Abstract
The fractionalization of elementary excitations in quantum spin systems is a central theme in current condensed matter physics. The Kitaev honeycomb spin model provides a prominent example of exotic fractionalized quasiparticles, composed of itinerant Majorana fermions and gapped gauge fluxes. However, identification of the Majorana fermions in a threedimensional honeycomb lattice remains elusive. Here we report spectroscopic signatures of fractional excitations in the harmonichoneycomb iridates β and γLi_{2}IrO_{3}. Using polarizationresolved Raman spectroscopy, we find that the dynamical Raman response of β and γLi_{2}IrO_{3} features a broad scattering continuum with distinct polarization and composition dependence. The temperature dependence of the Raman spectral weight is dominated by the thermal damping of fermionic excitations. These results suggest the emergence of Majorana fermions from spin fractionalization in a threedimensional Kitaev–Heisenberg system.
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Introduction
The fractionalization of elementary excitations is a characteristic feature of quantum spin liquids. Such a liquid evades conventional magnetic order even at T=0 K and thereby preserves all symmetries of the underlying spin Hamiltonian. In the last decade, there has been significant progress in the experimental identification of quantum spin liquids in a class of geometrically frustrated Heisenberg magnets^{1} with elementary excitations that are given by chargeless spinons carrying spin s=1/2. For twodimensional (2D) triangular and Kagome lattices, however, a quantitative understanding of spinons remains unsatisfactory due to a lack of reliable theoretical methods of handling macroscopic degenerate ground states^{2,3}.
In this context, the exactly solvable Kitaev honeycomb model offers a genuine opportunity of exploring spin liquid physics on a more quantitative level as the spin response functions for spin liquids can be analytically computed^{4,5,6,7,8,9}. Until now, searching for Kitaev materials has been centred on the iridates αA_{2}IrO_{3} (A=Li, Na) and the ruthenates αRuCl_{3}, in which Ir^{4+} (5d^{5}) or Ru^{3+} (4d^{5}) ions create the J_{eff}=1/2 Mott state by the combined effects of strong spin–orbit coupling, electronic correlations and crystal field^{10,11,12,13,14,15,16,17,18,19}. In the tricoordinated geometry of edgesharing IrO_{6} or RuO_{6} octahedra, J_{eff}=1/2 moments interact via two 90° IrOIr exchange paths, giving rise to anisotropic bonddependent Kitaev interactions^{20,21}.
A potential drawback is the development of longrange order at low temperatures and the presence of Heisenberg interactions in real materials^{22}. Despite the detrimental effects of residual interactions, αRuCl_{3} shows an indication of the spin fractionalization through a continuumlike excitation in the Raman response and a highenergy Majorana excitation in inelastic neutron scattering^{15,17}. In contrast, only a subtle signature of Kitaev interactions exists for αA_{2}IrO_{3}. The absence of welldefined fractionalized excitations in the iridates is ascribed to the structural distortion of planar IrOIr bonds^{10}.
In search for a new platform for Kitaev magnetism, the harmonic series of hyperhoneycomb lattices, β−and γ −Li_{2}IrO_{3}, were discovered^{23,24}. These structural polytypes have the same tricoordinated network of Ir ions as the layered αA_{2}IrO_{3}, and thus are a threedimensional (3D) analogue of the honeycomb iridate materials. An ensuing question is whether quantum spin liquids are preserved in such a 3D generalization of the Kitaev model^{25,26,27,28,29}.
Specific heat and magnetic susceptibility data evidence longrange magnetic order at T_{N}=38 and 39.5 K for β and γLi_{2}IrO_{3}, respectively^{23,24}. Strikingly, the two distinct structural polytypes display a similar incommensurate spiral order with noncoplanar and counterrotating moments. This implies that Kitaev interactions dictate magnetism of both compounds^{30,31,32,33}. In this respect, these materials present promising candidates for attesting the elusive spin fractionalization in the 3D honeycomb lattice. Currently, it is difficult to detect Majorana fermions with Xray and neutronscattering technique, as only submillimetresized crystals are available. Therefore, Raman spectroscopy is the most suitable method for addressing this issue, because it directly probes Majorana fermion density of states. Moreover, detailed theoretical predictions of the polarization dependence of magnetic Raman scattering in the hyperhoneycomb lattice exist that can prove fractionalized fermionic excitations^{8}.
In this study, we provide Raman spectroscopic evidence for weakly confined Majorana fermions in 3D honeycomb iridate materials. The polarization and composition dependence of broad spinon continua point towards a different topology of spinon bands comparing β and γLi_{2}IrO_{3}. In addition, the temperature dependence of the integrated Raman intensity obeys the Fermi statistics, being in stark contrast to bosonic Raman spectra observed in conventional insulating magnets. These results demonstrate the emergence of fermionic excitations from the spin fractionalization in a 3D honeycomb lattice.
Results
Polarization dependence of Raman spectra
Figure 1a shows the crystal structures of β and γLi_{2}IrO_{3}. βLi_{2}IrO_{3} consists of the zigzag chains (blue and orange sticks), which alternate in orientation between the two basal plane diagonals and are connected via the bridging bonds (green stick) along the c axis. In γLi_{2}IrO_{3}, two interlaced honeycomb layers alternate along the c axis. Figure 1b,c presents the polarizationdependent Raman responses χ′′(ω) of β and γLi_{2}IrO_{3} measured at T=6 K in two different scattering geometries. Here the notation (xy) with x=a and y=b, c refers to the incident and scattered light polarizations, which are parallel to the crystalline x and y axis, respectively. χ′′(ω) presents the dynamical properties of collective excitations and is obtained from the raw Raman spectra I(ω) using the relation I(ω)∝[1+n(ω)]χ′′(ω) where is the Bose thermal factor.
Within the Fleury–Loudon–Elliott theory^{34}, the magnetic Raman scattering intensity of a 3D Kitaev system is given by the density of states of a weighted twoMajorana spinon, I(ω)=π∑_{m,n;k}δ(ω−ɛ_{m,k}−ɛ_{n,k})B_{mn,k}^{2}, where ɛ_{m,k} is a Majorana spinon dispersion with the band indices m, n=1,2(1,2,3) for β(γ)Li_{2}IrO_{3} and B_{mn,k} is the matrix element creating two Majorana excitations^{8}. The observed χ′′(ω) is composed of sharp phonon excitations superimposed on a broad, featureless continuum extending up to 200 meV. The Ramanactive phonon modes are presented in Supplementary Fig. 1 and Supplementary Tables 1 and 2 (see also Supplementary Note 1 for details). The magnetic continuum arises mainly from twoMajorana spinon excitations. This assignment is analogue to observations in the 2D honeycomb lattice αRuCl_{3}, in which a broad continuum is taken as evidence of fractionalized excitations^{15}. The striking similarity of the magnetic response between αRuCl_{3} and β and γLi_{2}IrO_{3} suggests that the 3D honeycomb iridates and the 2D honeycomb ruthenate realize Kitaev magnetism to a similar extent.
Thanks to the multiple spinon bands in the 3D harmonic honeycomb system, the Raman response of β and γLi_{2}IrO_{3} will be polarization and compositiondependent, emulating a number of band edges and van Hove singularities^{8}. As seen in Fig. 1b,c, the iridate compounds show commonly an asymmetric magnetic response towards lower energy. The polarization dependence is mostly evident in the ωdependence of χ′′(ω). Compared with χ′′(ac), χ′′(ab) with green shading becomes systematically suppressed as ω→0. Examining its composition dependence, χ′′(ac) of β and γLi_{2}IrO_{3} is plotted together in Fig. 1d after subtracting phonon modes. χ′′(ac) of βLi_{2}IrO_{3} shows a round maximum at ∼33 meV, whereas its spectral weight is depressed to zero as ω→0. In contrast, χ′′(ac) of γLi_{2}IrO_{3} has two maxima at 26 and 102 meV along with a finite excitation gap of Δ=5−6 meV marked by the arrows in Fig. 1c,d. Here, the extracted gap is estimated by a linear extrapolation of χ′′(ω). The slightly richer spectrum of γLi_{2}IrO_{3} than βLi_{2}IrO_{3} is linked to the increasing number of Majorana spinon bands. Thus, these results establish a subtle yet discernible polarization and composition dependence of χ′′(ω) in the 3D hyperhoneycomb compounds.
A related question is to what extent the hyperhoneycomb iridate materials retain the characteristic of Majorana fermions inherent to the 3D Kitaev model. For this purpose, we first compare the experimental and theoretical Raman response of βLi_{2}IrO_{3}, which lies at the nearisotropic point with J^{x}=J^{y}≈J^{z} (see Supplementary Fig. 2 and Supplementary Note 2 for details). Similar trends are observed in the polarization dependence of the scattering intensity; the (ac) polarization spectrum has a much stronger intensity than the (ab) polarization spectrum, being in line with the theoretical calculations^{8}. However, the lowenergy spectrum does not open an excitation gap in the (ab) scattering channel and the fine spectral features anticipated in the bare twoMajorana spinon density of states do not show up. There is not much difference in the polarization dependence for the case of γLi_{2}IrO_{3}, which possesses three Majorana spinon bands and is at the anisotropic point with J^{x}≠J^{y}≠J^{z} (see Supplementary Fig. 2 and Supplementary Note 2 for the local bond geometry). The absence of the sharp spectral features and polarizationdependent spectral widths is ascribed to the unwanted spinexchange terms including Heisenberg, offdiagonal and longerrange interactions. These subdominant terms on the one hand lead to a weak confinement of Majorana spinons, rendering the smearing out of the vanHove singularities and the softening of spectral weight. On the other hand, they give rise to a bosonic (magnon) contribution to the magnetic continuum at low energies. In this regard, the excitation gap in γLi_{2}IrO_{3} corresponds to an energy gap in the lowenergy spin waves. As the pseudospin s=1/2 has a negligible single ion anisotropy, the anisotropic Kitaev interactions of γLi_{2}IrO_{3} are responsible for opening the large gap. Notably, no obvious energy gap is present in the lowenergy excitations of βLi_{2}IrO_{3} with nearly isotropic Kitaev interactions.
Before proceeding, we estimate the Kitaev exchange interaction J_{z}=17 meV from the upper cutoff energy of the magnetic continuum. The extracted value is almost two times bigger than J_{z}=8 meV of αRuCl_{3} (ref. 15), being consistent with larger spatial extent of Ir orbitals.
Evolution of fermionic excitations
The temperature dependence of the Raman spectra was measured for both β and γLi_{2}IrO_{3} in the (cc) and (ac) scattering symmetries, respectively. The representative spectra are shown in Fig. 2a,b. The broad magnetic continuum marked with pink shading develops progressively into a quasielastic response at low energies on heating through T_{N}. The lowenergy magnetic scattering grows more rapidly in β than γLi_{2}IrO_{3}, because the latter has the large excitation gap. The magnetic Raman scattering at finite temperatures arises from dynamical spin fluctuations in a quantum paramagnetic state and can provide a good measure of the thermal fractionalization of quantum spins. The integrated Raman intensity in the energy range of 1.5 J_{z}<ℏω<3 J_{z} is plotted as a function of temperature in Fig. 2c,d. The temperature dependence of the integrated I(ω) is well fitted by a sum of the Bose and the twofermion scattering contribution (1−f(ω))^{2} with the Fermi distribution function (ref. 35). The Bose contribution describes bosonic excitations such as magnons, whereas the twofermion contribution is related to the creation or annihilation of pairs of fermions. The deduced energy ℏω=0.76−79 J_{z} of fermions for β and γLi_{2}IrO_{3} validates the fitting procedure adopting a Fermi distribution function. Here we stress that the thermal fluctuations of fractionalized fermionic excitations are a Raman spectroscopic evidence of proximity to a Kitaev spin liquid. Essentially the same fermionic excitations were inferred from the Tdependence of the integrated spectral weight in αRuCl_{3} (ref. 35).
Figure 2e,f shows the Raman conductivity χ′′(ω)/ω versus temperature. The Raman conductivity features a pronounced peak centred at ω=0. The lowenergy Raman response exhibits a strong enhancement with increasing temperature. The intermediatetohigh energy χ′′(ω)/ω above 30 meV dampens hardly with temperature. From the Raman conductivity we can define a dynamic Raman susceptibility using Kramers–Kronig relation , that is, by first extrapolating the data from the lowest energy measured down to 0 meV and then integrating up to 200 meV. It is noteworthy to mention that χ^{dyn} is in the dynamic limit of χ^{static}=lim_{k→0}χ(k,ω=0)^{36}. Figure 2g,h plots the temperature dependence of χ^{dyn}(T) of β and γLi_{2}IrO_{3}. Irrespective of polarization and composition, χ^{dyn}(T) shows a similar variation with temperature. On heating above T_{N}, χ^{dyn}(T) increases rapidly and then saturates for temperatures above T*=220−260 K. Remarkably, the energy corresponding to T* is comparable to the Kitaev exchange interaction of J_{z}=17 meV. We further note that the 2D Heisenberg–Kitaev material αRuCl_{3} exhibits also a drastic change of magnetic dynamics through T∼J_{z}=100–140 K (ref. 15). For temperatures below T*, the power law gives a reasonable description of χ^{dyn}(T)∼T^{α} with α=1.58±0.05 and 2.64±0.09 in the respective (cc) and (ab) polarization for βLi_{2}IrO_{3} and α=1.77±0.06 for γLi_{2}IrO_{3}. As discussed in Supplementary Fig. 3 and Supplementary Note 3, χ^{dyn}(T) is temperature independent in the paramagnetic phase as paramagnetic spins are uncorrelated. This is contrasted to the powerlaw dependence of χ^{dyn}(T) in a spin liquid. This powerlaw is associated with slowly decaying correlations inherent to a spin liquid^{37} and the onset temperature T* heralds a thermal fractionalization of Kitaev spins^{9}.
We now compare the dynamic Raman susceptibility with the static spin susceptibility given by SQUID magnetometry. As evident from Fig. 2g,h, they behave in an opposite way. This discrepancy indicates that a large number of correlated spins are present in the limit ω→0.
Fano resonance of optical phonon and magnetic specific heat
The phonon Raman spectra unveil a strongly asymmetry lineshape at 24 meV in βLi_{2}IrO_{3} (see Fig. 3a) that is well fitted by a Fano profile I(ω)=I_{0}(q+ɛ)^{2}/(1+ɛ^{2}) (ref. 38). The reduced energy is defined by ɛ=(ω−ω_{0})/Γ where ω_{0} is the bare phonon frequency, Γ the linewidth and q the asymmetry parameter. In Fig. 3b,c, we plot the resulting frequency shift, the linewidth and the Fano asymmetry as a function of temperature. The errors are within a symbol size. Based on lattice dynamical calculations (see Supplementary Note 1), this phonon is assigned to an A_{g} symmetry mode, which involves contracting vibrations of Ir atoms along the c axis (see the sketch in the inset of Fig. 3a). Therefore, the observed anomalies could shed some light on the thermal evolution of Kitaev physics, because a Fano resonance has its root in strong coupling of phonons to a continuum of excitations.
With decreasing temperature, the Fano asymmetry, 1/q, increases continuously and then becomes constant below the magnetic ordering temperature. As clearly seen from Fig. 3b, the temperature dependence of 1/q follows the twofermion scattering form (1−f (ω))^{2}, which gives a nice description of the temperature dependence of the integrated I(ω) (see Fig. 2c,d). It is striking that the magnitude of the Fano asymmetry parallels a thermal damping of the fermionic excitations. In a Kitaev honeycomb system, spins are thermally fractionalized into the itinerant Majorana spinons^{9}. As a result, the continuum stemming from the spin fractionalization strongly couples to lattice vibrations that mediate the Kitaev interaction. It is noteworthy that the 24 meV mode involves the contracting motion of the bridging bonds between consecutive zigzag chains along the c axis. In addition, αRuCl_{3} shows a Fano resonance of a phonon, which reinforces our assertion that the Fano asymmetry is an indicator of the thermal fractionalization of spins into the Majorana fermions^{15}.
As the temperature is lowered, phonon modes usually increase in energy and narrow in linewidth due to a suppression of anharmonic phonon–phonon interactions. Indeed, as shown in Fig. 3c, the temperature dependence of ω and Γ is well described by conventional anharmonic decay processes (see also Supplementary Note 4). A small kink in Γ occurs at the onset temperature of the magnetic ordering. Unlike the 2D honeycomb lattice αRuCl_{3} (ref. 15), however, there appears to be no noticeable renormalization of the phonon energy and linewidth on crossing T_{N} and T*. This may be due to the large unit cell of the 3D network of spins and low crystal symmetry. In such a complex spin network, lattice vibrations involve simultaneous modulations of different magnetic exchange paths and thus spin correlation effects on the phonon are largely nullified. This scenario is supported by the lacking Fano resonance in γLi_{2}IrO_{3} having a lower symmetry and stronger trigonal distortion compared with βLi_{2}IrO_{3}.
Next, we turn to the magneticspecific heat C_{m} of a Kitaev system. Spin fractionalization into two types of the Majorana fermions leads to a twopeak structure^{9} and a rich phenomenology in its temperature dependence. A highT crossover is driven by the itinerant Majorana fermions and linked to the development of shortrange correlations between the nearestneighbour spins. A lowT topological transition is expected due to the Z_{2} fluxes. Quasielastic Raman scattering can be used to derive the magnetic specific heat using a hydrodynamic limit of the spin correlation function^{39}. Next, the Raman conductivity is associated with C_{m} by the relation χ′′(ω)/ω ∝ C_{m}TI_{L}(ω), where I_{L}(ω) is the Lorentzian spectral function (see the Methods for details)^{40,41,42}. A fit to this equation allows evaluating C_{m}(T) from the integration of χ′′(ω)/ω scaled by T. In Fig. 3d, the resulting C_{m} versus T is plotted. We confirm the two peaks at T_{N}=0.1 J and T*∼J. The highT peak at T*∼J is somewhat higher than that of the theoretical value of 0.6 J (ref. 9). In addition, the predicted topological transition at T∼0.005 J is preempted by the longrange magnetic order at T_{N}=0.1 J. We ascribe the discrepancy between experiment and theory to residual interactions, which lift the Raman selection rules of probing the Majorana fermions. In spite of the magnetic order, the persistent twopeak structure in C_{m} suggests that the hyperhoneycomb iridates are in proximity to a Kitaev spin liquid phase.
Discussion
Having established that β and γLi_{2}IrO_{3} have fractionalized fermionic excitations, it is due to compare them with spinon excitations in the wellcharacterized kagome Heisenberg antiferromagnet ZnCu_{3}(OH)_{6}Cl_{2} (refs 3, 43). In such a system, geometrical frustration is the key element.
Despite disparate sources of fractionalized excitations, a number of key features in the spectral shape and temperature dependence of magnetic scattering, as well as in the Fano (anti)resonance of optical phonons (see the asterisks in Fig. 4) are common to βLi_{2}IrO_{3} and ZnCu_{3}(OH)_{6}Cl_{2}. Both compounds show a broad continuum with a rounded maximum at low energies, as shown in Fig. 4. In ZnCu_{3}(OH)_{6}Cl_{2}, the lowenergy response decreases linearly down to zero frequency and the magnetic continuum extends up to a highenergy cutoff at 6 J with J≈16 meV. The former property suggests the formation of a gapless spin liquid and the latter the existence of multiple spinon scattering processes^{43,44}. In a similar manner, the lowenergy spectral weight of βLi_{2}IrO_{3} drops to zero with a steeper slope. The similar behaviour observed in the two compounds with different spin and lattice topologies may be due the fact that the bare spinon density of states is modified due to Dzyaloshinskii–Moriya interactions and antisite disorder in ZnCu_{3}(OH)_{6}Cl_{2} and other spinexchange interactions in βLi_{2}IrO_{3}. The resemblance becomes less clear for γLi_{2}IrO_{3}, mainly because the anisotropic Kitaev exchange interactions open a large excitation gap in the lowenergy excitations.
Next, we discuss the temperature dependence of the magnetic continuum. Irrespective of the spinon topology and spinexchange type, the three studied compounds share essentially the same phenomenology. The key feature is the evolution of a spinon continuum into a quasielastic response with increasing temperature. This is well characterized by the powerlaw dependence of χ^{dyn}(T)∼T^{α}. The exponent of α=1.58−2.64 is not much different comparing the three compounds^{43}. As discussed in Supplementary Note 3, this powerlaw behaviour is inherent to a longrange entangled spin liquid and is completely different from what is expected for conventional magnets. Despite the distinct spinon band structure, the spinon correlations may be not very different between the 2D kagome and the 3D hyperhoneycomb lattice.
The last remark concerns that χ^{dyn}(T) of the 3D hyperhoneycomb materials starts to deviate from a powerlaw behaviour at 220 K. At the respective temperature, the magnetic specific heat shows as a broad peak identified as a thermal crossover from a paramagnet to a Kitaev paramagnet. This anomaly is absent in ZnCu_{3}(OH)_{6}Cl_{2} with a single type of spinon and thus unique to β and γLi_{2}IrO_{3} having two species of Majorana fermions.
In summary, a Raman scattering study of the 3D honeycomb materials β and γLi_{2}IrO_{3} provides evidence for the presence of Majorana fermionic excitations. A polarization, temperature and composition dependence of a magnetic continuum indicates a distinct topology of spinon bands between β and γLi_{2}IrO_{3}. The temperature dependence of an integrated Raman response and the twopeak structure in specific heat demonstrate that a thermal fractionalization of spins brings about fermionic excitations and that the 3D harmonichoneycomb iridates realize proximate spin liquid at elevated temperatures. These results expand the concept of fractionalized quasiparticles to a 3D Kitaev–Heisenberg spin system.
Methods
Samples
Single crystals of βLi_{2}IrO_{3} were grown by a flux method. The starting materials Li_{2}CO_{3}, IrO_{2} and LiCl with ratio 10:1:100 were mixed together and pressed into a pellet. The pellet was placed in an alumina crucible, heated to 1,100 °C for 24 h and then cooled down to 700 °C for 14 h. Black powderlike crystal grains appeared at a bottom of the crucible. The collected grains were washed by distilled water for the removal of the LiCl flux and filtered. The obtained crystals are of a size of 30−50 μm. To grow single crystals of γLi_{2}IrO_{3}, polycrystalline pellets of αLi_{2}IrO_{3} were first prepared. The prepared αphase pellet was heated to 1,170 °C for 72 h and slowly cooled down to 900 °C in air. Shinny black crystals with a size of 100 μm were obtained on the surface of the pellet. The phase purity and composition of β and γLi_{2}IrO_{3} were confirmed via powder Xray diffraction. Their bulk magnetic susceptibility is presented in Fig. 2g,h of the main text.
Raman scattering experiment
A polarized, resolved Raman spectroscopy was employed to detect spin and phonon excitations of single crystals of β and γLi_{2}IrO_{3}. Raman scattering experiments were performed in backscattering geometry with the excitation line λ=532.1 nm of a Nd:YAG (neodymiumdoped yttrium aluminium garnet) solidstate laser. The scattered spectra were collected using a microRaman spectrometer (Jobin Yvon LabRam) equipped with a liquidnitrogencooled chargecoupled device. A notch filter and a dielectric edge filter were used to reject Rayleigh scattering to a lower cutoff frequency of 60 cm^{−1}. The laser beam was focused to a fewmicrometrediameter spot on the surface of the crystal using a × 50 magnification microscope objective. The samples were mounted onto a liquidHecooled continuous flow cryostat, while varying a temperature between 6 and 300 K. All Raman spectra were corrected for heating.
Analysis of quasielastic Raman scattering
Quasielastic light scattering arises from either diffusive fluctuations of a fourspin time correlation function or fluctuations of the magnetic energy density. According to Reiter^{40} and Halley^{41}, a twospin process leads to scattering intensity for temperatures above the critical temperature;
where E(k,t) is a magnetic energy density given by the Fourier transform of E(r)=−〈∑_{i>j}J_{ij}S_{i}·S_{j}δ(r−r_{i})〉 with the position of the ith spin r_{i}. Applying the fluctuation–dissipation theorem in the hydrodynamic limit^{41}, equation (1) is simplified to
where β=1/k_{B}T, C_{m} is the magnetic specific heat and D is the thermal diffusion constant D=K/C_{m} with the magnetic contribution to the thermal conductivity K. Equation (2) can be rewritten in terms of a Raman susceptibility χ′′ (ω),
This relation is employed to extract the magnetic specific heat from the Raman conductivity in the main text.
Data availability
The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information files.
Additional information
How to cite this article: Glamazda, A. et al. Raman spectroscopic signature of fractionalized excitations in the harmonichoneycomb iridates β and γLi_{2}IrO_{3}. Nat. Commun. 7:12286 doi: 10.1038/ncomms12286 (2016).
References
Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).
Yamashita, M. et al. Highly mobile gapless excitations in a twodimensional candidate quantum spin liquid. Science 328, 1246–1248 (2010).
Han, T.H. et al. Fractionalized excitations in the spinliquid state of a kagomelattice antiferromagnet. Nature 492, 406–410 (2012).
Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).
Baskaran, G., Mandal, S. & Shankar, R. Exact results for spin dynamics and fractionalization in the Kitaev model. Phys. Rev. Lett. 98, 247201 (2007).
Knolle, J., Kovrizhin, D. L., Chalker, J. T. & Moessner, R. Dynamics of a twodimensional quantum spin liquid: signatures of emergent majorana fermions and fluxes. Phys. Rev. Lett. 112, 207203 (2013).
Knolle, J., Chern, G.W., Kovrizhin, D. L., Moessner, R. & Perkins, N. B. Raman scattering signatures of Kitaev spin liquids in A2IrO3 iridates with A=Na or Li. Phys. Rev. Lett. 113, 187201 (2014).
Perreault, B., Knolle, J., Perkins, N. B. & Burnell, F. J. Theory of Raman response in threedimensional Kitaev spin liquids: Application to β and γLi2IrO3 compounds. Phys. Rev. B 92, 094439 (2015).
Nasu, J., Udagawa, M. & Motome, Y. Vaporization of Kitaev spin liquids. Phys. Rev. Lett. 113, 197205 (2014).
Singh, Y. et al. Relevance of the HeisenbergKitaev model for the honeycomb lattice iridates A2IrO3 . Phys. Rev. Lett. 108, 127203 (2012).
Choi, S. K. et al. Spin waves and revised crystal structure of honeycomb iridate Na2IrO3 . Phys. Rev. Lett. 108, 127204 (2012).
Chun, S. H. et al. Direct evidence for dominant bonddirectional interactions in a honeycomb lattice iridate Na2IrO3 . Nat. Phys. 11, 462–466 (2015).
Alpichshev, Z., Mahmood, F., Cao, G. & Gedik, N. Confinementdeconfinement transition as an indication of spinliquidtype behavior in Na2IrO3 . Phys. Rev. Lett. 114, 017203 (2015).
Plumb, K. W. et al. αRuCl3: a spinorbit assisted Mott insulator on a honeycomb lattice. Phys. Rev. B 90, 041112 (2014).
Sandilands, L. J., Tian, Y., Plumb, K. W., Kim, Y. J. & Burch, K. S. Scattering continuum and possible fractionalized excitations in αRuCl3 . Phys. Rev. Lett. 114, 147201 (2015).
Sears, J. A. et al. Magnetic order in αRuCl3: a honeycomb lattice quantum magnet with strong spinorbit coupling. Phys. Rev. B 91, 144420 (2015).
Banerjee, A. et al. Proximate Kitaev quantum spin liquid behavior in a honeycomb magnet. Nat. Mater. 15, 733–740 (2016).
Kim, B. J. et al. Novel J eff=1/2 Mott state induced by relativistic spinorbit coupling in Sr2IrO4 . Phys. Rev. Lett. 101, 076402 (2008).
Kim, B. J. et al. Phasesensitive observation of a spinorbital Mott state in Sr2IrO4 . Science 323, 1329–1332 (2009).
Jackeli, G. & Khaliullin, G. Mott insulators in the strong spinorbit coupling limit: from Heisenberg to a quantum compass and Kitaev models. Phys. Rev. Lett. 102, 017205 (2009).
Chaloupka, J., Jackeli, G. & Khaliullin, G. KitaevHeisenberg model on a honeycomb lattice: possible exotic phases in iridium oxides A2IrO3 . Phys. Rev. Lett. 105, 027204 (2010).
Lee, E. K.H., Rau, J. G. & Kim, Y. B. Two iridates, two models, two approaches: a comparative study on magnetism in 3D honeycomb materials. Phys. Rev. B 93, 184420 (2016).
Takayama, T. et al. Hyperhoneycomb iridate βLi2IrO3 as a platform for Kitaev magnetism. Phys. Rev. Lett. 114, 077202 (2015).
Modic, K. A. et al. Realization of a threedimensional spinanisotropic harmonic honeycomb iridate. Nat. Commun. 5, 4203 (2014).
Mandal, S. & Surendran, N. Exactly solvable Kitaev model in three dimensions. Phys. Rev. B 79, 024426 (2009).
Lee, E. K.H., Schaffer, R., Bhattacharjee, S. & Kim, Y. B. HeisenbergKitaev model on the hyperhoneycomb lattice. Phys. Rev. B 89, 045117 (2014).
Kimchi, I., Analytis, J. G. & Vishwanath, A. Threedimensional quantum spin liquids in models of harmonichoneycomb iridates and phase diagram in an infiniteD approximation. Phys. Rev. B 90, 205126 (2014).
Schaffer, R., Lee, E. K.H., Lu, Y.M. & Kim, Y. B. Topological spinon semimetals and gapless boundary states in three dimensions. Phys. Rev. Lett. 114, 116803 (2015).
Hermanns, M., O’Brien, K. & Trebst, S. Weyl Spin Liquids. Phys. Rev. Lett. 114, 157202 (2015).
Biffin, A. et al. Noncoplanar and counterrotating incommensurate magnetic order stabilized by Kitaev interactions in γLi2IrO3 . Phys. Rev. Lett. 113, 197201 (2014).
Biffin, A. et al. Unconventional magnetic order on the hyperhoneycomb Kitaev lattice in βLi2IrO3: full solution via magnetic resonant xray diffraction. Phys. Rev. B 90, 205116 (2014).
Lee, E. K.H. & Kim, Y. B. Theory of magnetic phase diagrams in hyperhoneycomb and harmonichoneycomb iridates. Phys. Rev. B 91, 064407 (2015).
Kimchi, I., Coldea, R. & Vishwanath, A. Unified theory of spiral magnetism in the harmonichoneycomb iridates α, β, and γ Li2IrO3 . Phys. Rev. B 91, 245134 (2015).
Hayes, W. & Loudon, R. Scattering of Light by Crystals Dover Books (2012).
Nasu, J., Knolle, J., Kovrizhin, D. L., Motome, Y. & Moessner, R. Fermionic response from fractionalization in an insulating twodimensional magnet. Nat. Phys. http://dx.doi.org/10.1038/NPHYS3809 (4 July 2016).
Gallais, Y. & Paul, I. Charge nematicity and electronic Raman scattering in ironbased superconductors. C.R. Phys. 17, 113 (2016).
Hermele, M., Ran, Y., Lee, P. A. & Wen, X.G. Properties of an algebraic spin liquid on the kagome lattice. Phys. Rev. B 77, 224413 (2008).
Fano, U. Effects of configuration interaction on intensities and phase shifts. Phys. Rev. 124, 1866–1878 (1961).
Halperin, B. I. & Hohenberg, P. C. Hydrodynamic theory of spin waves. Phys. Rev. 188, 898 (1969).
Reiter, G. F. Light scattering from energy fluctuations in magnetic insulators. Phys. Rev. B 13, 169 (1976).
Halley, J. W. Light scattering as a probe of dynamical critical properties of antiferromagnets. Phys. Rev. Lett. 41, 1605 (1978).
Choi, K.Y., Lemmens, P. & Berger, H. Critical spin dynamics of the S=1/2 spin chain compound CuSe2O5 . Phys. Rev. B 83, 174413 (2011).
Wulferding, D. et al. Interplay of thermal and quantum spin fluctuations in the kagome lattice compound herbertsmithite. Phys. Rev. B 82, 144412 (2010).
Ko, W.H., Liu, Z.X., Ng, T.K. & Lee, P. A. Raman signature of the U(1) Dirac spinliquid state in the spin1/2 kagome system. Phys. Rev. B 81, 024414 (2010).
Acknowledgements
This work was supported by the Korea Research Foundation (KRF) grant funded by the Korea government (MEST) (Grant Number 20090076079), as well as by GermanIsraeli Foundation (GIF, 1171486 189.14/2011), the NTHSchool Contacts in Nanosystems: Interactions, Control and Quantum Dynamics, the Braunschweig International Graduate School of Metrology and DFGRTG 1952/1, Metrology for Complex Nanosystems.
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A.G. set up and carried out the Raman scattering measurements and analysed data. S.H.D. and Y.S.C. synthesized and characterized the samples. K.Y.C. and P.L. planned and coordinated the project. A.G., P.L. and K.Y.C. wrote the paper. All authors discussed the results and commented on the manuscript.
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Supplementary Figures 13, Supplementary Tables 12, Supplementary Notes 14 and Supplementary References (PDF 880 kb)
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Glamazda, A., Lemmens, P., Do, SH. et al. Raman spectroscopic signature of fractionalized excitations in the harmonichoneycomb iridates β and γLi_{2}IrO_{3}. Nat Commun 7, 12286 (2016). https://doi.org/10.1038/ncomms12286
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DOI: https://doi.org/10.1038/ncomms12286
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