Abstract
Spin ice is a magnetic analog of H_{2}O ice that harbors dense static disorder. Dipolar interactions between classical spins yield a frozen frustrated state with residual configurational Pauling entropy and emergent magnetic monopolar quasiparticles. Introducing quantum fluctuations is of great interest as this could melt spin ice and allow coherent propagation of monopoles. Here, we report experimental evidence for quantum dynamics of magnetic monopolar quasiparticles in a new class of spin ice based on exchange interactions, Pr_{2}Zr_{2}O_{7}. Narrow pinch point features in otherwise diffuse elastic neutron scattering reflects adherence to a divergencefree constraint for disordered spins on long time scales. Magnetic susceptibility and specific heat data correspondingly show exponentially activated behaviors. In sharp contrast to conventional ice, however, >90% of the neutron scattering is inelastic and devoid of pinch points furnishing evidence for magnetic monopolar quantum fluctuations.
Introduction
Classical Ising spins with dipolar interactions on the pyrochlore lattice possess a macroscopically degenerate ensemble of ground states^{1,2}. Satisfying the twoin twoout ice rule on each tetrahedron^{3,4}, these states of rareearth spins in magnets such as Dy_{2}Ti_{2}O_{7} and Ho_{2}Ti_{2}O_{7} are in onetoone correspondence with the disordered configurations of protons in H_{2}O ice (Fig. 1a). Mimicking the formation of an H_{3}O^{+}–OH^{−} electric dipole in water ice, a spin flip from the spin ice manifold fractionalizes into a pair of emergent magnetic monopolar quasiparticles with Coulomb attraction^{5,6,7,8,9}.
In classical spin ice, monopole dynamics is diffusive—only activated thermally or by external magnetic field^{5,6,10} as solitons in semiclassical spin chains^{11}. The classical nature of Ising spins precludes attainment of thermal equilibrium at temperatures (T) below the effective nearest neighbor energy scale J_{ff}. Quantum fluctuations enhance dynamics and might allow coherent propagation of magnetic charge^{12,13,14,15}, much as spinons in onedimensional quantum magnets^{16}.
Contrary to H_{2}O ice, pyrochlorebased rareearth spin systems^{17} provide excellent opportunities to explore quantum effects in ice. Quantum fluctuations were theoretically proposed as the origin of a putative spin liquid in Tb_{2}Ti_{2}O_{7} (ref. 18). More recently, experimental work on Pr_{2}Ir_{2}O_{7} shows the possibility of a chiral spin liquid phase when quantum fluctuations melt spin ice^{19,20}. Strong spin fluctuations were observed for Pr_{2}Sn_{2}O_{7} (ref. 21) and Yb_{2}Ti_{2}O_{7} (refs 22, 23, 24, 25), which have spin ice correlations. There, superexchange interactions for the smaller moment Pr^{3+} and Yb^{3+} systems are much stronger than dipolar interactions; so, the materials are referred to as exchange(based) spin ice (Fig. 1c)^{24,26}. In this class of spin ice, the twoin twoout states should no longer be static, but dynamic—described by a quantum superposition of the spin ice manifold. Here, we report the experimental observation of spin ice correlations and quantum dynamics in Pr_{2}Zr_{2}O_{7}.
Results
Magnetic properties
Thermomagnetic measurements on Pr_{2}Zr_{2}O_{7} provide key insights particularly when compared with the classical spin ice compounds (Fig. 1). We have succeeded in growing highquality single crystals, which are stoichiometric to the 1–2% level, as determined by chemical analyses and single crystal synchrotron Xray diffraction measurements (Supplementary Figures S1 and S2, Supplementary Tables S1 and S2, and Supplementary Note 1). The temperaturedependence of the inverse susceptibility of single crystalline Pr_{2}Zr_{2}O_{7} exhibits Curie–Weiss behavior for T <10 K with an effective moment μ_{eff}=2.5(1)μ_{B} (Fig. 1d, inset) that is four times smaller than for dipolar spin ice Dy_{2}Ti_{2}O_{7} where μ_{eff}=10μ_{B}. Correspondingly, Pr_{2}Zr_{2}O_{7} displays an antiferromagnetic Weiss temperature Θ_{W}=−1.4(1) K compared with the ferromagnetic Θ_{W} ≈ +0.5 K for dipolar Dy_{2}Ti_{2}O_{7} (ref. 2).
While no anomaly indicating longrange ordering was found down to 20 mK, ACχ(T) exhibits a frequencydependent peak at T_{0}(f), which is the onset temperature for dynamics at the frequency, f (Fig. 1d). The data for T_{0}(f) (Fig. 1e, inset) are consistent with thermal activation over at least three decades. For classical spin ice, the corresponding activation energy Δ_{χ}=1.62(3) K, is the spin flip energy, 2J_{ff}, from the ice manifold. Though the activation energies are similar, the limiting (attempt) frequency f_{0}~1 MHz for Pr_{2}Zr_{2}O_{7} is three orders of magnitude larger than for Dy_{2}Ti_{2}O_{7} (refs 10, 27). Moreover, upon cooling to the lowest T, the real part of ACχ(T) approaches 50% of the peak height while it vanishes for Dy_{2}Ti_{2}O_{7} (refs 27, 28). These indicators of enhanced fluctuations at low temperatures are consistent with general expectations for a quantum spin system.
Crystal electric field
To determine the crystal electric field (CEF) scheme, we used inelastic neutron scattering and found five magnetic excitations at energy transfers 9.5 meV, 57.1 meV, 81.9 meV, 93.2 meV and 109 meV (Fig. 2). Simultaneous fitting of a crystal field model to neutron and magnetic susceptibility data (Fig. 2) shows the ground state is a magnetic doublet with (111) Ising anisotropy, as in classical spin ice systems^{1} (Supplementary Tables S3, S4, and Supplementary Note 2). This is, however, not a Kramers doublet; so, although the specific heat data (Fig. 1f) show there is no conventional JahnTeller transition, magnetoelastic effects may have an important role in Pr_{2}Zr_{2}O_{7}.
Elastic and inelastic neutron scattering
Lowenergy transfer neutronscattering data provide a direct view of dynamics in the spin ice state. A 40 mm × 20 mm^{2} cylindrical single crystal was used to achieve sufficient magnetic scattering intensity from the relatively small Pr^{3+} moments. Nominally elastic magnetic scattering covering broad regions of momentum space develops upon cooling from 22 K to 0.1 K, indicating shortrange quasistatic spin correlations (Fig. 3b). Sharp pinch point features near (111) and particularly (002), bear evidence of a divergencefree twoin twoout spin configuration on each tetrahedron^{9,25,29,30}. Indeed, the elastic Qmap resembles a classical Monte Carlo simulation for an exchangeonly model^{9,29}, which indicates dominant ferromagnetic superexchange interactions in Pr_{2}Zr_{2}O_{7}. Θ_{W} is, however, negative and this suggests—consistent with recent theoretical predictions^{31,32}—the exchange Hamiltonian includes antiferromagnetic transverse terms that induce quantum dynamics. Though the ground state doublet is Jahn–Teller active, the observed scattering does not resemble that predicted for a model where the local three fold axis is broken^{30}.
The transverse width of the pinch points in elastic neutron scattering is a measure of the density of Pr tetrahedra that violate the ice rule^{7}, hence the monopole density. To determine this density, we measured the pattern of elastic magnetic neutron scattering near the Q=(002) pinch point with a highresolution configuration. Two configurations of the MACS instrument were used for this experiment. For rapid high statistical mapping of Qdependent scattering, we used a fixed final neutron energy E_{f}=5 meV. The corresponding measured energy resolution δE=0.38 meV and the calculated principal axes of the Q resolution near (002) are δQ_{1}=0.093 Å^{−1} and δQ_{2}=0.041 Å^{−1}. For highresolution measurements, we used E_{f}=2.7 meV, where δE=0.12 meV, δQ_{1}=0.093 Å^{−1}, δQ_{2}=0.025 Å^{−1}, and the incoherent elastic count rate is reduced by a factor 5. The pinch point width was extracted by fitting the following phenomenological model of scattering to twodimensional intensity maps in the (HHL) reciprocal lattice plane^{7,9} (see Supplementary Note 3 for details):
Both the quasistatic (τ>/δE=6 ps) total moment S(0), and the spin ice correlation length ξ_{ice} increase on cooling (Fig. 3c and Supplementary Figure S3). The low T limit, ξ_{0}, indicates a quasistatic monopole density of 1.2%. This can be compared with the ~1% concentration of Zr on Pr sites determined by single crystal synchrotron Xray diffraction (Supplementary Figure S2 and Supplementary Note 1). The fitting function, , (black solid line, Fig. 3c) describes the data well with the activation energy fixed at the value of Δ_{χ}=1.6 K extracted from ACχ(T) data (Fig. 1e, inset).
The wave vector map of inelastic scattering at ω=0.25 meV and T=0.1 K is shown in Fig. 3a. As ħω>>k_{B}T~0.01 meV, the inelastic scattering is evidence for quantum spin dynamics and the frequency scale of ν=ω/2π=60 GHz sets it apart from the thermally activated spin dynamics probed by AC susceptibility measurements on the kHz frequency scale. While the overall pattern resembles the nominally elastic scattering (Fig. 3b), the pinch points have vanished. Excited states thus differ from the ground state by the appearance of tetrahedra that violate the ice rule or in other words by the presence of magnetic monopoles. Combined, the elastic scattering that does show pinch points and the inelastic scattering that does not, constitute experimental evidence for magnetic monopoles with quantum dynamics. By normalizing the intensity data (Supplementary Note 3 and Supplementary Figure S3), we find that such inelastic scattering accounts for >90% of the magnetic scattering cross section of the lowenergy CEF doublet at T=0.1 K. This shows magnetism in Pr_{2}Zr_{2}O_{7} is dominated by quantum fluctuations.
To characterize the monopole fluctuation spectrum, we carried out constant Q=(003) scans at various Ts, subtracting T=15 K data as a background (Fig. 3d). The low T spectrum places an upper limit of ΔE=0.2 meV on any excitation gap. Upon heating the spectrum broadens and to quantify this effect, we fitted the Tdifference data to the scattering associated with a single imaginary pole response, χ(ω)=χ_{0}Γ/(Γ−iω), accounting for the background subtraction with a consistent T=15 K relaxation rate as detailed in Supplementary Note 3. Figure 3c shows the Tdependence of the relaxation rate, Γ. At high temperatures, Γ might be interpreted as a monopole relaxation rate. The red solid line shows , where C=1.4(2). The crossover to linearity and thus ω/T scaling is evidence of a regime for T>J_{ff}, where T is the only relevant energy scale. In the low T limit, Γ~0.17 meV is similar to the spin flip energy 2J_{ff}=Δ_{χ}~1.6 K inferred from ACχ(T). The observation of a broad spectrum rather than a sharp spin flip mode indicates monopole quantum dynamics^{12,14,15}. A possible lowenergy regime of photons predicted for a U(1) spin liquid^{12,14,15}, however, cannot be resolved in the present neutron scattering experiment.
Heat capacity
Complementary information about the excitation spectrum is provided by specific heat measurements (Fig. 1f) (see Supplementary Note 4 for details). After subtraction of the lattice and CEF contributions, the temperaturedependence of the magnetic and nuclearspecific heat C_{MN}(T) does not exhibit sharp features that would result from longrange ordering or a conventional Jahn–Teller transition. Instead, a broad peak at T~2 K ~Δ_{χ}=2J_{ff}~1.6 K (Fig. 1f) can be associated with the proliferation of thermally activated monopole pairs. While the position of the peak is similar to that for Dy_{2}Ti_{2}O_{7}, the greater width is consistent with the broad spectrum of inelastic neutron scattering and monopole quantum dynamics in Pr_{2}Zr_{2}O_{7}.
The upturn in C_{MN}(T) below ~200 mK provides valuable information about electronic dynamics on the nuclear spin time scale. A fully frozen state of ~3.0 μ_{B} Pr moments would generate a nuclear Schottky peak at ~0.17 K with a height of ~7 J mol^{−1} K^{−1}, which is inconsistent with the data. Instead, as shown in Fig. 1f, the upturn can be fitted by C_{N}(T)=f × C_{Sch}(T), where C_{Sch}(T) is a full expression of nuclear Schottkyspecific heat with a hyperfine coupled Pr moment and f=0.37 is a dimensionless factor representing the fraction of Pr sites with the dipole moment (Supplementary Note 4). The corresponding elastic neutron scattering cross section barn is not inconsistent with the low T magnetic scattering cross section of 0.02 barn within the 0.1 meV energy resolution of the experiment (Supplementary Figure S3). Here g_{J}=4/5, r_{0}=−0.54 × 10^{−12} cm and =2/3 is the spherical average of the polarization factor for Ising spins. The reduced frozen moment indicates strong quantum fluctuations^{33} as does the large value of the real part of the ACsusceptibility χ′(T) in the low T limit (Fig. 1d). The apparent inhomogeneity indicated by f=0.37<1 suggests the weak frozen moment could be an extrinsic feature resulting from extreme sensitivity to chemical inhomogeneity.
Assigning in this way the low T upturn to nuclear spins, the electronic entropy obtained by integrating C_{M}/T=(C_{MN}−C_{N})/T up to 20 K is , which is close to the value of for classical ice^{2,4} (Fig. 1f) and inconsistent with a full static Jahn–Teller distortion^{34}. The inferred zerofield magnetic component C_{M} shows a decade of activated Tdependence between 2 K and 0.2 K (Fig. 1f, inset). The corresponding activation energy of Δ_{C}=0.72(1) K is approximately half of that associated with ACsusceptibility data Δ_{χ}=1.62(3) K. Note, however, that the above interpretation is not unique and we cannot exclude the possibility that the low T upturn in C_{MN}(T) involves both nuclear and electronic spin entropy.
Discussion
From the multiple streams of distinct but consistent data presented, the following key characteristics of Pr_{2}Zr_{2}O_{7} emerge. (1) Strong quantum fluctuations: At T=0.1 K, >90% of the magnetic scattering cross section is inelastic, the attempt frequency associated with the activated AC susceptibility is three orders of magnitude larger than for classical spin ice, and the nuclear Schottky anomaly is suppressed below that for a classical frozen state both in terms of moment density and magnitude. (2) Weak static correlations: while exponentially activated specific heat and AC susceptibility data, elastic magnetic neutron scattering, and the nuclear Schottky anomaly consistently indicate some spinfreezing, this could be an extrinsic consequence of weak chemical inhomogeneities. (3) Spinicelike correlations at long times: the wave vector dependence of the lowest energy scattering detected features pinch points, the width of which is consistent with the inferred chemical defect density. While further experimental work is required to fully determine the ground state of Pr_{2}Zr_{2}O_{7} in the clean limit, it is safe to say that it features quantum fluctuations of magnetic monopoles and is unlike any previously documented in an insulating crystalline magnet. The interplay between monopolar quantum dynamics and itinerant electrons may have an important role in the isostructural Kondo lattice system Pr_{2}Ir_{2}O_{7} (refs 19, 20).
Methods
Sample preparation
Single crystalline samples of Pr_{2}Zr_{2}O_{7} were grown by a floating zone method^{35}. The orientation of each crystals was established by Xray Laue diffraction. Energy dispersive Xray analysis found no impurity phases and determined the Pr/Zr ratio to be 0.94(3). Synchrotron Xray diffraction measurements were performed at SPring8. Structure analysis, detailed in Supplementary Note 1, confirmed the pyrochlore crystal structure with symmetry. The crystal structure information is provided in Supplementary Tables S1 and S2. Further analysis of the diffraction data indicates no significant site mixing beyond the detection limit, but Prdeficiency at the ~1% level (Supplementary Figures S1, S2, and Supplementary Note 1). A single crystal of the nonmagnetic analog La_{2}Zr_{2}O_{7} was also grown by a floating zone method, and used to approximate the lattice specific heat of Pr_{2}Zr_{2}O_{7}.
Measurement of magnetic properties
The temperaturedependence of the DC magnetization at a field of 1000 Oe above 2 K was measured using a commercial SQUID magnetometer. The temperature dependence of the AC susceptibility at an excitation field of 0.3 Oe below 5 K was measured down to 20 mK in a dilution refrigerator through a mutual inductance method. For both measurements, the field was applied along the [111] direction. The AC susceptibility data sets were scaled to data collected at 1000 Oe for temperatures above 2 K. For all data, a demagnetization correction was made with a demagnetization factor N=0.2 obtained by approximating the sample shape as a rectangular prism^{36}. The temperaturedependence of the specific heat was measured by a thermal relaxation method down to 0.07 K. A thin platelet single crystal with a (111) surface normal was used to achieve good thermal contact. Details regarding the analysis of specific heat date are provided in Supplementary Note 4.
Neutron scattering measurements
Neutron scattering measurements were carried out on a 0.8 cm^{3} single crystal sample using the MACS spectrometer at the NIST Center for Neutron Research for energy transfer below 2 meV and the ARCS spectrometer at the Spallation Neutron Source, Oak Ridge National Laboratories for energy transfer up to 120 meV. The detailed methods of data collection and analysis employed for ARCS and MACS are described in Supplementary Notes 2 and 3 respectively.
Additional information
How to cite this article: Kimura, K. et al. Quantum fluctuations in spinicelike Pr_{2}Zr_{2}O_{7}. Nat. Commun. 4:1934 doi: 10.1038/ncomms2914 (2013).
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Acknowledgements
We thank Y. Karaki, Y. Shimura, T. Sakakibara, K. Kuga, M. Takigawa, T. M. McQueen, S. Koohpayeh, K. Matsuhira, Y. Nambu, A.A. Nugroho, Y. Ohta, S. Onoda, R. Satake, O. Tchernyshyov and Y. Wan for useful discussions. This work is partially supported by GrantinAid for Scientific Research (Nos. 21684019 and 23740259) from JSPS, by GrantinAid for Scientific Research on Priority Areas (No. 19052003) and GrantinAid for JSPS Fellows from MEXT, Japan, by Global COE Program ‘the Physical Sciences Frontier’, MEXT, Japan, by PRESTO of JST, by a Toray Science and Technology Grant and by USJapan Cooperative Program, ISSP. Work at IQM was supported by the US DoE, office of Basic Energy Sciences, Division of Material Sciences and Engineering under DEFG0208ER46544. This work utilized facilities supported in part by the NSF under DMR0944772. Research conducted at ORNL’s Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy. Part of this work was conducted while CB was a JSPS fellow at ISSP. The use of the Materials Design and Characterization Laboratory at ISSP is gratefully acknowledged.
Author information
Affiliations
Institute for Solid State Physics (ISSP), University of Tokyo, Kashiwa, Chiba 2778581, Japan
 K. Kimura
 & S. Nakatsuji
PRESTO, Japan Science and Technology Agency (JST), 418 Honcho Kawaguchi, Saitama 3320012, Japan
 S. Nakatsuji
Institute for Quantum Matter and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA
 JJ. Wen
 & C. Broholm
NIST Center for Neutron Research, NIST, Gaithersburg, Maryland 20899, USA
 C. Broholm
Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
 C. Broholm
 & M. B. Stone
Department of Applied Physics, Graduate School of Engineering, Nagoya University, Nagoya 4648603, Japan
 E. Nishibori
 & H. Sawa
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Contributions
S.N. conceived the project. S.N. and C.B. coordinated experiments. K.K. and S.N. performed synthesis, crystal growth, and magnetic susceptibility measurements. J.W., C.B., S.N., and K.K. collected and analyzed thermodynamic and neutron diffraction data. M.B.S. collected crystal field inelastic scattering data, E.N. and H.S. collected and analyzed synchrotron Xray diffraction data. S.N. C.B. K.K. and J.W. wrote the paper. All authors critically reviewed the paper.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to S. Nakatsuji.
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