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 nearest-neighbour bonds describe the high-energy part of the excitation spectrum in YbMgGaO4, the effective spin-1/2 frustrated antiferromagnet on the triangular lattice, as originally considered by Anderson. By a thorough single-crystal inelastic neutron scattering study, we demonstrate that nearest-neighbour RVB excitations account for the bulk of the spectral weight above 0.5 meV. This renders YbMgGaO4 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 many-body state.
Quantum spin liquid (QSL) is a long-sought exotic phase in condensed matter physics. It is intimately related to the problem of high-temperature superconductivity and may be instrumental in realizing topological quantum computation1,2,3,4,5,6. In a QSL, spins are highly entangled up to long distances and times without symmetry breaking down to zero temperature due to strong quantum fluctuations3. Experimental systems exhibiting QSL behaviour are actively sought after. However, most of the existing materials are suffering from magnetic defects7,8, spatial coupling anisotropy8,9,10 and (or) antisymmetric Dzyaloshinsky–Moriya anisotropy11. Recently, a triangular QSL candidate YbMgGaO4 attracted much interest12,13,14,15, because it seems to be free from all of the above effects. Neither spin freezing nor long-range ordering were detected by muon spin relaxation (μSR) down to 0.048 K (ref. 14). Together with the absence of any residual spin entropy12, this renders YbMgGaO4 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 (spin-0 singlet pairs)3, as proposed by Anderson back in 1973 (refs 1, 2). Such valence bonds can be formed between nearest-neighbour spins and between spins beyond nearest neighbours. The longer the bond, the weaker the respective singlet pairing energy. Low-energy excitations arise from breaking long-range valence bonds or rearranging the short bonds into longer ones3,16. High-energy excitations result from breaking nearest-neighbour valence bonds. Therefore, for characterizing a QSL, the detailed investigation of both high- and low-energy excitations is required.
In YbMgGaO4, 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 Mott-insulator regime of the Hubbard model. Strong localization of the 4f electrons of Yb3+ should restrict magnetic interactions to nearest neighbours (S1 and S2), but these interactions are anisotropic13,
owing to the strong spin-orbit coupling, where the local moment S=1/2 is a pseudospin, that is, a combination of spin and orbital moments15,17,18,19. The lowest-energy eigenstate of a dimer formed by such anisotropic pseudospins is, nevertheless, a pure singlet, , with the energy −3/4J0 for the antiferromagnetic isotropic coupling, J0 ≡ (4J±+Jzz)/3=0.13(1) meV (ref. 13), as observed experimentally. In contrast to Heisenberg spins, the Yb3+ pseudospins do not form a three-fold degenerate triplet state and feature three non-degenerate excited states separated by 0.809J0, 1.012J0 and 1.179J0 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 YbMgGaO4 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 proposed20. However, given the nearest-neighbour magnetic energy of J0=0.13(1) meV only13, the excitations were observed at energies between 2J0 and 10J0. Therefore, they are high-energy magnetic excitations of YbMgGaO4.
In this paper, we propose a different interpretation of these high-energy excitations and also endeavour to probe YbMgGaO4 at lower energies. This task is extremely challenging, owing to the low energy scale of J0 and the limits of instrumental energy resolution for neutron spectrometers. We report a thorough INS investigation of a single crystal of YbMgGaO4 at energies between 0.02 and 3.5 meV, that is, 0.15–27 in units of J0. We present the data collected at the low temperature of 0.1 K, which is well inside the gapless ground-state regime defined by the saturation of the μSR rate14, and at a much higher temperature of 35 K corresponding to 23J0. The high-energy excitations observed previously20,21 are confirmed and ascribed to nearest-neighbour RVB correlations. At low temperatures, these excitations are suppressed at energies below J0, 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.
High energy nearest-neighbour RVB correlations
The INS data for YbMgGaO4 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 J0. In fact, there are no qualitative differences between the high-energy 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 wave-vector 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 Q-independent 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 Ei=5.5 meV. Assuming uncorrelated nearest-neighbour valence bonds on a triangular lattice, the equal-time INS intensity can be expressed as ref. 22
Here, f(Q) is the magnetic form factor of free Yb3+, and N is the total number of nearest-neighbour 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 solid23 and glass24,25, is excluded by our previous μSR study14, and the RVB scenario turns out to be most plausible, as supported by the following arguments:
First, the Q-dependence of the INS signal at 0.1 and 35 K (after the subtraction of the background term b) is well described by the uncorrelated nearest-neighbour 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 Q-dependence cannot be understood by short distance correlations in an arbitrary ground state on the triangular lattice. For example, the 120° long-range order would produce spin-wave excitations26 and a qualitatively different Q-dependence even at high energies (Supplementary Note 5 and Supplementary Figs 21–26).
Second, the antiferromagnetic nature of the isotropic nearest-neighbour coupling, J0 ≡ (4J±+Jzz)/3=0.13(1) meV (ref. 13), allows the formation of spin singlet in a pair of the Yb3+ spins (Supplementary Note 1 and Supplementary Fig. 1).
Third, temperature dependence of the pre-factor 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 Yb3+ dimer. With increasing temperature, a larger fraction of nearest-neighbour 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 fluctuation-dissipation theorem and the Kramers−Kronig transformation22, 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 high-energy 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 previously14.
The above six arguments suggest that the whole excitation continuum at energies above J0 may be due to the nearest-neighbour RVB-type correlations. We prove this explicitly above 0.5 meV, while below 0.5 meV the Q-dependent data measured with the incident energy Ei=5.5 meV are contaminated by the elastic signal (Supplementary Figs 14–19). Lower energies can be probed with Ei=1.26 meV (the energy resolution σ∼20 μeV (ref. 27)), but these data cover a limited Q-range only. Nevertheless, we find no qualitative differences between the spectra at ∼0.3 and ∼0.7 meV in all measured Q space (Ei=1.26 meV) apart from an overall increase in the intensity. This indicates same, nearest-neighbour nature of spin–spin correlations across the whole excitation continuum above J0 that was previously ascribed to the spinon Fermi surface.
It is crucial, though, that this continuum and the associated nearest-neighbour spin–spin correlations do not persist down to zero energy, because the nearest-neighbour RVBs are gapped, whereas YbMgGaO4 clearly shows gapless behaviour12,14. Therefore, the RVB scenario holds at high energies only. The presence of a distinct low-energy regime is supported by the analysis of the energy-dependent spectra integrated over all measured Q space.
Low energy long-range spin correlations
For energy transfer below J0, excitations related to the breaking of nearest-neighbour spin singlets must freeze out as long as thermal energy is insufficient to overcome J0, 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 downward-pointing 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 J0 (Fig. 3d).
Further information is obtained from the INS spectra at finite magnetic fields applied along the crystallographic c-direction. At 8.5 T, which fully polarizes the moments at low temperatures13,15, a clear boundary is observed in the low-energy 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 energy15,21 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 J0 and the Zeeman energy μ0μBg||H||=0.39 meV of spin-wave excitations for this field. When the field is reduced to zero, the crossing of intensities shifts to J0 (Fig. 3c). We, therefore, associate this effect with an energy gap for the continuum of nearest-neighbour RVB-type excitations3. These excitations seem to be unrelated to the gapless spinon Fermi surface, in contrast to recent expectations based on the INS measurements at higher energies20.
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 Er2Ti2O7 (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 low-temperature intensity sets in. This lower energy is roughly the same as the energy resolution σ∼20 μeV (0.15J0) (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 Lorentzian-Gaussian 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 temperature-dependent 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 low-energy excitations related to longer-range correlations in YbMgGaO4 (refs 12, 14). The most conspicuous effect of this change is the shift of the intensity maxima from the K-points in the high-energy regime to the M-points in the low-energy regime (Supplementary Note 4 and Supplementary Fig. 18), as also seen in the diffuse scattering reported by Paddison et al.21
The clear separation between the low-12,14 and high-energy excitations in the spectrum of YbMgGaO4 (see Fig. 3d) is interesting and unique, rendering YbMgGaO4 distinct from QSL materials known to date, such as herbertsmithite7,29, organic charge transfer salts9,10 and Ca10Cr7O28 reported recently30. The RVB scenario on the triangular-lattice was also discussed for the cluster magnet LiZn2Mo3O8, where a spin-liquid state with both nearest-neighbour and next-nearest-neighbour correlations is formed31,32,33. It is also worth noting that the continuum of nearest-neighbour RVB excitations goes back to the original idea by Anderson1 who argued that Heisenberg spins on the regular triangular lattice evade long-range magnetic order and form the nearest-neighbour RVB QSL state. Although Anderson’s conjecture was not confirmed in later studies34, the formation of a QSL on a triangular lattice with spatial anisotropy35, next-nearest-neighbour couplings36 and multiple-spin exchange37 was identified in the recent literature. Whereas the multiple-spin exchange can be clearly excluded due to the strongly localized nature of the 4f electrons of Yb3+, two other effects are potentially relevant to YbMgGaO4.
The presence of next-nearest-neighbour couplings is currently debated based on the modelling of the magnetic diffuse scattering21,38. Spatial anisotropy of nearest-neighbour couplings can be, at first glance, excluded, based on the three-fold symmetry of the crystal structure12. However, recent experiments15,21, including our INS study15 of crystal-field excitations of Yb3+, pinpoint the importance of the Mg/Ga disorder that leads to variations in the local environment of Yb3+. An immediate effect of this structural disorder is the distribution of g-values 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 too15,18,21.
Our result suggests that the broad excitation continuum in YbMgGaO4 reflects nearest-neighbour 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 conductivity39. On the other hand, gapless nature of YbMgGaO4 evidenced by the non-zero low-temperature susceptibility12,14 and the power-law behaviour of the magnetic specific heat12 are indicative of a distinct low-energy regime that has been glimpsed in our experiment. These low-energy excitations are likely to contain crucial information on whether the ground state of YbMgGaO4 is indeed a QSL, or a special case of the disorder-induced mimicry of a spin liquid, as proposed recently40,41.
Large single crystals (∼1 cm) of YbMgGaO4 were grown by the floating zone technique reported previously13. The as grown rod (∼50 g) was cut into slices along the ab-plane (the easily cleavable direction). Ten best-quality ab-slices of the single-crystal (total mass ∼10 g) were selected for the neutron scattering experiment on LET by Laue X-ray 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 multi-chopper 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, respectively27. 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 long-range magnetic order (Supplementary Figs 4–6). All neutron scattering data were processed and analysed using Horace-Matlab42 on the ISIS computers. Asymmetry of the intensities was observed due to the macro-scale non-rotational 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 (D3d) 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 c-axis. 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.
The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
How to cite this article: Li, Y. et al. Nearest-neighbour resonating valence bonds in YbMgGaO4. Nat. Commun. 8, 15814 doi: 10.1038/ncomms15814 (2017).
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We thank Gang Chen, Haijun Liao, Changle Liu, Sasha Chernyshev and Mike Zhitomirsky for helpful discussions. Y.L. was supported by the start-up 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 TRR-80 and the German Federal Ministry for Education and Research through the Sofja Kovalevskaya Award of the Alexander von Humboldt Foundation.
The authors declare no competing financial interests.
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Li, Y., Adroja, D., Voneshen, D. et al. Nearest-neighbour resonating valence bonds in YbMgGaO4. Nat Commun 8, 15814 (2017). https://doi.org/10.1038/ncomms15814
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