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Neutron scattering in the proximate quantum spin liquid α-RuCl3

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Science  09 Jun 2017:
Vol. 356, Issue 6342, pp. 1055-1059
DOI: 10.1126/science.aah6015

Sighting of magnetic Majorana fermions?

Quantum spin liquids—materials whose magnetic spins do not settle into order even at absolute zero temperature—have long captured the interest of physicists. A particularly lofty goal is finding a material that can be described by the so-called Kitaev spin model, a network of spins on a honeycomb lattice that harbors Majorana fermions as its excitations. Banerjee et al. present a comprehensive inelastic neutron scattering study of single crystals of the material α-RuCl3, which has been predicted to a host a Kitaev spin liquid. The unusual dependence of the data on energy, momentum, and temperature is consistent with the Kitaev model.

Science, this issue p. 1055

Abstract

The Kitaev quantum spin liquid (KQSL) is an exotic emergent state of matter exhibiting Majorana fermion and gauge flux excitations. The magnetic insulator α-RuCl3 is thought to realize a proximate KQSL. We used neutron scattering on single crystals of α-RuCl3 to reconstruct dynamical correlations in energy-momentum space. We discovered highly unusual signals, including a column of scattering over a large energy interval around the Brillouin zone center, which is very stable with temperature. This finding is consistent with scattering from the Majorana excitations of a KQSL. Other, more delicate experimental features can be transparently associated with perturbations to an ideal model. Our results encourage further study of this prototypical material and may open a window into investigating emergent magnetic Majorana fermions in correlated materials.

Quantum spin liquids (QSLs) are collective magnetic states that can form in the networks of atomic moments (“spins”) in materials. The spins fail to enter an ordinary static ordered state, such as a ferromagnet, as the temperature approaches zero and instead become highly entangled and fluctuate quantum mechanically (1, 2). A defining feature of QSLs, connected to their topological nature, is excitations that carry fractional quantum numbers (3, 4)—a phenomenon underpinning the physics of the fractional quantum Hall effect (5), magnetic monopoles (6), and spin-charge separation (7). Fractionalization can be seen experimentally by momentum-energy space reconstruction. Inelastic neutron scattering directly probes magnetic correlations in space and time. As discussed below, our experiments provide a comprehensive image of the collective magnetic fluctuations in a topological quantum magnet.

Kitaev QSLs (KQSLs) (813) are currently the focus of intense interest. The Kitaev model offers distinctive insight into spin-liquid physics, as its exact solubility permits a detailed analysis of its properties, including those of its fractionalized Majorana fermion and gauge flux excitations. Beyond their importance in fundamental physics, KQSLs are particularly noteworthy because a magnetic field turns them into non-Abelian anyons, which can underpin a quantum computing architecture topologically protected against decoherence (14, 15). The Kitaev model consists of an extremely simple spin network (8, 9) with localized S = ½ spins on a honeycomb lattice with an interaction HamiltonianEmbedded Image(1)for either ferromagnetic (FM) or antiferromagnetic (AF) coupling K. Here, Embedded Image runs over the lattice sites, and the index γ (= x, y, z in spin space) denotes the relevant interacting spin component for the nearest-neighbor bond joined by vector Embedded Image, with Ising interaction strength Kγ (Fig. 1A).

Fig. 1 Structure and magnetism in single-crystal α-RuCl3.

(A) Honeycomb lattice of Ru3+ magnetic ions in one plane of α-RuCl3, showing the projections of the three mutually competing Ising bonds corresponding to the Kitaev terms in Eq. 1. (B) The intensity of the magnetic Bragg Peak occurring at the M point of the 2D honeycomb lattice corresponding to a zigzag structure with three-layer stacking [Embedded Image in trigonal or Embedded Image in monoclinic notation]. The single sharp magnetic transition is characteristic of crystals with few or no stacking faults (26). The solid line is a power-law fit yielding TN = 6.96 ± 0.02 K and a critical exponent β = 0.125 ± 0.015, suggesting 2D Ising behavior. Error bars indicate 1 SD, assuming Poisson counting statistics. (Inset) The 490-mg single crystal of α-RuCl3 used for the neutron measurements. [For more sample details, see the materials and methods (27).]

Insulating materials comprising weakly coupled honeycomb layers of strongly spin-orbit–coupled transition metal ions in edge-sharing cubic octahedra (16) are promising candidates for realizing KQSLs. These have included iridates containing Ir4+ (1720) and, most recently, the Ru3+-based honeycomb magnet α-RuCl3 (2126). Here, we present inelastic neutron scattering on a single crystal of α-RuCl3, providing a complete measurement of the magnetic response function in four-dimensional (4D) energy-momentum space. From a technical perspective, our findings demonstrate a qualitative advance over the polycrystalline samples studied to date (25), as well as over single-crystal Raman studies (23), which are unable to distinguish between different directions in momentum space.

We used a 490-mg single crystal grown by vapor transport of phase-pure α-RuCl3 (27). This crystal has a low incidence of stacking faults and exhibits a single magnetic ordering transition at TN = 7 K, where TN is the Néel temperature (Fig. 1B and fig. S1). Below TN, the magnetic order is zigzag in the individual honeycomb layers, with a three-layer periodicity out of plane. The ordered moment <μ> ~ 0.5μB/Ru3+ (where μB is a unit Bohr magneton) is only about one-third of the net paramagnetic moment (22, 25, 26). The details of the ordering can vary in different samples depending on the precise stacking of the layers; in any case, the ordering is incidental to the 2D QSL physics of interest here.

Figure 2 contains a first set of central results. It depicts the temperature and momentum dependence of a magnetic scattering continuum for two energy ranges: 4.5 to 7.5 meV and 7.5 to 12.5 meV. The most salient feature is the robust response centered at the Γ point: It is present from low (T = 5 K < TN) all the way to high (T = 120 K >> TN) temperatures corresponding to the thermal energy scales of the Kitaev coupling, which is estimated to be Kγ ≈ 70 to 90 K (24, 25). On passing from below to above TN, the central portion of the scattering strengthens. The overall intensity, although weaker, is still readily visible at very high T. At all temperatures, this dynamic scattering extends through a large fraction of the Brillouin zone (BZ), indicating short-ranged liquid correlations [see (27) and figs. S2 and S3 for BZ definitions]. The energy dependence of the scattering at the Γ point is illustrated in Fig. 3, A and B, at temperatures below and above TN, respectively. Above TN, the broad scattering continuum (denoted “C”) extends nearly to 15 meV, in keeping with expectations for a pure Kitaev model with Kγ ≈ 5.5 to 8 meV (24, 25). Below TN, a fraction of the spectral weight shifts into sharp (i.e., energy-resolution–limited) spin-wave (SW) peaks arising from the small zigzag-ordered moments. The 2D nature of the response is shown by the rodlike dependence on the out-of-plane momentum component L of the scattering illustrated in Fig. 3, C and D.

Fig. 2 Momentum and temperature dependence of the scattering continuum.

Neutron scattering measurements using fixed incident energy Ei = 40 meV, projected on the reciprocal honeycomb plane defined by the perpendicular directions (H, H, 0) and (K, –K, 0), integrated over the interval L = [–2.5, 2.5]. Intensities are denoted by color, as indicated in the scale at right. Measurements integrated over the energy range [4.5, 7.5] meV are shown on the top row at temperatures (A) 5 K, (B) 10 K, and (C) 120 K. The corresponding measurements integrated over the interval [7.5, 12.5] meV are shown on the bottom in (D), (E), and (F). The white regions lack detector coverage. See fig. S11 for orientationally averaged data.

Fig. 3 Detailed features of the Γ point scattering.

(A and B) Energy dependence of the scattering at (A) 5 K and (B) 10 K shows a broad peak. The data shown are integrated over constant momentum volumes defined by the following ranges: Embedded Image over the range Embedded Image over the range Embedded Image. The solid lines are visual guides produced by modeling the elastic component as a Gaussian peak and the inelastic features using damped harmonic oscillator (DHO) functions: E, elastic component; S, spin-wave (SW) peaks appearing below TN; C, continuum. Fit parameters and the DHO function are presented in table S1. Error bars indicate 1 SD, assuming Poisson counting statistics. (C) Scattering symmetrized in the (H, H, L) plane and over positive and negative L, integrated over the intervals Embedded Image and E = [4.5, 7.5] meV at T = 10 K. (D) Same scattering, but in the (K, –K, L) plane integrated over ξ = [–0.1, 0.1] and E = [4.5, 7.5] meV. (E) Representative low-energy scattering expected from spin-wave theory (SWT) for a zigzag-ordered phase (25). (F) Scattering at T = 5 K integrated over L = [–2.5, 2.5] and E = [2, 3] meV. The white regions lack detector coverage.

The persistent energy continuum at the Γ point is incompatible with conventional SW physics. Fig. 3E shows the generic low-energy SW response for a zigzag-ordered state. This takes the form of dispersive energy-momentum cones centered about each M point magnetic Bragg peak (Fig. 3F). In SW theory, the Γ point scattering is present only at certain fixed-energy values, unlike the experimentally observed broad energy column (Figs. 2 and 3F). Moreover, SW scattering at long wavelengths is strongly sensitive to cooling through TN (25), in stark contrast to the continuum. The latter is very broad in energy and almost independent of temperature up to around 100 K (~Kγ >> TN), consistent with the thermodynamics of the Kitaev model (23, 2830) (Γ point scattering at T = 120 K is shown in fig. S6D). The energy breadth and temperature dependence of the continuum are consistent with fractionalized excitations (13, 2830).

Whether extensions of SW theory—for example, based on a sequence of multiple SW contributions (31)—might be able to account for such a phenomenology is an interesting question that involves technical and conceptual challenges in accounting for the temperature dependence of the inelastic neutron scattering data. In particular, as the low-energy single-SW response is reconstructed upon heating through the ordering temperature, one would naturally expect this to imply a strong renormalization of the high-energy multi-SW signal, which is in contrast to experimental observations.

Figure 4A shows an extended zone picture of the T = 5 K data integrated in energy over the range E = [4.5, 7.5] meV, symmetrized along the (H, H, 0) direction. In addition to the strong scattering at H = K = 0, features are now visible near adjacent Γ points ±(1, 1), showing that the continuum spectrum repeats every second BZ. Additional scattering at larger momentum transfer (Q) arises from phonons. In the following paragraphs, we show that a Kitaev QSL description reproduces the main qualitative features of the data—in particular, the broad energy width and T dependence of the scattering continuum, as well as its periodicity and relative orientation in the BZ, which encodes the orientational bond dependence of the spin anisotropy in Kitaev systems.

Fig. 4 Comparison of the scattering with T = 0 Kitaev model calculations.

(A) Data at Ei = 40 meV and T = 5 K, integrated over the range E = [4.5, 7.5] meV and L = [–2.5, 2.5] and symmetrized along the (H, H) direction. (B) Expected scattering from an isotropic antiferromagnetic (AF) Kitaev model at an energy E = 1.2Kγ, taking into account the neutron polarization and the spherically approximated Ru3+ form factor. (C) Plot of the nonsymmetrized data (points with error bars) along (H, H, 0) at T = 5 K, integrated over the same L and E intervals as in panel (A), as well as Embedded Image. The solid red line is the calculated scattering for an AF Kitaev model with R = 2, as discussed in the text. The solid purple line represents the corresponding unmodified AF Kitaev model, and the green line denotes the ferromagnetic (FM) Kitaev model. Some of the scattering at larger Q near (H, H) = ±(1, 1) is due to phonons. Error bars indicate 1 SD, assuming Poisson counting statistics.

The momentum dependence of the scattering for a pure Kitaev model at T = 0 is exactly known (12, 13, 27) (fig. S5). The dynamical structure factor consists of two energy-dependent correlations, those for onsite (S0) and nearest-neighbor (S1) spins (27) (fig. S4). For simplicity, we compare the scattering to calculations for an isotropic Kitaev model. Although a slightly spatially anisotropic Kitaev exchange is likely in α-RuCl3 (22), averaging over the in-plane structural domains (26) reduces the visibility of the anisotropy in experiments. Moreover, it is not expected to have a major effect on the higher-frequency portion of the collective dynamics discussed here (25). However, in a real material, the effective Hamiltonian includes non-Kitaev terms (10) that extend the liquid correlations and, in particular, lead to the long-range order observed below TN. To date, there is no comparably reliable theory available for the response of such an extended Hamiltonian (on which there is not yet a universal agreement for α-RuCl3). As a first, phenomenological attempt to account for the effect of additional terms, we consider minimally modifying the response function of the pure Kitaev model by varying the ratio of S1/S0 by a factor R that, for simplicity, is taken to be momentum-independent. As shown below, treating this ratio as an adjustable parameter yields an excellent account of the overall momentum dependence of the scattering.

Figure 4B illustrates the scattering for R = 2 at fixed E = 1.2Kγ. This calculation captures the overall extent, orientation, and periodicity of the scattering in reciprocal space. A direct comparison is made in the bottom panel (Fig. 4C), showing a cut of the intensity as a function of momentum along the (H, H) direction, integrated over a narrow band around Embedded Image. Also shown are three model calculations for an isotropic Kitaev model at fixed E = 1.2Kγ: AF (purple), FM (green), and AF response modified using R = 2 (red). In the absence of any further terms in the Hamiltonian, the high-energy part of the zero-temperature FM model is clearly incompatible with the data, as it shows a local minimum at the Γ point. The unmodified zero-temperature AF Kitaev response provides a reasonable description of the data but fails to capture the full intensity variation (figs. S5 and S6, A to C). The modified AF Kitaev response fits the data best, with R ≈ 2 indicating a relative enhancement of the spatial correlations.

The results reported here provide a distinctive picture of the magnetic response function of α-RuCl3 in momentum-energy space and demonstrate unequivocally the presence of an extended continuum of magnetic excitations centered at the Γ point. The continuum response is incompatible with SW theory and defies any known explanation in terms of conventional dispersive spin flip, single-particle, or simple dimer magnetic excitations (27) (figs. S6 and S7). Instead, the central features of the continuum are well described by the scattering from the high-energy part of an AF KQSL; with one phenomenological fitting parameter, nearly quantitative agreement is obtained. The exact calculation of the response function of the pure Kitaev model is based on fractionalized degrees of freedom: free Majorana fermions scattering off a pair of static emergent fluxes (12, 13). In such a scattering process, the demands of energy and momentum conservation impose only weak kinematic constraints. This provides a natural and intuitive picture for the broadness of the experimentally observed continuum.

One feature of the data that is not well described by a pure Kitaev model is the six-pointed star shape of the scattering in reciprocal space. However, it can be shown (27) (figs. S8 to S10) that modest amounts of additional neighbor correlation or simple perturbations based on mean-field approaches (32) away from the integrable model can yield a similar shape, even in the disordered state.

The data presented here constitute an important step in developing a complete understanding of the low- and high-energy dynamics in α-RuCl3. The good agreement of the continuum scattering with the simple AF KQSL is complementary to current density functional theory calculations relating the low-energy spin-½ description of the material to details of the electronic structure (33, 34). Further effort is needed to converge on an explanation of the sign of the Kitaev interaction and to determine the magnitude of additional interactions. It would be useful to develop a theory that describes both the low-energy response of the ordered state and the broad quantum fluctuation continuum. At the same time, the seeming proximity of the system to a true KQSL is a strong incentive for exploring the effects of doping, pressure, and field to determine a full picture of the ground and excited states. With this work, a comprehensive measurement of the high-energy excitations is now available to the community in a potential proximate Kitaev material and may open up the opportunity to investigate the magnetic version of the elusive Majorana fermions in two dimensions.

Supplementary Materials

www.sciencemag.org/content/356/6342/1055/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S11

Table S1

References (3553)

References and Notes

  1. See supplementary materials
Acknowledgments: We thank C. Batista, K. Burch, H. Cao, B. Chakoumakos, G. Jackeli, G. Khalliulin, J. Leiner, P. Kelley, R. Valenti, and S. Winter for valuable discussions and O. Garlea for assistance with the measurement on HYSPEC. J.K. and R.M. particularly thank J. Chalker and D. Kovrizhin for collaboration on closely related work. The work at Oak Ridge National Laboratory’s Spallation Neutron Source was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Scientific User Facilities Division. Part of the research was supported by the DOE, Office of Science, BES, Materials Sciences and Engineering Division (J.Y. and C.A.B.). D.G.M. acknowledges support from the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4416. The work at the Max Planck Institute for the Physics of Complex Systems, Dresden, Germany, and the collaboration as a whole were partially supported by Deutsche Forschungsgemeinschaft grant SFB 1143 (J.K. and R.M.) and by a fellowship within the postdoc program of the German Academic Exchange Service (DAAD) (J.K.). The data presented in this manuscript are available from the coauthors upon request.
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