Abstract
Since its proposal by Anderson, resonating valence bonds (RVB) formed by a superposition of fluctuating singlet pairs have been a paradigmatic concept in understanding quantum spin liquids. Here, we show that excitations related to singlet breaking on nearestneighbour bonds describe the highenergy part of the excitation spectrum in YbMgGaO_{4}, the effective spin1/2 frustrated antiferromagnet on the triangular lattice, as originally considered by Anderson. By a thorough singlecrystal inelastic neutron scattering study, we demonstrate that nearestneighbour RVB excitations account for the bulk of the spectral weight above 0.5 meV. This renders YbMgGaO_{4} the first experimental system where putative RVB correlations restricted to nearest neighbours are observed, and poses a fundamental question of how complex interactions on the triangular lattice conspire to form this unique manybody state.
Introduction
Quantum spin liquid (QSL) is a longsought exotic phase in condensed matter physics. It is intimately related to the problem of hightemperature superconductivity and may be instrumental in realizing topological quantum computation^{1,2,3,4,5,6}. In a QSL, spins are highly entangled up to long distances and times without symmetry breaking down to zero temperature due to strong quantum fluctuations^{3}. Experimental systems exhibiting QSL behaviour are actively sought after. However, most of the existing materials are suffering from magnetic defects^{7,8}, spatial coupling anisotropy^{8,9,10} and (or) antisymmetric Dzyaloshinsky–Moriya anisotropy^{11}. Recently, a triangular QSL candidate YbMgGaO_{4} attracted much interest^{12,13,14,15}, because it seems to be free from all of the above effects. Neither spin freezing nor longrange ordering were detected by muon spin relaxation (μSR) down to 0.048 K (ref. 14). Together with the absence of any residual spin entropy^{12}, this renders YbMgGaO_{4} a unique material that may exhibit a gapless U(1) QSL ground state.
A QSL state can be represented by a superposition of many different partitions of a system into valence bonds (spin0 singlet pairs)^{3}, as proposed by Anderson back in 1973 (refs 1, 2). Such valence bonds can be formed between nearestneighbour spins and between spins beyond nearest neighbours. The longer the bond, the weaker the respective singlet pairing energy. Lowenergy excitations arise from breaking longrange valence bonds or rearranging the short bonds into longer ones^{3,16}. Highenergy excitations result from breaking nearestneighbour valence bonds. Therefore, for characterizing a QSL, the detailed investigation of both high and lowenergy excitations is required.
In YbMgGaO_{4}, excellent transparence with the optical gap exceeding ∼3 eV and the robust insulating behaviour with the unmeasurably high resistance suggest a large charge gap, placing the material deep in the Mottinsulator regime of the Hubbard model. Strong localization of the 4f electrons of Yb^{3+} should restrict magnetic interactions to nearest neighbours (S_{1} and S_{2}), but these interactions are anisotropic^{13},
owing to the strong spinorbit coupling, where the local moment S=1/2 is a pseudospin, that is, a combination of spin and orbital moments^{15,17,18,19}. The lowestenergy eigenstate of a dimer formed by such anisotropic pseudospins is, nevertheless, a pure singlet, , with the energy −3/4J_{0} for the antiferromagnetic isotropic coupling, J_{0} ≡ (4J_{±}+J_{zz})/3=0.13(1) meV (ref. 13), as observed experimentally. In contrast to Heisenberg spins, the Yb^{3+} pseudospins do not form a threefold degenerate triplet state and feature three nondegenerate excited states separated by 0.809J_{0}, 1.012J_{0} and 1.179J_{0} from the singlet state instead. Excitations of a system can be viewed as the transitions between the singlet ground state and one of the excited states. Therefore, the resonating valence bond (RVB) picture holds, albeit with minor quantitative modifications due to the different structure of the excited states.
Two very recent inelastic neutron scattering (INS) studies reported a continuum of spin excitations in YbMgGaO_{4} in the energy range between 0.25 and 1.5 meV (refs 20, 21), and a phenomenological interpretation of these excitations in terms of a spinon Fermi surface has been proposed^{20}. However, given the nearestneighbour magnetic energy of J_{0}=0.13(1) meV only^{13}, the excitations were observed at energies between 2J_{0} and 10J_{0}. Therefore, they are highenergy magnetic excitations of YbMgGaO_{4}.
In this paper, we propose a different interpretation of these highenergy excitations and also endeavour to probe YbMgGaO_{4} at lower energies. This task is extremely challenging, owing to the low energy scale of J_{0} and the limits of instrumental energy resolution for neutron spectrometers. We report a thorough INS investigation of a single crystal of YbMgGaO_{4} at energies between 0.02 and 3.5 meV, that is, 0.15–27 in units of J_{0}. We present the data collected at the low temperature of 0.1 K, which is well inside the gapless groundstate regime defined by the saturation of the μSR rate^{14}, and at a much higher temperature of 35 K corresponding to 23J_{0}. The highenergy excitations observed previously^{20,21} are confirmed and ascribed to nearestneighbour RVB correlations. At low temperatures, these excitations are suppressed at energies below J_{0}, which suggests their gapped nature. Our results imply that distinct gapless excitations should exist at much lower energies, and we indeed observe traces of such excitations at the lowest energies accessible in our experiment.
Results
High energy nearestneighbour RVB correlations
The INS data for YbMgGaO_{4} are shown in Figs 1 and 2. A continuum of excitations broadly distributed in both momentum (Q) (see Fig. 1) and energy (0.1≤ħω≤2 meV) space (see Fig. 2) is clearly visible. At 0.1 K, external field shifts the spectral weight towards higher energies (see Fig. 2), thus indicating the magnetic origin of these excitations. Remarkably, the excitation continuum persists up to 35 K, that is, at a temperature that is 23 times higher than J_{0}. In fact, there are no qualitative differences between the highenergy parts of the INS spectra measured at 0.1 and 35 K apart from a 2.57(4)fold reduction in the intensity near the hump centre ∼0.7 meV (see Fig. 2) when the temperature is increased to 35 K. The wavevector and temperature dependence of the excitation continuum clearly indicates its spin–spin correlation origin and excludes other possible interpretations, such as CEF excitations, which are Qindependent and observed at energies larger than 39 meV (refs 13, 15, 21).
We first focus on the wave vector dependence of the INS intensity measured with the incident neutron energy of E_{i}=5.5 meV. Assuming uncorrelated nearestneighbour valence bonds on a triangular lattice, the equaltime INS intensity can be expressed as ref. 22
Here, f(Q) is the magnetic form factor of free Yb^{3+}, and N is the total number of nearestneighbour valence bonds probed in the INS measurement. This expression accounts for the experimental spectral weight above 0.5 meV, thus suggesting that at high energies spin–spin correlations are restricted to nearest neighbours. Any static state, such as valence bond solid^{23} and glass^{24,25}, is excluded by our previous μSR study^{14}, and the RVB scenario turns out to be most plausible, as supported by the following arguments:
First, the Qdependence of the INS signal at 0.1 and 35 K (after the subtraction of the background term b) is well described by the uncorrelated nearestneighbour valence bond model on a triangular lattice (see Fig. 1c–f). No signatures of spin–spin correlations beyond nearest neighbours are observed (Supplementary Note 2 and Supplementary Figs 10 and 11). This Qdependence cannot be understood by short distance correlations in an arbitrary ground state on the triangular lattice. For example, the 120° longrange order would produce spinwave excitations^{26} and a qualitatively different Qdependence even at high energies (Supplementary Note 5 and Supplementary Figs 21–26).
Second, the antiferromagnetic nature of the isotropic nearestneighbour coupling, J_{0} ≡ (4J_{±}+J_{zz})/3=0.13(1) meV (ref. 13), allows the formation of spin singlet in a pair of the Yb^{3+} spins (Supplementary Note 1 and Supplementary Fig. 1).
Third, temperature dependence of the prefactor a in the RVB expression, a(35 K)/a(0.1 K) ∼0.3 (Supplementary Table 1), is consistent with the expected ratio,
based on the thermal distribution of the eigenstates of the Yb^{3+} dimer. With increasing temperature, a larger fraction of nearestneighbour singlets is excited.
Fourth, the uniform spin susceptibility, χ′(E), which is obtained from the INS spectrum measured around the Gamma point (Q=0) via the fluctuationdissipation theorem and the Kramers−Kronig transformation^{22}, is almost zero at 0.1 K above ∼0.5 meV, in agreement with the proposed RVB state (Supplementary Note 3 and Supplementary Figs 12 and 13).
Fifth, the energy dependence of the integrated INS signal reveals gapped nature of the highenergy excitations (see below for the details), which is consistent with the aforementioned suppression of the uniform susceptibility above ∼0.5 meV.
Last, both spin and valence bond freezing are excluded by our μSR measurement reported previously^{14}.
The above six arguments suggest that the whole excitation continuum at energies above J_{0} may be due to the nearestneighbour RVBtype correlations. We prove this explicitly above 0.5 meV, while below 0.5 meV the Qdependent data measured with the incident energy E_{i}=5.5 meV are contaminated by the elastic signal (Supplementary Figs 14–19). Lower energies can be probed with E_{i}=1.26 meV (the energy resolution σ∼20 μeV (ref. 27)), but these data cover a limited Qrange only. Nevertheless, we find no qualitative differences between the spectra at ∼0.3 and ∼0.7 meV in all measured Q space (E_{i}=1.26 meV) apart from an overall increase in the intensity. This indicates same, nearestneighbour nature of spin–spin correlations across the whole excitation continuum above J_{0} that was previously ascribed to the spinon Fermi surface.
It is crucial, though, that this continuum and the associated nearestneighbour spin–spin correlations do not persist down to zero energy, because the nearestneighbour RVBs are gapped, whereas YbMgGaO_{4} clearly shows gapless behaviour^{12,14}. Therefore, the RVB scenario holds at high energies only. The presence of a distinct lowenergy regime is supported by the analysis of the energydependent spectra integrated over all measured Q space.
Low energy longrange spin correlations
For energy transfer below J_{0}, excitations related to the breaking of nearestneighbour spin singlets must freeze out as long as thermal energy is insufficient to overcome J_{0}, that is, T<1.5 K. We, therefore, expect that below 0.13 meV the INS intensity at 0.1 K falls below that at 35 K. As indicated by the downwardpointing arrow in Fig. 3c, this expected crossing of the overall scattering intensity is observed indeed. Respectively, the intensity difference I(0.1 K)–I(35 K) at zero magnetic field changes sign and becomes negative at energy transfer below J_{0} (Fig. 3d).
Further information is obtained from the INS spectra at finite magnetic fields applied along the crystallographic cdirection. At 8.5 T, which fully polarizes the moments at low temperatures^{13,15}, a clear boundary is observed in the lowenergy magnetic excitations, leading to a crossing of I(0.1 K, 8.5 T) with I(35 K) near 1 meV, as indicated by the arrow in Fig. 2. This gap is related to the Zeeman energy^{15,21} in the applied field of 8.5 T. In the same vein, under a moderate applied field of 1.8 T, which polarizes the spins only partially, negative values of I(0.1 K)–I(35 K) occur below 0.27 meV (see Fig. 3d). This energy lies in between J_{0} and the Zeeman energy μ_{0}μ_{B}g_{}H_{}=0.39 meV of spinwave excitations for this field. When the field is reduced to zero, the crossing of intensities shifts to J_{0} (Fig. 3c). We, therefore, associate this effect with an energy gap for the continuum of nearestneighbour RVBtype excitations^{3}. These excitations seem to be unrelated to the gapless spinon Fermi surface, in contrast to recent expectations based on the INS measurements at higher energies^{20}.
It is worth noting that a qualitatively similar crossing of the INS intensities measured at low and high temperatures has been recently observed in the frustrated pyrochlore Er_{2}Ti_{2}O_{7} (ref. 28), where magnetic excitations are gapped. In our case, the relation I(0.1 K, 0 T)<I(35 K, 0 T) is also clearly detected in zero field at transfer energies from 0.13 meV down to 0.018 meV, below which a rapid increase of the lowtemperature intensity sets in. This lower energy is roughly the same as the energy resolution σ∼20 μeV (0.15J_{0}) (ref. 27) of the LET spectrometer at the incident neutron energy of 1.26 meV. We emphasize that the inelastic signal does not become featureless at this energy (σ), as otherwise a smooth convoluted LorentzianGaussian peak profile would be expected (see the raw data in Fig. 3c and Supplementary Fig. 20). At low transfer energies, the inelastic signal is found on top of the elastic background (see Fig. 3c). Assuming a weakly temperaturedependent elastic signal at T≤35 K, we expect that it cancels out when analysing I(0.1 K)–I(35 K). Therefore, the intensity difference observed in zero field (see Fig. 3d) is intrinsic, as further confirmed by its tangible field dependence, and should reflect the onset of lowenergy excitations related to longerrange correlations in YbMgGaO_{4} (refs 12, 14). The most conspicuous effect of this change is the shift of the intensity maxima from the Kpoints in the highenergy regime to the Mpoints in the lowenergy regime (Supplementary Note 4 and Supplementary Fig. 18), as also seen in the diffuse scattering reported by Paddison et al.^{21}
Discussion
The clear separation between the low^{12,14} and highenergy excitations in the spectrum of YbMgGaO_{4} (see Fig. 3d) is interesting and unique, rendering YbMgGaO_{4} distinct from QSL materials known to date, such as herbertsmithite^{7,29}, organic charge transfer salts^{9,10} and Ca_{10}Cr_{7}O_{28} reported recently^{30}. The RVB scenario on the triangularlattice was also discussed for the cluster magnet LiZn_{2}Mo_{3}O_{8}, where a spinliquid state with both nearestneighbour and nextnearestneighbour correlations is formed^{31,32,33}. It is also worth noting that the continuum of nearestneighbour RVB excitations goes back to the original idea by Anderson^{1} who argued that Heisenberg spins on the regular triangular lattice evade longrange magnetic order and form the nearestneighbour RVB QSL state. Although Anderson’s conjecture was not confirmed in later studies^{34}, the formation of a QSL on a triangular lattice with spatial anisotropy^{35}, nextnearestneighbour couplings^{36} and multiplespin exchange^{37} was identified in the recent literature. Whereas the multiplespin exchange can be clearly excluded due to the strongly localized nature of the 4f electrons of Yb^{3+}, two other effects are potentially relevant to YbMgGaO_{4}.
The presence of nextnearestneighbour couplings is currently debated based on the modelling of the magnetic diffuse scattering^{21,38}. Spatial anisotropy of nearestneighbour couplings can be, at first glance, excluded, based on the threefold symmetry of the crystal structure^{12}. However, recent experiments^{15,21}, including our INS study^{15} of crystalfield excitations of Yb^{3+}, pinpoint the importance of the Mg/Ga disorder that leads to variations in the local environment of Yb^{3+}. An immediate effect of this structural disorder is the distribution of gvalues that manifests itself in the broadening of excitations in the fully polarized state, yet randomness of magnetic couplings resulting in local spatial anisotropy seems to be relevant too^{15,18,21}.
Our result suggests that the broad excitation continuum in YbMgGaO_{4} reflects nearestneighbour spin correlations and bears no obvious relation to the gapless spinon Fermi surface, a conclusion consistent with the absence of the Fermi spinon or any other magnetic contribution to the thermal conductivity^{39}. On the other hand, gapless nature of YbMgGaO_{4} evidenced by the nonzero lowtemperature susceptibility^{12,14} and the powerlaw behaviour of the magnetic specific heat^{12} are indicative of a distinct lowenergy regime that has been glimpsed in our experiment. These lowenergy excitations are likely to contain crucial information on whether the ground state of YbMgGaO_{4} is indeed a QSL, or a special case of the disorderinduced mimicry of a spin liquid, as proposed recently^{40,41}.
Methods
Sample preparation
Large single crystals (∼1 cm) of YbMgGaO_{4} were grown by the floating zone technique reported previously^{13}. The as grown rod (∼50 g) was cut into slices along the abplane (the easily cleavable direction). Ten bestquality abslices of the singlecrystal (total mass ∼10 g) were selected for the neutron scattering experiment on LET by Laue Xray diffractions on all surface (Supplementary Figs 2 and 3). The slices were fixed to the copper base by Cytop glue to avoid any shift in an applied magnetic field up to 8.5 T.
Neutron scattering measurements
Systematic neutron scattering experiments were carried out on a cold neutron multichopper spectrometer LET at the ISIS pulsed neutron and muon source. Incident energies of 26.8, 5.5, 2.3 and 1.26 meV were chosen for both elastic and inelastic scattering with the energy resolution of 1,400, 160, 48 and 20 μeV, respectively^{27}. The sample temperature of 0.1 K was achieved using dilution refrigerator. The neutron diffraction (elastic signal) showed that the alignment of the single crystals was sufficient for the INS study of the continuous excitations. No additional diffraction peaks were observed down to 0.1 K, compatible with the absence of longrange magnetic order (Supplementary Figs 4–6). All neutron scattering data were processed and analysed using HoraceMatlab^{42} on the ISIS computers. Asymmetry of the intensities was observed due to the macroscale nonrotational symmetry of the sample around the rotation axis. For the sake of clarity, the raw data have been symmetrized and averaged using the point symmetry (D_{3d}) in the reciprocal lattice space (see Fig. 1a,b). The corresponding raw data can be found in Supplementary Figs 7–9.
External magnetic fields of 1.8 and 8.5 T were applied along the caxis. The data sets in Fig. 1a,b were integrated over the momentum space, −0.9≤η≤0.9 in [0, 0, −η], and over a small energy range, 0.65≤E≤0.75 meV. The data sets in Fig. 1c–e were integrated over the momentum space, −1.03≤ξ≤−0.97 in [ξ, −ξ/2, 0], −0.03≤ξ≤0.03 in [ξ/2, −ξ, 0], and −0.03≤ξ≤0.03 in [0, ξ, 0], respectively. All data sets in Fig. 1c–e were integrated over the same momentum range, −0.9≤η≤0.9 in [0, 0, −η], and over the same energy range, 0.5≤E≤1.5 meV. a and b are fitted constants for the proportionality and background, respectively (see Fig. 1c–e and Supplementary Table 1). The data sets in Figs 2 and 3 were integrated over all measured momentum space.
Data availability
The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Additional information
How to cite this article: Li, Y. et al. Nearestneighbour resonating valence bonds in YbMgGaO_{4}. Nat. Commun. 8, 15814 doi: 10.1038/ncomms15814 (2017).
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Acknowledgements
We thank Gang Chen, Haijun Liao, Changle Liu, Sasha Chernyshev and Mike Zhitomirsky for helpful discussions. Y.L. was supported by the startup funds of Renmin University of China. Q.Z. was supported by the Fundamental Research Funds for the Central Universities, and by the Research Funds of Renmin University of China. This work was supported by the NSF of China (No. 11474357) and the Ministry of Science and Technology of China (973 Project No. 2016YFA0300504). The work in Augsburg was supported by the German Science Foundation through TRR80 and the German Federal Ministry for Education and Research through the Sofja Kovalevskaya Award of the Alexander von Humboldt Foundation.
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Y.L., D.A. and Q.Z. planned the experiments. Y.L. synthesized and characterized the sample. Y.L., D.V., R.I.B. and Q.Z. collected the neutron scattering data. Y.L. analysed the data. Y.L., A.A.T. and P.G. wrote the manuscript with comments from all coauthors. The manuscript reflects the contributions of all authors.
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Li, Y., Adroja, D., Voneshen, D. et al. Nearestneighbour resonating valence bonds in YbMgGaO_{4}. Nat Commun 8, 15814 (2017). https://doi.org/10.1038/ncomms15814
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DOI: https://doi.org/10.1038/ncomms15814
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