⇦Chinese Physics Letters, 2022, Vol. 39, No. 2, Article code 027501Express Letter Evidence for Magnetic Fractional Excitations in a Kitaev Quantum-Spin-Liquid Candidate $\alpha$-RuCl$_3$ Kejing Ran (冉柯静)1,2†, Jinghui Wang (王靖珲)1,2†, Song Bao (鲍嵩)2, Zhengwei Cai (蔡正蔚)2, Yanyan Shangguan (上官艳艳)2, Zhen Ma (马祯)2,3, Wei Wang (王巍)4, Zhao-Yang Dong (董召阳)5, P. Čermák6,7, A. Schneidewind6, Siqin Meng (孟思勤)8,9, Zhilun Lu (陆智伦)8,10, Shun-Li Yu (于顺利)2,11*, Jian-Xin Li (李建新)2,11*, and Jinsheng Wen (温锦生)2,11* Affiliations 1School of Physical Science and Technology, and ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai 200031, China 2National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China 3Institute for Advanced Materials, Hubei Normal University, Huangshi 435002, China 4School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China 5Department of Applied Physics, Nanjing University of Science and Technology, Nanjing 210094, China 6Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Forschungszentrum Jülich GmbH, Lichtenbergstr. 1, 85748 Garching, Germany 7Charles University, Faculty of Mathematics and Physics, Department of Condensed Matter Physics, Ke Karlovu 5, 121 16, Praha, Czech Republic 8Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Hahn-Meitner-Platz 1D-14109, Berlin, Germany 9China Institute of Atomic Energy (CIAE), Beijing 102413, China 10The Henry Royce Institute and Department of Materials Science and Engineering, The University of Sheffield, Sir Robert Hadfield Building, Sheffield, S1 3JD, United Kingdom 11Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China Received 1 December 2021; accepted 24 December 2021; published online 27 December 2021 †These authors contributed equally to the work.
*Corresponding authors. Email: slyu@nju.edu.cn; jxli@nju.edu.cn; jwen@nju.edu.cn Citation Text: Ran K J, Wang J H, Bao S et al. 2022 Chin. Phys. Lett. 39 027501    Abstract It is known that $\alpha$-RuCl$_3$ has been studied extensively because of its proximity to the Kitaev quantum-spin-liquid (QSL) phase and the possibility of approaching it by tuning the competing interactions. Here we present the first polarized inelastic neutron scattering study on $\alpha$-RuCl$_3$ single crystals to explore the scattering continuum around the $\varGamma$ point at the Brillouin zone center, which was hypothesized to be resulting from the Kitaev QSL state but without concrete evidence. With polarization analyses, we find that, while the spin-wave excitations around the $M$ point vanish above the transition temperature $T_{\rm N}$, the pure magnetic continuous excitations around the $\varGamma$ point are robust against temperature. Furthermore, by calculating the dynamical spin-spin correlation function using the cluster perturbation theory, we derive magnetic dispersion spectra based on the $K$–$\varGamma$ model, which involves with a ferromagnetic Kitaev interaction of $-7.2$ meV and an off-diagonal interaction of $5.6$ meV. We find this model can reproduce not only the spin-wave excitation spectra around the $M$ point, but also the non-spin-wave continuous magnetic excitations around the $\varGamma$ point. These results provide evidence for the existence of fractional excitations around the $\varGamma$ point originating from the Kitaev QSL state, and further support the validity of the $K$–$\varGamma$ model as the effective minimal spin model to describe $\alpha$-RuCl$_3$.
cpl-39-2-027501-fig1.png
cpl-39-2-027501-fig2.png
cpl-39-2-027501-fig3.png
cpl-39-2-027501-fig4.png
 DOI:10.1088/0256-307X/39/2/027501 © 2022 Chinese Physics Society Article Text Quantum spin liquids (QSLs) with fractionalized spin excitations and long-range entanglement have drawn a great deal of attention since Anderson first proposed the QSL state within the resonating-valence-bond (RVB) model[1] and used it to explain the high-temperature superconductivity.[2] Typically, QSL states have been proposed in triangular and kagome lattices where antiferromagnetic interactions are highly frustrated due to the geometrical constrain, and therefore there remain strong quantum fluctuations in these systems that prevent spins from establishing a long-range order as observed in a usual magnet at low temperatures.[3–7] Besides the RVB model which was built upon geometrical frustration, the Kitaev model defined on the ideal two-dimensional honeycomb lattice with spin $S= 1/2$ is an exactly solvable model with the QSL ground state.[8] On a honeycomb lattice, there is no geometrical frustration, and it is the anisotropic bond-dependent Kitaev interactions that lead to the frustration on a single site and give rise to the “Kitaev” QSL state.[8,9] A Kitaev QSL can host fractional excitations represented by, e.g., Majorana fermions, which behave as anyons obeying the non-Abelian statistics under magnetic field, and allow to be braided for fault-tolerant topological quantum computation.[8,10–12] In general, for a spin-only system, it is unrealistic to find the bond-dependent Kitaev interactions which underlie the Kitaev QSL, as the spin interactions ought to be isotropic along the three symmetry-equivalent bonds in a honeycomb lattice. One plausible approach of realizing the anisotropic Kitaev interaction is to resort to spin-orbital-coupling (SOC) assisted Mott insulators, where the effective moment $J_{\rm eff}=1/2$ is an entanglement of the orbital moment $L=1$ and spin moment $S=1/2$.[9,13,14] This idea was initially applied to iridates such as Na$_2$IrO$_3$ where iridium and oxygen ions form edge-shared IrO$_6$ octahedra, and the 5$d$ iridium ions with strong SOC and $J_{\rm eff}=1/2$ form a two-dimensional honeycomb network.[13,15–22] The geometrical configuration suppresses the isotropic Heisenberg interaction while on the other hand promotes the anisotropic Kitaev interaction in the aid of the large SOC and strong spatial anisotropy of the $d$ orbitals.[13,23] These features should have made iridates a promising platform to investigate the Kitaev physics. However, these materials suffer from the following problems: (i) The presence of monoclinic and trigonal distortions renders the applicability of the localized $J_{\rm eff}$ picture to these materials questionable.[22,24,25] (ii) Iridium has a large neutron absorption cross section and thus it is difficult to carry out inelastic neutron scattering (INS) measurements to extract the magnetic interactions. (iii) Large-size and high-quality single crystals are unavailable.[22] Therefore, more efforts have been devoted into $\alpha$-RuCl$_3$ with the honeycomb lattice recently, which can also realize the SOC-assisted $J_{\rm eff}=1/2$ Mott insulating state, despite the weaker SOC effect of the 4$d$ Ru$^{3+}$.[26,27] Although the low-temperature phase is not the long-sought Kitaev QSL but zigzag ordered phase instead,[27–31] it is shown to exhibit salient Kitaev interactions due to the close-to-ideal bond configuration in the RuCl$_6$ octahedron.[29,31–39] Notably, the zigzag order is close to the Kitaev QSL phase in the phase diagram and is fragile subject to external magnetic field[28,40–46] or pressure.[47–49] More intriguingly, a moderate in-plane magnetic field may drive the system into a QSL phase.[50–53] In the zigzag phase, INS experiments show that the excitations are mostly concentrated around the $M$ and $\varGamma$ points of the two-dimensional Brillouin zone, as sketched in Fig. 1(a). While it is generally believed the gapped sharp excitations around the $M$ point are spin waves associated with the magnetic order, the origin of the broad continuum around the $\varGamma$ point is still controversial.[29,31,37,54–60] On the one hand, various spectroscopic results, such as Raman scattering,[55,56] optical THz spectroscopy,[57–59] and INS spectra,[29,37,60] show a nearly temperature-independent excitation continuum around the $\varGamma$ point, which is reminiscent of the fractionalized excitations originating from the Kitaev QSL state. On the other hand, because of the small scattering angle around the $\varGamma$ point, the broad continuous feature in the INS experiment could be due to the nuclear scattering,[37] which is also temperature independent. In addition, it has been argued that the continuum feature may originate from the incoherent excitations caused by strong magnetic anharmonicity.[54] Therefore, it is critical to distinguish whether the continuum is fractional magnetic excitation or nuclear scattering. Polarized neutron scattering experiment, from which pure magnetic scattering can be obtained, will be helpful for solving this problem.[61,62] However, because of the much reduced scattering cross section in a polarized neutron experiment, it is very challenging to obtain meaningful results for thin single crystals like $\alpha$-RuCl$_3$.
cpl-39-2-027501-fig1.png
Fig. 1. (a) Schematic of the magnetic excitations in the first Brillouin zone for $\alpha$-RuCl$_3$ in the zigzag magnetic order state. Thick solid lines denote the Brillouin zone boundary. Excitations around the $\varGamma$ point represent the scattering continuum. The six-fold symmetric excitations around the $M$ point denote spin-wave excitations. (b) Schematic of the experimental setup for the polarized neutron scattering measurements for the case of $\boldsymbol{P}\|\boldsymbol{Q}$. $\boldsymbol{P}$ is the neutron polarization direction, and $\boldsymbol{Q}$ is the scattering vector, obtained by $\boldsymbol{Q}=\boldsymbol{k}_{\rm{i}}-\boldsymbol{k}_{\rm{f}}$ with $\boldsymbol{k}_{\rm{i}}$ and $\boldsymbol{k}_{\rm{f}}$ being the initial and final wave vector of neutrons, respectively. Neutron polarization directions parallel, up and perpendicular, with respect to $\boldsymbol{Q}$ are labeled by $\boldsymbol{u}$, $\boldsymbol{w}$ and $\boldsymbol{v}$, respectively. Spin-flip (SF) and non-spin-flip (NSF) denote spin-flip and non-spin-flip of neutrons, respectively.
In this Letter, we report polarized neutron scattering study on 2-g high-quality $\alpha$-RuCl$_3$ single crystals. The data allow us to distinguish magnetic scattering from nuclear scattering. Our polarization results show that the broad continuous excitations around the $\varGamma$ point are of magnetic origin, which are persistent at temperatures far above the magnetic ordering temperature $T_{\rm{N}}$. In contrast, the spin-wave excitations around the $M$ point disappear above $T_{\rm{N}}$. Moreover, by comparing the calculation results using Kitaev-off-diagonal ($K$–$\varGamma$) and Kitaev–Heisenberg ($K$–$J$) models with the unpolarized neutron experimental data, we find that the $K$–$\varGamma$ model with a ferromagnetic $K=-7.2$ meV and $\varGamma=5.6$ meV can describe both the low-energy response of the spin-wave excitations associated with the zigzag ordered state around the $M$ point, and the continuous magnetic excitations around the $\varGamma$ point. These results are consistent with the presence of fractional magnetic excitations in $\alpha$-RuCl$_3$ and further support the $K$–$\varGamma$ model to be the minimal effective spin model in describing the system.
cpl-39-2-027501-fig2.png
Fig. 2. (a) and (b) Energy scans of $\alpha$-RuCl$_3$ at the $\varGamma$ point (0, 0, 1.5) measured at 1.8 and 25 K with polarized neutrons in the spin-flip (SF) and non-spin-flip (NSF) modes. The neutron polarization used was $\boldsymbol{P}\|\boldsymbol{Q}$. (c) and (d) Same scans as in (a) and (b) but at the $M$ point (0.5, 0, 0). Lines through data are guides to the eyes. Errors represent one standard deviation throughout the article.
High-quality single crystals of $\alpha$-RuCl$_3$ were grown using the chemical-vapor-transport method with $\alpha$-RuCl$_3$ powders. The sheet-like crystals had a natural $a$–$b$ plane as the cleavage plane and had a typical size of $10\times10\times1$ mm$^3$, which weighed about 60 mg a piece. Both susceptibility measurements and specific heat measurements showed a sharp antiferromagnetic phase transition at $\sim $7.5 K.[31,44,51] Polarized neutron scattering measurements were performed on PANDA, a cold neutron triple-axis spectrometer with the polarized option located at FRM II.[63] The $XYZ$-difference method for neutron polarization analysis[64] was used to separate the magnetic scattering from nuclear coherent, nuclear spin incoherent, isotopic incoherent and background contributions. Unpolarized neutron scattering measurements were carried out on a cold neutron triple-axis spectrometer FLEXX located at Helmholtz-Zentrum Berlin (HZB). We used a fixed final energy mode with $E_{\rm f}=5$ meV, and a double-focusing condition for both the monochromator and analyzer for all these experiments. For the experiment on PANDA, about 45 single crystals weighed about 2 g in total were co-aligned together on two aluminum plates using a Laue x-ray diffractometer. Heusler alloys with the (1, 1, 1) reflection plane were used as the monochromator and analyzer. Guide magnetic fields were used to rotate the neutron polarization direction along ($\boldsymbol{P}\|\boldsymbol{Q}$) or vertical to the scattering wave vector ($\boldsymbol{P}\perp\boldsymbol{Q}$). A Mezei coil flipper was used between the sample and analyzer to flip the polarization of neutrons. The flipping ratios of the three directions $\boldsymbol{u}$, $\boldsymbol{v}$ and $\boldsymbol{w}$ we labeled in Fig. 1(b) were 18.1, 7.5, 16.9, respectively. We denote a scattering event in which the spin makes a 180$^\circ$ rotation as spin flip (SF), and scattering with no spin direction change as non spin flip (NSF). For the measurements on FLEXX, we used 24 single crystals with a total mass of $\sim $2.2 g. All the experiments were conducted in the $(H0L)$ plane. The sample mosaic was about 3.6 degrees at PANDA and 1.93 degrees at FLEXX scanned around the Bragg peak (0, 0, 3). We used the $P3_112$ notation with $a=b=5.96$ Å, and $c=17.2$ Å as the lattice constants. The wave vector ${\boldsymbol Q}$ was described by ($HKL$) reciprocal lattice units (r.l.u.) of $(a^{*}, b^{*}, c^{*})=(4\pi/\sqrt3a, 4\pi/\sqrt3b, 2\pi/c)$.
cpl-39-2-027501-fig3.png
Fig. 3. (a) and (b) Energy scans at the $\varGamma$ point measured at 1.8 and 25 K with polarized neutrons in the SF mode. The neutron polarization used was $\boldsymbol{P}\perp\boldsymbol{Q}$. ${I}_{v}^{\rm{SF}}$ and ${I}_{w}^{\rm{SF}}$ are scattering intensities for SF mode in directions of perpendicular and up with respect to $\boldsymbol{Q}$. (c) and (d) Pure magnetic intensities around the $\varGamma$ point measured at 1.8 and 25 K, obtained by ${I}_{\rm{mag}}^{w+v}={I}_{\rm{mag}}^{w}+{I}_{\rm{mag}}^{v}= 2{I}_{u}^{\rm{SF}}-{I}_{v}^{\rm{SF}}-{I}_{w}^{\rm{SF}}$. (e) and (f) Same as in (c) and (d) but at the $M$ point. Solid lines through data are guides to the eyes. Dashed lines denote the background.
According to the polarized neutron scattering rules,[65] when the neutron polarization direction $\boldsymbol{P}$ is parallel to the scattering vector $\boldsymbol{Q}$, magnetic and nuclear scatterings are detected in the SF and NSF channels, respectively. In the experiment, we polarized neutrons along the scattering vector first. Here we take (0, 0, 1.5) as the $\varGamma$ point since neutron scattering experiment cannot access the $\varGamma$ point at $\boldsymbol{Q}=0$. Figure 2 shows energy scans at the $\varGamma$ and $M$ points (0.5, 0, 0) at 1.8 K (below the $T_{\rm{N}}$ of 7.5 K) and 25 K (well above the $T_{\rm{N}}$). As shown in Fig. 2(a), pronounced magnetic excitations are observed in the SF channel at $T=1.8$ K at the $\varGamma$ point. There is a clear peak centered at about 1.8 meV. On the other hand, the data in the NSF channel is essentially a sloping background. These results indicate that the excitations at the $\varGamma$ point detected in the unpolarized INS experiments previously are of magnetic origin.[37,45,60] When we increase the temperature up to 25 K, in the SF channel, there are still clear magnetic excitations on top of the sloping background, as shown in Fig. 2(b). The magnetic excitations are significantly broader than those in the magnetically ordered state. This broad and continuum-like feature is a characteristic of the fractional excitations.[37,60] We suspect that the reason why intensities at low temperature are stronger than those at high temperature is that the low-temperature excitations have some interactions with the spin-wave excitations. For comparison, we also measured the excitations at the $M$ point, which is the wave vector of the zigzag magnetic order.[27,31] As shown in Fig. 2(c), a magnetic peak is observed at $T=1.8$ K in the SF channel on top of the sloping background obtained in the NSF channel. As expected, this peak disappears when the temperature is above the $T_{\rm{N}}$, consistent with it being the spin-wave excitations arising from the zigzag magnetic order.[29,31,37] The different temperature dependence of the excitations at the $\varGamma$ and $M$ points and the persistence of the magnetic excitations at the $\varGamma$ point further prove that these excitations are magnetic, fractionalized, and robust within a certain temperature window.[60,66] Although all the magnetic scattering is present in the SF channel when $\boldsymbol{P} \| \boldsymbol{Q}$, the total intensity contains some background resulting from the incoherent nuclear spin scattering with an intensity contribution of $\frac{2}{3}{{I}_{\rm{inc}}^{\rm{spin}}}$.[65] In order to deduct this term and obtain pure magnetic scattering intensities, we polarized the neutron beam to the two directions $\boldsymbol{v}$ and $\boldsymbol{w}$ perpendicular to $\boldsymbol{Q}$ as illustrated in Fig. 1(b) and performed the same scans as in Fig. 2 to obtain ${I}_{v}^{\rm{SF}}$ and ${I}_{w}^{\rm{SF}}$, as shown in Figs. 3(a) and 3(b). Pure magnetic scattering intensities are given by ${I}_{\rm{mag}}^{w+v}={I}_{\rm{mag}}^{w}+{I}_{\rm{mag}}^{v} =2{I}_{u}^{\rm{SF}}-{I}_{v}^{\rm{SF}}-{I}_{w}^{\rm{SF}}$.[65] The so-obtained intensities at the $\varGamma$ point are shown in Figs. 3(c) and 3(d). It can be seen that they exist at both low and high temperatures. The temperature-stable magnetic excitations at the $\varGamma$ point are evidently the fractional excitations which are originated from the Kitaev QSL state. For comparison, the pure magnetic intensities at the $M$ point are shown in Figs. 3(e) and 3(f). In contrast to those at the $\varGamma$ point, the magnetic excitations at the $M$ point are only observed at the low temperature. The observations that the pure magnetic intensities ${I}_{\rm{mag}}^{w+v}$ at the $\varGamma$ point remain constant against temperature while those of spin waves at the $M$ point are completely vanishing above the $T_{\rm N}$ indicate the emergence of fractional excitations at a higher temperature above the $T_{\rm N}$.[60,66]
cpl-39-2-027501-fig4.png
Fig. 4. (a) Magnetic dispersion along [$H$00] at $T=1.8$ K obtained by plotting the points marked with white cross dots. Left side of the dispersion from $H=-0.5$ to 0 r.l.u. is the mirror symmetrical figure on the right from $H=0$ to 0.5 r.l.u. (b) Magnetic excitation spectrum of the $K$–$\varGamma$ model for $\varGamma/K=-0.73$ with $K < 0$. (c) Theoretical magnetic excitation spectra for the zigzag phases of the $K$–$J$ model for $J/K=-0.36$ with $K>0$.
After pinning down the origin of the excitations at the $\varGamma$ point, we next discuss the effective spin model to describe the magnetic excitations at both the $\varGamma$ and $M$ points, in contrast to previous attempts where the excitations around the two momenta were treated separately using different models.[29,31,35,37] To do this, we first performed unpolarized INS measurements at $T=1.8$ K to obtain the magnetic dispersion as shown in Fig. 4(a). There are clear magnetic excitations weakly dispersing from both the $\varGamma$ and $M$ points. A broad continuum with a gap of $\sim $1.8 meV appears around the $\varGamma$ point, consistent with the polarization data and earlier results.[37,45,60] To describe these data, we first consider the $K$–$\varGamma$ model that we used to fit the spin-wave excitations in our previous works.[31,35] Its Hamiltonian can be written as $$ H_{K-\varGamma}=\sum_{\langle ij \rangle }[KS^{\gamma}_{i}S^{\gamma}_{j}+\varGamma(S^{\alpha}_{i}S^{\beta}_{j}+S^{\beta}_{i}S^{\alpha}_{j})].~~ \tag {1} $$ Here, $\gamma=x,y,z$ labels the three nearest-neighbor bonds on the honeycomb lattice, and $(\alpha,\,\beta,\,\gamma)$ is a permutation of $(x,y,z)$ with $\alpha$ and $\beta$ labeling the two remaining directions on a $\gamma$ bond. We calculated the dynamical spin-spin correlation function using the cluster perturbation theory, which has been successfully applied to other magnetically frustrated systems.[67] The calculated results with one typical parameter set of ferromagnetic $K=-7.2$ meV and $\varGamma=5.6$ meV are shown in Fig. 4(b). These parameters ensure that the system is in the zigzag order phase,[19,35] and are consistent with our previous works.[31,35] Compared to the experimental spectra shown in Fig. 4(a), it is clear that both the excitations at the $M$ and $\varGamma$ points can be well captured by this model. As we demonstrated in Refs. [31,35], the spin-wave excitations should only be observed to disperse from the $M$ point around 2 meV and reach the band top at the $\varGamma$ point at $E\sim4$ meV within the $K$–$\varGamma$ model. In Figs. 4(a) and 4(b), the continuous excitation spectra around the $\varGamma$ point drop to a much lower energy of $\sim $2 meV, therefore they should not be coherent spin waves but represent fractional excitations, supporting our polarization analyses. For comparison, we also consider the widely proposed $K$–$J$ model for Kitaev materials:[13,15,18,19] $$ H_{K-J}=\sum_{\langle ij \rangle }(J\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{j}+KS^{\gamma}_{i}S^{\gamma}_{j}).~~ \tag {2} $$ The calculated results with $K=9.8$ meV and $J=-3.6$ meV using the cluster perturbation theory are shown in Fig. 4(c). Although the parameter set places the system in the correct zigzag order phase in the phase diagram constructed using the $K$–$J$ model,[13,15,18,19] and produces the spin-wave excitations dispersing from the $M$ point, the calculation deviates from experimental observations in various aspects. Firstly, the spin-wave excitations are gapless, in contrast to the gapped spectra shown in Fig. 4(a). Secondly, the low-energy excitations near the $\varGamma$ point are absent. Thirdly, the Kitaev interaction is antiferromagnetic, contradicting previous reports on the ferromagnetic Kitaev interaction in $\alpha$-RuCl$_3$.[31,35,37,39] Therefore, the comparison between the experimental spectra and the theoretical results from different models not only confirms that the $K$–$\varGamma$ model is a more appropriate minimal spin model to describe $\alpha$-RuCl$_3$,[31,35,37] but also represents as evidence that there present continuous fractional magnetic excitations near the $\varGamma$ point. We note, however, in addition to the dominant $K$ and $\varGamma$ terms, some other finite longer-range terms, though small, may be necessary to stabilize the zigzag order state.[36,66] To conclude, we have studied the magnetic excitations by carrying out polarized INS measurements on high-quality $\alpha$-RuCl$_3$ single crystals. It is found that there exist pure continuous magnetic excitations near the $\varGamma$ point, which are robust against temperature, as opposed to the spin waves near the $M$ point that vanish above $T_{\rm N}$. In addition, our calculation using the cluster perturbation theory shows that the $K$–$\varGamma$ model with a ferromagnetic $K=-7.2$ meV and a comparable $\varGamma=5.6$ meV can reproduce not only the spin-wave excitations near the $M$ point, but also the continuous excitations near the $\varGamma$ point. These results are evident that there exist exotic fractional excitations in $\alpha$-RuCl$_3$ due to its proximity to the Kitaev QSL state. However, considering the weak cross sections of the polarized experiment, further investigations with better statistics by using more single crystals, increasing the counting time, or optimizing the polarization options, should be helpful to solidify the conclusion. Potentially, one can learn details about the anisotropic interactions and magnetizations with more serious polarization analysis because the scattering intensities contain both an interference and a Chiral magnetic scattering term.[68] In Ref. [66], it was proposed that the bond-dependent fractional excitations in $\alpha$-RuCl$_3$ can be manifested in the equal-time structure factor $S^{\gamma}({\boldsymbol Q})$ ($\gamma=x,y,z$ represents different spin components) in the intermediate temperature range. We believe this is an interesting proposal worth of future efforts. Furthermore, for systems like $\alpha$-RuCl$_3$, strongly spin-orbital-coupling may give rise to complex spin excitations like hybrid modes, the existence of frustrated anisotropic interactions will naturally lead to noticeable anharmonic effects.[54] These may better explain the enhancement of the low-temperature excitations at the $\varGamma$ point as observed experimentally. The work was supported by National Key Research and Development Program of China (Grant No. 2021YFA1400400), the National Natural Science Foundation of China (Grant Nos. 11822405, 12074174, 12074175, 11774152, 11904170, 12004249, 12004251, and 12004191), the Natural Science Foundation of Jiangsu Province (Grant Nos. BK20180006, BK20190436 and BK20200738), the Shanghai Sailing Program (Grant Nos. 20YF1430600 and 21YF1429200), the Fundamental Research Funds for the Central Universities (Grant No. 020414380183), and the Office of International Cooperation and Exchanges of Nanjing University. The experiment at FLEXX was carried out under proposal No. 172-05993CR using beam time allocated in the HZB-CIAE collaboration on the scientific use of instruments. References Resonating valence bonds: A new kind of insulator?The Resonating Valence Bond State in La2CuO4 and SuperconductivitySpin liquids in frustrated magnetsQuantum spin liquid statesQuantum spin liquids: a reviewExperimental identification of quantum spin liquidsQuantum spin liquidsAnyons in an exactly solved model and beyondKitaev MaterialsFault-tolerant quantum computation by anyonsNon-Abelian anyons and topological quantum computationCoherent transmutation of electrons into fractionalized anyonsMott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev ModelsSpin-Orbit Physics Giving Rise to Novel Phases in Correlated Systems: Iridates and Related MaterialsKitaev-Heisenberg Model on a Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides A 2 IrO 3 Kitaev-Heisenberg- J 2 - J 3 model for the iridates A 2 IrO 3 Relevance of the Heisenberg-Kitaev Model for the Honeycomb Lattice Iridates A 2 IrO 3 Zigzag Magnetic Order in the Iridium Oxide Na 2 IrO 3 Generic Spin Model for the Honeycomb Iridates beyond the Kitaev LimitImportance of anisotropic exchange interactions in honeycomb iridates: Minimal model for zigzag antiferromagnetic order in Na 2 IrO 3 Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate Na2IrO3Models and materials for generalized Kitaev magnetismChallenges in design of Kitaev materials: Magnetic interactions from competing energy scales Na 2 IrO 3 as a Molecular Orbital CrystalAb initio analysis of the tight-binding parameters and magnetic interactions in Na 2 IrO 3 α − RuCl 3 : A spin-orbit assisted Mott insulator on a honeycomb latticeMagnetic order in α − RuCl 3 : A honeycomb-lattice quantum magnet with strong spin-orbit couplingMonoclinic crystal structure of α − RuCl 3 and the zigzag antiferromagnetic ground stateProximate Kitaev quantum spin liquid behaviour in a honeycomb magnetLow-temperature crystal and magnetic structure of α − RuCl 3 Spin-Wave Excitations Evidencing the Kitaev Interaction in Single Crystalline α − RuCl 3 Kitaev magnetism in honeycomb RuCl 3 with intermediate spin-orbit couplingSpin-orbit excitations and electronic structure of the putative Kitaev magnet α − RuCl 3 Crystal structure and magnetism in α − RuCl 3 : An ab initio studyTheoretical investigation of magnetic dynamics in α − RuCl 3 Magnetization processes of zigzag states on the honeycomb lattice: Identifying spin models for α − RuCl 3 and Na 2 IrO 3 Neutron scattering in the proximate quantum spin liquid α-RuCl 3Quantum spin liquid signatures in Kitaev-like frustrated magnetsFerromagnetic Kitaev interaction and the origin of large magnetic anisotropy in α-RuCl3Successive magnetic phase transitions in α − RuCl 3 : XY-like frustrated magnet on the honeycomb latticeAnisotropic Ru 3 +   4 d 5 magnetism in the α − RuCl 3 honeycomb system: Susceptibility, specific heat, and zero-field NMRPhase diagram of α − RuCl 3 in an in-plane magnetic fieldAnisotropic magnetodielectric effect in the honeycomb-type magnet α − RuCl 3 Ultralow-Temperature Thermal Conductivity of the Kitaev Honeycomb Magnet α − RuCl 3 across the Field-Induced Phase TransitionExcitations in the field-induced quantum spin liquid state of α-RuCl3Finite field regime for a quantum spin liquid in α − RuCl 3 High-pressure magnetization and NMR studies of α − RuCl 3 Pressure-induced dimerization and valence bond crystal formation in the Kitaev-Heisenberg magnet α − RuCl 3 Pressure-induced melting of magnetic order and emergence of a new quantum state in α − RuC l 3 Evidence for a Field-Induced Quantum Spin Liquid in α - RuCl 3 Gapless Spin Excitations in the Field-Induced Quantum Spin Liquid Phase of α − RuCl 3 Dirac and Chiral Quantum Spin Liquids on the Honeycomb Lattice in a Magnetic FieldMajorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquidBreakdown of magnons in a strongly spin-orbital coupled magnetScattering Continuum and Possible Fractionalized Excitations in α − RuCl 3 Fermionic response from fractionalization in an insulating two-dimensional magnetAntiferromagnetic Resonance and Terahertz Continuum in α − RuCl 3 Magnetic Excitations and Continuum of a Possibly Field-Induced Quantum Spin Liquid in α − RuCl 3 Electronic and phonon excitations in α − RuC l 3 Majorana fermions in the Kitaev quantum spin system α-RuCl3The use of polarized neutrons in determining the magnetic scattering by iron and nickelPolarization Analysis of Thermal-Neutron ScatteringPANDA: Cold three axes spectrometerTheXYZ-Difference Method with Polarized Neutrons and the Separation of Coherent, Spin Incoherent, and Magnetic Scattering Cross Sections in a MultidetectorUniversal thermodynamics in the Kitaev fractional liquidDeconfinement of spinons in frustrated spin systems: Spectral perspectiveXYZ-polarisation analysis of diffuse magnetic neutron scattering from single crystals
[1] Anderson P W 1973 Mater. Res. Bull. 8 153
[2] Anderson P W 1987 Science 235 1196
[3] Balents L 2010 Nature 464 199
[4] Zhou Y, Kanoda K, and Ng T K 2017 Rev. Mod. Phys. 89 025003
[5] Savary L and Balents L 2017 Rep. Prog. Phys. 80 016502
[6] Wen J, Yu S L, Li S, Yu W, and Li J X 2019 npj Quantum Mater. 4 12
[7] Broholm C, Cava R J, Kivelson S A, Nocera D G, Norman M R, and Senthil T 2020 Science 367 eaay0668
[8] Kitaev A 2006 Ann. Phys. 321 2
[9] Trebst S 2017 arXiv:1701.07056 [cond-mat.str-el]
[10] Kitaev A Y 2003 Ann. Phys. 303 2
[11] Nayak C, Simon S H, Stern A, Freedman M, and Sarma S D 2008 Rev. Mod. Phys. 80 1083
[12] Barkeshli M, Berg E, and Kivelson S 2014 Science 346 722
[13] Jackeli G and Khaliullin G 2009 Phys. Rev. Lett. 102 017205
[14] Rau J G, Lee E K H, and Kee H Y 2016 Annu. Rev. Condens. Matter Phys. 7 195
[15] Chaloupka J, Jackeli G, and Khaliullin G 2010 Phys. Rev. Lett. 105 027204
[16] Kimchi I and You Y Z 2011 Phys. Rev. B 84 180407
[17] Singh Y, Manni S, Reuther J, Berlijn T, Thomale R, Ku W, Trebst S, and Gegenwart P 2012 Phys. Rev. Lett. 108 127203
[18] Chaloupka J, Jackeli G, and Khaliullin G 2013 Phys. Rev. Lett. 110 097204
[19] Rau J G, Lee E K H, and Kee H Y 2014 Phys. Rev. Lett. 112 077204
[20] Sizyuk Y, Price C, Wölfle P, and Perkins N B 2014 Phys. Rev. B 90 155126
[21] Chun S H, Kim J W, Kim J, Zheng H, Stoumpos C C, Malliakas C D, Mitchell J F, Mehlawat K, Singh Y, Choi Y, Gog T, Al-Zein A, Sala M M, Krisch M, Chaloupka J, Jackeli G, Khaliullin G, and Kim B J 2015 Nat. Phys. 11 462
[22] Winter S M, Tsirlin A A, Daghofer M, van den Brink J, Singh Y, Gegenwart P, and Valentí R 2017 J. Phys.: Condens. Matter 29 493002
[23] Winter S M, Li Y, Jeschke H O, and Valentí R 2016 Phys. Rev. B 93 214431
[24] Mazin I I, Jeschke H O, Foyevtsova K, Valentí R, and Khomskii D I 2012 Phys. Rev. Lett. 109 197201
[25] Foyevtsova K, Jeschke H O, Mazin I I, Khomskii D I, and Valentí R 2013 Phys. Rev. B 88 035107
[26] Plumb K W, Clancy J P, Sandilands L J, Shankar V V, Hu Y F, Burch K S, Kee H Y, and Kim Y J 2014 Phys. Rev. B 90 041112
[27] Sears J A, Songvilay M, Plumb K W, Clancy J P, Qiu Y, Zhao Y, Parshall D, and Kim Y J 2015 Phys. Rev. B 91 144420
[28] Johnson R D, Williams S C, Haghighirad A A, Singleton J, Zapf V, Manuel P, Mazin I I, Li Y, Jeschke H O, Valentí R, and Coldea R 2015 Phys. Rev. B 92 235119
[29] Banerjee A, Bridges C A, Yan J Q, Aczel A A, Li L, Stone M B, Granroth G E, Lumsden M D, Yiu Y, Knolle J, Bhattacharjee S, Kovrizhin D L, Moessner R, Tennant D A, Mandrus D G, and Nagler S E 2016 Nat. Mater. 15 733
[30] Cao H B, Banerjee A, Yan J Q, Bridges C A, Lumsden M D, Mandrus D G, Tennant D A, Chakoumakos B C, and Nagler S E 2016 Phys. Rev. B 93 134423
[31] Ran K, Wang J, Wang W, Dong Z Y, Ren X, Bao S, Li S, Ma Z, Gan Y, Zhang Y, Park J T, Deng G, Danilkin S, Yu S L, Li J X, and Wen J 2017 Phys. Rev. Lett. 118 107203
[32] Kim H S, Vijay S V, Catuneanu A, and Kee H Y 2015 Phys. Rev. B 91 241110
[33] Sandilands L J, Tian Y, Reijnders A A, Kim H S, Plumb K W, Kim Y J, Kee H Y, and Burch K S 2016 Phys. Rev. B 93 075144
[34] Kim H S and Kee H Y 2016 Phys. Rev. B 93 155143
[35] Wang W, Dong Z Y, Yu S L, and Li J X 2017 Phys. Rev. B 96 115103
[36] Janssen L, Andrade E C, and Vojta M 2017 Phys. Rev. B 96 064430
[37] Banerjee A, Yan J, Knolle J, Bridges C A, Stone M B, Lumsden M D, Mandrus D G, Tennant D A, Moessner R, and Nagler S E 2017 Science 356 1055
[38] Gohlke M, Wachtel G, Yamaji Y, Pollmann F, and Kim Y B 2018 Phys. Rev. B 97 075126
[39] Sears J A, Chern L E, Kim S, Bereciartua P J, Francoual S, Kim Y B, and Kim Y J 2020 Nat. Phys. 16 837
[40] Kubota Y, Tanaka H, Ono T, Narumi Y, and Kindo K 2015 Phys. Rev. B 91 094422
[41] Majumder M, Schmidt M, Rosner H, Tsirlin A A, Yasuoka H, and Baenitz M 2015 Phys. Rev. B 91 180401
[42] Sears J A, Zhao Y, Xu Z, Lynn J W, and Kim Y J 2017 Phys. Rev. B 95 180411
[43] Aoyama T, Hasegawa Y, Kimura S, Kimura T, and Ohgushi K 2017 Phys. Rev. B 95 245104
[44] Yu Y J, Xu Y, Ran K J, Ni J M, Huang Y Y, Wang J H, Wen J S, and Li S Y 2018 Phys. Rev. Lett. 120 067202
[45] Banerjee A, Lampen-Kelley P, Knolle J, Balz C, Aczel A A, Winn B, Liu Y, Pajerowski D, Yan J, Bridges C A, Savici A T, Chakoumakos B C, Lumsden M D, Tennant D A, Moessner R, Mandrus D G, and Nagler S E 2018 npj Quantum Mater. 3 8
[46] Balz C, Lampen-Kelley P, Banerjee A, Yan J, Lu Z, Hu X, Yadav S M, Takano Y, Liu Y, Tennant D A, Lumsden M D, Mandrus D, and Nagler S E 2019 Phys. Rev. B 100 060405
[47] Cui Y, Zheng J, Ran K, Wen J, Liu Z X, Liu B, Guo W, and Yu W 2017 Phys. Rev. B 96 205147
[48] Bastien G, Garbarino G, Yadav R, Martinez-Casado F J, Rodríguez R B, Stahl Q, Kusch M, Limandri S P, Ray R, Lampen-Kelley P, Mandrus D G, Nagler S E, Roslova M, Isaeva A, Doert T, Hozoi L, Wolter A U B, Büchner B, Geck J, and van den Brink J 2018 Phys. Rev. B 97 241108
[49] Wang Z, Guo J, Tafti F F, Hegg A, Sen S, Sidorov V A, Wang L, Cai S, Yi W, Zhou Y, Wang H, Zhang S, Yang K, Li A, Li X, Li Y, Liu J, Shi Y, Ku W, Wu Q, Cava R J, and Sun L 2018 Phys. Rev. B 97 245149
[50] Baek S H, Do S H, Choi K Y, Kwon Y S, Wolter A U B, Nishimoto S, van den Brink J, and Büchner B 2017 Phys. Rev. Lett. 119 037201
[51] Zheng J, Ran K, Li T, Wang J, Wang P, Liu B, Liu Z X, Normand B, Wen J, and Yu W 2017 Phys. Rev. Lett. 119 227208
[52] Liu Z X and Normand B 2018 Phys. Rev. Lett. 120 187201
[53] Kasahara Y, Ohnishi T, Mizukami Y, Tanaka O, Ma S, Sugii K, Kurita N, Tanaka H, Nasu J, Motome Y, Shibauchi T, and Matsuda Y 2018 Nature 559 227
[54] Winter S M, Riedl K, Maksimov P A, Chernyshev A L, Honecker A, and Valentí R 2017 Nat. Commun. 8 1152
[55] Sandilands L J, Tian Y, Plumb K W, Kim Y J, and Burch K S 2015 Phys. Rev. Lett. 114 147201
[56] Nasu J, Knolle J, Kovrizhin D L, Motome Y, and Moessner R 2016 Nat. Phys. 12 912
[57] Little A, Wu L, Lampen-Kelley P, Banerjee A, Patankar S, Rees D, Bridges C A, Yan J Q, Mandrus D, Nagler S E, and Orenstein J 2017 Phys. Rev. Lett. 119 227201
[58] Wang Z, Reschke S, Hüvonen D, Do S H, Choi K Y, Gensch M, Nagel U, Rõõm T, and Loidl A 2017 Phys. Rev. Lett. 119 227202
[59] Reschke S, Mayr F, Wang Z, Do S H, Choi K Y, and Loidl A 2017 Phys. Rev. B 96 165120
[60] Do S H, Park S Y, Yoshitake J, Nasu J, Motome Y, Kwon Y S, Adroja D T, Voneshen D J, Kim K, Jang T H, Park J H, Choi K Y, and Ji S 2017 Nat. Phys. 13 1079
[61] Nathans R, Shull C G, Shirane G, and Andresen A 1959 J. Phys. Chem. Solids 10 138
[62] Moon R M, Riste T, and Koehler W C 1969 Phys. Rev. 181 920
[63] Schneidewind A and Čermák P 2015 J. Large-Scale Res. Facil. 1 A12
[64] Schaerpf O and Capellmann H 1993 Phys. Status Solidi A 135 359
[65]Shirane G, Shapiro S M, and Tranquada J M 2002 Neutron Scattering with a Triple-Axis Spectrometer: Basic Techniques (Cambridge: Cambridge University Press)
[66] Li H, Qu D W, Zhang H K, Jia Y Z, Gong S S, Qi Y, and Li W 2020 Phys. Rev. Res. 2 043015
[67] Yu S L, Wang W, Dong Z Y, Yao Z J, and Li J X 2018 Phys. Rev. B 98 134410
[68] Schweika W 2010 J. Phys.: Conf. Ser. 211 012026