Abstract
The honeycomb lattice is one of the simplest lattice structures. Electrons and spins on this simple lattice, however, often form exotic phases with nontrivial excitations. Massless Dirac fermions can emerge out of itinerant electrons, as demonstrated experimentally in graphene^{1}, and a topological quantum spin liquid with exotic quasiparticles can be realized in spin1/2 magnets, as proposed theoretically in the Kitaev model^{2}. The quantum spin liquid is a longsought exotic state of matter, in which interacting spins remain quantumdisordered without spontaneous symmetry breaking^{3}. The Kitaev model describes one example of a quantum spin liquid, and can be solved exactly by introducing two types of Majorana fermion^{2}. Realizing a Kitaev model in the laboratory, however, remains a challenge in materials science. Mott insulators with a honeycomb lattice of spin–orbitalentangled pseudospin1/2 moments have been proposed^{4}, including the 5delectron systems αNa_{2}IrO_{3} (ref. 5) and αLi_{2}IrO_{3} (ref. 6) and the 4delectron system αRuCl_{3} (ref. 7). However, these candidates were found to magnetically order rather than form a liquid at sufficiently low temperatures^{8,9,10}, owing to nonKitaev interactions^{6,11,12,13}. Here we report a quantumliquid state of pseudospin1/2 moments in the 5delectron honeycomb compound H_{3}LiIr_{2}O_{6}. This iridate does not display magnetic ordering down to 0.05 kelvin, despite an interaction energy of about 100 kelvin. We observe signatures of lowenergy fermionic excitations that originate from a small number of spin defects in the nuclearmagneticresonance relaxation and the specific heat. We therefore conclude that H_{3}LiIr_{2}O_{6} is a quantum spin liquid. This result opens the door to finding exotic quasiparticles in a strongly spin–orbitcoupled 5delectron transitionmetal oxide.
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Acknowledgements
We thank Y. Motome, M. Udagawa, R. Valentí, A. Gibbs, Y. B. Kim, A. Smerald and N. Shannon for discussions, and U. Wedig, Y. Ishikuro, T. Nishioka and S. Nakatsuji for experimental support and discussions. This work was partly supported by the Japan Society for the Promotion of Science (JSPS) KAKAENHI (numbers 24224010, 26707018, 15K13523, JP15H05852, JP15K21717 and 17H01140) and the Alexander von Humboldt foundation.
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T.T. and A.K. prepared the sample and performed the bulk experiments. K.K., R.T. and Y.K. carried out the NMR measurements. Y.M. carried out the lowtemperature specific heat measurements. S.B. and R.D. performed structural analysis. G.J. gave theoretical inputs. T.T., K.K., Y.M. and H.T. wrote manuscript and all authors commented on it. H.T. designed and supervised the experiments.
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Extended data figures and tables
Extended Data Figure 1 Powder Xray diffraction pattern.
The measured powder Xray diffraction pattern of H_{3}LiIr_{2}O_{6} at room temperature was recorded with Ag Kα_{1} radiation; selected reflection indices based on the C2/m space group symmetry are also shown. The reflections that are vastly broadened owing to stacking faults are indicated in pink.
Extended Data Figure 2 Temperature and magneticfield dependence of magnetization at low temperatures.
a, Temperature dependence of magnetization M/B under magnetic fields up to B = 14 T for the sample used in the specific heat measurements shown in Fig. 4. The Curielike contribution that is absent in K(T) is clearly seen, originating from magnetic defects. The dashed line represents a powerlaw behaviour with T^{−1/2} dependence. The susceptibility in the lowfield limit appears to follow the T^{−1/2} dependence better than the conventional Curie–Weiss dependence. At very low temperatures, the Curielike contribution becomes independent of T for high B fields, implying the saturation of moment from magnetic defects. In the inset, M(T) at B = 7.0 T (green line) is compared with that calculated by integrating ∂M(T, B)/∂T, which we obtained from S(T, B) using the Maxwell relation (red pluses). b, M(B)–B curves at low temperatures. From the offset from the linear magnetization at high fields, we estimate the saturation moment that originates from magnetic defects to be 0.022μ_{B} (as indicated by the arrow). This corresponds to 2% of the magnetic defects with g = 2 and a S = 1/2 moment. The Zeeman gap in the densityofstates model in Fig. 4 is 2αμ_{B}B with α = 2.9, suggesting a gfactor of 5.8 for the moment that comes from the magnetic defects. Incorporating g = 5.8, the best estimate of the number of magnetic defects with S = 1/2 moments is 0.8%. This estimate is consistent with the estimate from the specific heat in Fig. 4, which indicates an entropy S(T) of 1%–2% of Rln(2) at T = 5 K, where R is the gas constant. c, B^{1/2} × ∂M(T,B)/∂T shows a scaling with T/B similar to that for C and T_{1}, indicating that the three probes capture the same excitations.
Extended Data Figure 3 Resistivity versus temperature.
The dependence of the resistivity ρ on the insulating temperature T is measured on a polycrystalline pellet of H_{3}LiIr_{2}O_{6}. The inset shows the Arrhenius plot of the same data, indicating the transport activation energy of approximately 0.12 eV.
Extended Data Figure 4 NMR spectral parameters and estimate of the hyperfine coupling constant in H_{3}LiIr_{2}O_{6}.
a, HWHM obtained by performing a Gaussian fit near the top of each peak. The open symbols are for B perpendicular to the honeycomb plane (⊥); the filled symbols are for B parallel to the plane (‖). The length of the arrow corresponds to a hyperfine field at a Li site when a moment of 0.002μ_{B} is placed on the Ir atoms. b, The integrated NMR signal intensities after T_{1} and T_{2} corrections, which is proportional to the number of nuclei under observation. c, Bulk magnetic susceptibility χ(T) of aligned powder at 7 T after the subtraction of core diamagnetism, with B parallel (purple crosses) and perpendicular (green pluses) to the honeycomb plane. The solid lines represent the intrinsic susceptibility χ_{i} calculated from χ by subtracting the Curie contribution that originates from the impurities and/or defects. The dotted lines are hightemperature (above 250 K) Curie–Weiss fits, which yield a Curie–Weiss temperature and effective moment in a parallel field of and , respectively, and in a perpendicular field of and . NMR Knight shifts K for ^{7}Li measured at 2 T (purple triangles and green diamonds) are superposed for comparisons. d, K for ^{7}Li plotted against χ_{i}. The data in a temperature region T > 150 K are well described by the linear relation K(T) = [A_{hf}/(N_{A}μ_{B})]χ_{i} (solid line), from which we determine the isotropic hyperfine coupling constant A_{hf} = 0.44/μ_{B} T.
Extended Data Figure 5 T/B scaling of C and (T_{1}T)^{−1}, and their fitting with the model density of excitations.
B^{1/2}(C/T) versus T/B (filled symbols, left axis) and B^{1/2}(T_{1}T)^{−1/2} versus T/B (open symbols, right axis) under various magnetic fields. All of the C(T, B) data points (closed symbols) and T_{1}(T, B) data points (open symbols) fall onto the respective universal curves at low T/B, indicating a scaling behaviour. Because , the plot for T_{1} is another way of representing the same (T/B) scaling as the inset to Fig. 3b. The physical meaning of B^{1/2}(T_{1}T)^{−1/2} is the Fermi average of D(E) with a renormalization factor B^{1/2}, which is closely related to B^{1/2}(C/T). C/T and (T_{1}T)^{−1} are given as follows using the standard equations, which express a Fermi averaging of D(E) and D(E)^{2}, respectively:
Here, f(T, E) = 1/{exp[(E/(k_{B}T)] + 1} is Fermi distribution function. The solid and dashed lines indicate B^{1/2}(C/T) and B^{1/2}(T_{1}T)^{−1/2}, respectively, calculated using the above equations for the model for D(E) shown in Fig. 4c. With α = 2.9 and Γ = 4.3 × 10^{8} J^{−1/2} per Ir atom, the two scaling curves observed experimentally for B^{1/2}(C/T) and B^{1/2}(T_{1}T)^{−1/2} are well reproduced by the calculations. A_{hf} = 0.44/μ_{B} T and γ_{n}/ 2π = 16.54680 MHz T^{−1} were used as known parameters for ^{7}Li. The calculated B^{1/2}(C/T) and B^{1/2}(T_{1}T)^{1/2} show different behaviour at high T/B (greater than about 1), which reflects the different methods of thermal averaging. This difference reasonably accounts for the difference between the two universal curves at high T/B that was observed experimentally.
Extended Data Figure 6 NMR relaxation details.
a, Ratio of for ^{1}H and ^{7}Li, and anisotropy of T_{1} for ^{7}Li with perpendicular and parallel fields. The error bars are calculated using the errors in the estimate of T_{1} (see Fig. 3b). b, Exponent β in the stretchedexponential function describing the NMR relaxation curve as a function of temperature. c, Comparison between the relaxation curves of ^{6}Li and ^{7}Li at 10 K and 5 T, yielding a ratio of for ^{6}Li and ^{7}Li of 0.14. P_{1} is the time period between a saturation pulse and the echo sequence in the relaxation measurements. This value agrees well with the squares of the gyromagnetic ratios. β = 0.6 was used in the calculation.
Extended Data Figure 7 Estimate of the intrinsic specific heat C_{i} by subtracting the specific heat due to magnetic defects.
Fitting the specific heat C in Extended Data Fig. 5 with the model for D(E) in Fig. 4c gives an estimate of the specific heat due to magnetic defects. The bestfitting function, represented dashed line in Extended Data Fig. 5, is subtracted from the total specific heat C(B, T) shown in Fig. 4a. The residual specific heat ΔC is almost independent of B and gives a measure of intrinsic specific heat C_{i}. ΔC/T appears to extrapolate almost to zero as the temperature approaches 0 K, within the uncertainty of the estimate of the contribution from magnetic defects. We do not attempt to separate the possible contribution from the bulk spin liquid to ΔC because of the difficulty in estimating the lattice contribution. The inset shows the plot of ΔC/T as a function of T^{2}.
Extended Data Figure 8 NMR spectra of H_{3}LiIr_{2}O_{6} before and after the alignment of the powder sample.
Before the alignment (top), an asymmetric line shape in obtained (reflecting the anisotropy in the Knight shift K_{‖} versus K_{⊥}; Fig. 2a), which represents a powder pattern. The middle and bottom curves shown the line shapes obtained after the alignment for B along the honeycomb plane (the magnetic easy plane) and perpendicular to the plane, respectively. Because of the easyplane anisotropy of H_{3}LiIr_{2}O_{6}, the alignment for the direction perpendicular to the plane is not as complete as that for the direction parallel to the plane.
Extended Data Figure 9 Subtraction of the nuclear Schottky contribution from the raw specific heat data.
a, Open circles represent the raw specific heat C_{tot}/T at B = 0, 1 T and 3 T. Filled circles represent the nonnuclear specific heat C/T obtained after subtracting the nuclear contribution C_{NS}/T ∝ T^{−3} (dashed lines) from C_{tot}/T. Filled triangles and pluses indicate the nonnuclear C/T obtained directly from a timedomain measurement, either by heating for a short period of 100–200 s and then measuring the short relaxation for a period of 100–200 s (triangles) or by heating for a long period of 1,500–2,500 s and then measuring only the short relaxation for a period of 100–200 s (pluses). The nonnuclear specific heat C/T at B = 3 T obtained by the three different methods agree reasonably well, which strongly supports the validity of our analyses. The C_{NS}/T ∝ T^{−3} term for B = 0 T is slightly larger than that for B = 1 T. The origin of the extra T^{−3} contribution observed in the B = 0 data remains elusive. b, Example of the relaxation curve at B = 3 T and T = 124 mK, which indicates the clear separation in the time domain between the fast relaxation due to the nonnuclear contribution and the slow relaxation due to the nuclear contributions. The inset shows the temperature dependence of the slow relaxation time constant at B = 3 T.
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Kitagawa, K., Takayama, T., Matsumoto, Y. et al. A spin–orbitalentangled quantum liquid on a honeycomb lattice. Nature 554, 341–345 (2018). https://doi.org/10.1038/nature25482
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DOI: https://doi.org/10.1038/nature25482
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