Abstract
A quantum spin liquid (QSL) is an exotic state of matter in which electrons’ spins are quantum entangled over long distances, but do not show magnetic order in the zerotemperature limit^{1}. The observation of QSL states is a central aim of experimental physics, because they host collective excitations that transcend our knowledge of quantum matter; however, examples in real materials are scarce^{2}. Here, we report neutronscattering experiments on YbMgGaO_{4}, a QSL candidate in which Yb^{3+} ions with effective spin1/2 occupy a triangular lattice^{3,4,5,6}. Our measurements reveal a continuum of magnetic excitations—the essential experimental hallmark of a QSL^{7}—at very low temperature (0.06 K). The origin of this peculiar excitation spectrum is a crucial question, because isotropic nearestneighbour interactions do not yield a QSL ground state on the triangular lattice^{8}. Using measurements in the fieldpolarized state, we identify antiferromagnetic nextnearestneighbour interactions^{9,10,11,12}, spinspace anisotropies^{4,10,13,14}, and chemical disorder^{15} between the magnetic layers as key ingredients in YbMgGaO_{4}.
Similar content being viewed by others
Main
The phenomenon of entanglement is one of the fundamental results of quantum mechanics. Among its most extraordinary manifestations are quantum spin liquids, in which a macroscopic number of spins are entangled but do not show conventional magnetic order^{1}. The earliest examples of QSLs were found in quasionedimensional systems such as Heisenberg spin chains^{16} and in the distorted triangularlattice antiferromagnet Cs_{2}CuCl_{4} (ref. 17). The search for QSLs in two and threedimensional systems has focused on frustrated magnets, in which the lattice geometry prevents all magnetic interactions from being satisfied simultaneously^{1,2} or in which spin–orbit coupling leads to interactions that depend on bond directions^{18}. In two dimensions, the prototype of a QSL is Anderson’s ‘resonating valence bond’ model^{19}. Its ground state is a superposition of all possible tilings of dimers on the triangular lattice, where each dimer is built from an entangled pair of spin1/2; its excitations are unpaired spin1/2, which are delocalized. The presence of delocalized excitations with fractional quantum numbers leads to a necessary experimental signature of a QSL: a magnetic excitation spectrum that is continuous in energy but structured in momentum space^{7}. Such a spectrum has indeed been observed using neutronscattering measurements of ZnCu_{3}(OD)_{6}Cl_{2} (‘herbertsmithite’)^{7,20}, in which spins occupy a kagome lattice. However, the stabilization of a continuous excitation spectrum by quantum fluctuations in a structurally perfect triangularlattice magnet—the scenario originally proposed by Anderson^{19}—has remained an open question.
Recent experiments have identified the triangularlattice magnet YbMgGaO_{4} as an exciting QSL candidate^{3,4}. The crystal structure (space group ) contains undistorted triangular planes of magnetic Yb^{3+} (Fig. 1a), separated by two triangular planes occupied in a disordered manner by Mg^{2+} and Ga^{3+} (Fig. 1b)^{3}. Thermodynamic and muonspin relaxation (μSR)^{5} measurements show the absence of conventional magnetic order and spin freezing to T < 0.1 K, far below the Weiss temperature θ_{W} ≈ −4 K; moreover, the apparent absence of zeropoint entropy indicates that the system essentially occupies a single quantum state at 0.06 K (ref. 3). The crystalfield ground state of Yb^{3+} is a Kramers doublet^{4}, separated by a large energy gap of 38 meV from the first excited state (Fig. 1c and Supplementary Fig. 1); hence, an effective spin1/2 description is appropriate at low temperature, as for ‘quantum spin ice’ Yb_{2}Ti_{2}O_{7} (ref. 21). Electronspin resonance (ESR) measurements indicate that the nearestneighbour magnetic interaction is anisotropic and depends on the bond orientation^{4}. This provides one potential mechanism to stabilize a QSL ground state on the triangular lattice^{13,14}. However, several alternative mechanisms for QSL behaviour are known, including furtherneighbour interactions^{9,10,11,12}, disorderinduced entanglement^{15} and multispin interactions^{22}.
Experiments to reveal the nature of the potential triangularlattice QSL in YbMgGaO_{4} are therefore crucial. To enable such experiments, we have grown a large single crystal of YbMgGaO_{4} using the floatingzone technique (Supplementary Fig. 2). We characterized our crystal using specific heat, magnetization, and singlecrystal neutrondiffraction measurements. In addition, we compared specificheat and Xray diffraction measurements of crushed crystal pieces with the same measurements of a powder sample prepared by the solidstate method of ref. 3. Singlecrystal neutrondiffraction measurements (Supplementary Fig. 3) reveal that our single crystal is predominantly a single grain and is therefore suitable for inelastic neutronscattering measurements. The magnetic specific heat shows a broad peak at T ≈ 3 K in zero magnetic field, and a T^{γ} dependence with γ = 0.70(1) below the peak (Fig. 1d), consistent with previous experimental results^{3}. Our magnetic entropy (Fig. 1e) and magnetization (Fig. 1f) measurements fully support the effective spin1/2 scenario (with g_{∥} ≈ 3.7) for Yb^{3+} at low temperatures. Rietveld refinements to our Xray diffraction data (Supplementary Fig. 4) yield a good fit for the published structural model^{3} with no Yb^{3+}/Ga^{3+} site mixing observed within our experimental accuracy and with disordered occupancy on the interstitial layers by Mg^{2+} and Ga^{3+} (Supplementary Table 1). Such disorder can affect magnetism by modifying the charge environment around Yb^{3+} ions. Our broadband neutronscattering experiment (Fig. 1c and Supplementary Fig. 1) reveals three intense crystal electricfield excitations at 38(1), 61(1) and 97(1) meV, consistent with the four Kramers doublets expected for Yb^{3+}. These excitations are broader than the instrumental resolution and an additional weak mode is observed around 134(8) meV, hinting at a distribution of local environments across Yb^{3+} ions. In all our measurements, we observe no significant differences between powder and singlecrystal samples, suggesting that the sample dependence that has afflicted QSL candidates such as Yb_{2}Ti_{2}O_{7} (ref. 23) is not evident in YbMgGaO_{4}.
Neutronscattering experiments measure spin correlations as a function of energy and reciprocal space, yielding direct information about correlated quantum states. Singlecrystal neutronscattering data measured in zero field are shown in Fig. 2, and provide strong evidence for QSL behaviour. Throughout, we plot neutronscattering intensity as (1 + e^{−βE}) I(Q, E), where I(Q, E) is the measured intensity as a function of scattering wavevector Q = h a^{∗} + k b^{∗} + l c^{∗} and energy transfer E, and the factor 1 + e^{−βE} corrects for detailed balance^{24}. Figure 2a shows the energy dependence of the scattering intensity along highsymmetry reciprocalspace directions. The scattering at E ≳ 0.2 meV is magnetic, as shown by its dependence on applied magnetic field (discussed below). At both 0.06 K and 14 K, our data reveal a broad continuum of excitations, in contrast to the spinwave scattering observed in the spin1/2 triangularlattice compound Ba_{3}CoSb_{2}O_{9} (ref. 25). The excitation continuum has a bandwidth of 1.3 meV and is gapless within the experimental energy resolution of approximately 0.1 meV. This scattering spans a much smaller energy scale but is otherwise qualitatively similar to herbertsmithite^{7,20}—a surprising result, because the kagome lattice of herbertsmithite is considered more highly frustrated than the triangular lattice of YbMgGaO_{4} (refs 8,26).
The wavevector dependence of the 0.06 K scattering intensity is shown in Fig. 2b (see also Supplementary Fig. 5). The elastic scattering is much broader than the instrumental resolution (δQ/Q ≈ 2%) and reveals no sharp magnetic Bragg peaks, confirming the absence of longrange magnetic order at 0.06 K. The lowenergy scattering (E ≤ 0.5 meV) shows a broad maximum at the point of the hexagonal Brillouin zone, whereas at higher energies (0.5 ≤ E ≤ 1.0 meV) the scattering is isotropic around the zone boundary. The energy dependence of the magnetic signal at the M point (Fig. 2c) indicates that at most 16(3)% of the total spectral weight is elastic. Correspondingly, the inelastic contribution (≥84(3)%) is large compared with the 66% expected for a spin1/2 glass^{27}. This rules out a totally frozen state for YbMgGaO_{4}, in agreement with μSR^{5}. Further insight into the Mpoint scattering maximum is obtained from line plots at different energies (Fig. 2d). At 0.06 K, these plots reveal a weak dispersive excitation originating from the M point, superimposed on the continuum shown in Fig. 2a. Strikingly, some structure persists in the excitation spectrum at 14 K (≳3θ_{W}), but the location of the intensity maximum at low energy (∼0.25 meV) shifts to between Γ ≡ (000) and points (Fig. 2c).
What is the probable origin of the QSL behaviour of YbMgGaO_{4}? To answer this question, we measured its spectrum with a large magnetic field applied along the c axis. The field polarizes the spins, yielding excitations that can be modelled by conventional linear spinwave theory to determine the values of the exchange interactions^{21}. The starting point for our analysis is the effective spin1/2 Hamiltonian^{4,21,28},
where spin operators S^{±} = S^{x} ± iS^{y}, g_{∥} and g_{⊥} are the components of the gtensor parallel and perpendicular to the applied field, and the complex numbers γ_{ij} are defined in ref. 4. Brackets 〈〉 and 〈〈〉〉 denote nearest and nextnearestneighbour pairs, respectively. The exchange interactions J^{zz} and J^{±} define an XXZ model^{29}, where we include nearest and nextnearestneighbour interactions (denoted by subscripts 1 and 2, respectively). We also include a symmetryallowed bonddependent (also known as pseudodipolar) interaction, J_{1}^{±±}; however, we neglect the other bonddependent interaction (J_{1}^{z±} in ref. 4), because our current experimental data are insensitive to this term and ESR measurements indicate that it takes a very small value, 0.003(9) meV (ref. 4). For a large applied field along c, it follows from equation (1) that the spinwave dispersion is given by^{13,14}
where r_{1, i} and r_{2, i} label the nearest and nextnearestneighbour vectors, respectively.
Our experimental data measured in a 7.8 T field close to the ≈ 8 T saturation are shown in Fig. 3a. As anticipated from equation (2) and from the field dependence of the specific heat (Fig. 1b), the applied field induces a spin gap of 1.0 meV, and a single dispersive signal with a bandwidth of 1.1 meV dominates the spectrum. A continuum of excitations is also visible and carries a significant spectral weight. This is a surprising observation that requires further investigation; we postulate that the continuum originates from exchange disorder induced by a distribution of local environments across Yb^{3+} ions. Nevertheless, two observations confirm that the system is essentially polarized: the integrated intensity of the inelastic scattering at 7.8 T is relatively constant throughout the Brillouin zone (Fig. 3b), and the overall intensity is significantly reduced compared with 0 T, as expected for a fieldinduced ferromagnetic state^{30}.
To model the dispersion, we first fit the Edependence at each Q with a double (that is, asymmetric) Lorentzian to determine the peak position; a representative fit is shown in Fig. 3c. We then fit equation (2) to the extracted dispersion curve by varying the exchange parameters J_{1}^{zz} + J_{2}^{zz}, J_{1}^{±}, J_{1}^{±±}, and J_{2}^{±}. The choice of a purely twodimensional model is justified by the absence of visible dispersion along l (Fig. 3a). Throughout, we fix g_{∥} = 3.721 from magnetization data^{4}. Initially, we also fix J_{1}^{±±} = 0.013 meV from ESR measurements^{4}. We obtain an excellent fit with J_{1}^{zz} + J_{2}^{zz} = 0.154(3), J_{1}^{±} = 0.109(4) and J_{2}^{±} = 0.024(1) meV . The ratio J^{zz}/J^{±} = 1.16(4) is comparable to the result from magnetization measurements (J^{zz}/J^{±} = 1.09(13); ref. 4), locating the interactions of YbMgGaO_{4} between isotropic and planar limits (J^{zz}/J^{±} equal to 2 and 0, respectively). In the limit of planar nearestneighbour interactions, a gapless Dirac QSL may be stabilized^{31}; however, this phase does not persist for J_{1}^{zz}/J_{1}^{±} > 0 (ref. 29). The parameters J_{1}^{±±} and J_{2}^{±} are therefore of particular interest, because both may in principle stabilize a QSL^{12,13,14}. Figure 3d shows the dependence of the goodnessoffit on these parameters. Our dispersion curve is relatively insensitive to J_{1}^{±±} : the best fit is for J_{1}^{±±} = 0, but the previously reported value of 0.013 meV (ref. 4) yields visually indistinguishable results. Crucially, however, our data are highly sensitive to J_{2}^{±}. Matching the shallow dispersion minimum at the M point yields a bestfit ratio J_{2}^{±}/J_{1}^{±} ≈ 0.22 (Fig. 3a, solid red line). It is possible to reproduce this minimum without J_{2}^{±} (ref. 13,14), but for fixed planar anisotropy J^{zz}/J^{±} = 1.1 this yields a large and dominant J_{1}^{±±} = 0.15(1) meV (Fig. 3a, blue dashed line), contradicting the ESR result (J_{1}^{±±} = 0.013(1) meV) (ref. 4). Remarkably, the ratio J_{2}^{±}/J_{1}^{±} lies close to the QSL regime of the spin1/2 J_{1}J_{2} Heisenberg triangularlattice antiferromagnet, which is predicted to occur for 0.06 ≲ J_{2}/J_{1} ≲ 0.19 (refs 9,10,11,12). Taken together with the published ESR results^{4}, our 7.8 T results thus point towards moderate planar anisotropy with small antiferromagnetic nextnearestneighbour J_{2}^{±} as a minimal model for YbMgGaO_{4}.
Instantaneous spinpair correlations in zero applied field provide an independent check on our results. We obtain the experimental diffuse intensity I(Q) = ∫ _{0}^{E′}(1 + e^{−βE})I(Q, E) dE, where E′ = 1.6 meV, and compare these data with classical Monte Carlo simulations in Fig. 3e, f. Our calculations, driven by equation (1), use our fitted values of the XXZ exchange interactions and assume J_{1}^{zz}/J_{1}^{±} = J_{2}^{zz}/J_{2}^{±}, where relevant. At a simulation temperature of 1.3 K, the calculated diffuse intensity for J_{2}^{±} = J_{1}^{±±} = 0 has its maximum at the K point, contrary to experiment. However, including J_{2}^{±} = 0.22J_{1}^{±} reproduces the Mpoint maximum observed experimentally (Fig. 3e)—a result also observed in quantum calculations^{12}. A dominant J_{1}^{±±} also produces Mpoint scattering^{4} but introduces an additional intensity modulation that depends on the sign of J_{1}^{±±}, and is not observed in our experimental data (Fig. 3f and Supplementary Fig. 6). Crucially, after cooling our Monte Carlo simulations below 1.3 K, the calculated diffuse scattering shows much sharper features than the experimental data (Supplementary Fig. 7). Quantum fluctuations are thus a possible contender to explain the suppressed spin correlation length of 2.8 Å we observe at 0.06 K.
Our observations in YbMgGaO_{4} set it apart from other inorganic triangularlattice magnets with quantum spins, in which magnetic order or spin freezing typically occurs at a temperature ∼θ_{W}/10 (see, for example, refs 25,32). In contrast, YbMgGaO_{4} shows a continuous excitation spectrum without magnetic order to temperatures below ∼θ_{W}/60. Our results identify magnetic scattering at the M point of the hexagonal Brillouin zone and provide strong constraints on spinspace anisotropies and nextnearestneighbour interactions to guide further theoretical investigations. Our work also calls for further studies to elucidate the effect of interstitial Mg^{2+}/Ga^{3+} disorder on exchange interactions. Compared with moleculebased triangular QSL candidates^{33,34,35}, the strength of the magnetic signal and the availability of large singlecrystal samples make YbMgGaO_{4} an exceptional candidate for neutronscattering experiments, suggesting that mapping the response of a twodimensional QSL candidate to temperature, applied magnetic field, and chemical perturbations is now a practical prospect.
Methods
Sample preparation.
A polycrystalline sample of YbMgGaO_{4} was synthesized by a solidstate method. Stoichiometric ratios of Yb_{2}O_{3}, MgO and Ga_{2}O_{3} fine powder were carefully ground and reacted at a temperature of 1,450 °C for 3 days with several intermediate grindings. The singlecrystal sample of YbMgGaO_{4} (Supplementary Fig. 2) was grown using the optical floatingzone method under a 5 atm oxygen atmosphere^{4}. The best single crystal was obtained with a pulling speed of 1.5 mm h^{−1}, and showed cliffed [001] surfaces after several hours of growth.
Xray diffraction measurements and refinements.
Roomtemperature powder Xray diffraction were carried out on powder and crushed singlecrystal samples using a Panalytical X’pert Pro Alpha1 diffractometer with monochromatic CuKα radiation (λ = 1.540598 Å). Preliminary measurements in flatplate geometry on a loose powder using a Huber Xray diffractometer showed preferred orientation, especially for the crushedcrystal sample. To minimize the extent of preferred orientation, we loaded our samples into a glass capillary that was rotated at 30 r.p.m. Due to the large absorption crosssection of Yb, these measurements were limited to small sample sizes. Measurements were taken between 5 ≤ 2θ ≤ 140° with Δ2θ = 0.016°. Rietveld refinement was carried out using the FULLPROF program^{36}. Peak shapes were modelled by pseudoVoigt functions, and the remaining preferred orientation was treated within the March model^{37}. Fits to data are shown in Supplementary Fig. 4, and refined values of structural parameters are given in Supplementary Table 1.
Thermomagnetic measurements.
Heatcapacity measurements were carried out on a Quantum Design physical property measurement system (PPMS) instrument using dilution fridge (0.06 ≤ T ≤ 4 K) and standard (1.6 ≤ T ≤ 100 K) probes in a range of measuring magnetic fields, 0 ≤ μ_{0}H ≤ 14 T. Singlecrystal measurements were made on a flat thin piece polished to ≈ 1 mg and oriented with the c axis parallel to the applied magnetic field. To ensure sample thermalization at low temperatures, powder measurements were made on pellets of YbMgGaO_{4} mixed with an approximately equal mass of silver powder, the contribution of which was measured separately and subtracted to obtain the specific heat C_{p}. The magnetic specific heat C_{m} was obtained by subtracting a modelled lattice contribution C_{L} with two Debye temperatures, 480 K and 142 K. The change in magnetic entropy ΔS(T) was subsequently obtained by integrating C_{m}/T from 0.06 K to T. Isothermal magnetization measurements M(H) were performed on the above singlecrystal piece using a PPMS vibrating sample magnetometer in a range of magnetic fields 0 ≤ μ_{0}H ≤ 14 T and temperatures 1.7 ≤ T ≤ 10 K.
Lowenergy inelastic neutronscattering measurements.
Inelastic neutronscattering^{24} experiments were performed on the cold neutron chopper spectrometer (CNCS) at the Spallation Neutron Source (SNS), Oak Ridge National Laboratory^{38}. The sample was a 2.2 g rodshaped crystal cut into two shorter pieces to fit in the bore of a cryomagnet. The two pieces (total dimensions 16 × 16 × 4 mm^{3}) were coaligned to within 1.5° using a Multiwire Xray Laue backscattering machine, and mounted in the (hk0) scattering plane on a oxygenfree copper holder (Supplementary Fig. 2). The mount was attached to the bottom of a dilution refrigerator reaching a base temperature of 0.06 K at the mixing chamber, indicating a sample temperature ≲0.1 K. The sample stick was inserted in an 8 T superconducting cryomagnet, and measurements were performed in zero field and in a field of 7.8 T applied along the c axis. Two incident neutron energies were used, E_{i} = 3.32 and 12.0 meV, yielding elastic energy resolutions (fullwidth at halfmaximum) of 0.11 and 0.75 meV, respectively. The sample was rotated in steps of 1°, with a range of 270° for E_{i} = 12 meV and 180° for E_{i} = 3.32 meV . For E_{i} = 3.32 meV, the background scattering from the sample environment was measured and subtracted from the data. For E_{i} = 12 meV, the background and nonmagnetic scattering at low energy (E ≤ 0.9 meV) was subtracted using the 7.8 T measurement, taking advantage of the spin gap induced by applied field.
Broadband neutronscattering measurements.
Broadband neutronscattering experiments were performed on a sintered powder sample and on one of the above single crystals using the SEQUOIA spectrometer at ORNL’s SNS^{39}. The powder sample was mounted in a cylindrical aluminium can sealed under one atmosphere of ^{4}He at room temperature. The single crystal was mounted with its ab plane horizontal and rotated continuously, yielding an effective twodimensional powder averaging. Both samples were cooled using a closedcycle refrigerator reaching a base temperature of T = 4 K. The spectrometer was operated with three distinct configurations to cover a large range of momentum and energy transfers: E_{i} = 300 meV with a Fermi chopper frequency of f = 420 Hz, E_{i} = 160 meV with f = 600 Hz, and E_{i} = 22 meV with f = 300 Hz. Background and sample holder contributions were subtracted from the data using empty can measurements. The contribution from the lattice was subtracted by extracting the phonon density of states at Q = 6.0 Å^{−1} (160 meV) and Q = 9.0 Å^{−1} (300 meV) and extrapolating to other momenta using the known Q^{2} intensity dependence for singlephonon scattering^{40}. The position and width of the observed excitations were obtained through fits to Gaussian profiles. The energy resolution of the spectrometer was estimated using simplified geometrical considerations yielding a correct elastic energy resolution but known to underestimate the energy resolution at finite energy transfer. Thirteen electrons reside in the 4f shell of the Kramers ion Yb^{3+}. Following ref. 41, the crystal electric field can maximally split the local electronic manifold into four Kramers doublets yielding three excited doublets above the doublet ground state.
Elastic neutronscattering measurements.
Elastic neutronscattering experiments were performed on the CORELLI spectrometer at ORNL’s SNS. One of the two above crystal pieces was attached to a copper pin at the bottom of a ^{3}He cryostat reaching a base temperature of 0.3 K. The sample was aligned in the (h0l) scattering plane to assess crystal quality and stacking of the triangularlattice planes along c. Neutronabsorbing Cd was used to shield the sample holder and an empty cryostat measurement was used to remove the background contribution. The sample was rotated in steps of 6° over a 360° range, and measurements were taken at temperatures of 0.3, 4.0 and 40 K. Elastic neutronscattering data measured at 0.3 K are shown in Supplementary Fig. 3.
Data analysis.
Initial data reduction was performed in MANTID^{42} for CNCS, SEQUOIA and CORELLI data sets. For the CNCS data, subsequent analysis was performed in HORACE^{43} on a dedicated node within Georgia Tech’s Partnership for Advanced Computing infrastructure. To increase counting statistics, inelasticscattering data were symmetrized into an irreducible 60° wedge of the hexagonal reciprocal lattice. The CORELLI data were normalized to absolute units in MANTID^{44}.
Monte Carlo simulations.
To perform classical Monte Carlo simulations, we rewrite the spin Hamiltonian, equation (1), for zero field in terms of spin components S^{x}, S^{y} and S^{z}:
where the phase factors for nearestneighbour bonds along the directions a, b and a + b, respectively (where a and b are shown in Fig. 1a). In Fig. 3e, we keep J_{1}^{zz} = 0.126 and J_{1}^{±} = 0.109 meV with either J_{2}^{zz} = 0 and J_{2}^{±} = 0 (bottom left) or J_{2}^{zz} = 0.027 and J_{2}^{±} = 0.024 meV (bottom right). In Fig. 3f, we take J_{1}^{±±} = ± 0.15 meV with J_{1}^{zz} = 0.12 and J_{1}^{±} = 0.11 meV . The effect of small nonzero J_{1}^{±±} = ± 0.013 meV is discussed in the Supplementary Information. We take the length of spin vectors as . Simulations were performed on a threedimensional spin configuration consisting of nine triangular layers, each containing 504 spins. A proposed spin move consists of rotating a spin by a small amount (chosen so that 50–70% of proposed moves were accepted). The simulations were initialized at a high temperature and cooled in gradual increments. At each temperature, the simulation was run for at least 10t_{0} proposed moves, where t_{0} is the number of proposed moves required to decorrelate the system. The singlecrystal diffusescattering intensity was calculated as^{40}
where f(Q) is the Yb^{3+} magnetic form factor^{45}, and S_{i}^{⊥} = S_{i} − Q(S_{i} ⋅Q)/Q^{2} is the component of spin S_{i} perpendicular to Q. The calculated pattern was slightly smoothed to allow calculation on an arbitrary Qgrid. To increase statistical averaging, the calculated I(Q) was averaged over 100 spin configurations and diffraction symmetry was applied.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author on request.
References
Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).
Lee, P. A. An end to the drought of quantum spin liquids. Science 321, 1306–1307 (2008).
Li, Y. et al. Gapless quantum spin liquid ground state in the twodimensional spin1/2 triangular antiferromagnet YbMgGaO4 . Sci. Rep. 5, 16419 (2015).
Li, Y. et al. Rareearth triangular lattice spin liquid: a singlecrystal study of YbMgGaO4 . Phys. Rev. Lett. 115, 167203 (2015).
Li, Y. et al. Muon spin relaxation evidence for the U(1) quantum spinliquid ground state in the triangular antiferromagnet YbMgGaO4 . Phys. Rev. Lett. 117, 097201 (2016).
Shen, Y. et al. Spinon Fermi surface in a triangular lattice quantum spin liquid YbMgGaO4. Preprint at http://arXiv.org/abs/1607.02615 (2016).
Han, T.H. et al. Fractionalized excitations in the spinliquid state of a kagomelattice antiferromagnet. Nature 492, 406–410 (2012).
Capriotti, L., Trumper, A. E. & Sorella, S. Longrange Néel order in the triangular Heisenberg model. Phys. Rev. Lett. 82, 3899–3902 (1999).
Manuel, L. O. & Ceccatto, H. A. Magnetic and quantum disordered phases in triangularlattice Heisenberg antiferromagnets. Phys. Rev. B 60, 9489–9493 (1999).
Li, P. H. Y., Bishop, R. F. & Campbell, C. E. Quasiclassical magnetic order and its loss in a spin1/2 Heisenberg antiferromagnet on a triangular lattice with competing bonds. Phys. Rev. B 91, 014426 (2015).
Zhu, Z. & White, S. R. Spin liquid phase of the spin1/2 J1–J2 Heisenberg model on the triangular lattice. Phys. Rev. B 92, 041105 (2015).
Iqbal, Y., Hu, W.J., Thomale, R., Poilblanc, D. & Becca, F. Spin liquid nature in the Heisenberg J1–J2 triangular antiferromagnet. Phys. Rev. B 93, 144411 (2016).
Li, Y.D., Wang, X. & Chen, G. Anisotropic spin model of strong spinorbitcoupled triangular antiferromagnets. Phys. Rev. B 94, 035107 (2016).
Li, Y.D., Shen, Y., Li, Y., Zhao, J. & Chen, G. The effect of spinorbit coupling on the effectivespin correlation in YbMgGaO4. Preprint at http://arXiv.org/abs/1608.06445 (2016).
Savary, L. & Balents, L. Disorderinduced entanglement in spin ice pyrochlores. Preprint at http://arXiv.org/abs/1604.04630 (2016).
Tennant, D. A., Perring, T. G., Cowley, R. A. & Nagler, S. E. Unbound spinons in the spin1/2 antiferromagnetic chain KCuF3 . Phys. Rev. Lett. 70, 4003–4006 (1993).
Coldea, R., Tennant, D. A., Tsvelik, A. M. & Tylczynski, Z. Experimental realization of a 2D fractional quantum spin liquid. Phys. Rev. Lett. 86, 1335–1338 (2001).
Banerjee, A. et al. Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet. Nat. Mater. 15, 733–740 (2016).
Anderson, P. W. Resonating valence bonds: a new kind of insulator? Mater. Res. Bull. 8, 153–160 (1973).
de Vries, M. A. et al. Scalefree antiferromagnetic fluctuations in the spin1/2 kagome antiferromagnet herbertsmithite. Phys. Rev. Lett. 103, 237201 (2009).
Ross, K. A., Savary, L., Gaulin, B. D. & Balents, L. Quantum excitations in quantum spin ice. Phys. Rev. X 1, 021002 (2011).
Misguich, G., Lhuillier, C., Bernu, B. & Waldtmann, C. Spinliquid phase of the multiplespin exchange Hamiltonian on the triangular lattice. Phys. Rev. B 60, 1064–1074 (1999).
Yaouanc, A., Dalmas de Réotier, P., Marin, C. & Glazkov, V. Singlecrystal versus polycrystalline samples of magnetically frustrated Yb2Ti2O7: specific heat results. Phys. Rev. B 84, 172408 (2011).
Marshall, W. & Lowde, R. D. Magnetic correlations and neutron scattering. Rep. Prog. Phys. 31, 705–775 (1968).
Ma, J. et al. Static and dynamical properties of the spin1/2 equilateral triangularlattice antiferromagnet Ba3CoSb2O9 . Phys. Rev. Lett. 116, 087201 (2016).
Yan, S., Huse, D. A. & White, S. R. Spinliquid ground state of the spin1/2 kagome Heisenberg antiferromagnet. Science 332, 1173 (2011).
Ross, K. A., Krizan, J. W., RodriguezRivera, J. A., Cava, R. J. & Broholm, C. L. Static and dynamic XYlike shortrange order in a frustrated magnet with exchange disorder. Phys. Rev. B 93, 014433 (2016).
Onoda, S. Effective quantum pseudospin1/2 model for Yb pyrochlore oxides. J. Phys. Conf. Ser. 320, 012065 (2011).
Yamamoto, D., Marmorini, G. & Danshita, I. Quantum phase diagram of the triangularlattice XXZ model in a magnetic field. Phys. Rev. Lett. 112, 127203 (2014).
Mourigal, M. et al. Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain. Nat. Phys. 9, 435–441 (2013).
Alicea, J., Motrunich, O. I. & Fisher, M. P. A. Algebraic vortex liquid in spin1/2 triangular antiferromagnets: scenario for Cs2CuCl4 . Phys. Rev. Lett. 95, 247203 (2005).
Nakatsuji, S. et al. Spin disorder on a triangular lattice. Science 309, 1697–1700 (2005).
Shimizu, Y., Miyagawa, K., Kanoda, K., Maesato, M. & Saito, G. Spin liquid state in an organic Mott insulator with a triangular lattice. Phys. Rev. Lett. 91, 107001 (2003).
Pratt, F. L. et al. Magnetic and nonmagnetic phases of a quantum spin liquid. Nature 471, 612–616 (2011).
Sheckelton, J. P., Neilson, J. R., Soltan, D. G. & McQueen, T. M. Possible valencebond condensation in the frustrated cluster magnet LiZn2Mo3O8 . Nat. Mater. 11, 493–496 (2012).
RodríguezCarvajal, J. Recent advances in magnetic structure determination by neutron powder diffraction. Physica B 192, 55–69 (1993).
Dollase, W. A. Correction of intensities for preferred orientation in powder diffractometry: application of the March model. J. Appl. Crystallogr. 19, 267–272 (1986).
Ehlers, G., Podlesnyak, A. A., Niedziela, J. L., Iverson, E. B. & Sokol, P. E. The new cold neutron chopper spectrometer at the Spallation Neutron Source: design and performance. Rev. Sci. Instrum. 82, 085108 (2011).
Granroth, G. E. et al. SEQUOIA: a newly operating chopper spectrometer at the SNS. J. Phys. Conf. Ser. 251, 12058 (2010).
Squires, G. L. Introduction to the Theory of Thermal Neutron Scattering 129–145 (Cambridge Univ. Press, 1978).
Gaudet, J. et al. Neutron spectroscopic study of crystalline electric field excitations in stoichiometric and lightly stuffed Yb2Ti2O7 . Phys. Rev. B 92, 134420 (2015).
Arnold, O. et al. Mantid—data analysis and visualization package for neutron scattering and μSR experiments. Nucl. Instrum. Methods Phys. Res. A 764, 156–166 (2014).
Ewings, R. A. et al. HORACE: software for the analysis of data from single crystal spectroscopy experiments at timeofflight neutron instruments. Nucl. Instrum. Methods Phys. Res. A 884, 132–142 (2016).
MichelsClark, T. M., Savici, A. T., Lynch, V. E., Wang, X. P. & Hoffmann, C. M. Expanding Lorentz and spectrum corrections to large volumes of reciprocal space for singlecrystal timeofflight neutron diffraction. J. Appl. Crystallogr. 49, 497–506 (2016).
Brown, P. J. International Tables for Crystallography Vol. C, 454–460 (KluwerAcademic, 2004).
Acknowledgements
We are very grateful to L. Ge for his help with heatcapacity measurements and J. Carruth, S. Elorfi, M. Everett and C. Fletcher for sample environment and instrument support during our neutronscattering experiments. It is our pleasure to thank S. Chernyshev, R. Coldea, K. Ross, M. Waterbury, Y. Wan and M. Zhitomirsky for insightful discussions. The work and equipment at the Georgia Institute of Technology (J.A.M.P., M.D. and M.M.) was supported by the College of Sciences and the Executive VicePresident for Research. The work at the University of Tennessee (Z.D. and H.Z.) was supported by the National Science Foundation through award DMR1350002. The research at Oak Ridge National Laboratory’s Spallation Neutron Source was sponsored by the US Department of Energy, Office of Basic Energy Sciences, Scientific User Facilities Division.
Author information
Authors and Affiliations
Contributions
J.A.M.P., M.D., Z.D., G.E., Y.L., M.B.S. and M.M. performed neutronscattering experiments. J.A.M.P., M.D. and M.M. analysed the data. Z.D. and H.Z. made the sample. Z.D. and M.M. characterized the sample. M.D. and M.M. aligned the sample. M.M. made the figures and J.A.M.P. wrote the paper with input from all authors. H.Z. and M.M. designed and supervised the project.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary information
Supplementary information (PDF 1709 kb)
Rights and permissions
About this article
Cite this article
Paddison, J., Daum, M., Dun, Z. et al. Continuous excitations of the triangularlattice quantum spin liquid YbMgGaO_{4}. Nature Phys 13, 117–122 (2017). https://doi.org/10.1038/nphys3971
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys3971
This article is cited by

Candidate spinliquid ground state in CsNdSe2 with an effective spin1/2 triangular lattice
Communications Materials (2024)

Proximate spin liquid and fractionalization in the triangular antiferromagnet KYbSe2
Nature Physics (2024)

Complete fieldinduced spectral response of the spin1/2 triangularlattice antiferromagnet CsYbSe2
npj Quantum Materials (2023)

Informing quantum materials discovery and synthesis using Xray microcomputed tomography
npj Quantum Materials (2022)

Evidence for a spinon Kondo effect in cobalt atoms on singlelayer 1TTaSe2
Nature Physics (2022)