Continuum of quantum fluctuations in a three-dimensional S=1 Heisenberg magnet
K. W. Plumb, Hitesh J. Changlani, A. Scheie, Shu Zhang, J. W. Kriza, J. A. Rodriguez-Rivera, Yiming Qiu, B. Winn, R. J. Cava, C. L. Broholm
CContinuum of quantum fluctuations in a three-dimensional S = 1 Heisenberg magnet
K. W. Plumb, Hitesh J. Changlani, A. Scheie, Shu Zhang, J. W. Krizan, J. A.Rodriguez-Rivera,
3, 4
Yiming Qiu, B. Winn, R. J. Cava, and C. L. Broholm
1, 3, 61
Institute for Quantum Matter and Department of Physics and Astronomy,The Johns Hopkins University, Baltimore, MD 21218, USA Department of Chemistry, Princeton University, Princeton, NJ 08544 NIST Center for Neutron Research,National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Department of Materials Science and Engineering,University of Maryland, College Park, MD 20742, USA NScD Division, Oak Ridge National Laboratory,Oak Ridge, Tennessee 37831-6473, USA Quantum Condensed Matter Division, Oak Ridge National Laboratory,Oak Ridge, Tennessee 37831-6473, USA (Dated: November 22, 2017) a r X i v : . [ c ond - m a t . s t r- e l ] N ov onventional crystalline magnets are characterized by symmetry breakingand normal modes of excitation called magnons with quantized angular mo-mentum (cid:126) . Neutron scattering correspondingly features extra magnetic Braggdiffraction at low temperatures and dispersive inelastic scattering associatedwith single magnon creation and annihilation. Exceptions are anticipated in so-called quantum spin liquids as exemplified by the one-dimensional spin-1/2 chainwhich has no magnetic order and where magnons accordingly fractionalize intospinons with angular momentum (cid:126) / . This is spectacularly revealed by a con-tinuum of inelastic neutron scattering associated with two-spinon processes andthe absence of magnetic Bragg diffraction. Here, we report evidence for thesesame key features of a quantum spin liquid in the three-dimensional Heisenbergantiferromagnet NaCaNi F . Through specific heat and neutron scattering mea-surements, Monte Carlo simulations, and analytic approximations to the equaltime correlations, we show that NaCaNi F is an almost ideal realization of thespin-1 antiferromagnetic Heisenberg model on a pyrochlore lattice with weakconnectivity and frustrated interactions. Magnetic Bragg diffraction is absentand 90% of the spectral weight forms a continuum of magnetic scattering notdissimilar to that of the spin-1/2 chain but with low energy pinch points indicat-ing NaCaNi F is in a Coulomb phase. The residual entropy and diffuse elasticscattering points to an exotic state of matter driven by frustration, quantumfluctuations and weak exchange disorder. Since then, it hasbeen established that the classical ( S → ∞ ) Heisenberg antiferromagnet does not undergoany magnetic ordering transition. The magnetic interaction energy is minimized by allspin configurations with vanishing magnetization on every tetrahedron and the ensemble ofthese configurations forms a macroscopically degenerate, but highly correlated, ground-statemanifold. Such a collective state is termed a Coulomb phase because coarse-grained spinconfigurations within the manifold form a divergence free vector-field that implies dipolarcorrelations.
Experiments probing magnetic correlations, and hence the solenoidal field,should include sharp pinch point features as in related classical spin ice materials whereferromagnetic Ising interactions dominate. Both classical spin ice and the classical Heisen-berg antiferromagnet may be classified as Coulomb phases but, while there is much activityand progress in exploring quantum spin ice, much less is understood about the quantumlimit of the Heisenberg model. There is theoretical evidence that pinch point correlationssurvive, but the specific character of the ground state and of the magnetic excitationsare unknown.The experimental challenge lies in realizing the pyrochlore Heisenberg model in a realmaterial. The highly degenerate manifold of the Coulomb phase is susceptible to smallperturbations and lattice instabilities such that at low temperatures the spin liquid phaseis more often than not supplanted by a broken symmetry phase. So far, the closest re-alizations of a Heisenberg antiferromagnet on a pyrochlore lattice have been found in thecubic-spinels. Many of these materials exhibit significant exchange interactions extendingto the second and third nearest neighbours. Magnetic frustration is manifest through self-organized independent hexagonal clusters, but a magneto-structural transition severelyimpacts almost half of the magnetic bandwidth.Extrinsic disorder, in the form of impurity ions, or variations in magnetic exchange in-teractions caused by chemical disorder may also disrupt the spin liquid. Generally, theseperturbations result in a spin freezing transition at low temperatures.
For example, inthe Heisenberg pyrochlore Y Mo O weak disorder results in a fully frozen, disordered state,with isotropic short range spin correlations. Here, we demonstrate that disorder is notnecessarily fatal to the search for quantum spin liquids and can act to only freeze the lowestenergy magnetic degrees of freedom. At higher energies a magnetic excitation continuum3haracteristic of fractionalized excitations persists.NaCaNi F is one member of a family of recently discovered transition metal pyrochloreflourides where charge balance in the neutral chemical structure requires an equal mixture ofNa and Ca . Diffraction measurements probing the average crystal structure indicatethat Na and Ca are uniformly and randomly distributed on the A-site of the pyrochlorelattice. Magnetic susceptibility measurements reveal Curie-Weiss behaviour, with an effec-tive moment of p eff = 3 . µ B , consistent with S = 1, and a Curie-Weiss temperature of θ CW = 129(1) K. A spin-glass like freezing transition is observed at T f = 3 . This freezing may result from the chargedisorder that can be expected to generate a random variation in the magnetic exchangeinteractions. For the Heisenberg pyrochlore antiferromagnet described by the Hamiltonian H = (cid:80) ij J ij S i · S j the freezing temperature provides an estimate of the strength of bonddisorder δJ = (cid:112) / k B T f = 0 .
19 meV, for S = 1. Notwithstanding the glassy features of NaCaNi F , we will provide evidence that a quan-tum spin liquid (QSL) remains a very realistic possibility. First, the co-existence of a lowenergy frozen component and the intrinsic excitations of a QSL at higher energies is man-ifest in the magnetic specific heat C m ( T ). Second, we use theoretical tools including theself consistent Gaussian approximation and classical Monte Carlo to perform extensive fitsto our neutron scattering data and determine the relevant Hamiltonian. We find that it ispredominantly characterized by a Heisenberg model with small additional exchange terms.Third, the presence of a continuum of magnetic excitations coupled with the unusually largeinelastic spectral weight suggest that this S = 1 magnet is in the strongly quantum regime.Finally, in the absence of a definitive theoretical understanding of the quantum version of thepyrochlore Heisenberg antiferromagnet, we explore several scenarios that may be consistentwith our experimental findings.Fig. 1a shows the magnetic specific heat C m ( T ). Beginning with the high temperatureregime for T >
18 K, C m ( T ) very closely follows the form expected for the classical spinliquid – Villain’s cooperative paramagnet – phase of the Heisenberg antiferromagnet on apyrochlore lattice. Indeed, our classical Monte-Carlo simulation of the Heisenberg model,using exchange parameters extracted from analysis of inelastic neutron scattering measure-ments to be discussed below, aligns very closely with the data. In the second regime, where T is of the order of the Heisenberg coupling, C m ( T ) falls below the classical model and the4 onte Carlo Monte CarloData Data HeisenbergNaCaNi F Figure 1: Spin freezing in NaCaNi F . a , Magnetic specific heat. Dashed line is aclassical Monte-Carlo simulation. Solid line is a fit to C m ( T ) = AT α , with A = 0 . α = 2 . b , Magnetic entropy obtainedby integration of C/T between T = 150 K and 100 mK corresponding to 84% of R ln(3). c , Diffuse elastic ( E = 0) magnetic scattering, integrated over the resolution window of ± .
37 meV and obtained by subtracting T = 40 K data from that at 1.6 K. Lower quadrantsdisplay disorder and configuration averaged ground state Monte-Carlo structure factors. d ,Temperature dependent intensity of the diffuse elastic scattering around q = (0 , , − T /T f ) β , with T f = 8 . β = 0 .
5. Inset shows the T = 1 . − . < ( h, h, < .
1, the horizontal dash denotesthe instrumental resolution. Error bars in all figures represent one standard deviation. e ,Histogram of bond vector order parameter components ( f , f ) from classical Monte-Carlosimulations for Heisenberg and exchange model relevant to NaCaNi F including exchangedisorder. Extremal spin configurations corresponding to collinear spin arrangements areshown. 5road maximum at 18 K signals the onset of a collective quantum state. Finally, a thirddistinct regime is identified below T f = 3 . C m ( T ) occurs. The approximately quadratic power law T dependence below this anomalyis interpreted as a consequence of static, or frozen, magnetism below T f . C m ∝ T for T < T f is characteristic of dense frustrated magnets where some disorder ispresent; the exponent appears to be independent of the dimensionality of the interactingsystem. This quadratic temperature dependence generally indicates gapless, linearlydispersing modes in two dimensions or along nodal lines in momentum space. While thelack of translational symmetry implies these do not manifest as coherent modes in neutronscattering measurements, the corresponding density of states should be reflected there, albeitbelow the range of energies that we have accessed spectroscopically. The low temperaturespecific heat exponent of α = 2 . although the presence of linenodes in the dispersion relation is non-trivial.In Fig. 1b we show the magnetic entropy recovered between 100 mK and 150 K whichsaturates at 84% of the available R ln(3) for S = 1. We interpret the 0.176 R /spin residualentropy at 100 mK as indicating broken ergodicity. Specifically, we propose that below T f ,a metastable spin configuration within the Coulomb phase manifold is kinetically arrestedby the disorder potential so the material no longer explores all states of a given energy.However, most of the magnetic entropy is associated with higher energy states. Thus, thereis an energy scale above k B T f where excitations are unaffected by exchange disorder andreflect the site averaged spin Hamiltonian of NaCaNi F . This notion is indeed verifiedthrough momentum and energy resolved neutron scattering measurements, which enableus to explicitly separate these two components of the spin correlation function. We firstinvestigate the frozen component at low energies and then the high energy continuum ofexcitations.Figure 1c shows the elastic neutron intensity in two high-symmetry reciprocal latticeplanes of the cubic lattice. The elastic magnetic signal is dominated by extended diffuseintensity arising from short range correlated spin configurations that are static within the10 ps time window of our measurement. Neutron intensity is concentrated in lobes centeredon (2 n ± .
6, 2 n ± .
6, 0) positions, where n is an integer. Near (002) and (220), where sharp6inch point features representing long-range correlations of the pure Heisenberg model areexpected, the momentum distribution of the scattering is broader than the experimentalresolution. The inverse momentum width corresponds to a real-space correlation length of ξ = 6 ˚A, or just two nearest neighbour lattice spacings.Figure 1d shows the onset of elastic scattering upon cooling below 8 K. This tempera-ture is significantly higher than the 3.6 K T f extracted from susceptibility measurements. Inelastic neutron scattering probes the imaginary part of the magnetic susceptibility in theTHz frequency range, a timescale orders of magnitude faster than AC susceptibility, andthe upward shift in apparent freezing temperature with the characteristic measurement fre-quency indicates a glass-like transition. We find the momentum width of the elastic signalis independent of temperature indicating that spatial correlations are unaffected by thefreezing transition. The observation of a time-scale dependent T f , temperature independentspatial correlations, and residual entropy are consistent with kinetically arrested magnetismin NaCaNi F . Below T f low energy spin configurations become trapped by the disorderpotential, resulting in an out-of-equilibrium frozen configuration that is a snap-shot of thenear degenerate manifold of states. Integrating the elastic ( E = 0) intensity over momen-tum we find that the frozen moment accounts for only |(cid:104) S (cid:105)| /S = 44% of the saturationmagnetization. Thus, magnetism in NaCaNi F at T = 1 . but in three-dimensional magnets is unique to NaCaNi F .To better understand the nature of the frozen low temperature state, we have carried outclassical Monte-Carlo simulations of the Heisenberg Hamiltonian relevant to NaCaNi F .Random bond disorder was included by sampling from a box distribution, with a half widthof δJ = 0 .
19 meV and exchange parameters extracted from an independent analysis ofinelastic neutron scattering data. In figure 1c we compare the measured elastic scatteringwith the corresponding numerically modeled signal. The high fidelity fit gives confidencein our optimized magnetic Hamiltonian. To gain additional insight we complement theseresults with a study of local metrics for individual tetrahedra on the pyrochlore lattice.In the absence of disorder, the energy of the classical Heisenberg Hamiltonian is min-imized by all states with zero total spin per tetrahedra, S tot = (cid:80) i =1 S i = 0. We find thelowest energy states for the bond-disordered Heisenberg Hamiltonian with small anisotropicexchanges relevant to NaCaNi F also fall within the S tot = 0 manifold [see supple-7ental information]. This manifold is parameterized by the order parameters f =[( S + S ) · ( S + S ) − S · S − S · S ] / √
12 and f = ( S · S + S · S − S · S − S · S ) / The statistical distribution of f and f over a Monte Carlo ensemble of tetrahedra providesa local characterization of the particular S tot = 0 spin configuration. Such histograms of( f , f ) extracted from our Monte-Carlo simulations are shown in figure 1e where possiblevalues span an equilateral triangle in the ( f , f ) plane. Tetrahedra with pairs of antiparallelspins lie along the triangular edges while collinear spin configurations are at the vertices.The classical Heisenberg (only) model with weak bond disorder is glassy with a tendencyto form locally collinear states; this is confirmed by the results in the top half of Fig. 1e. The enhanced density along the boundaries, and away from the corners, of the lower partof the triangle in Fig.1e indicates the tendency to form configurations of pairwise collinearspins when additional small anisotropic interactions specific to NaCaNi F are added.In Fig. 2 we present the momentum and energy dependence of inelastic magnetic scat-tering for NaCaNi F . In contrast to the distinct maxima in the elastic scattering (Fig. 1),the dynamic structure factor forms a bow tie pattern with pinch points characteristic ofdipolar spin correlations. The scattering closely resembles expectations for the Heisenbergantiferromagnet on the pyrochlore lattice but with important deviations, including aslight momentum broadening and reduction of intensity around the pinch points. Magneticscattering evolves into a continuum with a well-defined momentum structure at higher en-ergies. The highest energy magnetic excitations are spread everywhere in momentum spaceexcept at the Γ point where neutron intensity is precluded for a Heisenberg model.Fig. 3a shows the equal time structure factor S ( q ) obtained from the energy integratedmagnetic neutron scattering intensity. More detailed information is provided by polarizedneutron scattering in the ( h, h, (cid:96) ) plane which is sensitive to spin components within the( h, h, (cid:96) ) reciprocal lattice plane for the non-spin-flip (NSF) channel, and along (1 , − ,
0) forthe spin-flip channel (SF). The similarity of SF and NSF magnetic neutron intensities infigure 3a is evidence of a near spin-space isotropic manifold and immediately rules out singleion-anisotropy terms. Weakly anisotropic interactions are revealed by two features of thepolarized intensity. First, the SF scattering exhibits a pronounced asymmetry of the lobesof intensity centered on ( ± . , ± . ,
2) positions about the dashed line parallel to (1 , , , ,
2) indicated in figure 3b. Second, the NSF intensity is diminishedaround the (0 , ,
2) pinch point positions. 8
40 80 ( , , ) ( , , ) -2 0 2( h , h , 0)4202 ( , , )
11 meV ×4 ×2 -4 -2 0 2( h , 0, 0) ×4 k i k f d d dE (barn/eV sr Ni) Figure 2: Separation of magnetic energy scales in NaCaNi F . Momentum andenergy dependence of inelastic magnetic scattering in NaCaNi F for the ( h, h, (cid:96) ) and ( h, k, T = 1 . ± , , , ,
0) positions indicate that the net magnetisation per tetrahedron vanish in theCoulomb phase. Above energies of 5 meV the scattering forms a broad continuum with nointensity around the Γ points.We have analyzed the energy integrated neutron spectra using a self-consistent Gaussianapproximation (SCGA) for the equal time structure factor using the full symmetry allowednearest-neighbour Hamiltonian H = 1 / (cid:80) ij J µνij S µi S νj , where the 3 × J µν is parameterized by four independent terms: J , J , J , and J , in additional to nextnearest neighbour Heisenberg exchange J NNN . A symmetry allowed biquadratic exchange9erm was not included in our analysis. We find the best global fit of the measured equaltime factor with the SCGA using the exchange parameters: J = J = 3 . J =0 . J = − . J NNN = − . CW = −
150 K which may be compared with the experimentallydetermined value of Θ CW = − Details of the fitting procedure are containedin the supplementary information and the resulting modeled neutron intensity is shown infigure 3b. Although the SCGA is an approximate procedure, we find exceptional agreementbetween the model and data. Furthermore, these exchange parameters were directly inputinto the classical Monte-Carlo simulations which builds further confidence in the SCGA.Thus, the spin Hamiltonian for NaCaNi F very closely approximates the S = 1 Heisenbergantiferromagnet on the pyrochlore lattice, perturbed by small symmetric and antisymmetricexchange anisotropies as well as next nearest neighbour interactions.A number of theoretical investigations have shown that small perturbations in the classicalHeisenberg model, J (cid:48) , in the form of exchange anisotropies or further neighbour interactions,result in a magnetically ordered phase below temperatures of the order J (cid:48) S . InNaCaNi F these perturbations are significantly smaller than the freezing temperature, suchthat any lower temperature transition is pre-empted by spin freezing and inaccessible toexperiment. Indeed, our classical Monte-Carlo simulations for the anisotropic Hamiltonianrelevant to NaCaNi F , but in the absence of exchange disorder, do not indicate long-rangeordering above T = 500 mK. This temperature is well below the broad maximum in specificheat where quantum mechanical fluctuations become important and our classical simulationsare no longer strictly valid.In figure 4 we present the momentum and energy resolved spin-flip neutron scatteringcross-section. For our experiment this cross-section is sensitive to magnetic scattering andnuclear incoherent scattering, thus data in Fig. 4 are representative of the dynamic struc-ture factor uncontaminated by coherent non-magnetic scattering. The magnetic excitationsform a continuum that extends over an energy bandwidth of ∼ ∼ J S . Alongthe ( h, h,
2) direction, transverse to the pinch points, the inelastic neutron intensity is rel-atively featureless. However, along the (2,2, (cid:96) ) direction, longitudinal to the pinch points,the magnetic scattering is more structured and, importantly, does not factorize as foundtheoretically for the classical limit of the Heisenberg model on the pyrochlore lattice. Inthe constant momentum and energy transfer cuts plotted in Fig. 4 b and c very broad dis-10 igure 3: Equal time structure factor in NaCaNi F . a , Measured neutron cross-section integrated over the range 0 < E <
14 meV. Polarized neutron measurements arelabelled by SF, which measures components of the dynamic spin correlation function thatare perpendicular to the ( h, h, (cid:96) ) scattering plane, and NSF, which measures the componentof the dynamics spin correlation function polarized within the ( h, h, (cid:96) ) scattering plane andperpendicular to momentum transfer. b , Energy integrated neutron cross-section calculatedusing the self-consistent Gaussian approximation (SCGA) and exchange parameters J = J = 3 . J = 0 . J = − . J NNN = − . magnetic form factor was used when converting the calculated S ( q ) to a neutroncross-section.persive ridges are observed that are reminiscent of heavily damped spin-waves. While thespectrum is gapless down to the 0.17 meV scale set by our finest energy resolution measure-ments, the dynamic structure factor is peaked at finite energy transfers and can be fit withthe spectral form of an over-damped harmonic oscillator. The characteristic energy scaledisperses from E q = 4 . ∼ J at the pinch point q = (2 , , E q = 7 . q = (2 , , F from recent theoreticaltreatments of the semi-classical Heisenberg model which find a purely diffusive response at11 E n e r g y ( m e V ) hh <2.2 a h , h , 2) (r.l.u.) b a r n / e V N i b (2,2,0)(2,2,1) b a r n / s r N i c k i k f d dd E ( b a r n / e V s r N i ) Figure 4: Magnetic excitations in NaCaNi F . a , Energy-momentum cuts through thespin-flip portion of the polarized neutron scattering cross-section at T = 1 . b , Constantmomentum cuts of the spin-flip cross-section through a pinch point at q = (2 , ,
0) and nodalpoint at (2,2,1) integrated over (cid:96) ± .
2. Solid lines are a fit to the sum of a Lorentzian functioncentered on the elastic line and a damped oscillator form S ( E ) = ( n +1)2Γ E ( E − E q ) +(2Γ E ) where n isthe thermal population factor, Γ a relaxation rate, and E q the characteristic energy scale. c ,Constant energy transfer cuts, integrated over E ± .
25 meV, showing the energy evolutionof momentum dependent scattering which bifurcates above 5 meV and precludes a simplefactorization of the dynamic structure factor as S ( q , ω ) = S ( q ) f ( ω ).the pinch points . The absence of inelastic scattering at the Γ point and our polarizedneutron measurements rule out any sizable single-ion anisotropies that could explain thepeak in spectral weight at non-zero energy transfers. The only energy scale large enough toaccount for the resonance is the exchange interaction J .Disorder in NaCaNi F is small such that its effect is only to rearrange the low energypart of the spectrum for E < k B T f and the underlying translational invariance of the Heisen-berg spin Hamiltonian can be expected to prevail. Indeed, we find that NaCaNi F formsa Coulomb-like phase, with S tot ≈ (cid:126) .A conservative interpretation is that the dispersing modes are overdamped spin wavesof an underlying classical magnetic order, disrupted in NaCaNi F by exchange disorder.Since the frozen spin configurations feature non-collinear interacting spins, single particle S = 1 magnon excitations can decay from interactions with multimagnon states to form acontinuum of scattering. Such a scenario may be appropriate for the related pyrochloreXY antiferromagnet NaCaCo F . Elastic magnetic neutron scattering from NaCaCo F resembles that of an ordered antiferromagnet, consistent with the non-collinear magneticstructure favoured by an order-by-disorder mechanism. This order develops at a tempera-ture coincident with a broad peak in the magnetic specific heat, which constitutes the totalmagnetic entropy of the J = 1 / almost exactly as expected for anordered S = 1 / S /S ( S + 1) = 1 / F exchangedisorder truncates the magnetic correlations of the classical antiferromagnetic order favouredby the underling Hamiltonian. This is distinct from the Heisenberg Hamiltonian we infer forNaCaNi F . It does not favour a magnetically ordered state, consistent with the magneticspecific heat and the strong inelastic magnetic neutron scattering.In contrast to its Co counterpart, NaCaNi F does not show a full recovery of the fractionof elastic magnetic neutron intensity; rather, the significant proportion of inelastic spectralweight suggests a quantum fluctuating state. For a QSL the magnetic spectral weightat T = 0 must be entirely accounted for by the excitations and there can be no trulyelastic scattering. For a semi-classical state the elastic scattering should carry a fraction of S /S ( S + 1) of the spectral weight, which for S = 1 equals 1 /
2. By integrating the measureddynamic spin correlation function S ( q , E ) over momentum and energy, including the elasticdiffuse magnetic scattering, we recover the total spectral weight of 3 (cid:82) S ( q , E ) dEd q =13(1), which is consistent with the (3 . = 13 . µ B effective moment extracted from themagnetic susceptibility. Comparing the spectral weight for elastic,
E < . . < E <
14 meV, we find ∼
90% of the magnetic scattering is inelasticin the low T limit. This significantly exceeds the 50% mark for a semi-classical ground stateand is direct evidence of a spin system dominated by quantum fluctuations.13ur experimental and theoretical results admit the possibility that NaCaNi F is a QSLdriven to freezing by weak exchange disorder. At energies above E = k b T f the continuousspectrum indicates the absence of coherent quasiparticles carrying angular momentum (cid:126) and is consistent with the fractionalization of a spin flip excitation into weakly interactingquasiparticles with angular momentum (cid:126) /
2. The fact that the residual entropy as a fractionof the total spin entropy (∆
S/R ln(3) = 16(4)%) is within error bars of the fraction ofthe total spectral weight contained in elastic scattering ( (cid:104) m elastic (cid:105) /g S ( S + 1)) = 10(2)%)indicates the exchange disorder associated with the mixed Na/Ca site induces a non-ergodiclow energy landscape for these quasiparticles. Such a separation of energy scales betweenfrozen and fluctuating components is observed in other materials that support QSLs. Forexample in the one-dimensional S = 1 / , the spinon continuum is observableat high energies even in the N´eel ordered state. A QSL remains a realistic contender for the ground state of NaCaNi F , but preciouslylittle is known theoretically about the S = 1 Heisenberg model on the pyrochlore lattice.Our finding that, at the classical level, the frozen state involves tetrahedra with quasi-staticanti-parallel pairs of spins at low temperatures points to a quantum scenario where thesepairs correspond to a singlet covering of the pyrochlore lattice. Since there are exponentiallymany such coverings, effects analogous to those studied extensively for dimer models mayplay a role in explaining the specific value of the residual entropy. In addition, the S = 1Heisenberg model admits other rich possibilities. One such picture that might be pursuedinvolves fluctuating Haldane/AKLT loops decorating the pyrochlore lattice. In the AKLTconstruction, each S = 1 degree of freedom is built from two S = 1 / S = 1 subspace. Loops are constructed byjoining neighbouring spins into singlets across each bond. A single spin flip excitation willbreak this bond, fracturing the loop and leaving a chain with two free S = 1 /
2, one at eachend. These end states may then act as bulk fractionalized excitations that are deconfinedwithin the quantum superposition of fluctuating loop coverings. In the absence of anysignificant lattice distortion or cluster formation, such liquid like states remain a realisticpossibility on the pyrochlore lattice and could help to understand our observation of residualentropy and continuum scattering in NaCaNi F .14 ETHODS
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Specific Heat . Heat capacity measurements were conducted using a Quantum DesignPPMS with a dilution insert for temperatures between 100 mK and 4 K, and standard insertfor temperatures between 2 K and 270 K. All measurements were carried out on the same5 mg single crystal using the adiabatic pulse method. The non-magnetic contribution wasdetermined by scaling the measured specific heat of the iso-structural compound NaCaZn F by the relative Debye temperatures. Neutron Scattering.
All neutron scattering measurements were performed on the same3 g single crystal, grown as described elsewhere. Unpolarized neutron scattering mea-surements were preformed on the MACS spectrometer at the NIST Center for NeutronResearch. The neutron momentum transfer is indexed using the Miller indices of the cubicunit cell, ( h, k, (cid:96) ) = (2 π/a, π/a, π/a ), where a = 10 .
31 ˚A. Measurements were conductedwith the sample oriented in both the ( h, h, (cid:96) ) and ( h, k,
0) scattering planes. Elastic ( E = 0)measurements were conducted with the monochromator in a vertical focusing configurationusing a neutron energy of 5 meV. Two configurations were utilized for inelastic measure-ments, both with the monochromator in double focusing mode. For energy transfers below1.4 meV, MACS was operated with fixed final energy of 3.7 meV and Be BeO filters beforeand after the sample respectively. For energy transfers above 1.4 meV we used a fixed finalenergy of 5 meV with a Be filter after the sample and no incident beam filter. The data forenergy transfers above 1.4 meV was corrected for contamination from high-order harmonicsin the incident beam neutron monitor.Polarized neutron scattering measurements were carried out on the HYSPEC spectrometer at the Spallation neutron Source at Oak Ridge National Lab. An incident neutron energy of17 meV was selected using a Fermi chopper rotating at 240 Hz resulting in an energy resolu-tion of δE = 1 . h, h, l ) scattering plane, alongthe (1 , − ,
0) direction. In this configuration, spin-flip scattering measures the component15f magnetic cross section that is polarized within the scattering plane, while non-spin-flipmeasures the out-of-plane component. The flipping ratio measured on a (4 , ,
0) nuclearBragg peak was 16. All data reduction and analysis was carried out using the Mantidsoftware suite. Measured neutron count rates from both instruments were placed into absolute unitsof the neutron scattering cross-section using incoherent elastic scattering from the sample.The scale factor for conversion to absolute units was additionally cross-checked againstmeasurements from a Vanadium standard.
Numerical Methods.
We fit the static structure factor from the neutron scattering datato the corresponding prediction of the self-consistent Gaussian approximation (SCGA) at1.6 K, to obtain the parameter set in the main text. Details of the method, includingthe cost function and error analysis are discussed in the supplementary information. Theresults of the SCGA are complemented by classical Monte-Carlo (MC) calculations, forboth the specific heat and the structure factor. MC simulations used single spin updates forcontinuous spin on pyrochlore lattices (with 16 site cubic unit cells) of size N = 16 L for L = 3 to L = 10. For determining the classical ground state of the fitted spin Hamiltonian,parallel tempering MC was carried out with T min = 0 .
01 K and T max = 1 K with thenumber of replicas N r = √ N ln (cid:16) T max /T min (cid:17) (approximately 100 for L = 3 and 400 for L = 8, the two sizes studied extensively, see supplementary for more analyses) and thesimulation carried out for 10 total steps. With the lowest energy configurations encounteredin this finite Monte-Carlo run, further iterative minimization was performed to acceleratethe approach to the classical ground state. For these optimized spin configurations (many ofwhich are local minima) two-component local order parameters ( f and f ) are calculated onall N/ L = 8, was averaged to obtain an estimate of the elasticcross-section. Further details of all methods and algorithms employed are discussed in thesupplementary information. 16 CKNOWLEDGMENTS
We are grateful to Yuan Wan for enlightening discussions. This work benefited frommany insightfull comments from Oleg Tchernyshyov. We would also like to thank RoderichMoessner, John Chalker, and Senthil Todadri for critical reading of this manuscript. Work atthe Institute for Quantum Matter was supported by the U.S. Department of Energy, Officeof Basic Energy Sciences, Division of Material Sciences and Engineering under grant DE-FG02-08ER46544. Access to MACS was provided by the Center for High Resolution NeutronScattering, a partnership between the National Institute of Standards and Technology andthe National Science Foundation under Agreement No. DMR-1508249. A portion of thisresearch used resources at the Spallation/ Neutron Source, a DOE Office of Science UserFacility operated by the Oak Ridge National Laboratory. This work was supported by thePaul Scherrer Institut by providing the supermirror analyzer as a temporary loan to OakRidge National Laboratory. We gratefully acknowledge the Johns Hopkins Homewood HighPerformance Cluster (HHPC) and the Maryland Advanced Research Computing Center(MARCC), funded by the State of Maryland, for computing resources.
AUTHOR CONTRIBUTIONS
K. W. P., A. S., B. W., J. A. R., and Y. Q. performed the neutron scattering experiments.K. W. P. performed the specific heat measurements and analyzed the experimental data.J. W. K. and R. J. C. synthesized and characterized the single crystal sample. H. J. Cand S. Z. performed Monte-Carlo simulations and self consistent Gaussian approximationcalculations, along with assisting with the theoretical interpretation. K. W. P. wrote themanuscript with input from all co-authors. C. L. B. oversaw all aspects of the project. Jacques Villain, “Insulating spin glasses,” Zeitschrift f¨ur Physik B Condensed Matter , 31–42(1979). A. B. Harris, A. J. Berlinsky, and C. Bruder, “Ordering by quantum fluctuations in a stronglyfrustrated Heisenberg antiferromagnet,” Journal of Applied Physics , 5200–5202 (1991). B. Canals and C. Lacroix, “Pyrochlore antiferromagnet: A three-dimensional quantum spinliquid,” Phys. Rev. Lett. , 2933–2936 (1998). Benjamin Canals and Claudine Lacroix, “Quantum spin liquid: The Heisenberg antiferromagneton the three-dimensional pyrochlore lattice,” Phys. Rev. B , 1149–1159 (2000). R. Moessner and J. T. Chalker, “Properties of a classical spin liquid: The Heisenberg pyrochloreantiferromagnet,” Phys. Rev. Lett. , 2929–2932 (1998). R. Moessner and J. T. Chalker, “Low-temperature properties of classical geometrically frus-trated antiferromagnets,” Phys. Rev. B , 12049–12062 (1998). S. V. Isakov, K. Gregor, R. Moessner, and S. L. Sondhi, “Dipolar spin correlations in classicalpyrochlore magnets,” Phys. Rev. Lett. , 167204 (2004). C. L. Henley, “Power-law spin correlations in pyrochlore antiferromagnets,” Phys. Rev. B ,014424 (2005). Christopher L. Henley, “The Coulomb phase in frustrated systems,” Annual Review of Con-densed Matter Physics , 179–210 (2010). T. Fennell, P. P. Deen, A. R. Wildes, K. Schmalzl, D. Prabhakaran, A. T. Boothroyd, R. J.Aldus, D. F. McMorrow, and S. T. Bramwell, “Magnetic Coulomb phase in the spin iceHo Ti O ,” Science , 415–417 (2009). Makoto Isoda and Shigeyoshi Mori, “Valence-bond crystal and anisotropic excitation spectrumon 3-dimensionally frustrated pyrochlore,” J. Phys. Soc. Japan , 4022–4025 (1998). R. Moessner, S. L. Sondhi, and M. O. Goerbig, “Quantum dimer models and effective hamil-tonians on the pyrochlore lattice,” Phys. Rev. B , 094430 (2006). Yuan Huang, Kun Chen, Youjin Deng, Nikolay Prokof’ev, and Boris Svistunov, “Spin-ice stateof the quantum Heisenberg antiferromagnet on the pyrochlore lattice,” Phys. Rev. Lett. ,177203 (2016). Oleg Tchernyshyov, R. Moessner, and S. L. Sondhi, “Order by distortion and string modes inpyrochlore antiferromagnets,” Phys. Rev. Lett. , 067203 (2002). P. H. Conlon and J. T. Chalker, “Absent pinch points and emergent clusters: Further neighborinteractions in the pyrochlore Heisenberg antiferromagnet,” Phys. Rev. B , 224413 (2010). S.-H Lee, C. Broholm, W. Ratcliff, G. Gasparociv, Q. Huang, T. H. Kim, and S. W. Cheong,“Emergent excitations in a geometrically frustrated magnet,” Nat. , 856–858 (2002). K. Kamazawa, S. Park, S.-H. Lee, T. J. Sato, and Y. Tsunoda, “Dissociation of spin objectsin geometrically frustrated CdFe O ,” Phys. Rev. B , 024418 (2004). J.-H. Chung, M. Matsuda, S.-H. Lee, K. Kakurai, H. Ueda, T. J. Sato, H. Takagi, K.-P. Hong,and S. Park, “Statics and dynamics of incommensurate spin order in a geometrically frustratedantiferromagnet CdCr O ,” Phys. Rev. Lett. , 247204 (2005). K. Tomiyasu, H. Suzuki, M. Toki, S. Itoh, M. Matsuura, N. Aso, and K. Yamada, “Molecularspin resonance in the geometrically frustrated magnet MgCr O by inelastic neutron scattering,”Phys. Rev. Lett. , 177401 (2008). L Bellier-Castella, M JP Gingras, P CW Holdsworth, and R Moessner, “Frustrated order bydisorder: The pyrochlore anti-ferromagnet with bond disorder,” Canadian Journal of Physics , 1365–1371 (2001). T. E. Saunders and J. T. Chalker, “Spin freezing in geometrically frustrated antiferromagnetswith weak disorder,” Phys. Rev. Lett. , 157201 (2007). Arnab Sen and R. Moessner, “Topological spin glass in diluted spin ice,” Phys. Rev. Lett. ,247207 (2015). J. S. Gardner, B. D. Gaulin, S.-H. Lee, C. Broholm, N. P. Raju, and J. E. Greedan, “Glassystatics and dynamics in the chemically ordered pyrochlore antiferromagnet Y Mo O ,” Phys.Rev. Lett. , 211–214 (1999). H. J. Silverstein, K. Fritsch, F. Flicker, A. M. Hallas, J. S. Gardner, Y. Qiu, G. Ehlers, A. T.Savici, Z. Yamani, K. A. Ross, B. D. Gaulin, M. J. P. Gingras, J. A. M. Paddison, K. Foyevtsova,R. Valenti, F. Hawthorne, C. R. Wiebe, and H. D. Zhou, “Liquidlike correlations in single-crystalline Y Mo O : An unconventional spin glass,” Phys. Rev. B , 054433 (2014). J. W. Krizan and R. J. Cava, “NaCaCo F : A single-crystal high-temperature pyrochlore an-tiferromagnet,” Phys. Rev. B , 214401 (2014). J. W. Krizan and R. J. Cava, “NaCaNi F : A frustrated high-temperature pyrochlore antifer-romagnet with S=1 Ni ,” Phys. Rev. B , 014406 (2015). M. B. Sanders, J. W. Krizan, K. W. Plumb, T. M. McQueen, and Cava R. J., “NaSrMn F ,NaCaFe F , and NaSrFe F : novel single crystal pyrochlore antiferromagnets,” J. of Phys.Cond. Matter , 045801 (2017). A. P. Ramirez, B. Hessen, and M. Winklemann, “Entropy balance and evidence for local spinsinglets in a kagom´e-like magnet,” Phys. Rev. Lett. , 2957–2960 (2000). Satoru Nakatsuji, Yusuke Nambu, Hiroshi Tonomura, Osamu Sakai, Seth Jonas, Collin Bro-holm, Hirokazu Tsunetsugu, Yiming Qiu, and Yoshiteru Maeno, “Spin disorder on a triangularlattice,” Science , 1697–1700 (2005). B. I. Halperin and W. M. Saslow, “Hydrodynamic theory of spin waves in spin glasses and othersystems with noncollinear spin orientations,” Phys. Rev. B , 2154–2162 (1977). Daniel Podolsky and Yong Baek Kim, “Halperin-saslow modes as the origin of the low-temperature anomaly in NiGa S ,” Phys. Rev. B , 140402 (2009). S.-H. Lee, C. Broholm, M. F. Collins, L. Heller, A. P. Ramirez, Ch. Kloc, E. Bucher, R. W.Erwin, and N. Lacevic, “Less than 50% sublattice polarization in an insulating s = kagom´eantiferromagnet at T ≈ , 8091–8097 (1997). Gia-Wei Chern, R. Moessner, and O. Tchernyshyov, “Partial order from disorder in a classicalpyrochlore antiferromagnet,” Phys. Rev. B , 144418 (2008). Kate A. Ross, Lucile Savary, Bruce D. Gaulin, and Leon Balents, “Quantum excitations inquantum spin ice,” Phys. Rev. X , 021002 (2011). P. J. Brown,
International Tables for Crystallography , Vol. C (Springer, Berlin, 2006) Chap.4.4.5, pp. 454–461. J. N. Reimers, A. J. Berlinsky, and A.-C. Shi, “Mean-field approach to magnetic ordering inhighly frustrated pyrochlores,” Phys. Rev. B , 865–878 (1991). Maged Elhajal, Benjamin Canals, Raimon Sunyer, and Claudine Lacroix, “Ordering in thepyrochlore antiferromagnet due to Dzyaloshinsky-Moriya interactions,” Phys. Rev. B , 094420(2005). P. H. Conlon and J. T. Chalker, “Spin dynamics in pyrochlore Heisenberg antiferromagnets,”Phys. Rev. Lett. , 237206 (2009). M. E. Zhitomirsky and A. L. Chernyshev, “Colloquium: Spontaneous magnon decays,” Rev.Mod. Phys. , 219–242 (2013). K. A. Ross, J. W. Krizan, J. A. Rodriguez-Rivera, R. J. Cava, and C. L. Broholm, “Static anddynamic xy -like short-range order in a frustrated magnet with exchange disorder,” Phys. Rev.B , 014433 (2016). Bella Lake, D. Alan Tennant, Chris D. Frost, and Stephen E. Nagler, “Quantum criticality anduniversal scaling of a quantum antiferromagnet,” Nat Mater , 329–334 (2005). Chong Wang, Adam Nahum, and T. Senthil, “Topological paramagnetism in frustrated spin-1mott insulators,” Phys. Rev. B , 195131 (2015). Lucile Savary, “Quantum loop states in spin-orbital models on the honeycomb lattice,” ArXive-prints (2015), arXiv:1511.01505 [cond-mat.str-el]. J. A. Rodriguez, D. M. Adler, P. C. Brand, C. Broholm, J. C. Cook, C. Brocker, R. Hammond,Z. Huang, P. Hundertmark, J. W. Lynn, N. C. Maliszewskyj, J. Moyer, J. Orndorff, D. Pierce,T. D. Pike, G. Scharfstein, S. A. Smee, and R. Vilaseca, “MACS a new high intensity coldneutron spectrometer at NIST,” Measurement Science and Technology , 034023 (2008). Igor A Zaliznyak, Andrei T. Savici, V. Ovidiu Garlea, Barry Winn, Uwe Filges, John Schneeloch,John M. Tranquada, Genda Gu, Aifeng Wang, and Cedomir Petrovic, “Polarized neutronscattering on HYSPEC: the hybrid spectrometer at SNS,” Journal of Physics: Conference Series , 012030 (2017). O. Arnold, J.C. Bilheux, J.M. Borreguero, A. Buts, S.I. Campbell, L. Chapon, M. Doucet,N. Draper, R. Ferraz Leal, M.A. Gigg, V.E. Lynch, A. Markvardsen, D.J. Mikkelson, R.L.Mikkelson, R. Miller, K. Palmen, P. Parker, G. Passos, T.G. Perring, P.F. Peterson, S. Ren,M.A. Reuter, A.T. Savici, J.W. Taylor, R.J. Taylor, R. Tolchenov, W. Zhou, and J. Zikovsky,“Mantid data analysis and visualization package for neutron scattering and µ SR experiments,”Nucl. Instr. Meth. Phys. Res. Section A: Accelerators, Spectrometers, Detectors and AssociatedEquipment , 156 – 166 (2014). Koji Hukushima and Koji Nemoto, “Exchange monte carlo method and application to spinglass simulations,” Journal of the Physical Society of Japan , 1604–1608 (1996)., 1604–1608 (1996).