Abstract
The Ising model—in which degrees of freedom (spins) are binary valued (up/down)—is a cornerstone of statistical physics that shows rich behaviour when spins occupy a highly frustrated lattice such as kagome. Here we show that the layered Ising magnet Dy_{3}Mg_{2}Sb_{3}O_{14} hosts an emergent order predicted theoretically for individual kagome layers of inplane Ising spins. Neutronscattering and bulk thermomagnetic measurements reveal a phase transition at ∼0.3 K from a disordered spinicelike regime to an emergent charge ordered state, in which emergent magnetic charge degrees of freedom exhibit threedimensional order while spins remain partially disordered. Monte Carlo simulations show that an interplay of interlayer interactions, spin canting and chemical disorder stabilizes this state. Our results establish Dy_{3}Mg_{2}Sb_{3}O_{14} as a tuneable system to study interacting emergent charges arising from kagome Ising frustration.
Introduction
The kagome lattice—a twodimensional (2D) arrangement of cornersharing triangles—is at the forefront of the search for exotic states generated by magnetic frustration. Such states have been observed experimentally for Heisenberg^{1,2,3,4} and planar^{5,6,7} spins. If Ising spins lie within kagome planes and point either towards or away from the centre of each triangle, the potential for emergent behaviour is shown by considering a spin (magnetic dipole) as two separated + and −magnetic charges: the emergent charge of a triangle is defined as the algebraic sum over the three charges it contains (Fig. 1a)^{8}. Ferromagnetic nearestneighbour interactions favour states, yielding six degenerate states on each triangle. This macroscopic groundstate degeneracy leads to a zeropoint entropy S_{0}≈ ln R per mole of Dy (where R is the molar gas constant), and suppresses spin order^{9}, in analogy to threedimensional (3D) spinice materials^{10,11}. The longrange magnetic dipolar interaction generates an effective Coulomb interaction between emergent charges, driving a transition to an emergent charge ordered (ECO) state that is absent for nearestneighbour interactions alone^{8,12}. In this state, + and − charges alternate, but the remaining threefold degeneracy of spin states for each charge means that spin order is only partial (Fig. 1b). The ECO state has two bulk experimental signatures: nonzero entropy S_{0}≈0.11R per mole of Dy^{12}, and the presence of both Bragg and diffuse magnetic scattering in neutronscattering measurements^{13,14}. Experimentally, kagome ECO states have been observed in spinice materials under applied magnetic field^{15,16} and nanofabricated systems in the 2D limit^{14,17,18,19}. However, a crucial experimental observation has remained elusive—namely, observation of the spatial arrangement of emergent charges in a bulk kagome material.
In this article, we show that an ECO state exists at low temperature in the recentlyreported bulk kagome magnet Dy_{3}Mg_{2}Sb_{3}O_{14} (ref. 20). Our experimental evidence derives from neutronscattering and thermodynamic measurements, while Monte Carlo (MC) simulations reveal that this ECO state is stabilized by a combination of interactions between kagome layers, spin canting out of kagome layers and chemical disorder.
Results
Structural and magnetic characterization
Structural and magnetic characterization suggests that Dy_{3}Mg_{2}Sb_{3}O_{14} (ref. 20) is an ideal candidate for an ECO state. The material crystallizes in a variant of the pyrochlore structure (space group Rm^{20}) in which kagome planes of magnetic Dy^{3+} alternate with triangular layers of nonmagnetic Mg^{2+} (Fig. 1c). Xray and neutron powder diffraction measurements confirm the absence of a structural phase transition to ≲0.2 K (Supplementary Figs 1 and 2 and Supplementary Tables 1 and 2) and reveal a small amount of site disorder in our sample, with 6(2)% of Dy kagome sites occupied by Mg (and 18(6)% of Mg sites occupied by Dy). CurieWeiss fits to the magnetic susceptibility (Fig. 1d) yield a CurieWeiss constant θ_{CW}=−0.1(2) K for fitting range 5≤T≤50 K, consistent with ref. 20 (however, the value depends strongly on fitting range). Demagnetization effects may also be significant—increasing θ_{CW} by 1.4 K in spinice materials^{21}—but cannot be quantitatively determined for a powder sample. The local Dy environment in Dy_{3}Mg_{2}Sb_{3}O_{14} is similar to the cubic spin ice Dy_{2}Ti_{2}O_{7} (ref. 22) (Supplementary Fig. 3), suggesting that Dy^{3+} spins have an Ising anisotropy axis directed into or out of the kagome triangles with an additional component perpendicular to the kagome planes. Experimentally, we confirm Ising anisotropy at low temperatures using isothermal magnetization measurements, which are ideally described by paramagnetic Ising spins with magnetic moment μ=10.17(8) μ_{B} per Dy (Fig. 1e). Moreover, our inelastic neutronscattering measurements show that the groundstate Kramers doublet is separated from the first excited crystalfield state by at least 270 K (Supplementary Fig. 4), indicating that crystalfield excitations are negligible at the low temperatures (≤50 K) we consider.
Lowtemperature spin correlations
The magnetic specific heat C_{m}(T) shows that spin correlations start to develop below 5 K and culminate in a large anomaly at T*=0.31(1) K that we attribute to a magnetic phase transition (Fig. 2a and Supplementary Fig. 5). Below 0.20 K, the spins fall out of equilibrium, as is also reported in spinice materials^{23}. In zero applied field, the entropy change ΔS_{m}(T) from 0.2 K to T=10 K is slightly less than the expected Rln2 for random Ising spins; however, the full Rln2 entropy is recovered in a small applied field of 0.5 T. The 0.05(3)R difference between ΔS_{m}(10 K) in zero field and in a 0.5 T field could be explained either by ECO (with entropy 0.11R in the 2D case^{12}), or by the ∼6% randomlyoriented orphan Dy spins on the Mg site (with entropy 0.06 Rln2). Neutronscattering experiments on a powder sample of ^{162}Dy_{3}Mg_{2}Sb_{3}O_{14} distinguish these two scenarios by revealing the microscopic processes at play across T*. Figure 2b shows magnetic neutronscattering data at 0.5 K (above T*) and at the nominal base temperature of 0.03 K (below T*). At 0.5 K, our data show magnetic diffuse scattering only, with a broad peak centred at ≈0.65 Å^{−1} that is characteristic of icerule correlations in structurally related pyrochlore magnets^{24}. In contrast, at 0.03 K, strong magnetic diffuse scattering is observed in addition to magnetic Bragg peaks. These peaks develop at T≤0.35 K; that is, as T* is crossed. No additional peaks are observed on further cooling and the magnetic scattering does not change between 0.1 and 0.03 K. Between 0.03 and 50 K, the scattering is purely elastic within our maximum experimental resolution of ≈17 μeV (Supplementary Fig. 6), indicating that the spins fluctuate on a timescale longer than ∼0.2 ns. Our 0.03 K data suggest two immediate conclusions. First, the magnetic Bragg peaks are described by the propagation vector k=0; that is, order preserves the crystallographic unit cell below T*. Second, a large fraction of the magnetic scattering is diffuse; hence, correlated spin disorder persists below T* and involves the majority of spins. These results cannot be explained by only a small fraction of orphan spins, but are consistent with an ECO state^{13,14}.
Average magnetic structure
We use reverse Monte Carlo (RMC) refinement^{25,26} to fit spin microstates to data collected between 0.03 and 4 K. A single RMC microstate can capture both the average spin structure responsible for Bragg scattering and the local deviations from the average responsible for diffuse scattering (Fig. 2b and Supplementary Fig. 7). We determine the average spin structure by two methods: first, by averaging refined RMC microstates onto a single unit cell; second, by using a combination of symmetry analysis and Rietveld refinement to model the magnetic Bragg profile (obtained as the difference between 0.03 and 0.5 K data) (Fig. 2c). Details of the Rietveld refinements are given in Supplementary Note 1. Both approaches yield the same allin/allout average spin structure (inset to Fig. 2c and Supplementary Fig. 8). The ordered magnetic moment at 0.03 K, μ_{avg}=2.82(4) μ_{B} per Dy, is much less than the total moment of μ≈10 μ_{B}. These results are consistent with ECO: Fig. 2d shows that averaging over the three possible ECO microstates for a given triangle generates an allin/allout average structure, as observed experimentally; moreover, the expected ordered moment for ECO, μ/3≈3.3 μ_{B} per Dy^{13}, is in general agreement with the measured value of 2.82(4) μ_{B} per Dy.
Evidence for emergent charge order
To look for signatures of ECO in real space, we compare the temperature evolution of μ_{avg} with the percentage of charges (Fig. 3a). The latter quantity, f_{±3}, takes a value of 25% for random spins, 100% for an allin/allout microstate, and 0% for a microstate that fully obeys the ice rule. The value of f_{±3} extracted from RMC refinements decreases with lowering temperature to a minimum value of <5% below 1 K; these values represent upper bounds because RMC refinements were initialized from random microstates. Crucially, below T*, the rule is obeyed while μ_{avg} is nonzero (Fig. 3a); this coexistence of icerule correlations with an allin/allout average structure is a defining feature of the ECO state^{13,14}. We confirm ECO by calculating the chargecorrelation function , the average product of charges separated by radial distance r_{ab} on the honeycomb lattice formed by the triangle midpoints. At 0.5 K, this function decays with increasing r_{ab}, indicating that charges are disordered (Fig. 3b). At 0.03 K, shows two key features that indicate an ECO state: a diverging correlation length, and an alternation in sign with a negative peak at the nearestneighbour distance (Fig. 3c). The magnitude of found experimentally (≈0.6=(0.94 × 3μ_{avg}/μ)^{2}) is smaller than the value of unity corresponding to an ideal ECO state, which indicates that the alternation of charges contains some errors; we show below this is probably due to the presence of site disorder.
Explanation of emergent charge order
Why does Dy_{3}Mg_{2}Sb_{3}O_{14} show fundamentally the same ECO as predicted for a 2D kagome system of inplane Ising spins? This is far from obvious, because the real material differs from the existing model^{8} in three respects: (i) the spins are canted at an angle of 26(2)° to the kagome planes, (ii) the planes are layered in 3D and (iii) there is Dy/Mg site disorder (Fig. 2c). This puzzle is elucidated by Monte Carlo simulations for a minimal model containing the nearestneighbour exchange interaction J=−3.72 K determined for structurallyrelated Dy_{2}Ti_{2}O_{7} (refs 22, 27), and the longrange magnetic dipolar interaction D=1.28 K calculated from experimentally determined Dy–Dy distances. In 2D, spin canting interpolates between two limits—an ECO transition followed by lowertemperature spin ordering for inplane spins^{8}, and a single spinordering transition for spins perpendicular to kagome planes^{28}—and hence destabilizes ECO compared with the 2D inplane limit. In contrast, the stacking of kagome planes stabilizes 3D ECO—uniquely minimizing the effective Coulomb interaction between emergent charges—but leaves the spinordering transition temperature essentially unchanged. The effect of random site disorder is shown in Fig. 3d. Disorder broadens the specificheat anomalies and suppresses the ECO transition temperature. In spite of this, we find that a distinct ECO phase persists for 6% Mg on the Dy site; that is, the estimated level of disorder present in our sample of Dy_{3}Mg_{2}Sb_{3}O_{14}. Moreover, simulated magnetic specificheat (Fig. 3d) and powder neutronscattering (Supplementary Fig. 9) curves with ∼4 to 6% Mg on the Dy site show remarkably good agreement with experimental data, especially given that J is not optimized for Dy_{3}Mg_{2}Sb_{3}O_{14} .
Implications of emergent charge order
An ECO microstate can be coarsegrained into a magnetization field with two components: the allin/allout average spin structure with nonzero divergence, and the local fluctuations from the average that are captured by (divergencefree) dimer configurations on the dual honeycomb lattice^{13}. These two components are independent, which leads to descriptions of the ECO state in terms of spin fragmentation^{13,14}. Without site disorder, the fluctuating component yields pinchpoint features in singlecrystal diffusescattering patterns, the signature of a Coulomb phase^{13,29}. Figure 3e shows that the introduction of site disorder blurs the pinch points and reduces the magnitude of the ordered moment in the ECO phase. We find good overall agreement between patterns from model simulations with ∼4 to 6% Mg on the Dy site and from RMC microstates refined to powder data (Fig. 3e). These results suggest that pinchpoint scattering could be observed in singlecrystal samples of Dy_{3}Mg_{2}Sb_{3}O_{14} with low levels of disorder. Our simulations also suggest why a transition from ECO to spin ordering is not observed experimentally: singlespinflip dynamics (arguably more appropriate to real materials) become frozen in the ECO state and nonlocal (loop) dynamics are required to observe the spinordering transition in Monte Carlo simulations.
Discussion
The ECO state in Dy_{3}Mg_{2}Sb_{3}O_{14} is the first realization of ordering of emergent degrees of freedom in a solidstate kagome material. Phase transitions driven by emergent excitations are rare—related examples being the critical endpoint in spin ice^{11,30,31} and the recent report of spin fragmentation in pyrochlore Nd_{2}Zr_{2}O_{7} (ref. 32). Moreover, the unusually slow spin dynamics offer the exciting possibility of measuring finitetime (KibbleZurek) scaling at the ECO critical point^{31}. The ECO state in Dy_{3}Mg_{2}Sb_{3}O_{14} presents an intriguing comparison with other partially ordered magnets. In Gd_{2}Ti_{2}O_{7}, symmetry breaking yields two inequivalent Gd sites, only one of which orders^{33,34}; in contrast, in the ECO state, all spins possess both ordered and disordered components. In Ho_{3}Ga_{5}O_{12}, local antiferromagnetic correlations coexist with average antiferromagnetic order^{35}, whereas in the ECO state, the average order is antiferromagnetic (allin/allout) while the local correlations are ferromagnetic (twoin/oneout or vice versa). Whether the predicted spinordering^{8} eventually occurs in Dy_{3}Mg_{2}Sb_{3}O_{14} remains to be seen: spin freezing^{36,37} or site disorder may prevent its onset. We expect physical and/or chemical perturbations to control the properties of Dy_{3}Mg_{2}Sb_{3}O_{14} ; for example, application of magnetic field slightly tilted from the caxis should drive a Kastelyn transition towards spinordering^{15,16}; modified synthesis conditions may allow the degree of site mixing to be controlled^{20}; and application of chemical pressure may alter the spincanting angle and/or the distance between kagome layers, potentially generating a novel spinordering phase instead of ECO for sufficiently large canting^{28}. Substitution of Dy^{3+} by other lanthanide ions^{20,38,39,40} may increase the ratio of exchange to dipolar interactions, offering promising routes towards exotic spinliquid behaviour: dimensionality reduction by effective layer decoupling (when exchange dominates over dipolar interactions), and realization of quantum kagome systems with local spin anisotropies.
Methods
Sample preparation
Powder samples of Dy_{3}Mg_{2}Sb_{3}O_{14} were prepared from a stoichiometric mixture of dysprosium (III) oxide (99.99%, Alfa Aesar*), magnesium oxide (99.998%, Alfa Aesar*) and antimony (V) oxide (99.998%, Alfa Aesar*). For neutronscattering experiments a ∼5 g sample isotopically enriched with ^{162}Dy (94.4(2)% ^{162}Dy_{2}O_{3}, CK Isotopes*) was prepared. For all samples, starting materials were intimately mixed and pressed into pellets before heating at 1,350 °C for 24 h in air. This heating step was repeated until the amount of impurity phases as determined by Xray diffraction was no longer reduced on heating. The enriched sample contained impurity phases of MgSb_{2}O_{6} (6.4(5) wt%) and Dy_{3}SbO_{7} (0.97(8) wt%), the latter of which orders antiferromagnetically at T≈3 K (ref. 41).
*The name of a commercial product or trade name does not imply endorsement or recommendation by the National Institute of Standards and Technology (NIST).
Xray diffraction measurements
Powder Xray diffraction was carried out using a Panalytical Empyrean* diffractometer with Cu Kα radiation (λ=1.5418 Å). Measurements were taken between 5≤2θ≤120° with Δ2θ=0.02°.
*The name of a commercial product or trade name does not imply endorsement or recommendation by NIST.
Neutronscattering measurements
Powder neutron diffraction measurements were carried out on the General Materials (GEM) diffractometer at the ISIS Neutron and Muon Source, Harwell, UK^{42}, at T=0.50, 0.60, 0.90, 2.0, 4.0, 25 and 300 K. For T=25 and 300 K measurements, around 4.2 g of isotopically enriched powder was loaded into a ϕ=6 mm vanadium can and cooled in a flow cryostat. For measurements at T≤25 K, the same sample was loaded into a ϕ=6 mm vanadium can, which was attached directly to a dilution refrigerator probe and loaded within a flow cryostat. Inelastic neutronscattering experiments were carried out on the Disk Chopper Spectrometer (DCS) at the NIST Center for Neutron Research, Gaithersburg MD, USA^{43}, at T=0.03, 0.10, 0.20, 0.30, 0.35, and 0.50 K. Around 1.1 g of isotopically enriched powder was loaded into a ϕ=4.7 mm copper can and mounted at the base of a dilution refrigerator. The temperature was measured at the mixing chamber and does not necessarily reflect the sample temperature for 0.1 and 0.03 K, as the spins progressively fall out of equilibrium. On DCS, data were measured with incident wavelengths of 1.8, 5 and 10 Å. The 1.8 Å data were used to look for crystalfield excitations (Supplementary Fig. 4). The 10 Å data were used to look for lowenergy quasielastic scattering (Supplementary Fig. 6). The 5 Å data were integrated over the energy range −0.15≤E≤0.15 meV to obtain the total scattering (Supplementary Fig. 10). Data reduction was performed using the MANTID and DAVE^{44} programs. All data were corrected for detector efficiency using a vanadium standard, normalized to beam current (GEM) or incident beam monitor (DCS), and corrected for absorption by the sample.
Crystalstructure refinements
Combined Rietveld analysis of the 300 K Xray and neutron (GEM) diffraction data was carried out using the FULLPROF suite of programs^{45}. The individual patterns were weighted so that the total contribution from Xray and neutron diffraction was equal; that is, data from each of the five detector banks on GEM was assigned 20% of the weighting of the single Xray pattern. The neutron scattering crosssection for Dy was fixed to b_{Dy}=−0.6040, fm, to reflect the isotopic composition as determined by inductively coupled plasma mass spectrometry. Peak shapes were modelled using a pseudoVoigt function, convoluted with an IkedaCarpenter function or an axial divergence asymmetry function for neutron and Xray data, respectively. Backgrounds were fitted using a Chebyshev polynomial function. At 25 K, Rietveld analysis of only the neutron diffraction data was carried out. In addition to the impurity phases observed in Xray diffraction, a small amount (<1 wt%) of vanadium (IV) oxide from corrosion of the vanadium sample can was also observed in the neutrondiffraction data. The fit to 300 K data is shown in Supplementary Fig. 1, refined values of structural parameters are given in Supplementary Table 1, and selected bond lengths are given in Supplementary Table 2.
Magnetic measurements
Magnetic susceptibility measurements, χ(T)=M(T)/H, were made using a Quantum Design* Magnetic Properties Measurement System (MPMS) with a superconducting interference device (SQUID) magnetometer. Measurements were made after cooling in zero field (ZFC) and in the measuring field (FC) of μ_{0}H=0.1 T over the temperature range 2≤T≤300 K. Isothermal magnetization M(H) measurements were made using a Quantum Design* Physical Properties Measurement System (PPMS) at selected temperatures 1.6≤T≤80 K between −14≤μ_{0}H≤14 T. A global fit to the M(H) data for T≥5 K (Fig. 1e) was performed using the powderaveraged form for free Ising spins,
where H is applied magnetic field, and magnetic moment μ is the only fitting parameter^{21}. The fitted value μ=10.17(8) μ_{B} per Dy is in close agreement with the expected value of 10.0 μ_{B} for a Kramers doublet ground state with g=4/3 and m_{J}=±15/2; in particular, the reduced value of the saturated magnetization, M_{sat}≈μ/2, is as expected for powderaveraged Ising spins^{21}.
*The name of a commercial product or trade name does not imply endorsement or recommendation by NIST.
Heatcapacity measurements
Heatcapacity measurements were carried out on a Quantum Design* Physical Properties Measurement System instrument using dilution fridge (0.07≤T≤4 K) and standard (1.6≤T≤250 K) probes in a range of measuring fields, 0≤μ_{0}H≤0.5 T. To ensure sample thermalization at low temperatures, measurements were made on pellets of Dy_{3}Mg_{2}Sb_{3}O_{14} mixed with an equal mass of silver powder, the contribution of which was measured separately and subtracted to obtain C_{p}. The magnetic specific heat C_{m} was obtained by subtracting modelled lattice C_{l} and nuclear C_{n} contributions from C_{p}. We obtained C_{l} by fitting an empirical Debye model to the 10<T<200 K data, with θ_{D}=272(13) K. To obtain a lower bound on the contact hyperfine and electronic quadrupolar contributions to C_{p}^{23,46}, we used previous experimental results on dysprosium gallium garnet^{47}, a related material for which these contributions are known down to T=0.037 K. Correcting for the larger static electronic moment ≈4.2 μ_{B} of dysprosium gallium garnet compared with 〈μ〉≥2.5 μ_{B} below 0.2 K for Dy_{3}Mg_{2}Sb_{3}O_{14}, we obtained the hightemperature tail of the nuclear hyperfine contributions as C_{p}=A/T^{2} with A=0.0032 J K (Supplementary Fig. 5).
*The name of a commercial product or trade name does not imply endorsement or recommendation by NIST.
Average magnetic structure analysis
The magnetic Bragg profile was obtained by subtracting data collected at T<0.5 K from the 0.5 K data. Refinements were carried out using the Rietveld method within the FULLPROF suite of programs^{45}, as described above. For the magneticstructure refinement shown in Fig. 2c, candidate magnetic structures were determined using symmetry analysis^{48} via the SARAH^{49} and ISODISTORT^{50} programs, as described in Supplementary Note 1. The average magnetic structure is described by the irreducible representation Γ_{3}, in Kovalev’s notation^{51}. The basis vectors of the magnetic structure are given in Supplementary Table 3 and refined values of structural parameters are given in Supplementary Table 4.
Magnetic total scattering
To isolate the total magnetic contribution to the neutronscattering data, data collected at a high temperature T_{high}>>θ_{CW} were subtracted from the lowtemperature data of interest, where T_{high}=25 K (GEM data) or 50 K (DCS data). For the data obtained below the magnetic ordering temperature of the Dy_{3}SbO_{7} impurity phase (≈3 K (ref. 41)), a refined model of the magnetic Bragg scattering of Dy_{3}SbO_{7} was subtracted, as described in Supplementary Note 2 (we note that the orthorhombic crystal structure of Dy_{3}SbO_{7} (ref. 52) allowed the impurity Bragg peaks to be readily distinguished from sample peaks). The fit to neutron data of the Dy_{3}SbO_{7} magneticstructure model is shown in Supplementary Fig. 11, the magnetic basis vectors are given in Supplementary Table 5, and refined values of structural parameters are given in Supplementary Table 6. The data were placed on an absolute intensity scale (barn sr^{−1} Dy^{−1}) by normalization to the calculated nuclear Bragg profile at T_{high}.
Reverse Monte Carlo refinements
Refinements to the total (Bragg+diffuse) magnetic scattering were performed using a modified version of the SPINVERT program^{53} available from J.A.M.P. In these refinements, a microstate was generated as a periodic supercell containing N=7776 Dy^{3+} spin vectors , where μ=10.0 μ_{B} is the fixed magnetic moment length, the unit vector specifies the local Ising axis determined from Rietveld refinement, and the Ising variable σ_{i}=±1. A random sitedisorder model with 6% nonmagnetic Mg on the Dy site was assumed, and S_{i}≡0 for atomic positions occupied by Mg. Ising variables were initially assigned at random, and then refined against experimental data in order to minimize the sum of squared residuals,
where I(Q) is the magnetic totalscattering intensity at Q, subscripts ‘calc’ and ‘expt’ denote calculated and experimental intensities, respectively, σ(Q) is an experimental uncertainty, and W is an empirical weighting factor. For data collected on GEM, a refined flatinQ background term was included in the calculated I(Q). For data collected at T≤0.35 K, we obtain I_{calc}(Q)=I_{Bragg}(Q)+I_{diffuse}(Q)−I_{random}(Q), where subscripts ‘Bragg’, ‘diffuse’ and ‘random’ indicate magnetic Bragg, magnetic diffuse and hightemperature contributions, respectively. Here, I_{random}(Q)=C[μf(Q)/μ_{B}]^{2}, where the constant C=(γ_{n}r_{e}/2)^{2}=0.07265 barn and f(Q) is the Dy^{3+} magnetic form factor^{54}. The Bragg and diffuse contributions were separated by applying the identity S_{i}≡〈S_{i}〉+ΔS_{i} to each atomic position^{55}, where the average spin direction 〈S_{i}〉 is obtained by vector averaging the supercell onto a single unit cell, and the local spin fluctuation ΔS_{i}≡S_{i}−〈S_{i}〉. The Bragg contribution is given by
in which G is a reciprocal lattice vector with length G, V is the volume of the unit cell, N_{c} is number of unit cells in the supercell, R(Q−G) is the resolution function determined from Rietveld refinement^{56}. The magnetic structure factor F^{⊥}(G)=∑_{i}〈S_{i}〉^{⊥} exp(iG·r_{i}), where supercript ‘⊥’ indicates projection perpendicular to G, and the sum runs over all atomic positions in the unit cell. The diffuse contribution is given by
where sums run over all atomic positions in the supercell, r_{ij} is the radial distance between positions i and j, and the correlation coefficients A_{ij}=ΔS_{i}·ΔS_{j}−(ΔS_{i}·r_{ij})(ΔS_{j}·r_{ij})/ and B_{ij}=3(ΔS_{i}·r_{ij})(ΔS_{j}·r_{ij})/−ΔS_{i}·ΔS_{j} (refs 53, 57). For data collected at T≥0.5 K, which show no magnetic Bragg scattering, we obtain I_{calc}(Q)=I_{diffuse}(Q)−I_{random}(Q), where S_{i} replaces ΔS_{i} everywhere. All refinements employed the Metropolis algorithm with singlespin flip dynamics, and were performed for 200 proposed flips per spin, after which no significant reduction in χ^{2} was observed. Fitstodata at T=0.03, 0.20, 0.50, 0.60, 0.90, 2.0 and 4.0 K are shown in Supplementary Fig. 7.
Monte Carlo simulations
Simulations were performed for the dipolar spin ice model^{27,58}, extended to the geometry of interest in this work. The model is defined for Ising spins , which are constrained to point along the local easyaxis directions and can thus be described by the Ising pseudospin variables, σ_{i}=±1. The Hamiltonian comprises an exchange term of strength J between nearestneighbour spins 〈i, j〉, and longrange dipolar interactions of characteristic strength D=(μ_{0}/4π)μ^{2}/ between all pairs of spins, where μ≈10 μ_{B} is the magnitude of the Dy^{3+} spin and r_{nn} is the nearestneighbour distance of the lattice. The Hamiltonian is thus given by
where r_{ij} is the vector of length r_{ij} connecting spins i and j. We use D=1.28 K as calculated from experimentally determined DyDy distances, and J=−3.72 K from Dy_{2}Ti_{2}O_{7} (ref. 27), which has a similar Dy environment to Dy_{3}Mg_{2}Sb_{3}O_{14} (ref. 22) (Supplementary Fig. 3). We treat the longrange dipolar interactions using Ewald summation^{58,59} with tinfoil boundary conditions at infinity. In simulations including site disorder, nonmagnetic ions are simulated by setting the corresponding σ_{i} to zero. Our unit cell comprises three stacked kagome layers, each layer made from four kagome triangles. The whole system comprises N=7776 spins in total, commensurate with the possible × spinordered state found in 2D (ref. 8). We use both singlespin flip and loop dynamics^{58,60}, with Metropolis weights. Loop dynamics are necessary to ensure ergodicity at low temperatures and explore possible longrange spinordered states. We use the short loop algorithm^{58,60}. One Monte Carlo sweep is defined as N single spinflip attempts, followed by the proposal of loop moves until the cumulative number of proposed spinflips (in the loops) is at least N. We use an annealing protocol, initializing the system at high temperature with ∼10^{4}N single spinflip attempts, then decrease the temperature incrementally. After each temperature decrement, the system is updated with ∼10^{3} Monte Carlo sweeps to ensure equilibration before collecting data every ∼10 Monte Carlo sweeps. Powderaveraged magnetic neutronscattering patterns calculated from Monte Carlo are shown in Supplementary Fig. 9.
Data availability
The underlying research materials can be accessed at the following location: http://dx.doi.org/10.17863/CAM.4902.
Additional information
How to cite this article: Paddison, J. A. M. et al. Emergent order in the kagome Ising magnet Dy_{3}Mg_{2}Sb_{3}O_{14}. Nat. Commun. 7, 13842 doi: 10.1038/ncomms13842 (2016).
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Acknowledgements
Work at Cambridge was supported through the Winton Programme for the Physics of Sustainability. The work of J.A.M.P., X.B. and M.M. and facilities at Georgia Tech were supported by the College of Sciences through M.M. startup funds. J.A.M.P. gratefully acknowledges Churchill College, Cambridge for the provision of a Junior Research Fellowship. H.S.O. acknowledges a Teaching Scholarship (Overseas) from the Ministry of Education, Singapore. J.O.H. is grateful to the Engineering and Physical Sciences Research Council (EPSRC) for funding. C.C. was supported by EPSRC Grant No. EP/G049394/1, and the EPSRC NetworkPlus on ‘Emergence and Physics far from Equilibrium’. Experiments at the ISIS Pulsed Neutron and Muon Source were supported by a beamtime allocation from the Science and Technology Facilities Council. This work utilized facilities at the NIST Center for Neutron Research. Monte Carlo simulations were performed using the Darwin Supercomputer of the University of Cambridge High Performance Computing Service (http://www.hpc.cam.ac.uk/) and the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk/, for which access was provided by an ARCHER Instant Access scheme). We thank G.W. Chern, J. Goff, A. L. Goodwin, G. Lonzarich, G. Moller, D. Prabhakaran, J. R. Stewart and A. Zangwill for valuable discussions, and M. Kwasigroch for preliminary theoretical work.
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H.S.O., P.M. and S.E.D prepared the samples. H.S.O., P.M., X.B., M.M. and S.E.D. performed and analysed the thermomagnetic measurements. J.A.M.P., P.M., X.B., M.G.T., N.P.B. and S.E.D. performed the neutronscattering measurements and J.A.M.P., M.M. and S.E.D. analysed the data. J.A.M.P. carried out the RMC refinements. J.O.H. and C.C. carried out the Monte Carlo simulations. C.C. and S.E.D. conceived the project, which was supervised by C.C., M.M. and S.E.D. J.A.M.P. wrote the paper with input from all authors.
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Paddison, J., Ong, H., Hamp, J. et al. Emergent order in the kagome Ising magnet Dy_{3}Mg_{2}Sb_{3}O_{14}. Nat Commun 7, 13842 (2016). https://doi.org/10.1038/ncomms13842
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DOI: https://doi.org/10.1038/ncomms13842
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