Beyond magnons in Nd2ScNbO7: An Ising pyrochlore antiferromagnet with all in all out order and random fields
A. Scheie, M. Sanders, Yiming Qiu, T.R. Prisk, R.J. Cava, C. Broholm
BBeyond magnons in Nd ScNbO : An Ising pyrochlore antiferromagnet with all in allout order and random fields A. Scheie,
1, 2
M. Sanders, Yiming Qiu, T.R. Prisk, R.J. Cava, and C. Broholm
2, 4, 5 Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Institute for Quantum Matter and Department of Physics and Astronomy,Johns Hopkins University, Baltimore, MD 21218 Department of Chemistry, Princeton University, Princeton, NJ 08544 NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899 Department of Materials Science and Engineering,Johns Hopkins University, Baltimore, MD 21218 (Dated: March 1, 2021)We report the low temperature magnetic properties of Nd pyrochlore Nd ScNbO . Suscepti-bility and magnetization show an easy-axis moment, and heat capacity reveals a phase transitionto long range order at T N = 371(2) mK with a fully recovered ∆ S = R ln(2) , 53% of it recoveredfor T > T N . Elastic neutron scattering shows a long-range all-in-all-out magnetic order with low- Q diffuse elastic scattering. Inelastic neutron scattering shows a low-energy flat-band, indicating amagnetic Hamiltonian similar to Nd Zr O . Nuclear hyperfine excitations measured by ultra-high-resolution neutron backscattering indicates a distribution of static electronic moments below T N ,which may be due to B-site disorder influencing Nd crystal electric fields. Analysis of heat capacitydata shows an unexpected T -linear or T / term which is inconsistent with conventional magnonquasiparticles, but is consistent with fractionalized spinons or gapless local spin excitations. We uselegacy data to show similar behavior in Nd Zr O . Comparing local static moments also reveals asuppression of the nuclear Schottky anomaly in temperature, evidencing a fraction of Nd sites withnearly zero static moment, consistent with exchange-disorder-induced random singlet formation.Taken together, these measurements suggest an unusual fluctuating magnetic ground state whichmimics a spin-liquid—but may not actually be one. I. INTRODUCTION
Quantum spin liquids are a long-sought but elusivestate of matter in which magnetic spins form a many-body entangled state [1–4]. The quantum fluctuatingstate supports emergent topological quasiparticles withunusual properties, like magnetic charge, fractionalizedvalues of electron spin, and non-locality [3–6]. Althoughthe quest for a definitive 3D quantum spin liquid has beenunsuccessful so far [3, 4], there are proposals that thefeatures of interest—exotic quasiparticles—appear evenin materials which have long-range magnetic order butmagnetic exchange Hamiltonians close to a quantum spinliquid (called "proximate" or "condensed" quantum spinliquids) [7–10]. Such behavior has been observed in 1Dspin chains [11, 12], and is theorized to exist for higherdimensional materials as well. Thus, even magneticallyordered materials can be relevant to the search for exoticquasiparticles of quantum spin liquids.The pyrochlore lattice, with its magnetically frus-trating geometry of corner-sharing tetrahedra, featuresprominently in proposals for spin-liquids [5, 6, 13–17] andproximate spin liquids [9, 10]. A recent class of com-pounds that has received much attention is pyrochloremagnets based on the Nd ion [18–33]. Nd Zr O , Nd Hf O , and Nd ScNbO show many similar featureswhich defy conventional expectations: a pinch-point inthe Q -dependence of the inelastic magnetic neutron scat-tering cross section [25, 29] and a strongly reduced mag-netic ordered moment [18, 23]. The behavior of these materials has been attributed to "moment fragmenta-tion" [25]: a theoretically proposed magnetic state witha crystallized lattice of magnetic monopoles, forming athree-in-one-out order on a pyrochlore lattice [17, 34–36].However, despite much use of the fragmentation language[20, 31], the three-in-one-out order is absent from thesematerials and thus it is not the ground state but the ex-citations which are considered fragmented [37].In this article we focus on Nd ScNbO , a recently re-ported Nd pryochlore with a disordered Nb and Sc B -site lattice [31, 38]. We use magnetization, susceptibil-ity, heat capacity, and elastic, quasielastic, and inelasticneutron scattering to characterize its ground state mag-netism and show that Nd ScNbO exhibits many of thesame behaviors as the sister-compounds Nd Zr O and Nd Hf O . We report an all-in-all-out magnetic orderwith easy-axis anisotropy and a distribution of static or-dered moments possibly due to disordered crystal-electricfields. Comparing measures of the local ordered momentmagnitude indicates some sites with no static moment,and analysis of low-temperature heat capacity reveals anunexpected density of states. We offer two possible ex-planations of these results: a moment modulated quan-tum fragmented phase with exotic quasiparticles, or lo-cal disorder-induced spin singlets within the long-rangeordered phase, possibly a result of proximity to a spin-liquid phase. We then use these results and analysis ofpreviously published Nd Zr O data to show that thissame behavior appears in other Nd pyrochlores, indi-cating a surprising similarity in multiple members of the a r X i v : . [ c ond - m a t . s t r- e l ] F e b Nd pyrochlore family.
II. EXPERIMENTS AND RESULTSA. Sample Synthesis
Polycrystaline Nd ScNbO was synthesized from a sto-ichiometric mixture of Nd O , Sc O , and Nb O , whichwe mixed and pre-reacted at 1000 °C for 60 hours withrepeated grinding and pelletizing. We confirmed the py-rochlore crystal structure with a Bruker D8 Advance Ecox-ray diffractometer with Cu K α radiation ( λ = 1 . Å) and a Lynxeye detector. Refinements showed the lat-tice parameter a = b = c = 10 . Å at 296 K, andequal mixing of Nb and Sc on the B site of the pyrochlorelattice. Further details are in ref. [38].
B. Susceptibility and Magnetization
We measured the susceptibility and magnetization ofa pressed pellet sample using a Quantum Design Physi-cal Properties Measurement System (PPMS) [39]. Mag-netic susceptibility was measured between T = 1 . Kand T = 300 K at 0.5 T, and magnetization at T = 2 Kwas measured between 0 T and 9 T. The data and fitsare shown in Fig. 1. The temperature dependence ofthe susceptibility shows a bend at T = 50 K typical ofrare-earth ions with low-lying CEF levels. We fit thesusceptibility data to a Curie-Weiss law at high tem-peratures (100 K < T <
300 K) and low temperatures(2 K < T <
10 K). The high-temperature Curie-Weissfit yields µ eff = 3 . µ B , close to the expected freeion value of g J µ B (cid:112) J ( J + 1) = 3 . µ B where J = 9 / and g j = 8 / . . The low-temperature fit (sensitive tothe lowest CEF doublet and thus the ground state mag-netism) yields µ eff = 2 . µ B and Θ cw = − . K,so the net magnetic interactions nearly cancel out but areslightly antiferromagnetic.We fit the magnetization versus applied field for thepressed pellet to a Brillouin function (Fig. 1(b)) whichindicates a saturated moment of 1.323(5) µ B . This valueis almost exactly half of the free-ion saturation moment,which is consistent with powder averaging of an easy-axismagnet (typical for Nd on the pyrochlore lattice [19, 23,40]). The slight deviations from the Brillouin functionlineshape are presumably due to magnetic exchange andhigher crystal field levels. C. Heat Capacity
We measured zero-field heat capacity of Nd ScNbO as a function of temperature using a Quantum DesignPPMS with a dilution refrigerator insert [39]. Heat ca-pacity at each temperature was measured three times ( / ) = . ( )= . ( ) ( / ) = . Figure 1. (a) Inverse susceptibility of powder Nd ScNbO and CW fits, showing high temperature and low temperatureregimes where the ground state crystal field doublet is ther-mally populated. (b) Powder averaged magnetization at 2 K,showing a saturation magnetization of roughly 1.3 µ B . with the semi-adiabatic relaxation method, and then av-eraged. The sample was a cold-pressed 1.3 mg powderpellet with equal masses of Nd ScNbO and silver powderfor thermal connection; we subtracted the heat capacityof silver (measured separately) from the raw data. Thedata are shown in Fig. 2, and reveal a peak at 370(10)mK indicative of a magnetic phase transition.The heat capacity below 100 mK shows an upturn thatwe associate with a nuclear Schottky anomaly (where theNd nuclear moments become polarized along the localfield-direction of the static electronic magnetism). Wefit this Schottky anomaly using PyNuclearSchottky [41](which uses the hyperfine coupling constants in ref. [42])to simulate the nuclear heat capacity for Nd and aphenomenological C electronic = aT n + γT for the low-temperature tail of the lambda anomaly. These fits yielda sample averaged ordered moment of (cid:104) µ (cid:105) = 1 . µ B and electronic exponent n = 2 . . (Possible rea-sons for the non-integer exponent are discussed in sec-tion III.) As Fig. 2(a) shows, the ordered moment iswell-constrained by the high-temperature Schottky tail,even though the entire peak was not measured.We calculated the entropy recovered across the phasetransition ∆ S = (cid:82) CT dT after subtracting the nuclearSchottky anomaly. As Fig. 2(b) shows, the electronicmagnetic entropy across the entire temperature rangeprobed converges to R ln 2 , the entropy of a Kramers dou- ( J / K m o l ) data(a) ..... ( J / K m o l ) ( ) (b) Figure 2. (a) Low temperature heat capacity of Nd ScNbO ,showing a phase transition at 370 mK. The colored lines showcalculated nuclear Schottky anomalies for different orderedelectronic Nd moment sizes. (b) Entropy calculated from heatcapacity after subtracting the nuclear Schottky term. blet. D. Neutron Scattering
We performed two neutron experiments on Nd ScNbO : one experiment using the MACS triple axisspectrometer at the NCNR to measure the diffractionpattern and the spin excitations, and one experimentusing the HFBS backscattering spectrometer at theNCNR to measure the nuclear hyperfine excitations.Both experiments were performed on the same 9.33 gpowder sample sealed under 10 bar He in a copper can,and mounted in a dilution refrigerator.
1. MACS experiment
In the MACS experiment, we measured the elastic scat-tering using a double-focusing configuration and E i = E f = 5 meV neutrons with beryllium filters in the inci-dent and scattered beams at sample temperatures 0.1 K(below the phase transition) and 3 K (where the magneticentropy is completely recovered). These data are shownin Fig 3. There are clear temperature- dependent peakswhich signal long-range magnetic order at low tempera-tures. The 5 meV neutron data have an energy resolutionof 0.35 meV, which is roughly the bandwidth of the ex-citations (see Fig. 4), which makes it functionally an ( c t s / m o n = ) (a) ( ) ( )( ) ( ) Å )0.500.51.0 ( c t s / m o n = ) (b) Data excludedFitDifference ( c t s / m o n = ) MACS (c) ( )
3D Isingcrit. exp. ( c t s / m o n ) HFBS (d)0.36 Å Figure 3. (a) Elastic neutron scattering of Nd ScNbO at0.1 K and 3 K, showing significant temperature dependenceon the (220) and (311) peaks. (b) Rietveld refinement oftemperature subtracted scattering data, showing a best fitall-in-all-out structure. Note also the temperature dependentlow-Q diffuse scattering. (c) Energy-integrated Nd ScNbO scattering as a function of temperature at the (220) Braggpeak. (d) Elastic ( ± . µ eV ) low- Q scattering as a functionof temperature. Both (c) and (d) order parameter curves areconsistent with a 3D Ising critical exponent, shown in black.Error bars indicate one standard deviation. energy-integrated diffraction configuration for this sam-ple.We measured the inelastic spectrum with E f =2 . meV neutrons (still with beryllium filters) for a fullwidth at half maximum (FWHM) energy resolution of0.08 meV, covering energy transfers from ¯ hω = 0 . meVto ¯ hω = 0 . meV. The inelastic signal was measured at T = 0 . K while a paramagnetic background was ac-quired at T = 6 K. These data are shown in Fig. 4.There is an intense flat band excitation at 0.1 meV, verysimilar to the sister-compound Nd Zr O [25] (ref. [31]also observed the Nd ScNbO flat-band excitation andplaced this band at 0.07 meV). This similarity suggeststhat whatever magnetic state exists in Nd Zr O alsoexists in Nd ScNbO .To determine the magnetic ordered structure, weperformed a Rietveld refinement on the temperature-subtracted diffraction data [Fig. 3(b)] using the Full-prof software package [43] and SARAh [44] to gener-ate the irreducible representations. The only strongly- Å )00.10.20.30.4 ( m e V ) I n t e n s i t y ( c t s / m o n = ) Figure 4. Neutron spectrum of Nd ScNbO at 0.1 K with6 K background subtracted, showing an intense flat band at0.1 meV and weaker scattering with a bandwidth of 0.4 meV.Note that the scattering comes down to ¯ hω = 0 at low Q . temperature dependent peaks are (220) and (311) ,but the fit nonetheless uniquely points to all-in-all-out(AIAO) order on the pyrochlore lattice with an or-dered moment 1.121(9) µ B . (There are perhaps smalltemperature-dependent features on the (111) and (200) peaks, but these are barely above the experimental errorbar associated with the T-difference measurement, andno single Irrep was able to account for their intensityalongside the stronger magnetic peaks.) The discoveryof AIAO order is not surprising: nearly all Nd py-rochlores order with this structure: cf. Nd Zr O [25], Nd Hf O [18], Nd Sn O [21], and Nd Ir O [45].One thing that is unique about Nd ScNbO is the exis-tence of temperature-dependent low- Q diffuse scatteringthat appears to be magnetic (see Fig. 3). Subsequentbackscattering measurements showed this diffuse scatter-ing to be real and elastic to within 0.4 µ eV, with an onsetat T N [Fig. 3(d)]. This diffuse scattering carries 9(2)% ofthe total integrated intensity of the (220) and (311) peaks(integrating (cid:82) d σd Ω dω Q dQ to account for the powder aver-age). Despite the low- Q diffuse scattering, the magneticBragg peaks are no wider than the nuclear Bragg peaks(the fitted correlation length from the (220) Bragg peakis > Å), indicating a very long magnetic correlationlength.Finally, we measured the temperature dependence ofthe (220) Bragg peak and found an order parameter curveconsistent with a 3D Ising order [ β = 0 . [46], Fig.3(c)] with a fitted critical temperature T N = 373(2) mK.
2. Backscattering experiment
In the HFBS experiment, we measured the excitationspectrum with a ± µ eV bandwidth and an energy reso-lution FWHM of 0.8 µ eV . At this high resolution, the Ndnuclear hyperfine excitations become visible just off thecentral elastic peak (cf. refs. [21, 47, 48]). These arise from a static electronic moment splitting the I + 1 = 8 degenerate nuclear spin levels via the nuclear hyperfineinteraction. This gives rise to a Q -independent peak inthe inelastic neutron scattering spectrum at an energythat corresponds to ± the level splitting. [49]. When thenuclear hyperfine energy levels are known, the local staticelectronic moment can be calculated from the energies ofthese peaks [21, 47, 48]. We measured the excitationspectrum as a function of temperature from 50 mK to 3K. These data are shown in Fig. 5.As is immediately evident from Fig. 5(a), the observednuclear hyperfine excitations are significantly broaderthan the resolution width (which is defined by a vana-dium scan—see Appendix A 1). We show this broaden-ing to be intrinsic by comparing Fig. 5(a) to the relatedcompound Nd Sb Mg O , shown in Fig. 5(b) (from ref.[47]). Both data sets were taken on the same instru-ment with the same sample environment and the sameexchange gas pressure but Nd ScNbO has much broadernuclear hyperfine excitations: this shows the broadeningis a property of Nd ScNbO . This broadening indicateseither a static distribution of ordered moments or a shortlifetime of the nuclear hyperfine levels that could resultfrom electronic spin fluctuations.The temperature dependence of these features is shownin Fig. 5(c). At the lowest temperatures, two humpsare visible on either side of the central elastic peak. Astemperature increases, the hyperfine peaks shift into thecentral elastic peak and become Lorentzian-like tails. Wefit the HFBS data to a model including both Lorentzianand Gaussian broadening (described in Appendix A 2),where the Gaussian is meant to account for a static mo-ment distribution and the Lorentzian is meant to accountfor dynamic moments. The fits are shown as the blacklines in Fig. 5(c) and the widths as a function of tem-perature are shown in Fig. 5(d). From these fits, weextract the mean ordered moment using the empiricalrelation between Nd nuclear hyperfine energies ∆ E andstatic magnetic moment µ in ref. [48], ∆ E = µ × (1 . ± . µ eV µ B . (1)At the lowest temperatures, the broadening appears tobe mostly Gaussian with an average ordered moment of1.47(6) µ B and a Gaussian spread of 0.8(1) µ B (see Fig.5(d)). Above T = 370 mK the peak standard deviationoverlaps with zero static moments and the broadeningshifts to Lorentzian. This model is very rough, but it isconsistent with static moments for T < T N and quasielas-tic scattering from fluctuating moments for T > T N .We also measured a high resolution elastic ( ± . µ eV )order parameter scan with the fixed-window mode, focus-ing on Q = 0 . Å − (Fig. 3(d)). This showed the low- Q diffuse scattering observed in Fig. 3(b) to be associatedwith the magnetic order: its onset is at T N = 370(2) mKand its temperature dependence is consistent with the3D Ising order parameter ( β = 0 . [46]), showing thelow- Q diffuse scattering to be magnetic in origin. Al- I n t e n s i t y ( a . u . ) fit50mK (a) 5 0 5 ( )1020 fit50mK (b) 10 5 0 5 10 ( )1015202530 I n t e n s i t y ( a . u . ) m o m e n t () (d) avg. static moment0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (K)01 ( m e V ) Gaussian Width Lorentzian Width(e)
Figure 5. (a) Nd ScNbO Nuclear hyperfine neutron scatter-ing on HFBS. (b) Nd Sb Mg O measured with the sameinstrument configuration. Comparison between (a) and (b)shows significant broadening of the nuclear hyperfine peaksin Nd ScNbO , indicating a distribution of local static mo-ments. (c) Hyperfine Nd ScNbO scattering as a function oftemperature, with the fitted model. (d) Average static mo-ment extracted from the model. (c) Gaussian and Lorentzianwidths to the nuclear hyperfine peaks. Below the transitiontemperature, Gaussian distributions dominate, but near andabove the transition temperature Lorentzian broadening dom-inates. Error bars indicate one standard deviation. lowing the critical exponent to vary in these order pa-rameter fits, we find β = 0 . for the MACS data and β = 0 . for the HFBS data—consistent with 3D Ising( β = 0 . ), 3D XY ( β = 0 . ), and 3D Heisenberg( β = 0 . ) magnetic order [46]; but the uncertaintyis too high to pinpoint the anisiotropy of the exchangeinteraction. III. ANALYSIS
These experiments have shown Nd ScNbO to haveeasy-axis anisotropy and AIAO long-range magnetic or-der. These features are similar to many other Nd py-rochlores. Combining the Neel temperatures measuredfrom heat capacity and the two order parameter curves,we find a mean Neel temperature T N = 371(2) mK.These experiments also reveal some unusual features:(A) an unusual temperature dependence in heat capac-ity, (B) a discrepancy between ordered moments obtainedfrom different experimental methods, and (C) a distri-bution in static magnetic moments evidenced by broad-ened hyperfine excitations. Considered together, theyindicate an unconventional magnetic ground state withstrong quantum fluctuations. A. Density of states in heat capacity
The first unusual feature is in the temperature depen-dence of the magnetic heat capacity below T N . Our ini-tial phenomenological fit showed that the electronic heatcapacity does not follow the expected T behavior of lin-ear dispersive modes in three dimensions, but a powerlaw closer to T . . The peculiarity becomes more appar-ent when we plot C/T vs T , as in Fig. 6. This showsa very nearly linear trend, but with a positive offset in C/T (i.e., the heat capacity seems to follow C = γT + αT with γ > ). This is opposite to expectations for gappedBosonic modes: in general, heat capacity from a gappedbosonic spectrum produces a negative offset in C/T [47].Several things can cause non- T heat capacity in a 3Dantiferromagnetic insulator: (i) a spin wave spectrumthat cannot be approximated by gapped linear disper-sive modes (such as in Gd Sn O [50, 51]). (ii) glassyspin disorder which produces a non-integer power law in T [52, 53]. (iii) Exotic excitations which can producepositive non- T terms in heat capacity [10, 54]. We testeach of these scenarios by building an appropriate modeland comparing it with the data. (Details of our fittingmethods are in Appendix D.)We test (i) by building a phenomenological spin-wavemodel. Having observed the low-energy flat mode (Fig.4), we infer the following magnetic Einstein term in thespecific heat: C flat = k B (cid:0) ¯ hω k B T (cid:1) e ¯ hω /k B T ( e ¯ hω /k B T − (2)with ¯ hω = 0 . meV, and we insert it into the followingmodel: C = A × C flat + B × C ld (∆ , c ) + C schottky ( µ ) (3)where C ld (∆ , c ) is the heat capacity of a gapped lineardispersive spectrum of the form η ( q ) = (cid:112) ∆ + ( cq ) (cal-culated as in ref. [47]), and C schottky ( µ ) is a Schottkyanomaly. A , B , ∆ , c , and µ are fitted parameters. Thebest fit is shown in Fig. 6(a) and the best fit parametersare given in Table A2. This fit underestimates heat ca-pacity around 0.1 K and does not match the temperature-dependence at higher temperatures. This indicates a low-energy density of states (DOS) below 0.07 meV not ac-counted for by this model. There are clearly some sig-nificant contributions below the flat band that are notlinear dispersive spin waves.We test (ii) by re-doing the fit, this time allowing thedispersive spin wave contribution to take on an arbitrarypower in temperature: C = A × C flat + B T n + C schottky ( µ ) (4)where A , B , n , and µ are fitted parameters. The result ofthis fit is shown in Fig. 6(b) and the best fit parametersare given in Table A3. The fit is much improved, but themodel still does not match the temperature dependenceabove 0.05 K .We test (iii) by trying two models. First, we add alinear term in the heat capacity to account for fermionicquasiparticles [54, 55]: C = A × C flat + B T + γT + C schottky ( µ ) (5)where A , B , γ , and µ are fitted parameters. The result ofthis fit are shown in Fig. 6(c) and Table A4. The matchis very good: there are virtually no deviations from theexperimental data points, and the reduced χ is improvedover the power-law fit by nearly two orders of magnitude.Second, we consider a T / term to account for con-densed spin-liquid spinons as in ref. [10]: C = A C flat + C ld (∆ , c ) + B T / + C schottky ( µ ) (6)where A , ∆ , c , B , and µ are fitted parameters. The re-sult of this fit are shown in Fig. 6(d) and Table A5. Thematch is again very good. This model included one ad-ditional fitting parameter as compared to the others, butthe reduced χ shows that the match is nearly equivalentto the linear term model.As a side-note, these fits better constrain the orderedmoment from the Schottky anomaly. If we take the threefits which most accurately model the electronic specificheat (the linear term fit and the T / fit) and the initialphenomenological fit, the weighted mean ordered mo-ment is µ = 1 . µ B with a standard deviation of . µ B . Here we take the standard deviation as uncer-tainty reflecting the uncertainty in which model is cor-rect, and so the ordered moment from the nuclear Schot-tky anomaly is µ = 1 . µ B . Comparison to other Nd pyrochlores This exercise in fitting shows that—given reasonableassumptions about collective bosonic excitations—thedensity of states in Nd ScNbO cannot be accounted forby such quasiparticles alone. This conclusion is bolstered by comparison to the heat capacity of Nd Zr O , whichis also quite unusual with an apparent low-temperature T dependence [29, 56]. Fortunately, the Nd Zr O spinwave spectrum has been measured and modeled in de-tail [29], so it is possible to directly calculate the magnonheat capacity in the low T limit (details are in AppendixF). The results are depicted in Fig. 7(b), and show thatthe measured specific heat is much higher than the cal-culated magnon specific heat at low temperatures wherehigher order effects associated with the phase transition(damping and softening of collective modes) can be ne-glected. This indicates an anomalous low-energy DOSthat cannot be accounted for through conventional spinwave theory —just like Nd ScNbO .Now this unusual heat capacity is not present inall Nd pyrochlores. Fig. 7(c)-(d) show twoNd pyrochlore-based materials with conventionalmagnon behavior: T heat capacity with − γ offsets.7(c) shows data from the Nd pyrochlore derivative Nd Sb Mg O (a kagome compound with AIAO orderand easy-axis anisotropy) which has heat capacity ex-actly matching gapped linear dispersive magnon modes[47]. Fig. 7(d) shows data from pyrochlore Nd Sn O [56] (another Nd pyrochlore with AIAO order andeasy-axis anisotropy [21]), is beautifully consistent withgapped dispersive magnons indicated by the solid redlines (further details are in Appendix F). Thus, notall Nd pyrochlores have the extra DOS observed in Nd ScNbO and Nd Zr O . Specifically, only the spe-cific heats of the two materials with gapped excitationspectra are well accounted for by thermal excitation ofbosonic collective modes. B. Static moment discrepancy
Another unusual feature of Nd ScNbO is the apparentdiscrepancy in the measured static moment. Specificallyunder the assumption of a static spin structure, three dif-ferent measurements (nuclear Schottky anomaly, diffrac-tion refinement, and hyperfine excitations) yield differentvalues for the ordered moment size, shown in Table I. Allare less than half of the single site saturation magneti-zation inferred from magnetization measurements on apowder sample. The Schottky anomaly and the neutrondiffraction refinement values agree to within uncertainty,but there is an apparent discrepancy with the value de-rived from the nuclear hyperfine excitations. C / T ( J / K m o l ) flat bandspin wavesschottky extra DOS (a) = magnon fit datafit . ( ) spin waves (b) = linear term flat bandspin waves(c) = . possible DOS origins / term flat bandspin waves(d) = . Figure 6. Low temperature heat capacity of Nd ScNbO compared to various models. (a) shows a fitted model based on an0.07 meV flat band spin wave excitation plus linear dispersive spin waves and a schottky anomaly. It fails to properly accountfor the density of states at low T, indicating some exotic behavior. (b) shows the same fitted model but the heat capacity fromspin wave excitations are raised to an arbitrary fitted power. (c) shows a fitted model which includes a linear term in specificheat. This linear term model fits the data two orders of magnitude better than the model in (b). (d) shows a fit including aspinon T / contribution as predicted by ref. [10].Table I. Low temperature ordered electronic moment µ of Nd ScNbO measured by nuclear Schottky anomaly, neutrondiffraction, and hyperfine excitations. The bottom row de-scribes what quantity each technique measures.Nuclear Neutron HyperfineSchottky diffraction excitations1.17(3) µ B µ B . µ B Local RMS µ Mean LRO µ Mean local µ To interpret these differences we must consider whatquantity each technique measures. The neutron diffrac-tion measurement is a spatially-averaging probe: it givesonly the static moment (on a time scale set by the en-ergy resolution) that participates in a long-range-ordered(LRO) state. Meanwhile, the measurements based onthe nuclear spin (Schottky anomaly and hyperfine ex-citations) are local probes: they give the value of theelectronic ordered moment without reference to spatialcorrelations. There are important differences betweenthe hyperfine excitation and nuclear Schottky probes too:the high-temperature tail of a nuclear Schottky anomalymeasures the root-mean-squared (RMS) local static mo-ment (at intermediate temperatures it measures a valuebetween the mean and RMS, see Appendix B) while thehyperfine excitation spectrum is a measure of the proba-bility distribution function for the magnitude of the localmoment.It is possible for the local static moment to be largerthan the LRO moment if there is static spin disorder [58].This is what one expects for the moment fragmentedstate on the pyrochlore lattice: in that case, the localstatic moment would be twice the LRO moment [17, 25].However, this is not what we observe: the local momentis only . . = 1 . times larger than the LRO mo- ment, and both are much smaller than the sublattice sat-uration magnetization inferred from the powder magne-tization measurements. Thus Nd ScNbO does not havea moment fragmented ground state (in the sense of refs.[17, 25, 35, 59] referring to a three-in-one-out crystal-lized lattice of magnetic monopoles). The fragmentationtheory notwithstanding, static spin disorder (from, forexample, a distribution of moment sizes) may explainwhy the static moment inferred from the nuclear spinpolarization is greater than the moment inferred frommagnetic neutron diffraction.The most curious discrepancy is in the difference be-tween the moment inferred from the nuclear Schottkyanomaly and the hyperfine spectrum. If all moments areuniform size, these values should be the same (cf. ref.[47]). With spin disorder, the nuclear Schottky momentshould exceed the mean value from hyperfine excitations(by definition, RMS ≥ mean). However, we see the op-posite: the RMS nuclear Schottky value is less than themean nuclear hyperfine value. The same three measure-ments were performed using the exact same equipment onthe related Nd magnet Nd Sb Mg O [47] (which has noexotic features in its ground state) and those three mea-surements agree beautifully—so the measurement tech-nique, calibration, and analysis methods seem to be inorder. Rather, this discrepancy and the reduction seenin all these moment measurements relative to the sublat-tice saturation magnetization suggests that some fractionof the Nd electronic moments is fluctuating.If the moment measured from the hyperfine spectrumis larger than the moment from the Schottky anomaly,this indicates that the hyperfine measurement missedsome sites with very small moments. A limitation offitting hyperfine scattering is that signals from vanish-ing static magnetism are hidden in the elastic scattering,and the normalization is not precise enough to identify C / T ( J / K m o l ) (a) dataSchottky sub. LSWT (b)0 0.2 0.4T (K )01020 C / T ( J / K m o l ) (c) 0 0.5 1.0T (K )01020 (d) Exotic ( > )
Conventional ( )
Figure 7. Heat capacity of Nd ScNbO plotted in C/T vs T (a) compared to three other Nd pyrochlore-based com-pounds: Nd Zr O (b) [56], Nd Sb Mg O (c) [57], and Nd Sn O (d) [56]. (c) and (d) show conventional spin-waveheat capacity, while (a) and (b) show excess heat capacityat low temperatures, even compared to heat capacity com-puted directly from the spin wave spectrum (b). Note thatthe nuclear hyperfine contribution to the specific heat wassubtracted from all data, including Nd Sn O shown in (d). the missing spectral weight. Thus, if the distribution ofNd moments is not Gaussian (as our model above as-sumed) and a significant fraction of Nd sites have van-ishing static moments (small enough to be hidden in theelastic channel), the fitted mean moment would be higherthan the moment revealed by a Schottky anomaly fit.This is demonstrated in Fig. 8, where two different dis-tributions reproduce the hyperfine spectrum but have dif-ferent mean moments and nuclear Schottky signals. Dis-tribution 1 ( (cid:104) µ (cid:105) = 1 . µ B , µ RMS = 1 . µ B ) follows theprofile of the positive energy transfer hyperfine spectrum,but distribution 2 ( (cid:104) µ (cid:105) = 1 . µ B , µ RMS = 1 . µ B ) has1/4 of moments clustered around zero. Distribution 2is consistent with both neutron and heat capacity data.(Note that the RMS moment is greater than the fit above:this is because at low temperatures the fitted moment de-viates from the RMS — see Appendix B.) Under this in-terpretation, the static moment discrepancy shows somefraction of Nd sites have fluctuating moments to the low-est temperatures, such that the static moment on somesites is nearly zero. % N d s i t e s distribution 1 = . distribution 2 = . (a)10 (K)0.515 ( J / K m o l ) dist. 1dist. 2 (b) 5 0 5 ( )1015 I n t e n s i t y ( a . u . ) (c) dist. 1dist. 2HFBS, 50mK Figure 8. (a) Two possible distributions of static momentsin Nd ScNbO . Distribution 1 (red) was modeled after thepositive-energy transfer hyperfine spectrum. Distribution 2(blue) is the same distribution, but with 1/4 of the momentshaving zero or nearly zero static moments. (b) CalculatedSchottky heat capacity from the two distributions; distribu-tion 2 matches the data very well. (c) Hyperfine spectrum cal-culated from distributions 1 and 2, normalized to match thehyperfine scattering. They are identical: the zero-momentsfrom model 2 are hidden in the elastic peak. C. Ordered moment distribution
A third question raised by these results is why thereis a distribution in ordered moment size indicated by thebackscattering measurements. Although we cannot sayfor certain, it is possible that this distribution is due tovariations in the local crystal fields of Nd from the Nb-ScB-site disorder. To demonstrate this, we simulated thecrystal field environment using a point charge model.Measurements by Mauws et al [31] show a distributionof crystal electric field (CEF) excitations in Nd ScNbO .Based on the relative peak weights, they estimate that14% of the sites have D symmetry. Assuming this tobe true, we performed Monte Carlo simulations of Nb Sc charge ice at finite temperature to determine therelative frequency of the other symmetry-unique sites(see Appendix E). Although there are = 64 pos-sible Nb-Sc arrangements around a Nd ion, there areonly 13 symmetry-unique arrangements. When 14% ofNd sites have D symmetry, the percentages of othersymmetry-unique CEF environments are shown in Fig.9(a).We then calculated the CEF Hamiltonians of the vari-ous CEF states using the point charge approximation andPyCrystalField [60] (see Appendix E for details), and thelevels of the four most probable states are plotted in Fig.9(b). Because Nd is a Kramers ion, every environmentyields five doublets no matter how low the symmetry. We Sc Nb (a) local environments 0204060 ( m e V ) Nd Nd Nd NdScNb
10 20 30 40 50 (meV)0123 ( a . u . ) total total (nocharge ice) (c) simulated neutron spectrum(b) CEF levels . . .. . . Figure 9. Simulated CEF spectrum for Nd ScNbO . (a) Distribution of symmetry-equivalent local Nd environments if 14% are D . (b) CEF levels of the four most common local environments ( C : 26.2%, D : 14%, and both D : 13%) calculated from apoint-charge model assuming modest oxygen displacement. (c) Total simulated neutron spectrum from all CEF environments,showing that observable peaks are from more than just D environments. The light grey dashed line shows the total scatteringassuming no charge ice correlations (totally random Nd and Sc placement). then simulated the neutron spectrum by summing overall CEF states with the appropriate weights [Fig. 9(c)]and found a spectrum which resembles what was mea-sured in ref. [31]. However, we find that many of thestrongest peaks are not from D sites. If instead we as-sume a completely random Nb and Sc distribution withno charge-ice correlations [shown by the grey line in Fig.9(c)] the calculated neutron spectrum is only moderatelychanged—and possibly corresponds to the data in ref.[31] even better. These simulations cast doubt on themethodology behind the 14% estimate in ref. [31].In any case, assuming that these calculations give atleast an approximate picture of the distribution of CEFstates, we find that the maximum static moments forthese sites have a mean value µ = 1 . µ B with a stan-dard deviation . µ B . This is very close to the ob-served . ± . µ B from the hyperfine excitations.Therefore the ordered moment distribution might be par-tially caused by disordered crystal field environments.It should be noted that the precise values of the calcu-lated static moments are dependent on the details of thepoint-charge simulations. Different assumptions aboutoxygen displacement produced a higher mean µ and alower standard deviation. Furthermore, the hyperfinespectrum simulated directly from the CEF model showsmore discrete peaks than the observed hyperfine spec-trum (see Appendix E), possibly because of the simplisticpoint-charge model and collective effects which affect thelow temperature static moments. Therefore, these cal-culations demonstrate the possibility (but do not prove)that the distribution of ordered moments is caused byB-site disorder.With this moment size distribution, one would expectto see some diffuse elastic scattering indicating a superpo-sition of random spin orientation with long-range AIAOorder: the disordered moment estimated from nuclearhyperfine excitations is (0 . µ B ) (1 . µ B ) = 26(7) %. This mayexplain the observed low-Q diffuse elastic signal in Fig. 3. However, the low- Q scattering decreases with Q muchmore dramatically than the Nd form factor, indicatingcorrelated spin disorder rather than random disorder. IV. DISCUSSION
The results from heat capacity and local moment mea-surements indicate a strongly fluctuating magnetic statewith an unusual DOS which seems to be common amongseveral Nd pyrochlores. It is clear that conventionalmagnons cannot account for the DOS in Nd ScNbO and Nd Zr O , but it is not clear what kind of excitationsthey are. In principle, the apparent linear offset could befrom a very low-energy bosonic flat mode. If the energyof the mode is ∼ µ eV , the heat capacity signal wouldlook very much like a linear term. However, such a stronglow-energy mode would have been visible in the neutronbackscattering experiment, so we can constrain any suchmode to have an energy greater than µ eV or less than . µ eV . Furthermore, such a low-energy mode is notpredicted in the spin-Hamiltonian of Nd Zr O [20, 29],rendering any such mode to be beyond LSWT magnons.We present two possible explanations for the low-energyDOS: fermionic quasiparticles and local spin degrees offreedom.Linear C/T offsets are ordinarily signals of fermionicquasiparticles, and the γ term (called a Sommerfeld co-efficient) is a measure of the density of states at thechemical potential, which is proportional to the effec-tive mass of the fermions. In Nd ScNbO we measure γ = 3 . Jmol K . This is three orders of magni-tude higher than Sommerfeld coefficients for most metals( γ Ag = 6 . × − , γ Nb = 8 . × − [55]),and is of the order of heavy-fermions systems such asYbBiPt, which has γ ≈ Jmol K [61]. This γ could pos-sibly be the signal of fermionic spinons of a quantumspin liquid [54], which is plausible given the theoretical0proximity of Nd Zr O to a U (1) quantum spin liquid[20]. Alternatively, it could be that the heat capacity sig-nal is from C ∝ T / spinons predicted for a condensedAIAO pyrochlore spin liquid phase [10]. In either case,this correspondence suggests that the mystery DOS in Nd ScNbO and Nd Zr O are spinons of a proximatequantum spin liquid.Under this interpretation, the distribution of momentsmay be due to a moment-modulated spin fragmentedstate, as proposed for Ho Sb Mg O [62]. In this phase,dipolar interactions induce a quantum moment frag-mented state where certain spins fluctuate more thanothers, leading to a distribution of static moments. Al-though the moment size for Nd is smaller than Ho, sim-ilar physics from asymmetric exchange may be relevant.If so, the specific fraction 1/4 of spins having nearly zeromoment, which is consistent with the nuclear hyperfinespectrum, could indicate the "one-out" spin fluctuateswhile the "three-in" spins are static.Alternatively, it could be that the mystery DOS is as-sociated with local spin degrees of freedom induced bythe Nd-Sc disorder. It is known that specific heat in dis-ordered spin glasses can have T -linear or T / specificheat [52, 63]. If such a heat capacity signal were super-imposed upon magnon specific heat, the result could lookvery much like the fits in Fig. 7. Under this interpreta-tion, the CEF and magnetic exchange disorder from theNd-Sc disorder create local soft spin degrees of freedomwithin a long-range ordered state. This may explain thelow-energy low- Q scattering in Fig. 4.This disorder hypothesis also naturally explains howa fraction of the spins could have no static moment. Ithas been shown theoretically that random bond disorderon the pyrochlore lattice can produce random singlet for-mation between spins [64], or even long-range entangledstates [65]. Given how close Nd Zr O is to a U quan-tum spin liquid phase [20] it is possible that Nd ScNbO is also close enough that bond and anisotropy disordercauses certain site pairs to be locally within that phaseand form singlets, such that there is no static momenton those sites and low-energy singlet-triplet excitationsappear in the DOS.If this interpretation is correct, Nd ScNbO hosts theunusual combination of long-range magnetic order, sin-glet formation, glassy local excitations, and a sharpphase transition. This would imply that the disorderis weak enough that the long-range order is preserved(with a resolution-limited correlation length) though theexchange and single ion anisotropy disorder produces lo-cal soft modes for the spins (some of which pair to formsinglets) and diffuse low Q scattering.Assuming this to be true, the exotic behavior of thesister-compound Nd Zr O may also be associated withdisorder-induced local spin degrees of freedom (perhapsthrough oxygen deficiency and "stuffing" problems whichplague other pyrochlores [66–68]). This disorder hypoth-esis may explain the dramatic sample-dependence re-ported in ref. [56] and why different studies with different samples report conflicting values of the ordered moment[23, 25, 32].Whether the mysterious magnetic DOS is from frac-tionalized quasiparticles or local excitations, the result-ing ground state is clearly dominated by strong quantumfluctuations. Comparison between nuclear hyperfine ex-citations and specific heat data indicates there are siteswhere the electronic moment continues to fluctuate fasterthan the nuclear spin relaxation time so that no hyperfinelevel splitting is generated. Thus, quantum fluctuationspersist to the lowest temperatures. An apparent sup-pression of ordering was also documented for Gd Sn O ,and it was proposed that quantum fluctuations in theelectronic moment suppress a nuclear Schottky anomaly[69]. The quantum fluctuations in Nd ScNbO could befrom an exchange-disordered proximate spin liquid [70],or localized excitations from a random singlet state. Oc-cam’s razor leads us to prefer the localized DOS hypoth-esis over the exotic quasiparticle hypothesis, but trans-port experiments [71] are needed to determine whetherthe excitations are localized spin degrees of freedom ordelocalized quasiparticles which can transport energy.These results have strong implications for other Ndpyrochlores. The many similarities between Nd ScNbO and Nd Zr O suggests a common magnetic state, andwe expect these same features to be present in Nd Hf O because of the close correspondence with Nd Zr O . Themagnetic ground state is strongly influenced by collectivequantum physics: The combination of frustration anddisorder in Nd ScNbO produces a state which fluctu-ates to the lowest temperatures and mimics spin-liquidbehavior—but may not be the long-sought many-bodyentangled spin liquid state. V. CONCLUSION
We have shown the Nd pyrochlore Nd ScNbO tohave an easy-axis moment with AIAO magnetic orderbelow T N = 371(2) mK. Susceptibility and magnetiza-tion indicate an easy-axis moment, heat capacity showsa fully-ordered system with R ln 2 entropy, and elasticneutron scattering shows AIAO order. Inelastic scatter-ing shows a flat-band excitation similar to Nd Zr O ,and all order parameter curves are consistent with 3DIsing magnetic order. Nuclear hyperfine excitations re-veal a distribution of static electronic moments below T N ,which we suggest is due to B-site disorder influencing theNd crystal electric field states.Analysis of our results has revealed two unconventionalbehaviors in Nd ScNbO : First, we used heat capacity toshow an anomalous density of states in both Nd ScNbO and Nd Zr O , that is inconsistent with conventionalmagnons. These excitations give a heat capacity sig-nal consistent with fractionalized spinons of a proximatespin liquid phase, or localized spin excitations from arandom-singlet phase that coexists with long range or-der. Second, by comparing the local and long-range or-1dered moments, we rule out the possibility of classicalmoment fragmentation (in the sense of refs. [17, 25, 34])in Nd ScNbO . This comparison of local ordered mo-ments reveals that a fraction of Nd sites have vanish-ing static moments in the low T limit, as in a randomsinglet state or an exotic moment-modulated quantumfragmented state. Taken together, these results indicatea spin system dominated by quantum fluctuations thatcannot be described in terms of conventional magnons inan otherwise long range ordered state. Disorder however,plays a significant role so that experiments probing thedegree of spatial coherence of the low energy excitationswill be needed to properly classify the unusual quantumstate of matter in Nd-based pyrochlore magnets that wehave drawn attention to in this paper. Acknowledgments
Thanks to Christian Balz, Sayak Dasgupta, GáborHalász, Mark Lumsden, and Steven Nagler for helpfuldiscussions. This research used resources at the Spal-lation Neutron Source, a DOE Office of Science UserFacility operated by the Oak Ridge National Labora-tory. Initial stages were supported as part of the Institutefor Quantum Matter, an Energy Frontier Research Cen-ter funded by the U.S. Department of Energy, Office ofScience, Basic Energy Sciences under Award No. DE-SC0019331. AS and CB were supported by the Gordonand Betty Moore foundation under the EPIQS programGBMF4532. The dilution refrigerator used for heat ca-pacity measurements was funded by the NSF throughDMR-0821005. Use of the NCNR facility was supportedin part by the NSF through DMR-1508249. Thanks toJianhui Xi for sharing his SpinW code.
Appendix A: Backscattering experiment andanalysis1. HFBS resolution function
The resolution function for HFBS is asymmetric in en-ergy [72], as is common for backscattering spectrometers[73]. This is clearly shown by the incoherent elastic scat-tering acquired for a vanadium standard sample in the ± µ eV configuration shown in Fig. A1. There are avariety of instrumental causes for this asymmetry [72],and we account for it with a phenomenological resolu-tion function that consists of five Gaussian peaks plus aLorentzian peak: R ( ω ) = (cid:88) i =1 G ( ω, ω i , A i , σ i ) + L ( ω, ω , A , Γ ) . (A1)This function is normalized so (cid:82) R ( ω ) dω = 1 . Here thepeak centers ω i , areas A i and widths σ i are fitted se-
10 0 10 ( )9101112131415 I n t e n s i t y ( a . u . ) (a) vanadium 1 0 1 ( )0100200300400 (b) Figure A1. HFBS scattering from a vanadium standard sam-ple, showing an asymmetric lineshape and our phenomeno-logical fitted resolution function. (a) and (b) show differentviews of the same data. quentially: we started with a single Lorentizan fit, addeda Gaussian term and refit, added another Gaussian andrefit, etc. As shown in Fig. A1, this provides a verygood reproduction of the lineshape. This normalizedlineshape is used to fit all the hyperfine HFBS data inthis paper, via convolution with models for the intrinsicphysical spectrum.
2. Hyperfine excitation model
The neutron cross section of powder-averaged nuclearhyperfine excitations is (cid:16) d σd Ω dE (cid:17) ± = 13 k f k i e − W ( Q ) (cid:88) i ρ i I i ( I i + 1) σ i π (cid:104) (cid:88) j p j ( T ) δ (∆ M ji − ∆ M j +1 i − ¯ hω )+ (cid:88) j p j +1 ( T ) δ (∆ M j +1 i − ∆ M ji − ¯ hω ) (cid:105) . (A2)Here ± is positive and negative energy transfer, ∆ M j are the hyperfine splitting energies, σ i is the Nd incoher-ent scattering cross section, I is the nuclear spin state, e − W ( Q ) is the Debye-Waller factor, and k i and k f are theincident and scattered neutron wave vectors. The sum i is over isotopes with their relative abundance ρ i [49], andthe sum j is over all energies. In this analysis, we approx-imate k f k i ≈ because of the small energy transfers and e − W ( Q ) ≈ because of the low temperatures. The neg-ative (positive) energy transfer scattering is suppressed(enhanced) by the Boltzmann weight p ( T ) = e ∓ β ∆ e − β ∆ + e β ∆ .The measured hyperfine peaks are broader than theresolution width defined above. We account for this witha Gaussian (which we use to model a random distributionof static moments) convolved with a Lorentzian (whichwe use to model finite lifetime of the excited nuclear hy-2perfine states), convoluted again with the resolution func-tion. We apply the Boltzmann factor to the Gaussian dis-tribution prior to convolution, because the nuclear stateswith large splitting will experience more thermal depop-ulation than those with smaller splitting (the effect ofthis is to suppress the negative energy tail of the − ¯ hω Gaussian and to enhance the positive energy tail of the +¯ hω Gaussian).The two stable isotopes of Nd with nuclear spin are143 and 145, with relative abundances 12.2% and 8.3%respectively, and both with I = 7 / . Thus, the equationused for fitting in Fig. 5 is (cid:16) d σd Ω dE (cid:17) ± = 13 (cid:20) (cid:16)
72 + 1 (cid:17)(cid:21)(cid:16) . σ π (cid:88) j C ( G, L, ∆ M j − ¯ hω, T )+ 0 . σ π (cid:88) j C ( G, L, ∆ M j − ¯ hω, T ) (cid:17) (A3)where C ( G, L, ∆ M − ¯ hω ) is the convoluted broadeningdefined by a Gaussian width G and a Lorentzian width L with the Boltzmann weight defined by temperature T .In fitting this model, we were also able to fit the effec-tive temperature using the Boltzmann factor. Curiously,the lowest temperature data (which had a sample ther-mometer reading of T = 0 . K) gave a fitted temper-ature of T = 0 . K. The difference is at least par-tially due to beam heating. Our previous measurementsof Nd Sb Mg O (which, in total, had 1.3 times moreabsorption cross section in the beam) showed a Boltz-mann fitted temperature of T = 0 . K for a samplethermometer reading of 45 mK [47].
Appendix B: Nuclear Schottky anomaly and theordered moment
The equation for a nuclear Schottky anomaly is C = 1 Zk B T (cid:104) (cid:88) i E i e − EikBT − Z (cid:0) (cid:88) i E i e − EikBT (cid:1) (cid:105) (B1)[74]. Here E i are the levels of the nuclear spin states,which are determined by the hyperfine nuclear spinHamiltonian H = a (cid:48) I z + P (cid:0) I z − I ( I + 1) (cid:1) (B2)(see eq. 1 in ref. [42]), where a (cid:48) = a (cid:104) J z (cid:105) , (cid:104) J z (cid:105) being theexpectation value of the effective electronic spin J , and P is a constant defining the hyperfine quadrupole cou-pling strength and the electronic quadrupole moment.For Nd , the hyperfine quadrupole coupling strength isthree orders of magnitude weaker than the dipole cou-pling [42, 75]. Our CEF calculations indicate a poten-tially large static electric quadrupole order in the ground state doublet (cid:104) + | O | + (cid:105) / [ J (2 J − . (here O is the quadrupole Stevens Operator [76] and J is thetotal Nd spin). Even assuming a completely satu-rated quadrupole moment, the quadrupolar hyperfinelevel splitting only varies by ≈ % relative to the dipole-only hyperfine splitting (0.014 µ eV assuming the con-stants in ref. [42] or 0.024 µ eV assuming the constantsin ref. [75]–two orders of magnitude smaller than the ob-served splitting). Thus, although we include it in our fits,the quadrupolar hyperfine coupling is negligible and theheat capacity is primarily from dipolar hyperfine effects.(These values also confirm that the inelastic neutron hy-perfine peak broadening cannot be explained by Nd quadrupolar order.) In passing, we note that it is alsopossible for nuclear levels to be split by crystal electricfields, but the nuclear quadrupole moment is small forNd, and these effects tend to be <0.1 µ eV for Nd [77];therefore we do not consider these effects in our analysis.The high temperature limit of Eq. B1 is C ∝ T . If theordered magnetic moment size varies from site to site butis the same ion, the high-temperature tail of the nuclearSchottky anomaly gives the root-mean-squared orderedmoment. This can be demonstrated with a sum over allsites indexed by k in the high-temperature limit: C net ∝ (cid:88) k E k T ∝ (cid:80) k (cid:104) J z k (cid:105) T = (cid:104) J z net (cid:105) T (B3)and thus J z net = (cid:115)(cid:88) k (cid:104) J z k (cid:105) . (B4)However, this approximation is only valid in the high-temperature limit. As one nears the Schottky peak, thisapproximation breaks down. If the distribution is verybroad, the peak height itself will be suppressed. In prac-tice, the ordered moment fitted from the upturn will bein between the mean and RMS moment.We finally note that there is an intrinsic disorder asso-ciated with the different nuclear spins of the 7 differentisotopes of Nd, which could play a role in the low tem-perature electronic magnetism. Experiments on nuclearspin free isotopic samples would be of interest to lookinto this. Appendix C: Heat capacity measurements
Heat capacity data were acquired using a Quantum de-sign PPMS with the semi-adiabatic relaxation method,wherein the time dependent thermal response T ( t ) of thesample stage to a heat pulse applied to it is monitored.With knowledge of the thermal conduction between thesample stage and the thermal reservoir the specific heatof the sample is determined from a fit to T ( t ) [78]. Weapplied sufficient heat to achieve a 3% temperature riseat each temperature (ie, the 95 mK data point had a ∼ t i m e ( h r s ) (a) measurement timeequilibration time0.05 0.06 0.07 0.08 0.09 0.10 (K)0.550.600.65 ( J / K m o l ) (b)0 0.1 0.2 0.3 0.4 0.5 (K)00.20.4 ( % ) (c) Figure A2. (a) Measurement and equilibration times for theheat capacity measurements of Nd ScNbO . The lowest tem-perature data points had a measurement time of ∼ hr andequilibration times of several hours. (b) Three lowest heatcapacity temperatures measured, showing the small spread inmeasured heat capacity and temperature for each data point.(c) Uncertainty σ net for heat capacity, as computed with eq.C2.
1. Temperature calibration
Whenever one observes an unusual experimental sig-nal (as we have in this paper), it is always important toconsider errors in the measurement technique. If temper-ature calibration were off by 100% at 100 mK (showing100 mK when the sample is really at 200 mK), this pos-itive γ term would disappear. However, we consider thisto be unlikely for four reasons: (a) The transition temper-ature T N = 370(10) mK measured from heat capacity isin agreement with T N = 371(2) mK from neutron diffrac-tion, suggesting that the low-temperature calibration is correct to within 10 mK at 300 mK. (b) The compoundmeasured immediately prior to this on the same PPMSwas Nd Sb Mg O , which gave a Schottky anomalyupturn beautifully consistent with the ordered momentfrom neutron measurements [47], indicating good cali-bration. The same equipment and calibration was usedhere. (c) Sample coupling—the measure of the expo-nential nature of the semi-adiabatic heating and coolingcurves on a PPMS—is > % for the entire tempera-ture range, indicating excellent exponential-like heatingand cooling behavior and reliable heat capacity values.(d) The same low-temperature density of states was ob-served in Nd Zr O heat capacity (which was measuredwith a completely different experimental setup), suggest-ing an intrinsic behavior to Nd pyrochlores. Therefore,we consider the positive offset in C/T to be a reliableexperimental results subject to the statistical error barsquoted.
2. Uncertainty
We measured heat capacity at each temperature threetimes, and this repetition revealed good temperature sta-bility and very reproducible values for every temperaturemeasured [Fig. A2(b)]. We combined the three datapoints by taking the mean of heat capacity and tem-perature. Uncertainty in heat capacity for each datapoint was computed as the standard error of the meanof the three data points σ std / √ n added in quadrature tothe standard error of the weighted mean ( (cid:80) i σ − i ) − / (this is to account for both the uncertainties generatedby the PPMS fitting routine values plus the spread ofdata points) so that uncertainty is σ = (cid:118)(cid:117)(cid:117)(cid:116) σ std (cid:16) (cid:88) i σ − i (cid:17) − . (C1)Here σ std is the standard deviation of the three datapoints, and σ i are the uncertanties of the individual datapoints (which comes from covariance matrix of the expo-nential fits). Uncertainty in temperature was computedas the standard error of the mean of the three averagedpoints. For every temperature the uncertanties are quitesmall: σ T < . mK and σ c < . below 1 K. Errorbars are always smaller than the data point markers inthe main text, and these uncertanties are used to definethe χ of the different fitted models.The fitting algorithm used for heat capacity mod-els (Scipy’s "curve_fit" [79], which uses a least-squaresmethod) does not account for uncertainty in the x -axis(in this case temperature), so we converted σ T to σ C by multiplying the uncertainty in temperature σ T by theslope of the data at that point s i . We then added it inquadrature to the heat capacity uncertainty: σ neti = (cid:113) σ C i + ( σ T i s i ) . (C2)4 C / T ( J / K m o l / ) silver Figure A3. Heat capacity of Nd ScNbO compared to the sil-ver powder it was mixed with. The difference in heat capacityis well over three orders of magnitude. These σ net , plotted in Fig. A2(c), were used in defining χ for the fits.
3. Silver subtraction
For the heat capacity measurement, Nd ScNbO wasmixed with silver powder with a 1:1 mass ratio, and thenthe contribution from silver was subtracted from the datapost-facto. Because the heat capacity of silver is so smallbelow 1 K (three orders of magnitude below Nd ScNbO )the effect of this subtraction is very slight as shown in Fig.A3. This figure also shows the Sommerfeld coefficient ofsilver to be γ = 7 . × − , which is far too smallto account for the large C/T offset in low temperature Nd ScNbO heat capacity. Appendix D: Heat capacity fits1. Models
As noted in the main text, we fit the heat capacityusing three different models. In this section we list thebest fit parameters for each model. a. Dispersive magnons + flat band
This model is given in Eq. eq. 3 in the main text C = A C flat + C ld (∆ , c ) + C schottky ( µ ) , which includes a a flat band excitation (eq. 2, with ¯ hω =0 . meV) plus a gapped linear dispersive spin wave modeplus a nuclear Schottky anomaly. The calculation for thelinear dispersive mode heat capacity C ld (∆ , c ) with spinwave velocity c is based off ref. [47]. Because we donot know the precise spin wave spectrum of Nd ScNbO , we allow the relative contributions of the flat band andthe spin wave mode to be fitted parameters. (This isequivalent to fitting the phonon specific heat to the sumof an Einstein mode and a Debye spectrum.) The bestfit parameters and reduced χ are given in Table A2. Table A2. Best fit parameters for eq. 3 χ A ∆ (meV) c (m/s) µ ( µ B )3896 0.40(12) 0.00(0) 24.2(1.3) 1.72(8) b. Power law + flat band This model is given in eq. 4 in the main text C = A C flat + B T n + C schottky ( µ ) , which includes a flat band excitation plus a spin wavemode with an arbitrary power plus a nuclear Schottkyanomaly. The best fit parameters are given in Table A3.The weight on the flat band excitation converged to zero.This indicates that this fit is not very reliable, becauseneutron scattering clearly shows the flat band to be quiteintense (at least % of the measured spectral weight)and not a tiny contribution like the fit indicates. Table A3. Best fit parameters for eq. 4 χ A B n µ ( µ B )140 0.00(4) 49(3) 2.13(3) 1.32(3) c. Dispersive magnons + linear term This model is given in Eq. 5 in the main text C = A C flat + C ld (∆ = 0 , c ) + γT + C schottky ( µ ) which includes a flat band plus gapless spin wave modeplus a nuclear Schottky anomaly. The best fit parametersare given in Table A4. Table A4. Best fit parameters for eq. 5 χ A c (m/s) γ µ ( µ B )7.58 0.072(7) 26.62(11) 3.23(4) 1.130(8) We also tried this fit allowing the gap ∆ to be nonzero,but the value converged to zero every time. Thus we tookit out of the fit and set ∆ = 0 so that the number ofparameters matched Eq. 2. d. Dispersive magnons + T / term It was theoretically predicted in ref. [10] that a mag-netically ordered pyrochlore can have spinons excitations5 C / T ( J / K m o l ) linear term flat bandspin waves = . (a)051015 C / T ( J / K m o l ) / term flat bandspin waves = . fitted (b)0 0.02 0.04 0.06 0.08 0.10T (K )051015 C / T ( J / K m o l ) / term flat bandspin waves = = .= (c) Figure A4. Low temperature heat capacity of Nd ScNbO fit-ted to (a) a model including a linear term, (b) a model includ-ing a T / contribution from spinons and a gapped magnonmode, and (c) a model including a T / contribution and agapless ( ∆ = 0 ) magnon mode. All models match the datawell, but as revealed by χ comparisons, the gapped magonwith T / heat capacity matches the best. which produce a T / power law dependence in heat ca-pacity. To test this, we fitted the model in eq. 6 twoways: once allowing ∆ to be a free fitted parameter, andonce fixing ∆ = 0 (to keep the same number of fittedparameters as eq. 3). The results are plotted in Fig.A4 and the best fit parameters are in Table A5. Thismodel fits as well as the linear term model, suggestingthat the anomalous low-energy density of states couldbe due to the C ∝ T / spinon excitations consideredin ref. [10]. This model also predicts a magnon gap of0.07 meV, which is almost exactly the energy of the flatband—suggesting that this model could be a better re-flection of the actual magnon behavior in Nd ScNbO . Table A5. Best fit parameters for eq. 6. The first top rowgives the best fits parameters for when the magnon gap was afitted parameter, the bottom row gives the best fit parametersfor gapless magnons. χ A ∆ c (m/s) B µ ( µ B )7.57 0.077(4) 0.070(16) 27.9(8) 12.7(18) 1.178(8)12.47 0.02(1) 0.0 28.5(18) 10.2(16) 1.216(9)
2. Varying maximum fitted temperature
The range of data fitted for each different model was
T < . K. This temperature was chosen because itis the highest temperature at which the heat capacityexhibits roughly T behavior. If we decrease the highestfitted temperature, the fits give the same general results:the model including only dispersive magnons plus a flatband fits the data poorly, a power law fits the data better,and a model including a linear heat capacity term fits thedata best.Figure A5 illustrates this by plotting reduced χ as afunction of maximum fitted T . Any reasonable maximumfitted T still causes the linear term model to fit the databest by over an order of magnitude. Curiously, the fittedtemperature range makes almost no difference for thefitted values for the linear term model, as demonstratedin Fig A5(d). Thus, the main conclusions of this paperare independent on which temperature range is chosenfor a fit. Appendix E: Monte Carlo charge ice simulations
We performed Monte Carlo (MC) simulations with alarge box of × × pyrochlore unit cells, allowing theNb and Sc sites to flip from one to another. Charge-icecorrelations were simulated by assigning an energy − J toevery Nb-Sc pair on a tetrahedron and + J to every Nb-Nb or Sc-Sc pair on a tetrahedron. This led to a two-Nd-two-Sc tetrahedron having an energy − J , a three-Nd-one-Sc or three-Sc-one-Nd tetrahedron having an energyof J , and a four-Nd or four-Sc tetrahedron having anenergy +6 J . The lowest energy state is a "charge ice",with two Nd and two Sc on each tetrahedron, but simula-tions at finite MC temperature will deviate from this. Weused a Metropolis algorithm to calculate the relative fre-quency of the 13 symmetry-unique local environments asa function of temperature, sweeping through the latticefour times between measuring and taking 250 samples ateach temperature. The percentage of sites with D sym-metry as a function of MC temperature is shown in Fig.A6, and the code for these MC simulations can be foundin the Supplemental Information [80].6 Dispersive magnonsPower lawLinear Term(a)0 0.05 0.10 0.15 (K )051015 C / T ( J / K m o l ) (b) Dispersive magnons
Power law
Linear term
Figure A5. (a) reduced χ for the three models as a function of maximum fitted temperature. (b) Dispersive magnon modelfits (solid lines) compared with data. The shade of the line indicates the maximum fitted temperature, which is indicated bythe shade of the data marker. (c) Power law model fits (solid lines) compared with data. (c) Linear term model fits (solid lines)compared with data. / % N d w i t h MC simulationMauws et al
Figure A6. The left panel shows the percentage of Nd siteswith D symmetry as a function of MC temperature. 14%of sites have D symmetry at approximately T /J = 1 . (Thelow-temperature jump is a finite-size effect and varies as thesystem size is changed.) The right panel shows the relativepopulations of the 13 symmetry-unique environments for Ndin Nd ScNbO where the orange dots are Sc and the blue-green dots are Nb. We then calculated a point charge model using Py-CrystalField [60] for every symmetry-unique ring aroundthe Nd sites (there are configurations but only thir-teen symmetry-unique rings). The point-charge model isan approximation, but it is usually close enough to give a qualitative picture of the CEF Hamiltonian [81, 82].To account for the different charges of the Nd and Scsites, we used different effective charges for each and as-sumed that the O − ligands shift towards the Nb by3% as compared to Sc . To constrain the model to fitbetter with the observed CEF transitions in ref. [31],we fit the effective charges in the point charge model tominimize the difference between eigenvalues of the D symmetric sites and the observed transitions in ref. [31].This fit yielded effective charges of − . e and − . e on the two symmetry-independent O sites, . e on theSc site, and . e on the Nb site. We then used theseeffective charges to simulate the CEF Hamiltonian of all13 symmetry-unique ligand environments for Nd , asshown in Fig. 9.To test the correspondence to the experimental data,we calculated the hyperfine spectrum and nuclear Schot-tky heat capacity, as shown in Fig. A7. This shows amostly bimodal distribution of static moments in the hy-perfine spectrum, which is not true of the actual data.We also (unsurprisingly) see that the nuclear Schottkyanomaly onset is at a much higher temperature than ex-periment. These discrepancies are partially due to thesimplistic assumptions we have made in modeling themoments with a point-charge model, and partially dueto inter-site exchange which affect the static moments7 I n t e n s i t y ( a . u . ) HFBS data (0.05 K) CEFmodel (a)Hyperfine spectrum 10 (K)0.515 ( J / K m o l ) CEFmodel (b) Heat capacity
Figure A7. Calculated hyperfine spectrum (a) and heat ca-pacity (b) from the CEF disorder point-charge model. TheCEF spectrum predicts some sites to have much larger staticmoments than is observed, and the distribution is not nearlyas continuous as in experiment. Heat capacity also shows theCEF model overestimates the mean ordered moment. size—either singlet formation or collective fluctuations.
Appendix F: Calculated Nd Zr O heat capacity We calculated heat capacity by simulating the Nd Zr O spin wave spectrum from the Hamiltonian inref. [29] using the SpinW software package [83]. Wesimulated the spin wave spectrum over the reciprocalspace range < h < RLU , < k < RLU , and < l < RLU , and then calculated heat capacity fol-lowing ref. [84] as C v = β (cid:88) k (cid:88) α [ η α ( k ) n B ( η α ( k ))] exp[ βη α ( k )] (F1)where n B ( η α ( k )) is the Bose factor. We then dividedthe calculated heat capacity by the number of spin wavemodes to get heat capacity per Yb ion, and then dividedagain by six because there are six symmetry-related re-gions in the box chosen. (This is a computationally expe-dient alternative to summing over only the first Broullinzone, but is formally equivalent.) 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