Nearest-neighbour resonating valence bonds in YbMgGaO4

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.

Q uantum 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 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 long-range ordering were detected by muon spin relaxation (mSR) 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 (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. Lowenergy excitations arise from breaking long-range valence bonds or rearranging the short bonds into longer ones 3,16 . 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 YbMgGaO 4 , excellent transparence with the optical gap exceeding B3 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 Yb 3 þ should restrict magnetic interactions to nearest neighbours (S 1 and S 2 ), but these interactions are anisotropic 13 , owing to the strong spin-orbit coupling, where the local moment S ¼ 1/2 is a pseudospin, that is, a combination of spin and orbital moments 15,[17][18][19] . The lowest-energy eigenstate of a dimer formed by such anisotropic pseudospins is, nevertheless, a pure singlet, ð1 ffiffi ffi 2 p Þ j "#i À j #"i ð Þ , with the energy À 3/4J 0 for the antiferromagnetic isotropic coupling, J 0 (4J ± þ J zz )/ 3 ¼ 0.13(1) meV (ref. 13), as observed experimentally. In contrast to Heisenberg spins, the Yb 3 þ pseudospins do not form a three-fold degenerate triplet state and feature three non-degenerate 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 nearest-neighbour magnetic energy of J 0 ¼ 0.13(1) meV only 13 , the excitations were observed at energies between 2J 0 and 10J 0 . Therefore, they are high-energy magnetic excitations of YbMgGaO 4 .
In this paper, we propose a different interpretation of these high-energy 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 ground-state regime defined by the saturation of the mSR rate 14 , and at a much higher temperature of 35 K corresponding to 23J 0 . The high-energy excitations observed previously 20,21 are confirmed and ascribed to nearest-neighbour 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 nearest-neighbour 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 (0.1r:or2 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 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 B0.7 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 E i ¼ 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 Yb 3 þ , 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 solid 23 and glass 24,25 , is excluded by our previous mSR study 14 , 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 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 excitations 26  Second, the antiferromagnetic nature of the isotropic nearestneighbour coupling, J 0 (4J ± þ J zz )/3 ¼ 0.13(1) meV (ref. 13), allows the formation of spin singlet in a pair of the Yb 3 þ 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) B0.3 (Supplementary Table 1), is consistent with the expected ratio, based on the thermal distribution of the eigenstates of the Yb 3 þ dimer. With increasing temperature, a larger fraction of nearest-neighbour singlets is excited. Fourth, the uniform spin susceptibility, w 0 (E), which is obtained from the INS spectrum measured around the Gamma point (Q ¼ 0) via the fluctuation-dissipation theorem and the Kramers À Kronig transformation 22 , is almost zero at 0.1 K above B0.5 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 B0.5 meV.
Last, both spin and valence bond freezing are excluded by our mSR measurement reported previously 14 .
The above six arguments suggest that the whole excitation continuum at energies above J 0 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  Calculated |F(Q)| 2 (from equation (2)) (f). The black lines represent Brillouin zone boundaries. Pink lines show the high-symmetry directions with special reciprocal-space points labelled. Note that the experimental data contain a Q-independent background, which is about the same at G 1 , G 2 and G 3 . This background is missing in the RVB calculation in f, where I G ¼ 0.  Supplementary Figs 14-19). Lower energies can be probed with E i ¼ 1.26 meV (the energy resolution sB20 meV (ref. 27)), but these data cover a limited Q-range only. Nevertheless, we find no qualitative differences between the spectra at B0.3 and B0.7 meV in all measured Q space (E i ¼ 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 J 0 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 YbMgGaO 4 clearly shows gapless behaviour 12,14 . 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 J 0 , excitations related to the breaking of nearest-neighbour spin singlets must freeze out as long as thermal energy is insufficient to overcome J 0 , that is, To1.5 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 J 0 (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 temperatures 13,15 , 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 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 m 0 m B g || H || ¼ 0.39 meV of spin-wave 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 nearest-neighbour RVB-type 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)oI(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 sB20 meV (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 (s), 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 Tr35 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 YbMgGaO 4 (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

Discussion
The clear separation between the low-12,14 and high-energy 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 triangular-lattice was also discussed for the cluster magnet LiZn 2 Mo 3 O 8 , where a spin-liquid state with both nearest-neighbour and next-nearest-neighbour correlations is formed [31][32][33] . It is also worth noting that the continuum of nearest-neighbour RVB excitations goes back to the original idea by Anderson 1 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 studies 34 , the formation of a QSL on a triangular lattice with spatial anisotropy 35 , next-nearest-neighbour couplings 36 and multiple-spin exchange 37 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 Yb 3 þ , two other effects are potentially relevant to YbMgGaO 4 .
The presence of next-nearest-neighbour couplings is currently debated based on the modelling of the magnetic diffuse scattering 21,38 . Spatial anisotropy of nearest-neighbour couplings can be, at first glance, excluded, based on the three-fold symmetry of the crystal structure 12 . However, recent experiments 15,21 , including our INS study 15 of crystal-field excitations of Yb 3 þ , pinpoint the importance of the Mg/Ga disorder that leads to variations in the local environment of Yb 3 þ . 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 too 15,18,21 .
Our result suggests that the broad excitation continuum in YbMgGaO 4 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 conductivity 39 . On the other hand, gapless nature of YbMgGaO 4 evidenced by the non-zero low-temperature susceptibility 12,14 and the power-law behaviour of the magnetic specific heat 12 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 YbMgGaO 4 is indeed a QSL, or a special case of the disorder-induced mimicry of a spin liquid, as proposed recently 40,41 .

Methods
Sample preparation. Large single crystals (B1 cm) of YbMgGaO 4 were grown by the floating zone technique reported previously 13 . The as grown rod (B50 g) was cut into slices along the ab-plane (the easily cleavable direction). Ten best-quality ab-slices of the single-crystal (total mass B10 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 meV, 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 long-range magnetic order ( Supplementary  Figs 4-6). All neutron scattering data were processed and analysed using Horace-Matlab 42 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 (D 3d ) in the reciprocal lattice space (see Fig. 1a,b). The corresponding raw data can be found in Supplementary Figs 7-9.
Data availability. The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.