Magnon-spinon dichotomy in the Kitaev hyperhoneycomb β-Li_2IrO_3
Alejandro Ruiz, Nicholas P. Breznay, Mengqun Li, Ioannis Rousochatzakis, Anthony Allen, Isaac Zinda, Vikram Nagarajan, Gilbert Lopez, Mary H. Upton, Jungho Kim, Ayman H. Said, Xian-Rong Huang, Thomas Gog, Diego Casa, Robert J. Birgeneau, Jake D. Koralek, James G. Analytis, Natalia B. Perkins, Alex Frano
MMagnon-spinon dichotomy in the Kitaev hyperhoneycomb β -Li IrO Alejandro Ruiz, ∗ Nicholas P. Breznay, Mengqun Li, Ioannis Rousochatzakis, AnthonyAllen, Isaac Zinda, Vikram Nagarajan,
5, 6
Gilbert Lopez,
5, 6
Mary H. Upton, JunghoKim, Ayman H. Said, Xian-Rong Huang, Thomas Gog, Diego Casa, Robert J.Birgeneau,
5, 6
Jake D. Koralek, James G. Analytis,
5, 6
Natalia B. Perkins, and Alex Frano † Department of Physics, University of California, San Diego, California 92093, USA Department of Physics, Harvey Mudd College, Claremont, CA 91711 School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55116, USA Department of Physics, Loughborough University, Loughborough, LE11 3TU, United Kingdom Department of Physics, University of California, Berkeley, California 94720, USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA SLAC National Accelerator Laboratory, Menlo Park, California, 94025, USA (Dated: February 8, 2021)The family of edge-sharing tri-coordinated iridates and ruthenates has emerged in recent years asa major platform for Kitaev spin liquid physics, where spins fractionalize into emergent magneticfluxes and Majorana fermions with Dirac-like dispersions. While such exotic states are usuallypre-empted by long-range magnetic order at low temperatures, signatures of Majorana fermionswith long coherent times have been predicted to manifest at intermediate and higher energy scales,similar to the observation of spinons in quasi-1D spin chains. Here we present a Resonant InelasticX-ray Scattering study of the magnetic excitations of the hyperhoneycomb iridate β -Li IrO undera magnetic field with a record-high-resolution spectrometer. At low-temperatures, dispersing spinwaves can be resolved around the predicted intertwined incommensurate spiral and field-inducedzigzag orders, whose excitation energy reaches a maximum of 16 meV. A 2 T magnetic field softensthe dispersion around Q = 0. The behavior of the spin waves under magnetic field is consistent withour semiclassical calculations for the ground state and the dynamical spin structure factor, whichfurther predicts that the ensued intertwined uniform states remain robust up to very high fields(100 T). Most saliently, the low-energy magnon-like mode is superimposed by a broad continuumof excitations, centered around 35 meV and extending up to 100 meV. This high-energy continuumsurvives up to at least 300 K – well above the ordering temperature of 38 K – and gives evidence forpairs of long-lived Majorana fermions of the proximate Kitaev spin liquid. I. INTRODUCTION
Recent years have seen a vigorous experimental re-search effort on identifying candidate materials for thecelebrated quantum spin liquid (QSL) [1–7], a topo-logical state of matter with long-range entanglementand fractionalized excitations, whose efficient control canlead to novel and transformative technological applica-tions [8–11]. A central focus in this effort are the tri-coordinated Kitaev materials [5, 7, 12–17], a family ofspin-orbit-assisted Mott insulators, in which spin-orbit-entangled j eff = 1 / ∗ Corresponding Author:[email protected] † Corresponding Author:[email protected]
QSL. Known examples include the collinear, commensu-rate zigzag state in Na IrO [27, 28] and α -RuCl [29],and the non-coplanar, counter-rotating incommensuratespiral order in α, β , γ -Li IrO [30–32]. Such orderedstates arise due to the presence of additional interac-tions, such as the Heisenberg exchange J and the sym-metric off-diagonal anisotropy Γ [12, 16, 33–38]. Whilesuch perturbations naturally impede the realization ofQSLs, a wealth of theoretical and experimental work hasshown that the Kitaev exchange is still the dominant in-teraction in all Kitaev materials, and that it stronglyinfluences their magnetic behavior [14, 16, 39]. Mostsaliently, it has been suggested that this separation ofenergy scales must give rise to an extended ‘proximatespin-liquid’ regime with signatures of long-lived fraction-alized excitations, separating the low- T ordered phasefrom the high- T paramagnetic regime [25, 26, 40–45].For example, inelastic neutron scattering measure-ments in RuCl have revealed a gapped, broad continuumof magnetic excitations far above the magnetic orderingtemperature [46, 47], while Raman spectroscopy has pro-vided evidence for the fermionic character of these exci-tations [48–51]. These observations signify a dichotomybetween the expected magnon-like response at low en-ergies, originating from the long-range ordering, and acontinuum multi-spinon-like response at higher energies, a r X i v : . [ c ond - m a t . s t r- e l ] F e b originating from pairwise excitations of long-lived Majo-rana fermions of the proximate Kitaev QSL model. Thisproximate spin-liquid regime is characterized by strong,short-range spin-spin correlations and disordered fluxes,and can be observed in thermodynamic measurements, ormore directly in dynamical probes such as inelastic neu-tron scattering, Raman and ultrafast spectroscopy, andResonant Inelastic x-ray Scattering (RIXS) [25, 26, 40–45].Here we report experimental RIXS evidence for such adichotomy of low-energy magnons and broad excitationsfrom the proximate Kitaev spin liquid regime in the 3Dhyperhoneycomb iridate β -Li IrO , with a characteristiccontinuum response extending up to at least 300 K, wellabove the ordering temperature of 38 K in this material. β -Li IrO is perhaps the most intriguing example ofthe complex interplay between the Kitaev exchange andexternal magnetic fields [52, 54, 55]. At zero field, the sys-tem orders at T I = 38 K into a complex incommensurateorder (IC) with counter-rotating spirals and propagationwavevector Q = (0 . , , Q − vectors are reportedin orthorhombic reciprocal lattice units [31]. Applyinga magnetic field along the crystallographic ˆ b -axis rapidlydestroys the IC order at a critical field µ o H ∗ b = 2 . a , similar to the magnetic order of Na IrO and RuCl [52]. In the following, we refer to this phase as a field-induced zigzag (FIZZ) phase.It is now understood [53, 56–58] that the IC order of β -Li IrO can be thought of as a long-wavelength twist-ing of the energetically closest commensurate state with Q = ( , , J - K -Γmodel [37, 38], with coupling parameters J (cid:39) . K (cid:39) −
18 meV, and Γ (cid:39) −
10 meV [57, 58]. Despite thisunderstanding, there is very little experimental informa-tion about the low-energy excitation spectra and theirresponse to magnetic fields, in part because the crystalsize and the large neutron absorption cross-section of Irimpede inelastic neutron scattering studies.The experiments reported below provide a comprehen-sive picture of the dynamic response of β -Li IrO un-der magnetic fields H b along the ˆ b -axis. We measuredits low-energy magnetic excitation spectrum by perform-ing Ir- L -edge RIXS experiments. Central to this ef-fort are complementary experiments with a conventionalspectrometer with a medium energy resolution (MER)∆ E ∼
25 meV as well as a state-of-the-art spectrome-ter with a high energy resolution (HER) ∆ E ∼
10 meV.These measurements were performed at 0 T as well asunder a 2 T magnetic field. The results of theses mea-surements show magnetic excitations branching from themagnetic zone centers corresponding to the intertwinedIC and FIZZ states. We observe that magnetic excita- tions naturally split into low-energy dispersive modes andhigh-energy continuum like excitations. The former reacha maximum energy around 16 meV, and an applied 2 Tmagnetic field softens the dispersion around Q = 0. Theexperimental data are in very good agreement with semi-classical calculations for the dynamical spin structuralfactor of β -Li IrO , lending strong support to the mi-croscopic J - K -Γ description. Furthermore, we identify abroad continuum of magnetic excitations whose intensityis insensitive to the low-temperature magnons, remainsconstant up to ∼
100 K (i.e., well above T I = 38 K), andslowly decreases at higher temperatures. The tempera-ture dependence of this continuum is consistent with thebehavior of nearest neighbor spin correlations emergingfrom the dominant Kitaev energy scale, and its persis-tence to high temperature alludes to the long coherencetimes of the fractionalized excitations of the proximateQSL phase [23–25].The paper is organized as follows: Section II describesthe crystal synthesis and the RIXS experiment set-up.Section III presents the experimental data and calcula-tions of the spin dynamical structural factor. Section IVdiscusses the strongly interwined nature of the IC andFIZZ states, and the origin of the multi-spinon contin-uum. Section V summarizes our findings. II. EXPERIMENTAL DETAILS
High-quality single crystals of β -Li IrO were grownby a vapor transport technique. Ir (99.9% purity, BASF)and Li CO (99.999 % purity, Alfa-Aesar) powders wereground and pelletized at 3,000 psi in the molar ratio of1:1.05. The pellets were placed in an alumina crucible,reacted for 12 h at 1,050 ° C, and then cooled to roomtemperature at 2 ° C/h to yield single crystals whichwere then extracted from the reacted powder. β -Li IrO crystallizes in the orthorhombic Fddd space group withsingle-crystal sizes ∼ × × µ m (more details inthe Supplementary Information).RIXS measurements were performed at beamline 27-ID of the Advanced Photon Source using π -polarized x-rays in the horizontal scattering plane at detector anglesclose to 2 θ ∼ ◦ , in order to suppress elastic scatter-ing. The RIXS spectra were obtained using two differ-ent setups: (1) a medium-resolution ∆ E MER ∼
25 meVwith a two-bounce Si(844) monochromator and a 2m-radius Si(844) diced spherical analyzer [59], and (2) ahigh-resolution ∆ E HER ∼
10 meV with a four-bounceSi(844) monochromator and a 2m-radium quartz(309)diced spherical analyzer [60, 61] at the Ir L absorptionedge (2 p / → d t g ∼ .
215 keV).To study the magnetic field dependence of the low-energy excitations, we devised the setup shown in Fig-ure 1 a. A β -Li IrO crystal with a surface normal inthe [001] crystallographic direction was mounted betweentwo Nd Fe B magnets separated by 400 µ m. The scat-tering plane is defined by the (001) × (100) reciprocal lat- FIG. 1. (Color online) (a) The β -Li IrO single crystal oriented with ˆ a − axis in the scattering plane and the ˆ b − axis perpendicularto it. The crystal was placed between two neodymium permanent magnets, with a 2 T magnetic field applied along theˆ b − axis and a π -polarized x-ray beam. (b) The field dependence of the scattered intensity at 5 K measured at the wave vectorscorresponding to IC and FIZZ ( Q = (0 . , ,
22) and Q = (0 , , µ o H b ≈ Q = (0 . , ,
22) obtained with incident energy E i = 11 . ≈ .
42 eV)corresponds to an intersite exciton formed by a particle-hole pair across the Mott gap, while B ( ≈ .
73 eV) and C ( ≈ .
84 eV)represent intrasite excitations of 3 λ SO / j eff = / and j eff = / states, indicating a spin-orbit coupling ∼ . tice vectors, and the magnetic field is applied parallel tothe [010] direction (ˆ b -axis). III. RESULTS
Measurements of the resonant elastic x-ray scattering(REXS) intensity taken with a tunable DC magnetic field[52] on the same sample serve as a check of the magneticfield experienced by the sample in the permanent mag-net set-up. Figure 1 b,c show the comparison of the low-temperature REXS intensity with the RIXS integratedintensity taken at the same reciprocal space points cor-responding to the IC ( Q = (0 . , , Q = (0 , ,
22) states.The magnetic field suppresses the non-coplanar ICstate that develops below T I = 38 K [30], while the sys-tem develops a coplanar FIZZ component [52]. The cal-culated field evolution of the Bragg peak intensities arealso shown in Figure 1 b. The low temperature inte-grated intensity values of both peaks measured in ourRIXS setup map directly onto the REXS intensity valuesfor a 2 T field, shown in Figure 1 c. Furthermore, thetemperature dependence of the integrated RIXS signal isplotted on Figure 1 c. The observed transition tempera-ture in the in-field RIXS data, T I (H) = 32 K, is consistentwith an applied 2 T field when compared to the reportedthermodynamic measurements [52, 54, 62].Figure 1 d shows the MER RIXS spectra as a func-tion of energy loss ( E loss = E i − E f ) up to 1.2 eV. Thesescans were performed at 5 K near Q = (0 . , , j eff = / band across the Mottgap. Features B and C correspond to intrasite excita-tions between the j eff = / and j eff = / states [63, 64],resulting from a spin-orbit coupling ∼ . j eff = / pseudo-spin provides a suitabledescription for the electronic ground state of β -Li IrO under an applied magnetic field. This is in stark contrastwith recent observations of the collapse of the relativistic j eff = / state under the application of remarkably smallhydrostatic pressure [65, 66].Having established that the field does not alter thespin-orbit entangled Mott state, we investigate its effecton the low-energy excitations. The top panel in Figure 2 acompares raw MER spectra for 0 and 2 T over a limitedenergy loss window. These data were taken at differentmomentum transfer positions with varying in-plane value Q a = ( h, , Q -vectors. The bottom panels show the integrated REXSsignal for both field values [52], revealing sharp Braggpeaks that span a much narrower range than that overwhich the inelastic features soften. Intriguingly, the 2 Tspectrum near Q = (0 , ,
22) has shifted to lower energy
FIG. 2. (Color online) (a) Upper panels show the momentum dependence of the RIXS spectra taken along the (h,0,22) directionat 5 K with 0 T and 2 T fields. Lower panels show the REXS scans around the zone centers of the INC and FIZZ states. Theapplied magnetic field suppresses the INC order while giving rise to FIZZ state [52]. (b) A comparison of the low-energy RIXSspectra collected at 5 K with two spectrometers. In the vicinity of the INC Bragg peak Q = (0 . , , E MER = 25 meV and ∆ E HER = 10 meV, respectively. Slightly away from the Bragg peak at Q = (0 . , , ∼
16 meV, and a broad continuum ofmagnetic excitations centered ∼
35 meV which is nearly momentum-independent. compared to the 0 T data, which constitutes one of thecentral findings of our work.To better understand these low-energy features, Figure2 b shows representative low-energy RIXS spectra for β -Li IrO , collected at 5 K using the MER and HER spec-trometers. The first panel shows the spectra taken at theIC Bragg peak position and was used to determine theworking resolution. The strong elastic signal has a full-width-at-half-maximum (FWHM) of 25 meV and 10 meVrespectively, consistent with previous calibration mea-surements [61, 67]. The second panel compares inelas-tic spectra taken at Q = (0 . , , Q -vector. In the MER scans,a clear shoulder of intensity is seen at non-zero energyloss. These spectra were fitted by a sum of pseudo-Voigtfunctions representing the elastic line (gray) and two in-elastic features (red and pink) ∼
15 meV and ∼
35 meV.The elastic line width was set by the energy resolutionand only its intensity and peak position was allowed tovary in the fit (more details in the Supplementary Infor-mation). Since the lowest energy inelastic feature (red) iswithin the energy resolution of the MER scan, we com-pare them to HER scans taken at the same reciprocalspace positions. The inelastic signal ∼
15 meV is mostclearly seen in the high resolution spectra. (The modeat ∼
35 meV is not visible in the HER scans because ofthe lower throughput of this detector.) In summary, thedata show (i) a resolution-limited, low-energy excitation( E loss <
16 meV) which is sensitive to momentum trans- fer, and (ii) a broad continuum centered around 35 meVwith width much larger than the experimental resolutionand insensitivity to variations along the Q a momentumtransfer direction. This dichotomy is the key experimen-tal finding of our work.These low-energy inelastic features can originate fromeither lattice or spin excitations within the ∼
400 meVMott gap [68]. Phonon excitations arise due to dynamicsin the short-lived intermediate state and are expected tobe very weak in the well-screened intermediate state of IrL-edge RIXS, which explains why no detectable phononcontributions have thus far been observed in other Mottinsulator 5 d iridates [69, 70]. Moreover, O K-edge RIXSexperiments on Li IrO showed that the majority of thephonon spectra resides around 70 meV [71], well abovethe inelastic features we observed.By contrast, magnetic excitations are strongly en-hanced by the resonance process. In particular, the mag-netic contribution to the RIXS response comes from twodistinct types of processes, the spin-conserving (SC) andthe non-spin-conserving (NSC) [23, 24]. In a SC process,the spin of the 5d valence shell does not change dur-ing RIXS, while in a NSC process the spin gets flipped.Furthermore, for the pure Kitaev model, there are threetypes of NSC channels, corresponding to the rotation ofthe spin by π around the x , y , and z axes, respectively.These three NSC channels and the SC channel are or-thogonal in the pure Kitaev spin liquid [23, 24], but theyare not fully orthogonal in the more complete, J - K -Γ FIG. 3. (Color online) (a) The momentum dependence of the low-energy RIXS excitations was extracted from the data inFigure 2 a. The magnon mode with a maximum energy ∼
16 meV disperses toward the zone centers corresponding to the IC( Q a = 0 . Q a = 0) states. The 2 T spectra softens around Q a = 0 compared to the 0 T data. The experimentalresults are well captured by the dynamical structure factor calculations presented in panel (b). On the other hand, the broadmagnetic continuum centered around 35 meV is insensitive to momentum transfer along Q a . (b) The diagonal componentsof the spin dynamical structure factor, S aa ( Q, ω ) , S bb ( Q, ω ) , S cc ( Q, ω ), and their sum, S ( Q, ω ), computed at H b = 0 T (leftpanels) and H b = 2 T (right panels) along the Q a direction. The colors and the width indicate the magnitude of the associatedcomponent after convolving with a Gaussian. Note that the color scale shown individually for each panel varies significantlybetween different components. (c) Exemplary temperature dependence of the 0 T RIXS excitation spectra above T I . (d) The0 T RIXS intensity integrated above 20 meV shows that the inelastic continuum hardly changes up to 100 K and persists up to300 K. This feature seems to be insensitive to the low-temperature INC transition at T I . description of β -Li IrO . Nevertheless, we can assumethat since the Kitaev interaction is dominant, part of theresponse is predominantly coming from the NSC chan-nel and part from the SC channel. In this case, theNSC RIXS response reduces to the corresponding diag-onal component of the dynamical spin structure factor(DSF) defined as S αβ ( Q , ω ) = (cid:90) dt e − iωt (cid:104) S α ( − Q , S β ( Q , t ) (cid:105) , (1)where Q and (cid:126) ω are the momentum and energy transfers, and S α ( Q , t ) = 1 N (cid:88) r e − i Q · r S α r ( t ) (2)is the Fourier transform of the spin density at time t ,with N being the total number of spins and r denotingthe physical positions of the spins. The SC RIXS channelwill manifest itself only if fractionalized excitations arepresent. It will pick up exclusively the Majorana fermionsof the fractionalized Kitaev spin liquid, and the overallenergy dependence of this contribution will be propor-tional to the two-fermion joint density of states [23, 24](more details in the Supplementary Information). FIG. 4. (Color online) Evolution of the total dynamical structure factor S ( Q , ω ) calculated in magnetic field along the b axis.The spectra is shown for H b = 0T , , , and 150T. Even for H b (cid:29) H ∗ b , the Q a = 2 / Let us return to our RIXS data. To probe the natureof the magnetic contribution to the RIXS response of β -Li IrO , we extract the full momentum and tempera-ture dependence. Figure 3 a shows the fitted dispersionsof the two inelastic features along Q = ( h, ,
22) = Q a for 0 T and 2 T. The error bars reflect the uncertaintyarising from the three-peak fitting procedure. The low-energy mode clearly disperses from the IC momentumvalue Q = (0 . , ,
22) and reaches a maximum energy ∼
16 meV away from it. A slight softening can be seennear the point corresponding to the FIZZ state, Q =(0 , , ∼
35 meV isalmost non-dispersive and does not show any significantchanges in the applied field. It is noteworthy that recentindependent studies of Ir L -edge RIXS on two relatedcompounds, Na IrO and α − Li IrO , have observed asimilar broad continuum around ∼ −
30 meV, whichhas been interpreted as magnetic in origin through a care-ful study of its momentum and temperature dependence[67, 70].The temperature dependence of the RIXS spectra isshown in Figure 3 c. The low-energy mode dispersingup to ∼
15 meV (indicated by the left arrow) monoton-ically decreases above T I and becomes featureless above ∼
150 K. Conversely, the broad excitation mode cen-tered around 35 meV remains largely unchanged with in-creasing temperature. Its intensity, integrated for ener-gies above 20 meV, remains constant up to ∼
100 K, andslowly decreases as 1 /T above that. Nonetheless, this ex-citation persists to very high temperatures, with a sizableintensity at room temperature.To better understand our data, we first model the Q -dependence of the RIXS response using a semiclassical approach. To this end, we follow previous theoreticalworks which employ a 1 /S -expansion around the clos-est commensurate approximant of the observed IC order,namely a period-3 state with counter-rotating momentsand the same irreducible representation as the observedIC order [53, 56–58]. Expanding around this period-3state leads to a magnon excitation spectrum and the spinDSF (1). Figure 3 b shows the diagonal components ofthe DSF S aa ( Q , ω ), S bb ( Q , ω ), S cc ( Q , ω ), as well as thetotal DSF defined as S ( Q , ω ) = S aa ( Q , ω ) + S bb ( Q , ω ) + S cc ( Q , ω ) along the orthorhombic Q a direction, com-puted at µ o H b = 0T (left panel) and 2T (right panel).Note that the off-diagonal components are non-zero, butthey are several orders of magnitude weaker comparedto the diagonal components, which is why we disregardthem in our analysis. In the background we have also su-perimposed the linear spin wave spectrum (thin whitelines). We can see that the most intense mode bothat zero-field and at 2T is the one near Q = (2 / , , cc polarization channel. The in-tensities of the soft modes at Q = 0 are much weaker andare dominated by the bb channel, consistent with the factthat the low-field Q = 0 zig-zag and ferromagnetic staticorders are much weaker compared to the IC components.The applied field causes a softening of the Q = 0 modecompared to the zero-field data, as seen experimentally(more details in the Supplementary Information).The magnetic continuum around ∼
35 meV is not seenin our semiclassical analysis of the DSF (see Figure 3 b).Moreover the fact that it survives at temperatures wellabove the ordering temperature T I with almost the samemomentum-integrated intensity as the one at very lowtemperatures (see Figure 3 c) suggests that magnons can-not account for this broad continuum. The dominanceof the Kitaev coupling in β -Li IrO offers an alterna-tive interpretation, whereby the high-energy continuumof magnetic excitations is predominantly coming fromlong-lived pairs of Majorana fermions, characteristic ofthe proximate QSL, which is seen in the predominantlySC response. In particular, according to theory calcula-tions for the pure Kitaev model [24], the SC response of β -Li IrO is given by a broad continuum, centered at anenergy around 6 K/ ∼
27 meV [72], in good agreementwith the observed position of the peak around 35 meV.The difference is likely accounted for by the additionalinteractions in β -Li IrO , and in particular by Γ, whichis as high as 55% of the dominant Kitaev coupling. IV. DISCUSSION
The low-energy excitation spectra hold important in-formation about the nature of the magnetic state in β -Li IrO . With our data in agreement with semiclassicalcalculations of the dynamical spin structure factor, wecan confidently use the model Hamiltonian to computethe expected excitation spectra at higher fields. In Fig-ure 4, we show the evolution of the low-energy spectrumat a wide range of magnetic field values H b . We can seethat the Q a = 2 / ∗ b , indi-cating strong spin fluctuations despite the absence of thecorresponding long-range magnetic order. This happensbecause the modulated components with Q = (2 / , , Q = (0 , ,
0) are inter-twined at all fields due to the spin-length constraints, aswas previously shown by Li et al. [73]. With increasingfield, this mode’s energy increases continuously. How-ever, only at fields above 100 T does the spectrum showa dispersion with a minimum at Q = 0 wavevector char-acteristic of a fully polarized state. It is remarkable howthe system resists full polarization due to the presence ofintertwined orders.On the other hand, the broad continuum around35 meV shows clear clues on the impact of the domi-nant Kitaev interaction on the spectrum of β -Li IrO at intermediate and high energy scales, and reflects thenear proximity of this system to the ideal quantum spinliquid state. The unusual temperature dependence ofthe observed broad continuum cannot be explained interms of conventional magnon or phonons since the spindegree of freedom contribution to the scattering inten-sity should rapidly decrease and vanish at a temperatureabove the characteristic exchange energy, while a latticecontribution would increase monotonically with temper-ature due to the Bose population factor [67]. However,the integrated intensity of this excitation shows Curie-like behavior (1 /T ) above 100 K and a constant value be-low that, without being affected by the low-temperaturelong-range order. The thermal characteristics of the JK Γmodel have been theoretically studied and it is predicted that above the low-temperature order (
T > T I ), itinerantMajorana fermions remain coherent up to a temperature T ∼ K where the fractional excitations recombine intospins and the nearest-neighbor spin correlations decaywith increasing temperature. This fractionalization isexperimentally observable in specific heat, thermal trans-port and spin correlation measurements, and its associ-ated excitations are very robust against temperature.Recently, Ruiz et al. [74] reported a magnetic anomalyin β -Li IrO around T η = 100 K with signatures appear-ing in magnetization, heat capacity, and muon spin re-laxation ( µ SR) measurements. A sharp onset was ob-served on the magnetization at T η as well as a crossoverin the heat capacity. Below 100 K, magnetic hystere-sis was detected which grows with decreasing temper-ature and penetrates the low-temperature magneticallyordered state with complete impunity. This temperaturealso coincides with previous report on the reordering ofthe principle magnetic axes in the three-dimensional iri-dates [52, 54, 75] and the onset of fermionic contribu-tions to the Raman spectra [49]. Moreover, µ SR dataindicate that the entity responsible for this transition-like anomaly is static and homogeneously distributedthroughout the sample without causing a true long-rangeordered state.Our observation of the temperature dependence of thebroad multi-spinon excitation centered at 35 meV pro-vides a connection to the previously reported magneticanomaly at T η = 100 K and suggests that T η may rep-resent a crossover between a high-temperature feature-less paramagnet and a proximate spin liquid regime gov-erned by the physics of Majorana fermions. Below T η ,the nearest neighbor spin correlations saturate to theirlow-temperature value. These spin-spin correlations areprecisely equivalent to the kinetic energy of the emer-gent Majorana fermions which release 1 / R ln 2 of thespin entropy as the system enters this thermal crossover[76]. In addition, we note that similar RIXS excitationshave been observed in the two-dimensional honeycombiridates α -Na IrO and α -Li IrO [67, 70], which hasbeen equated to the inelastic neutron scattering contin-uum reported on α − RuCl . Such ubiquity across Kitaevmaterials is another confirmation of the dominant Kitaevexchange, and its characteristic multi-spinon-like dynam-ical fingerprints at intermediate energy and temperaturescales, irrespective of the long range ordering taking placeat very low temperatures. V. CONCLUSION
We have studied the low-energy excitations of thethree-dimensional Kitaev magnet β -Li IrO using amedium- and a high-resolution RIXS spectrometers with25 meV and 10 meV resolution at the Ir L -edge, respec-tively. These measurements were carried out using 0 Tas well as a 2 T applied magnetic field in order to accessthe intertwined IC and FIZZ state. In contrast to thecase of hydrostatic pressure, an applied field does not dis-turb the relativistic j eff = / state. Moreover, the field-temperature dependence of the low-energy RIXS spec-tra reveal two distinct modes: (1) dispersive magnonsbranching from Q -vectors corresponding to IC and FIZZstates and reaching a maximum energy ∼
16 meV, com-parable with the reported Kitaev exchange energy in β -Li IrO [56, 73], and (2) a broad-continuum of magneticexcitations centered at 35 meV, which is insensitive tothe low-temperature order, remains constant up to 100 Kand slowly decreases above that. Indications for such acontinuum response around 30 meV have been previouslyreported by Raman experiments [49]. Also, this contin-uum contribution was not seen in the THz measurementsof Ref. [55] because these went only up to 80 cm − .We have compared the experimental low-temperaturedispersing magnons to the calculated spin dynamicalstructure factor above the closest period-3 approximantof the actual IC order and have shown that the domi-nant contribution to the Q a = 2 / cc po-larization channel, whereas the bb channel dominates the Q = 0, FIZZ mode. Experimentally, we observe that a2 T magnetic field softens the dispersion around Q = 0,in agreement with the calculation. The validity of theminimal microscopic Hamiltonian allows us to predictthe evolution of the dynamical structural factor for largevalues of H b . The persistent of the soft mode around Q a = 2 / T η = 100 K, and a slow decrease above that. This charac-teristic temperature has been previously reported as theonset of a magnetic anomaly affecting thermodynamicvariables without causing long-range magnetic order [74].Both findings point towards unconventional magnetismand suggest that T η represents the thermal fractionaliza-tion of spins into disordered fluxes and Majorana fermionexcitations which almost entirely dominate the thermo-dynamic response of β -Li IrO . VI. ACKNOWLEDGEMENTS
We thank Yi-Zhuang You, Tarun Grover, John Mc-Greevy, Dan Arovas, Ken Burch, Yiping Wang and Ga-bor Halasz for fruitful discussions. Use of the AdvancedPhoton Source was supported by the U. S. Departmentof Energy, Office of Science, Office of Basic Energy Sci-ences, under Contract No. DE-AC02-06CH11357. Mate-rial synthesis and experimental measurements were sup-ported by the Department of Energy, Office of Basic En-ergy Sciences, Materials Sciences and Engineering Divi-sion, under Contract No. DE-AC02-05CH11231. Thework at LBNL is funded by the US Department of En-ergy, Office of Science, Office of Basic Energy Sciences,Materials Sciences and Engineering Division under Con-tract No. DE-AC02-05-CH11231 within the QuantumMaterials Program (KC2202). Alejandro Ruiz acknowl-edges support from the University of California Pres-ident’s Postdoctoral Fellowship Program. 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