states. Red lines are fits to T − ∝ T exp( − ∆ m /T ) forgapped magnon excitations in the 3D ordered state. Blue andgray lines are fits to Eq. (1) for fractional spin excitations inthe Kitaev paramagnet valid up to 50 K. Inset demonstratesthe resulting linear dependence of ln( T − T ) on T − below50 K. (b) Temperature dependent Cl magnetic shift (i.e.,NMR shift with subtracted quadrupole shift [33]) of the dom-inant NMR peak (the inset shows the whole central line) in9 . J K = 190 K [21] and rescaled in verti-cal direction to match the B ⊥ c ∗ dataset between 6 K and50 K. Red and blue lines are phenomenological linear fits inthe semi-log scale. field of B ab = 4 . T − compo-nents develop below T N , both exhibiting a steep drop,one below T N and the other one below T N = 8 K.These two phase transitions were observed before andascribed to the presence of AB and ABC stackings, re-spectively [19, 31]. The analysis of the data below T N and T N using the expression T − ∝ T exp( − ∆ m /T )valid for gapped magnon excitations in the 3D orderedstate [33] gives comparable values of the magnon gap∆ m = 32 K and 35 K, respectively, implying the samelow-energy physics in both cases. The obtained valuesare compatible with the gap of 29 K determined by in-elastic neutron scattering [19, 40].To access the key information held by the T − ( T )datasets in the Kitaev paramagnetic state, we first ob-serve in Fig. 2(a) that the dataset for B ⊥ c ∗ below 100 Kexhibits the same shape as the theoretical dataset numer-ically calculated for the ferromagnetic Kitaev model inzero field [33, 42]. A characteristic broad maximum of thelatter is a sign of thermally excited pairs of gauge fluxesover the two-flux gap [42], whose exact value amountsto ∆ = 0 . J K [6, 11] where J K is the Kitaev ex-change coupling. As shown in Fig. 2(a), a large part ofthe theoretical dataset, up to around 0 . J K , well abovethe maximum, can be excellently described by the phe-nomenological expression T − ∝ T exp (cid:18) − n ∆ T (cid:19) , (1)where n is set to 0 .
61 in order for ∆ to match the re-quired value of ∆ . This expression has a useful property:its maximum appears at a temperature ∆ /n , which al-lows for a simple estimate of ∆ directly from the T − ( T )dataset. The B ⊥ c ∗ ( B ab = 9 . T − T ) on T − below 50 K. Meanwhile,even in the absence of the characteristic maximum, the B k c ∗ ( B ab = 4 . T N , and up to high temperatures matches thetheoretical zero-field dataset using the value J K = 190 Kdetermined by inelastic neutron scattering, also based onthe ferromagnetic Kitaev model [21]. This means thatthe gap for 4 . = 0 . J K = 12 . B ab )variation in the Kitaev paramagnetic state. Finally, thetemperature-independent part of both T − ( T ) datasetsabove 100 K indicates a crossover into the classical para-magnetic state [41], in line with the result of Ref. [21].The expression given by Eq. (1) is not merely phe-nomenological, but reveals the presence of gapped frac-tional spin excitations. Similar expressions are obtainedfor the T relaxation due to gapped magnons in mag-netic insulators at low temperatures T ≪ ∆ [33]. In thiscase, the prefactor T − is replaced by a more general T p originating from the magnon density of states g ( E ),which depends on the dimensionality D , while n is gener-ally the number of magnons involved in the process. For n = 1 (single-magnon scattering) and a quadratic disper- o l n ( T - T ) T -1 (K -1 )50 K FIG. 3:
Determination of the spin-excitation gap ∆ . (a,b) Cl T − as a function of temperature T in 2 .
35, 4 . . ϑ . Arrows mark the transition temperatures T N into the magnetically orderedstate, defined by a weakly pronounced onset of a T − decrease on decreasing T . Solid lines are fits to Eq. (1) for fractional spinexcitations in the Kitaev paramagnet, between the temperature slightly above T N and 50 K (blue background). These allowus to determine ∆( ϑ ) and ∆( B ab ) dependencies shown in Fig. 1(c). Dashed line is the curve T − ∝ T − defined by ∆ = 0showing the largest negative slope. Insets demonstrate the resulting linear dependence of ln( T − T ) on T − in the appropriate T − range (blue background). The ϑ = 180 ◦ dataset corresponds to the only field direction outside the red fan in Fig. 1(a). sion relation for magnons, one obtains g ( E ) ∝ E D/ − and thus p = D − ≥
0, while higher n (multi-magnonscattering) lead to even higher powers p [33]. At highertemperatures T ∼ ∆, the effective p changes, but al-ways remains positive. As the very unusual p = − T . ∆ cannot be obtained for magnons,fractional spin excitations should be involved. This is fur-thermore supported by a fractional n in Eq. (1), implyingthat fractions of a spin-flip excitation are involved in therelaxation process. In contrast to this unusual gapped T − ( T ) behavior, the temperature dependence of the lo-cal susceptibility monitored by the Cl NMR shift inFig. 2(b) is monotonic over the whole covered temper-ature range, as predicted for the ferromagnetic Kitaevmodel [42]. Such a dichotomy between the two observ-ables is a direct sign of spin fractionalization, as differentfractional quasiparticles enter the two observables in dif-ferent ways [42].To obtain the spin-excitation gap ∆ as a function of B ab in Fig. 1(c), the T − ( T ) datasets in Fig. 3 taken inmagnetic fields of different directions and magnitudes arefitted to Eq. (1) in the temperature range of the Kitaevparamagnetic phase. As the curve T − ∝ T − definedby ∆ = 0 is steeper than any dataset in this range, theobtained excitation gaps are apparently all nonzero. Theinset of Fig. 1(c) showing the symmetric ∆( ϑ ) depen-dence around 90 ◦ in 9 . ϑ traverses nonequiv- alent directions with respect to the Kitaev axes on bothsides [inset of Fig. 1(a)], demonstrates that the under-lying physics is indeed isotropic as assumed when intro-ducing B ab . The obtained ∆( B ab ) in Fig. 1(c) can beperfectly reproduced as a sum of two terms: the two-fluxgap ∆ and the gap acquired by Majorana fermions in aweak magnetic field, predicted to be proportional to thecube of the field [6, 16, 17, 33],∆ = ∆ + α e B ∆ , (2)using J K = 190 K [21] to evaluate ∆ , as before, while e B = g ab µ B B ab /k B is the field in kelvin units, k B is theBoltzmann constant, µ B the Bohr magneton, and α = 4 . e B term [6]. This result demonstratesthat a spin-flip excitation in α -RuCl indeed fractional-izes into a gauge-flux pair and a Majorana fermion.Focusing on the Kitaev paramagnetic region in thephase diagram of α -RuCl in Fig. 1(b) is essential for ourobservation of two types of anyons. Instead, other recentexperimental studies focused on the low-temperatureregion above B c , observing the spin-excitation contin-uum [46] with either a gapless behavior [37, 47] or thegap opening linearly with B − B c [36, 48–50], but withouta definite conclusion about the identity of the involvedquasiparticles. Such an ambiguous behavior likely orig-inates in the presence of additional, smaller non-Kitaevinteractions between the spins [19, 35, 40, 51], whose roleshould be pronounced particularly at low temperaturesand which are indeed predicted to drive the system to-wards a gapless QSL ground state [52]. Our result showsthat spin fractionalization into two types of anyons is ro-bust against these interactions in a broad range of tem-peratures and magnetic fields. This is the main practicaladvantage of α -RuCl with respect to all other anyon re-alizations, such as the fractional quantum Hall effect in2D heterostructures [1] or hybrid nanowire devices [53],where anyons are observed only at extremely low tem-peratures and at certain field values. Our discovery thusestablishes α -RuCl as a unique platform for future in-vestigations of anyons and braiding operations on them,which form the functional basis of a topological quantumcomputer [6].The work was partly supported by the Slovenian ARRSprogram No. P1-0125 and project No. PR-07587. ∗ Electronic address: [email protected][1] R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky,G. Bunin, and D. Mahalu, Direct observation of a frac-tional charge,
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SUPPLEMENTAL MATERIALCrystal growth
Crystals of α -RuCl were synthesized from anhydrousRuCl (Strem Chemicals). The starting material washeated in vacuum to 200 ◦ C for one day to remove volatileimpurities. In the next step, the powder was sealed ina silica ampoule under vacuum and heated to 650 ◦ Cin a tubular furnace. The tip of the ampoule was keptat lower temperature and the material sublimed to thecolder end during one week. Phase pure α -RuCl (with ahigh-temperature phase of C /m crystal structure) wasobtained as thin crystalline plates. The residual in thehot part of the ampoule was black RuO powder. Thepurified α -RuCl was sublimed for the second time inorder to obtain bigger crystal plates. The phase andpurity of the compounds was verified by powder X-raydiffraction. All handling of the material was done understrictly anhydrous and oxygen-free conditions in gloveboxes or sealed ampoules. Special care has to be takenwhen the material is heated in sealed-off ampoules. If gasevolves from the material, this may result in the explosionof the ampoule. Magnetic susceptibility
Magnetic susceptibility measurements of α -RuCl wereperformed using a Quantum Design MPMS. A powderedsample of the mass 22 . T N = 14 K (inset of Fig. 4) is almost identical to thecorresponding curve in Ref. [34]. Nuclear magnetic resonance
General.
The Cl nuclear magnetic resonance(NMR) experiments were performed on a foil-like α -RuCl single crystal of approximate dimensions 5 × × . in a continuous-flow cryostat allowing us toreach temperatures down to 4 . powder, which was allowed to harden,in order to ensure the rigidity of the coil. The coil wasthen mounted on a teflon holder attached to a rotator,which allowed us to vary the orientation of the samplewith respect to the external magnetic field. In order to FC ( - e m u / m o l ) T (K)T N2 FIG. 4:
Magnetic susceptibility.
Zero-field-cooled (ZFC)and field-cooled (FC) magnetic susceptibility of the powdered α -RuCl sample in a magnetic field of 1 . T N = 14 K. reduce the noise of an already weak Cl NMR signal,a consequence of the extremely broad Cl NMR spec-trum, we used a bottom-tuning scheme. With the outputradio-frequency power of around 20 W, the typical π/ µ s. The NMR signals wererecorded using the standard spin-echo, π/ − τ d − π pulsesequence with a typical delay of τ d = 70 µ s (much shorterthan the spin-spin relaxation time T ) between the π/ π pulses. T relaxation. The spin-lattice relaxation (i.e., T )experiment was carried out using an inversion recoverypulse sequence, ϕ i − τ − π/ − τ d − π , with an inver-sion pulse ϕ i < π (suitable for broad NMR lines) and avariable delay τ before the read-out spin-echo sequence.The spin-lattice relaxation datasets were typically takenat 20 increasing values of τ . The datasets were analyzedusing the model of magnetic relaxation for I = 3 / − / ←→ / m ( τ ) = 1 − (1 + s ) (cid:20) . (cid:18) − τT (cid:19) + 0 . (cid:18) − τT (cid:19)(cid:21) , (3)where T is the spin-lattice relaxation time and s is the in-version factor. In the region of the phase diagram outsidethe magnetically ordered phase [Fig. 2(b)], this expres-sion reproduces the experimental relaxation curves per-fectly. In the magnetically ordered phase, two T compo-nents appear, and the relaxation curves are reproducedas a sum of two terms of the form given by Eq. (3). Forinstance, the temperature dependence of the correspond-
35 40 45 500306090120150180 B = 9.4 TT = 20 K i n t en s i t y ( a r b . u . ) , ba s e li ne = J ( o ) (MHz) L = 39.18 MHz FIG. 5:
Orientation dependence of the NMR spec-trum.
Central line of the Cl NMR spectrum of the α -RuCl single crystal taken at 20 K in the field of 9 . ϑ with respect to the crystal ab plane [as in Fig. 1(a)].The plane of rotation is at an angle of 15 ◦ from the crystal b axis [as shown in Fig. 1(a)]. Dashed vertical line indicates theLarmor frequency ν L . Arrows mark the peak whose tempera-ture dependence is analyzed in Fig. 6 and where T displayedin Fig. 3 was measured. The ϑ = 180 ◦ spectrum correspondsto the only field direction outside the red fan in Fig. 1(a). ing two T values for B = 9 . B k c ∗ is givenin Fig. 2(a). In cases where only a narrow temperatureregion below the transition was covered, the two compo-nents in the relaxation curves were hard to identify, andwe used Eq. (3) with a stretched exponent instead. Orientation dependence of the NMR spectrum.
The Cl NMR spectra were recorded point by point infrequency steps of 50 or 100 kHz, so that the Fouriertransform of the signal was integrated at each step to ar-rive at the individual spectral point. The covered NMRfrequency range was from 34 MHz, the lower limit ofour setup, up to 50 MHz. The dependence of the cor- responding part of the NMR spectrum on the directionof the external magnetic field of 9 . ϑ from the crystal ab plane) at a temperature of20 K is shown in Fig. 5. The spectra are extremely broadbecause of large Cl (with I = 3 / / ←→ − / Cl NMR transition. As thistransition is observed to consist of at least three peaks(Fig. 5), even for the symmetric B k c ∗ orientation (with ϑ = 90 ◦ ), while there are only two inequivalent Cl sitesin the crystal structure, the splitting of the central lineis likely a consequence of stacking faults in the layeredcrystal structure or crystal twinning, or both. Relation bewteen orientation and field depen-dence of T . Measuring the T dependence on the direc-tion of the magnetic field (described by the angle ϑ fromthe crystal ab plane) instead of on its magnitude in the ab plane allows us to cover low B ab values, while keeping theapplied magnetic field B high. This is beneficial for tworeasons related to the strong quadrupole broadening ofthe Cl NMR spectrum (Fig. 5): to minimize an alreadylarge NMR linewidth and to keep the Larmor frequencywell above the quadrupole splitting, which is of the orderof 10 MHz as concluded in the following. The validityof this approach is supported by the fact that ∆( B ab )data points for various angles ϑ and field values 2 . . . B ab ) data points taken in lowerfields apparently exhibit much larger error bars. Namely,the corresponding T − ( T ) datasets in Fig. 3(b) are morescattered than the datasets taken in 9 . Temperature dependence of the NMR line.
Wemeasured the temperature dependence of the dominant Cl NMR peak in a field of 9 . B ab < B ab in the inset of Fig. 6, we obtain the phaseboundary of the magnetically ordered state, which per-fectly matches the result of the reference study [34]. Incontrast, the NMR shift does not exhibit any signs ofa magnetic transition, except for the ϑ = 90 ◦ ( B k c ∗ )dataset. We find the NMR shift to be a monotonic func-tion of temperature T , empirically following a log T de-pendence over a broad temperature range. Contributions to the NMR shift.
To separatethe magnetic contribution to the NMR frequency shiftfrom the temperature-independent quadrupole contribu-tion, we plot the relative NMR shift (i.e., the NMR shift o N M R s h i ft ( M H z ) T (K) T ( K ) B ab (T)magneticorder FIG. 6:
Temperature dependence of the NMR line.
The temperature ( T ) dependence of (a) the width δν and (b) theNMR frequency shift of dominant Cl NMR peaks (marked by arrows in Fig. 5) in the field of 9 . Cl Larmor frequency ν L = 39 .
18 MHz) for various sample orientations given by ϑ . The overlaping lines for some values of ϑ do not allow thedetermination of δν . For each dataset drawn in (a), straight lines are linear fits on both sides of the kink at T N (determinedas the temperature of intersection and marked by an arrow) indicating the onset of low-temperature magnetic ordering. Insetshows the obtained points of the phase boundary [same symbols as the corresponding δν ( T ) datasets] compared to the resultof Ref. [34] (gray line). Straight lines in (b) are phenomenological linear fits. Only the ϑ = 90 ◦ ( B k c ∗ ) dataset in (b) showssigns of magnetic transitions (marked by arrows). divided by the Cl Larmor frequency ν L = 39 .
18 MHzin a field of 9 . B ⊥ c ∗ , i.e., in the ab plane [the ϑ = 0 ◦ dataset inFig. 6(b)], against the rescaled magnetic susceptibility χ ab in the inset of Fig. 7. In Ref. [34], an experimentalratio between the susceptibility χ of the powdered sam-ple and the susceptibility χ ab of the single crystal witha field applied in the ab plane is obtained as (2 + r ) / r = 0 . χ ab = 3 χ/ (2 + r ). We usethis empirical relation to evaluate χ ab ( T ) from our field-cooled χ ( T ) dataset shown in Fig. 4. As we did notmeasure susceptibility in high magnetic fields, we rely onthe dataset taken in 1 . χ ab up to 20 · − emu/mol (i.e.,down to 35 K), we obtain the hyperfine coupling constant A = 2 . µ B and the zero-temperature relative shift − .
039 that, when multiplied by ν L , gives the quadrupoleshift ∆ ν Q = − .
53 MHz.From the obtained quadrupole shift ∆ ν Q , we can esti-mate the quadrupole splitting ν Q between the successive Cl NMR transitions. For the case of an axially sym-metric 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∆ ν Q = − ν Q (1 − cos ϑ ′ )(9 cos ϑ ′ − / (16 ν L ) for the I = 3 / ◦ of v ZZ from c ∗ ,so that ϑ ′ ∼ ◦ . From the previously evaluated ∆ ν Q wethen obtain ν Q ∼ . Cl NMR tran-sition and the satellite transitions. We can thus concludethat the NMR peaks in the covered frequency range ofFig. 5 all belong to the central transition.
Theory
Theoretical T − ( T ) curve. The theoretical temper-ature dependence of T − is numerically calculated forthe Kitaev model in zero field [42]. T − contains twocontributions, one coming from a single fluctuating spin(i.e., on-site) and the other one coming from fluctuatingnearest-neighboring (NN) spins in the Kitaev honeycomblattice. As the Cl nucleus in α -RuCl is located atequal distances from the closest two Ru S = 1 / T − contains both contributions with equalweights. Namely, as the hyperfine coupling constant A of Cl to both spins is the same, the relevant spin-spin0 ab ( - e m u / m o l ) r e l a t i v e N M R s h i ft ab (10 -3 emu/mol) FIG. 7:
NMR shift against susceptibility.
Temperaturedependences of the Cl NMR shift and magnetic suscepti-bility χ ab are proportional to each other down to 35 K. Insetshows the dependence of the relative NMR shift (i.e., NMRshift divided by ν L = 39 .
18 MHz) on χ ab . Line is a linear fitof the dataset up to 20 · − emu/mol, i.e., down to 35 K. correlation function can be generally written as D A (cid:8) S ( t ) ± S ( t ) (cid:9) · A ( S ± S ) E == A h(cid:10) S ( t ) S (cid:11) + (cid:10) S ( t ) S (cid:11) ±± (cid:16)(cid:10) S ( t ) S (cid:11) + (cid:10) S ( t ) S (cid:11)(cid:17)i , (4)for the involved components S and S of both Ru spins, where the plus (minus) sign is valid for ferro-magnetic (antiferromagnetic) fluctuations. The first twoterms on the right side of Eq. (4) represent the on-sitecontributions, while the last two represent the NN-sitescontributions, both with apparently equal weights. Ac-cordingly, the theoretical curve for the ferromagneticcase, plotted in Fig. 2(a), is the average of the on-siteand NN-sites contributions. T relaxation due to gapped magnons. When spinfluctuations in the magnetic lattice are due to excitedmagnons, the corresponding spin-lattice relaxation ratefor a single-magnon process is given by [45] T − ∝ Z g ( E ) n ( E ) (cid:2) n ( E ) (cid:3) d E, (5)where E is the energy of magnons, g ( E ) is their den-sity of states, n ( E ) = [exp( βE ) − − is the Bose-Einstein distribution function, β = 1 / ( k B T ), and k B is the Boltzmann constant. Denoting the magnon gapby ∆ (in kelvin units), we define ε = E − k B ∆ asthe energy measured from the bottom of the magnonband. The power-law dispersion relation ε ∝ k s in D dimensions, which includes the standard parabolic dis-persion ( s = 2) and the Dirac dispersion ( s = 1) asspecial cases, leads to g ( E ) ∝ ε D/s − . For low tem-peratures T ≪ ∆, the distribution function n ( E ) canbe approximated by the Boltzmann distribution, n ( E ) ≈ exp( − βE ) = exp( − ∆ /T ) exp( − βε ). Plugging these ex-pressions for g ( E ) and n ( E ) into Eq. (5), we obtain T − ∝ T D/s − exp (cid:18) − ∆ T (cid:19) Z ∞ exp( − x ) x D/s − d x. (6)The integral on the right side of Eq. (6) converges if s < D and evaluates to Γ(2 D/s −
1) where Γ is the gammafunction. We can thus rewrite Eq. (6) as T − ∝ T p exp (cid:18) − ∆ T (cid:19) (7)with the power of the prefactor p = 2 D/s −
1. In caseof D = 2, which is relevant for the Kitaev honeycombmagnet, p = 1 for s = 2 and p = 3 for s = 1, so that thepower p cannot be negative. Even in case of D = 1, p can only reach the lowest value of 0 precisely for s = 2[although care should be taken in this case, as the integralin Eq. (6) then formally diverges]. If more than a singlemagnon is involved in the T process, the power p is alsopositive and becomes even higher [45]. Gapped magnonsthus cannot lead to the T relaxation described by Eq. (7)with p < m . In this case D = 3 and s = 2,and this leads to T − ∝ T exp( − ∆ m /T ). We use thisexpression to analyze the T − ( T ) data [Fig. 2(a)] in thelow-temperature ordered state of α -RuCl .As a side observation, all these examples showthat a frequently used simple gapped model T − ∝ exp( − ∆ s /T ) with the gap ∆ s , which was used beforeto analyze the T − ( T ) datasets in α -RuCl [36], is actu-ally not justified in any region of the phase diagram of α -RuCl . Majorana fermion gap.
In the Kitaev model, Majo-rana fermions acquire a gap in the presence of an externalmagnetic field [6]. This is shown for a field applied per-pendicularly to the honeycomb plane, i.e., in the (1 , , x , y and z . The corresponding Zeeman term thenreads H Z = − h P j ( S xj + S yj + S zj ), where h = gµ B B/ √ B in energyunits, g is the g -factor and µ B is the Bohr magneton.When treated as a perturbation to the Kitaev Hamil-tonian, the Zeeman term contributes to the Majoranafermion gap only at third order [6]. The correspondingeffective Hamiltonian is thus proportional to h and canbe written as [6, 16, 17] H (3)eff = − α h k B ∆ X jkl S xj S yk S zl , (8)1 theory B (low-B limit) FIG. 8:
Spin-excitation gap.
The spin-excitation gap ∆as a function of the magnetic field B applied in the crystal ab plane obtained from T − ( T ) data in our work and in tworecent works [36, 37] using our model given by Eq. (1). Thedata are compared to the theoretical expression (using J K =190 K [21], α = 4 . g = g ab ), which simplifies to the B dependence given by Eq. (2) [i.e., ∆ added to ∆ f in Eq. (10)]in the low-field region. where ∆ is a two-flux gap (in kelvin units), while α (of the order of unity) accounts for the sum over theexcited states, and its exact value is not known. TheKitaev model extended with such a three-spin exchangeterm − κ P jkl S xj S yk S zl with κ = αh / ( k B ∆ ) is still ex-actly solvable and the dispersion relation of the Majoranafermions is calculated as [16] E k = 2 q k B J K | e i k · a + e i k · a | + κ sin ( k · a ) , (9)where J K is the Kitaev coupling (in kelvin units), while a and a are the unit vectors of the honeycomb lattice.The dispersion relation given by Eq. (9) is gapped for κ = 0, and the corresponding gap ∆ f can be calculatednumerically as a function of κ and thus as a function ofthe magnetic field. For small magnetic fields, i.e., for κ ≪ k B J K , the Majorana fermion gap (in kelvin units)simplifies to∆ f = √ κk B = α (cid:18) gµ B Bk B (cid:19) , (10)while for high magnetic fields it saturates to ∆ f = 2 J K .The total spin-excitation gap ∆ is obtained by adding∆ f to the two-flux gap ∆ . The field dependence of ∆is shown in Fig. 8 for J K = 190 K (taken from Ref. [21]and used in this work), g = g ab and α = 4 . B ab ) data points]. The cubicapproximation given by Eq. (10), which is also plotted inFig. 8, is apparently valid up to 15 T, well beyond thefield range covered in this work. Comparison with recent works
Recent NMR works.
Very recently, two Cl NMRstudies of α -RuCl appeared [36, 37]. The analysis ofboth studies is focused on the low-temperature regionbelow 15 K. Using a simple exponential model T − ∝ exp( − ∆ s /T ) in this region, Ref. [36] finds that the ex-citation gap ∆ s opens linearly with the field above thecritical field around 10 T. On the other hand, Ref. [37]extends the covered temperature range down to 1 . T − ( T ) in the covered high-field region above thecritical field around 8 T.As these results are very different from our results,also because of quite different analysis, we analyze the T − ( T ) datasets obtained in these two works also withour model given by Eq. (1). As in our work, we focuson the Kitaev paramagnetic region, to the temperaturerange from 50 K down to slightly above the transitiontemperature below 8 T, and down to 4 . T − ( T ) as a main feature. The data inRef. [37] were taken with a field applied in the crystal ab plane, while the data in Ref. [36] were taken with a fieldapplied at 30 ◦ and − ◦ with respect to the ab plane. Inthis case, we calculate the effective field values B ab in thesame way as in our work. The obtained field dependenceof the excitation gap ∆( B ab ) for both works is shown inFig. 8 together with our result. Applying our analysis tothe data in all three works apparently leads to relativelyconsistent results. Nevertheless, the results for the datafrom Refs. [36, 37] alone do not allow to conclude on thecubic field dependence of ∆, mostly due to the lack ofimportant low-field data points. Regarding Ref. [36], thetwo data points at the highest fields seem to deviate fromthe trend set by the other points. The corresponding two T − ( T ) datasets exhibit a suspicious plateau at low tem-peratures, not observed in any other dataset of the threeworks, which casts some doubts on their credibility. Re-garding Ref. [37], the obtained ∆( B abab