Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet

Journal name:
Nature Materials
Year published:
DOI:
doi:10.1038/nmat4604
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Abstract

Quantum spin liquids (QSLs) are topological states of matter exhibiting remarkable properties such as the capacity to protect quantum information from decoherence. Whereas their featureless ground states have precluded their straightforward experimental identification, excited states are more revealing and particularly interesting owing to the emergence of fundamentally new excitations such as Majorana fermions. Ideal probes of these excitations are inelastic neutron scattering experiments. These we report here for a ruthenium-based material, α-RuCl3, continuing a major search (so far concentrated on iridium materials) for realizations of the celebrated Kitaev honeycomb topological QSL. Our measurements confirm the requisite strong spin–orbit coupling and low-temperature magnetic order matching predictions proximate to the QSL. We find stacking faults, inherent to the highly two-dimensional nature of the material, resolve an outstanding puzzle. Crucially, dynamical response measurements above interlayer energy scales are naturally accounted for in terms of deconfinement physics expected for QSLs. Comparing these with recent dynamical calculations involving gauge flux excitations and Majorana fermions of the pure Kitaev model, we propose the excitation spectrum of α-RuCl3 as a prime candidate for fractionalized Kitaev physics.

At a glance

Figures

  1. Structure and bulk properties of 2D layered [alpha]-RuCl3.
    Figure 1: Structure and bulk properties of 2D layered α-RuCl3.

    ac, The structure of α-RuCl3 (space group no. 151, P3112). a, In-plane honeycomb structure showing edge-sharing RuCl6 octahedra and the unit cell of the honeycomb lattice. b, View along the c axis showing the stacking of honeycomb layers in the unit cell, with Ru atoms in each layer denoted by the colours red, blue or green. The different intralayer Ru–Ru bonds, corresponding to the index ‘m’ in equation (1), are labelled in the red layer as α, β, or γ, each with distance . The 2D zigzag magnetic structure is illustrated by the black spins on the red layer. c, Side view of the unit cell showing the offsets along the c axis. Values noted are for room-temperature lattice constants. d, Specific heat of powder α-RuCl3. The solid red line is a fit of the data following the 2D Debye model Cp(T) = ANk(T/θD)20θD/T(x2/ex − 1)dx for T > 16K, and for T < 16K an empirical function describing the anomaly associated with magnetic order. The inset in d shows a close-up of the anomaly associated with the low-temperature magnetic ordering transition at TN ≈ 14K in powder samples. (See Supplementary Fig. 1 for more details of thermodynamic measurements.) The error bars include statistical and systematic uncertainties of the physical property measurement system (PPMS) measurement. e, Order parameter plot of the (1/2 0 3/2) magnetic Bragg peak (Q = 0.81Å−1) in powder samples measured using neutron diffraction (see Methods). The solid blue line is a power-law fit to the data above 9K, yielding TN = 14.6(3) K, with β = 0.37(3). f, Similar plot for single crystals showing two coexisting ordering wavevectors (1/2 0 1), with TN1 = 7.6(2)K (green), and (1/2 0 3/2), with TN2 = 14.2(8)K (blue). Note that the (1/2 0 1) peak loses intensity sharply, as compared to the (1/2 0 3/2) peak. Inset: picture of the single crystal (22.5mg) used in these measurements. Signals in e,f, are normalized to countss−1 and the error bars represent 1 s.d. (σ), assuming Poisson counting statistics.

  2. Spin-orbit coupling mode in [alpha]-RuCl3, measured using inelastic neutron scattering at T = 5[thinsp]K with incident energy Ei = 1.5[thinsp]eV.
    Figure 2: Spin–orbit coupling mode in α-RuCl3, measured using inelastic neutron scattering at T = 5K with incident energy Ei = 1.5eV.

    a, Difference between data integrated over the ranges Q = [2.5,4.0] Å−1 and [4.5, 6.0] Å−1 shown in Supplementary Fig. 4, subtracted point by point, illustrating the enhanced signal at low Q. The solid line is a fit to a background plus a Gaussian peak centred at 195 ± 11meV with HWHM 48 ± 6meV. With the settings used for the measurement, the width is resolution limited. Error bars represent 1σ (see Methods). b, Intensity for various values of wavevector integrated over the energy range [150, 250] meV (each point represents a summation in Q over 0.5 Å−1, except for the first point, which is over 0.26Å−1). The solid line shows a two-parameter fit of the data to the equation A |fmag(Q)|2 + B, where fmag(Q) is the Ru3+ magnetic form factor in the spherical approximation. The shaded area represents the contribution arising from magnetic scattering. Inset: a schematic of the single-ion energy levels for d5 electrons in the strong octahedral field (that is, low spin) limit with spin–orbit coupling showing the J1/2 to J3/2 transition at energy 3λ/2.

  3. Collective magnetic modes measured with inelastic neutron scattering using 25[thinsp]meV incident neutrons.
    Figure 3: Collective magnetic modes measured with inelastic neutron scattering using 25meV incident neutrons.

    a, False colour plot of the data at T = 5K showing magnetic modes (M1 and M2) with band centres near E = 4 and 6meV. M1 shows an apparent minimum near Q = 0.62Å−1, close to the magnitude of the M point of the honeycomb reciprocal lattice. The white arrow shows the concave lower edge of the M1 mode. The yellow ‘P’ denotes a phonon that contributes to the scattering at an energy near that of M2, but at higher wavevectors of Q > 2Å−1. b, The corresponding plot above TN at T = 15K shows that M1 has disappeared, leaving strong quasi-elastic scattering at lower values of Q and E. c, Constant-Q cuts through the scattering depicted in a and b centred at wavevectors indicated by the dashed lines. The cuts A and C are summed over the range [0.5, 0.8]Å−1, which includes the M point of the 2D reciprocal lattice, whereas B and D span [1.0, 1.5]Å−1. The data from 2–8meV in cut B is fitted (solid blue line) to a linear background plus a pair of Gaussians, yielding peak energies E1 = 4.1(1)meV and E2 = 6.5(1)meV. d, Constant-E cuts integrated over the energy range [2.5, 3.0] meV, at 4K (E) and 15K (F). See text for detail. The intensity in all four panels, including the colour bars, is reported in the same arbitrary units. In c,d, the solid lines through all the cuts A–F are guides to the eye. The error bars represent 1σ (see Methods).

  4. Spin wave theory calculations.
    Figure 4: Spin wave theory calculations.

    a, Spin wave simulation for the H–K model with (K, J) = (7.0, −4.6) meV with a ZZ ground state. The lattice is the honeycomb plane appropriate for the P3112 space group. b, The calculated powder-averaged scattering including the magnetic form factor. The white arrow shows the concave nature of the edge of the lower mode in (Q, E) space, similar to the data in Fig. 3a. c, Cuts through the data of Fig. 3a integrated over 0.2Å−1 wide bands of wavevector centred at the values shown. Lines are guides to the eye. Note that actual data include a large elastic response from Bragg and incoherent scattering at E = 0meV. The error bars represent 1σ. d, The same cuts, through the calculated scattering shown in Fig. 4b. Inset: phase diagram of the H–K model, after ref. 33. The various phases are denoted by different colours: spin liquid (SL, blue), antiferromagnetic (AFM, light violet), stripy (ST, green), ferromagnetic (FM, orange) and zigzag (ZZ, red). The red dots represent the two solutions for α-RuCl3 as determined by the zone-centre spin wave mode energies.

  5. Disagreements with classical SWT and agreement with QSL calculations.
    Figure 5: Disagreements with classical SWT and agreement with QSL calculations.

    a, Scattering from mode M1 measured using INS at T = 5K using Ei = 8meV. Lower panel shows constant-energy cuts over the energy ranges shown, centred at the locations labelled (G, H) in the upper panel. The absence of structured scattering below 2meV confirms the gap in the magnetic excitation spectrum. b, Constant-E cuts of the data through the upper mode at four different temperatures, of which one curve at T = 5K is below TN (red squares) and rest above TN. The lines are guides to the eye. c, A constant-Q cut of the Ei = 25meV, T = 5K data in the Q range shown. The blue triangles show the M2 portion of the cut B in Fig. 3c, but with the linear background term subtracted, and the blue line is a fit to a Gaussian peak. As discussed in the text, the red line shows simulated SWT scattering and the green line shows the scattering calculated from a Kitaev QSL response function. The shaded area represents magnetic scattering that is not captured by the SWT. The double-ended arrow marked ‘R’ shows the full-width at half-maximum (FWHM) of the instrumental resolution of 0.5meV at 6.5meV. In panels ac, the error bars represent 1σ (see Methods). d, The powder-averaged scattering calculated from a 2D isotropic Kitaev model, with antiferromagnetic K, using the results of ref. 10, including the magnetic form factor. The upper feature is broad in energy and decreases in strength largely monotonically as Q increases.

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Author information

Affiliations

  1. Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA

    • A. Banerjee,
    • A. A. Aczel,
    • M. B. Stone,
    • G. E. Granroth,
    • M. D. Lumsden &
    • S. E. Nagler
  2. Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA

    • C. A. Bridges
  3. Material Sciences and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA

    • J.-Q. Yan &
    • D. G. Mandrus
  4. Department of Materials Science and Engineering, University of Tennessee, Knoxville, Tennessee 37996, USA

    • J.-Q. Yan &
    • D. G. Mandrus
  5. Department of Physics, University of Tennessee, Knoxville, Tennessee 37996, USA

    • L. Li &
    • Y. Yiu
  6. Neutron Data Analysis & Visualization Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA

    • G. E. Granroth
  7. Department of Physics, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0HE, UK

    • J. Knolle &
    • D. L. Kovrizhin
  8. Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany

    • S. Bhattacharjee &
    • R. Moessner
  9. International Center for Theoretical Sciences, TIFR, Bangalore 560012, India

    • S. Bhattacharjee
  10. Neutron Sciences Directorate, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA

    • D. A. Tennant
  11. Bredesen Center, University of Tennessee, Knoxville, Tennessee 37966, USA

    • S. E. Nagler

Contributions

S.E.N., A.B. and D.G.M. conceived the project and the experiment. C.A.B., A.B., L.L., J.-Q.Y., Y.Y. and D.G.M. made the sample. J.-Q.Y., L.L., A.B. and C.A.B. performed the bulk measurements, A.B., A.A.A., M.B.S., G.E.G., M.D.L. and S.E.N. performed INS measurements, A.B., S.E.N., C.A.B., M.D.L., M.B.S. and D.A.T. analysed the data. Further modelling and interpreting of the neutron scattering data was carried out by A.B., M.D.L., S.E.N., J.K., S.B., D.L.K. and R.M., where A.B., M.D.L., S.B. and S.E.N. performed SWT simulations, and J.K., S.B., D.L.K. and R.M. carried out QSL theory calculations. A.B. and S.E.N. prepared the first draft, and all authors contributed to writing the manuscript.

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