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Spin dynamics of the director state in frustrated hyperkagome systems

We present an experimental study of the magnetic structure and dynamics of two frustrated hyperkagome compounds, Gd3Ga5O12 and Gd3Al5O12. It has previously been shown that Gd3Ga5O12 exhibits long-range correlations of multipolar directors, that are formed from antiferromagnetic spins on loops of ten ions. Using neutron diffraction and Reverse Monte Carlo simulations we prove the existence of similar magnetic correlations in Gd3Al5O12, showing the ubiquity of these complex structures in frustrated hyperkagome materials. Using inelastic neutron scattering we shed further light on the director state and the associated low lying magnetic excitations. In addition we have measured quasielastic dynamics that show evidence of spin diffusion. Finally, we present AC susceptibility measurements on both Gd3Ga5O12 and Gd3Al5O12, revealing a large difference in the low frequency dynamics between the two otherwise similar compounds.

We present an experimental study of the magnetic structure and dynamics of two frustrated hyperkagome compounds, Gd3Ga5O12 and Gd3Al5O12. It has previously been shown that Gd3Ga5O12 exhibits long-range correlations of multipolar directors, that are formed from antiferromagnetic spins on loops of ten ions. Using neutron diffraction and Reverse Monte Carlo simulations we prove the existence of similar magnetic correlations in Gd3Al5O12, showing the ubiquity of these complex structures in frustrated hyperkagome materials. Using inelastic neutron scattering we shed further light on the director state and the associated low-lying magnetic excitations. In addition we have measured quasielastic dynamics that show evidence of spin diffusion. Finally, we present AC susceptibility measurements on both Gd3Ga5O12 and Gd3Al5O12, revealing a large difference in the low frequency dynamics between the two otherwise similar compounds.

I. INTRODUCTION
Emergence is the phenomenon of collective behavior that does not depend solely on the individual parts of a system, but rather through their interactions. This effect often results in novel and diverse states of matter. For example, in the field of magnetic frustration, emergent phenomena have been gracefully demonstrated by the observation of magnetic monopoles emerging in spin ice materials [1][2][3][4][5] .
A further example is the director state uncovered in the geometrically frustrated garnet, Gd 3 Ga 5 O 12 (GGG) 6 . The Gd 3+ ions in GGG form two interpenetrating hyperkagome lattices. In these structures, the resultant moment on connected loops of ten ions form a nematic director state at low temperatures 6 . Fig. 1 illustrates three such ten-ion loops. Taking the position of each director as the atom at the center of the ten-ion loop (illustrated in black), the directors also form a hyperkagome structure.
The directors in GGG are reminiscent of the emergent cluster state in the geometrically frustrated spinel ZnCr 2 O 4 7 , where groups of six spins self-organise into decoupled antiferromagnetic (AF) loops, also known as director protectorates.
It has been suggested that such directors are inherent in geometrically frustrated systems, and that they provide an organising principle in which emergent clusters form out of a manifold of ground states with the low temperature dynamics governed by the director state.
In Gd 3 Ga 5 O 12 the director state originates from the interplay between near-neighbor AF interactions and local xy anisotropy 6 . The directors are highly anisotropic and align along their local z-axis. The alignment of all the directors along their local z axis leads to long range correlations, in contrast to the decoupled directors in ZnCr 2 O 4 . The observed magnetic excitations in GGG range from the peV to the meV scale [8][9][10][11][12][13] , and the ten-ion loops seem to be central to understanding these dynamics in GGG 8,13 .
In this paper we delve further into the dynamics of the emergent behavior of spins on the hyperkagome lattice. We first reveal the existence of the director state in Gd 3 Al 5 O 12 (GAG), isostructural to GGG, thus showing that such emergent behavior is not unique to GGG. We further probe the low energy magnetic dynamics in GGG and GAG using AC magnetic susceptibility and inelastic neutron scattering with µeV and meV energy resolution. Finally, we discuss how the magnetic excitations can be understood in terms of a combination of single spin dynamics and collective dynamics on the ten-ion loops.

II. BACKGROUND
GGG is a well known magnetically frustrated compound with Gd 3+ ions positioned on a three dimensional hyperkagome lattice. The Gd 3+ ions have S = 7/2 and are usually regarded as spherically symmetric, although there are indications of a weak planar anisotropy stemming from crystal field effects 6,14 . The space group of GGG is Ia3d, with a = 12.39 Å. The Curie-Weiss temperature of GGG is θ CW ∼ −2.25 K and neighboring spins are coupled with an antiferromagnetic exchange of J 1 = 107 mK 15 . No conventional long-range magnetic order has been found down to 25 mK 16 . Neutron scattering revealed the onset of short-range correlations for T < 4 K with the director correlations developing for T 1.5 K 6, 16 . Below T GGG f = 175 mK, the spins freeze partially [17][18][19] . The freezing is a result of the interplay between the next nearest exchange interaction, −12 mK≤ J 2 ≤ 4 mK, the inter-sublattice exchange interaction, −3 mK< J 3 < 12 mK, and the long-range dipole interaction, D 20 . The magnetic behavior of GGG can be modeled using a standard Heisenberg model 13 : wherer ij is a unit vector between atoms i and j.
In comparison to GGG, GAG has a larger nearneighbor exchange constant of J 1 = 186 mK 19 , and the magnitude of the Curie-Weiss constant is larger as well, θ CW ∼ −3.9(3) K. The further-neighbor exchange constants are presently unknown for GAG. The (H, T ) phase diagram determined for GAG 19 is similar to that of GGG 17,21 with phase boundaries normalised to the magnitude of J 1 . A spin freezing in GAG happens below T GAG f ∼ 300 mK. The dipolar interaction strength, D, in GAG and GGG, calculated using the near-neighbor Gd distances in each compound, are similar, being slightly larger in GAG (48 mK) than in GGG (45 mK). The ratio D/J 1 is thus smaller in GAG than in GGG by a factor 0.6. The parameters for GGG and GAG are summarized in Table I.  and Refs. 9, 21, and 22 respectively. Both samples use isotopically enriched 160 Gd (99.98% for GGG, 98.4% for GAG) to reduce the neutron absorption cross-section.
We have performed a polarized neutron powder diffraction experiment on 1.6 g of GAG on the diffuse scattering spectrometer D7 at the Institut Laue Langevin (ILL) 23,24 . The sample was cooled using a dilution refrigerator, to access the temperature regime 60 mK< T < 5 K. The magnetic signal was separated from the nuclear and spin incoherent signal using the 10-point method 25 . We corrected for finite polarization using a quartz standard and for detector efficiency using a vanadium standard 26 . These corrections were carried out using the Large Array Manipulation Program, LAMP 27 . Leaked signals at Qvalues corresponding to strong nuclear Bragg peaks from the copper sample container were removed. The data were normalized to mbarns/sr/Gd 3+ by refining the nuclear structure using Fullprof 28 .
The magnetic dynamics of powdered GAG on the meV scale were measured using the cold neutron multichopper spectrometer LET at ISIS 29,30 using incident energies E i = 1.25, 2.0, 3.96 and 8.69 meV with an elastic energy resolution of 0.023(1), 0.042(1), 0.11 (1) and 0.42(1) meV, respectively (full width at half maximum (FWHM)), determined using an incoherent scatterer. The data were reduced using Mantid 31 . We measured the magnetic dynamics at three temperatures (50 mK, 500 mK and 1 K). The slow magnetic dynamics of powdered GGG on the µeV scale were measured using the neutron backscattering instrument IN16b of the ILL 32,33 , across a temperature range 60 mK< T < 1 K, using a dilution refrigerator. The neutron data were converted from acquisition channel to energy using standard treatments using LAMP 27 . The instrumental resolution function can be described by a Gaussian with weak Lorentzian tails, FWHM of 0.94(2) µeV, determined using a vanadium standard.
In all neutron scattering experiments the samples were cooled for 24 hours prior to measurements to ensure that base temperature was reached.
AC susceptibility measurements were performed using a superconducting quantum interference device magnetometer equipped with a miniature dilution refrigerator, developed at the Institut Néel, Grenoble 34 . Powders of GGG (17.13 mg) and GAG (5.7 mg) were cooled in zero applied magnetic field, and measured at temperatures down to 80 mK with the frequency varied between 0.057 Hz and 1110 Hz under an applied field of µ 0 H AC = 0.055 mT. In these low temperature measurements, the samples were mixed with Apiezon N grease to ensure thermalization. Furthermore, a single crystal of GGG was measured at frequencies down to 1 mHz with the AC field applied along the (110) direction.

IV. RESULTS AND DISCUSSION
A. Magnetic structure of GAG Fig. 2(a) shows the magnetic scattering cross section, S(Q), of powdered GAG as a function of wave vector transfer, Q, at 60 mK, measured at D7. It is worth noting that no energy analysis is performed on D7, and the observed signal is thus integrated over energy. This means that low energy magnetic excitations contribute to the observed diffraction signal. However, from the LET experiment on GAG presented in the next section we find that 97.4% of the energy integrated signal is within the elastic line, and the low energy magnetic excitations thus contribute only 2.6% to the total magnetic and nuclear scattering. At the peak near Q = 1.1 Å −1 more than half of the total signal observed at D7 is magnetic. Therefore, the observed magnetic signal at D7 is at least ∼ 95% elastic scattering, and the small contribution from inelastic scattering can safely be ignored.
The broad diffuse peaks around Q = 1.1 Å −1 , Q = 1.8 Å −1 and Q = 2.9 Å −1 are similar to those observed in GGG 16 , and indicate the presence of short-range magnetic order. In GGG, a detailed Reverse Monte Carlo (RMC) analysis further showed that the shape of the peaks at Q = 1.1 Å −1 and 1.8 Å −1 , is a signature of the director correlations 6 . We have carried out a similar analysis of our GAG data using the RMC program SPINVERT as explained below 35 .
The SPINVERT analysis was performed using a magnetic supercell of 6 × 6 × 6 unit cells with a spin on each of the magnetic sites. We found n = 6 (5184 spins) to be the smallest unit cell to yield a refinement consistent with our data. The spins were treated as classical vectors, an approximation which is justified by the large spin of Gd 3+ and the localized nature of the spins. Each simulation was initiated with every spin pointing along a random direction. In each Monte Carlo step a ran-dom spin was rotated slightly, and the powder averaged magnetic neutron diffraction signal from the spin configuration was calculated and scaled by a constant scale factor, s = g 2 S(S + 1) = 63, then compared with the experimental data. The new spin orientation was kept or rejected using the Metropolis algorithm 36 . We carried out 100 independent simulations for each temperature, with 100 steps per Gd 3+ ion, resulting in 518400 steps for each simulation. Data presented are an average of these 100 simulations. We note that unlike direct Monte Carlo, this procedure provides no information on the relevant exchange interactions, but does yield a spin structure consistent with the neutron scattering data.
The resulting RMC fit is shown as a continuous red line through the data points in Fig. 2(a), showing excellent agreement with the data. The finite system size makes the calculation unreliable for Q < 2π/(na/2) ≤ 0.17 Å −1 , indicated by the dashed black line.
We now look at the RMC spin structure to determine the local spin directions and any director correlations. Fig. 2(c) shows a three dimensional histogram of the local orientation of each Gd 3+ spin at 60 mK, normalized by the solid angle of the bin. Similar to GGG we find a strong local xy-anisotropy. It is possible that the anisotropy originates from the dipolar interaction 6 . Fig. 2(d) shows the magnitude of the spin-spin correlations averaged over shells at distance r, | S(0)·S(r) |. We find antiferromagnetic near-neighbor correlations which follow an approximate exponentially decay, S(0)·S(r) ≈ exp(−r/ξ), with ξ = 4.2(1) Å. For near neighbors we find S(0) · S(r) = −0.41 (1). This corresponds to an average angle between near-neighbor spins of 114 • , significantly lower than the 120 • expected for a pure Heisenberg spin state. In GGG a much smaller deviation was found, 118 • .
In GGG it was found that the spin structure depends strongly on the precise values of J 1 , J 2 and J 3 , and their interplay with the dipolar interaction 20 . The same is true for GAG, but since J 2 and J 3 presently are unknown in GAG, we cannot quantify their effect on the spin structure.
In order to look for evidence of the director state we follow Ref. 6 and define the 10-spin directors, L as the staggered magnetization of a loop The distribution of directors in GAG is shown in Fig. 2(e), revealing a strong local z anisotropy, as also found in GGG. The correlation between directors can be quantified as whereL = L/L. The director-director correlation is g = 1 for directors that are parallel, and g = −1 for directors that are orthogonal. We show g(r) in Fig. 2(f). The nearneighbor director-director correlation is strong, which is unsurprising, as neighboring directors share several spins. Importantly, the director-director correlations then falls to a constant value, g max , that extends beyond 30 Å. As such we find that the directors in GAG remain correlated, albeit weakly, over long distances. Fig. 2(b) shows the temperature dependence of g max for GGG and GAG. Both compounds show an increase in g max below 1.5 K, as the director state emerges. g max increases rapidly with decreasing temperatures, and peaks at ∼400 mK and 175 mK for GAG and GGG, respectively. Below this temperature, which coincides roughly with the onset of the spin freezing [17][18][19] , g max appears to remain constant. g max (GGG) ≈ 3g max (GAG) indicating that the director-director correlations at the lowest temperatures are stronger in GGG than in GAG. This difference is likely caused by a combination of variations in J 2 and J 3 , as well as the variation in D/J 1 , with D/J 1 for GAG being smaller by a factor 0.6. It would be interesting to study other Gd 3+ based garnets with different values of D/J. For example, Gd 3 Te 2 Li 3 O 12 (GTLG) has D/J = 0.36 18 , and we therefore predict a director state to be present in GTLG with 0.05 < g GTLG max < 0.15. Ref. 37 also suggests multiple Gd 3+ -based garnets with varying values of D/J 1 that would be interesting to study. It would also be valuable to determine J 2 and J 3 for GAG. For GGG, the most accurate determination of these parameters was based on an analysis of the longer-range correlations that set in in GGG for temperatures below 175 mK 20 . Such correlations are not present in GAG, making a similar analysis impossible at present.
The presence of director correlations in GAG as well as in GGG indicate that any hyperkagome system with antiferromagnetic near-neighbor interactions and relatively strong dipolar interactions should display a director state.
A recent study of the isostructural compound Yb 3 Ga 5 O 12 , (YbGG), also reveals a director state. However, in stark contrast to GGG and GAG, the director state in YbGG is derived from ferromagnetic nearneighbor exchange, with spins directed along the local z direction 38 , with directors correlated over short distances. Despite these differences, the director state on hyperkagome lattices thus appears to be rather robust. To form a director state, some form of anisotropy in the spin orientation seems to be required. In GGG and GAG, this anisotropy is most likely a result of the dipolar interaction, with the potential addition of small crystal field effects, whereas the anisotropy in YbGG originates almost entirely from crystal field effects,. The precise interactions which determine whether the director state is long-or short-range are unclear.
B. meV dynamics of GAG Fig. 3 shows S(Q, E), at 50 mK for GAG measured at LET using incident energies E i = 3.69 and 2.0 meV. We observe three distinct features, similar to previous results on GGG 9 . We label the three features INS1, INS2 and INS3 in line with the nomenclature employed for GGG.
INS3 is a weak non-dispersive excitation observed at 0.588(5) meV, shown in Fig. 3(a). The intensity of this excitation does not follow a single ion form factor, and is thus not a crystal field excitation. In GGG this excitation was found at 0.58(3) meV 9 . At lower energies, Fig. 3(b), we observe two further distinct features, labelled INS1 and INS2. Within the energy resolution and statistics afforded by the experiment INS1 is a low-lying dispersionless excitation at E = 0.05(1) meV with INS2, a much broader feature that is more dispersive in nature. At low Q, INS2 appears at E = 0.19 meV and falls into the elastic line near Q ∼ 1 Å −1 . All three excitations have slightly higher energies in GAG than in GGG, consistent with the stronger exchange interactions in GAG. The dotted vertical line in Fig. 3(b) shows the direction of cuts through the data for a range of temperatures T = 50 mK, 500 mK and 1 K. The cuts are presented in Fig. 4(b). At 50 mK the three excitations are well defined and can be modeled as gapped modes. Upon increasing the temperature, the signal broadens. The gap to INS1 remains visible, but INS2 appears to become overdamped and thus transforms into almost featureless quasi-elastic scattering. INS3, in contrast, remains well defined at least up to 1 K. dispersion of GGG in an applied magnetic field using linear spin-wave theory on the ten-ion loops, including nearneighbor interactions and the dipolar interaction. This study was carried out in a magnetic field large enough to create a ferromagnetic ground state, very different from the zero-field ground state. Still, the dispersion is similar in both applied and zero field, indicating that the underlying excitations are similar. Comparing our results with those of Ref. 13 suggests that INS1 originates from bands that are near dispersion-less across the entire Brillouin Zone. INS2, on the other hand, consists of multiple dispersive excitations that cross and overlap in reciprocal space.
No equivalent of INS3 is found by linear spin wave theory in the ordered state, and it may therefore have an entirely different origin. The mode is relatively sharp in energy, and has a strikingly similar position and width in the two compounds. The lack of scaling with J 1 suggests that near-neighbor interactions is not the main driver of this excitation. In GGG, the INS3 mode was modeled as a singlet-triplet excitation, based on the Q-dependence of the intensity 9 . Due to the weak signal we could not extract the Q-dependence of the intensity in GAG, and thus cannot verify if this model fits the present data.

C. µeV dynamics of GGG
An overview of the backscattering data, S(Q, E), for GGG is shown in Fig. 6 for 60 mK< T < 1 K. At all temperatures a strong elastic signal is present for all accessible wavevector transfers. At the lowest temperature probed, 60 mK, the signal appears almost entirely elastic. A broad quasielastic broadening is observed around 450 mK with a peak in intensity near wavevevector transfer Q = 1.1 Å −1 . At 1 K the quasi-elastic signal has broadened further and reaches the edge of the experimental energy window.
The energy dependence of the total scattering signal is well described, for all scattering vectors and temperatures, by a convolution of the resolution function with a temperature independent elastic term, represented by a delta function, and a temperature dependent quasielastic Lorentzian term where R is the resolution function, A el is the integrated intensity of the elastic signal, and A qel is the intensity of the quasielastic signal, 2Γ is the FWHM of the quasielastic signal. A linear sloping background is included, described by the parameters C 1 and C 2 . We have accounted for detailed balancing in the fitting 39 , although the effect is minimal. Fig. 7 shows an example of the data (black circles) and fit (red solid line) at 450 mK and Q = 1.1 Å −1 . We see a significant quasielastic broadening, compared with the instrumental resolution (dashed green line).
The inset of Fig. 7 shows the temperature dependence of the wavevector integrated intensity, which follows the trend seen in Fig. 6. The temperature dependence of the Lorentzian signal indicates its magnetic origin.
The most important fitted parameters are shown in Fig. 8, and the background parameters are discussed in the appendix. The Q-and temperature dependence of the integrated intensity of the Lorentzian signal, A qel , is shown in Fig. 8(a). The integrated intensity, with broad peaks at Q=1.1 and 1.8 Å −1 , follows the Q dependence of the elastic magnetic structure factor previously determined for GGG 22 , and shown for GAG in Fig. 2(a). The temperature dependence of the integrated intensity of the quasi-elastic component, A qel at Q = 1.1 Å −1 is shown in Fig. 8(c). It remains roughly constant for T < 0.2 K and decreases rapidly with increasing temperature, consistent with the decrease of magnetic correlations with increasing temperature.
The temperature and Q dependence of the FWHM of the quasi-elastic signal, 2Γ, is shown in Fig. 8(b). It is interesting to note that Γ is independent of Q in the accessible range (0.7 Å −1 < Q < 1.8 Å −1 ), but increases rapidly with increasing temperature. The lack of Q-dependence indicates that the spin fluctuations are uncorrelated on this energy scale, a surprising result given the existence of both director and static spin-spin correlations in GGG.
A single Lorentzian lineshape indicates that the temporal spin-spin correlations decay as where the decay rate, τ is related to the FWHM by 2Γ = 2 /τ 40 . The temperature dependence of τ at Q = 1.1 Å −1 is shown in Fig. 8(d) and follows τ = AT B , with B = −1.5 (3). Our results are consistent with earlier Mössbauer measurements on GGG 12 as well as muon spin rotation measurements 10,11 , who all found spin fluctuations in GGG on a similar time scale as presented here. In the Mössbauer measurements, the temperature dependence was quadratic 12 and the fluctuations were shown to be confined to a plane, consistent with the spin anisotropy mentioned earlier 6 .
Significant theoretical work has been performed on the spin fluctuation rate for a system of classical spins on frustrated lattices. Conlon and Chalker predicted, for Heisenberg spins on a pyrochlore lattice in a cooperative paramagnetic regime, a linear temperature dependence of the spin fluctuation rate (τ −1 ) with spin correlations that relax at a rate independent of the wavevector transfer 41 The hyperkagome lattice can be viewed as a depleted pyrochlore lattice, where each tetrahedron is missing a site. As such, we can expect the magnetic behavior on these two systems to be similar. In GGG, for T < 1 K, the spin correlations are indeed independent of Q, although τ follows a power law rather than being linear.
More recently, the diffusive dynamics on frustrated kagome lattices have been investigated. Taillefumier predicts a distribution of timescales in the cooperative paramagnetic regime representing loops diffusing in an entropically dominated free energy landscape. The signature of such diffusion can be observed for Q < 2π/d where d = a is the loop spatial scale. In this Q region, τ −1 increases linearly with Q 42 . Unfortunately, we do not access a wavevector transfer below 0.7 Å −1 , and thus do not directly observe this. However, in line with Taillefumier, at larger wavevector transfers the fluctuation rate is proportional to temperature and independent of Q 42 . We therefore suggest that we are probing the diffusive dynamics of the multipolar director state in GGG. Data at lower Q is required to confirm or reject this hypothesis. Such experiments will be challenging due to the vanishing intensity at low Q as seen in diffraction experiments.
Further progress on understanding the director dynamics on the µeV and meV energy scale will also require inelastic neutron scattering experiments on single crystal samples. Such experiments are only feasible on isotope enriched single crystals to minimize the absorption of cold neutrons required to achieve the relevant energy resolutions. The cost of such crystals is prohibitively high.
Finally, we note that some magnetic scattering is seen in the apparent background signal, as it depends on temperature (see the appendix for details). This indicates that part of the background originates from motions on a larger energy scale than the accessible energy window of IN16b (∼ ±30 µeV), as also found in similar backscattering measurements on Tb 2 Sn 2 O 7 40 . It is likely that this contribution originates from the tails of the lowest excitation, INS1, observed at 0.05(1) meV at the lowest temperature.

D. AC susceptibility
AC susceptibility is a very sensitive probe of low frequency magnetic dynamics 43 . In our AC susceptibility measurements we measure the real (χ ) and imaginary susceptibility (χ ), for both GGG and GAG as a function of temperature. Fig. 9(a) shows χ for a single crystal of GGG with the AC field applied along the [110] direction. An anomaly around T f ∼ 175 mK is seen, with a slight frequencydependence. Fig. 9(b) shows χ for the same single crystal, with (c) zooming in on selected frequencies. The temperature dependence of χ varies from a single peak for frequencies below ∼ 5 Hz to a double peak structure emerging for frequencies > 5 Hz. We fitted the peaks using Gaussian line shapes to extract the position of the peaks.  The frequency dependence of the peak position of these curves follows an Arrhenius temperature dependence, see  .
where i = 1, 2, E i is an activation energy and τ 0,i = 2π/f the relaxation lifetime of spin vector reversal. The excellent fit to the Arrhenius law indicates that the motion of the spins on this energy scale is governed by thermal activation across an energy barrier. Two distinct peaks indicate two distinct relaxation processes. The fits to the data using the Arrhenius law for each of the two peaks provide activation energies of E a,1 = 0.33(2) meV (3.8(3) K) and E a,2 = 0.11(1) meV (1.3(2) K) and relaxation lifetimes of τ 0,1 = 3 × 10 −14 s and τ 0,2 = 6×10 −8 s. It should be noted that the uncertainty on τ is about an order of magnitude. The energy barriers are the same order of magnitude as the excitation energies probed with neutron scattering.
The temperature dependence of χ for powdered GGG is given in Fig. 11(a) for f = 57, 111 and 211 Hz. Two anomalies are seen, near 350 mK and 100 mK, as also found previously 44 . The signal is much broader than for the single crystal, and the two anomalies are at different temperatures than the anomaly in χ for the single crystal of GGG. Fig. 9(b) shows the temperature dependence of χ for powdered GAG for a larger range of frequencies. Here, a single, broad peak is seen around 350 mK. The broadness of the peak indicates a broad distribution of time scales in the system.
The temperature dependence of χ is shown in Fig. 9(c) for GGG and Fig. 9(d) for GAG. In both cases a broad anomaly is seen around 250 mK. The peaks in the powdered data are substantially broader than the single crystal data on GGG, which indicates the existence of a broader distribution of relaxation times in the system. In both compounds the peak temperature of χ depends very weakly on frequency. However, fitting to an Arrhenius law, gives unphysical characteristic times, suggesting that the freezing mechanism is more complex, and cannot be explained with a single energy barrier.
Modifications to the Arrhenius law, such as the Vogel-Fulcher law 45 , can be made to fit the data with sensible parameters, but the correlations between fitted parameters become so large that interpretation of their values is meaningless. Instead, the frequency dependence of the χ peak can be analyzed through the Mydosh parameter K, frequently used to describe spin glasses, where T f is the temperature of the χ maximum. For the GGG powder, we have insufficient data to determine K. For GAG, we obtain K GAG = 0.03 − 0.04, which falls in the range of insulating spin-glasses, if a bit smaller 46 . Nevertheless, the broad shape of the χ features in GAG and of the ZFC-FC magnetization 19 seems to preclude a conventional spin-glass transition. In particular, we could not accurately determine the freezing temperature T g which would have allowed us to perform dynamical scaling and estimate possible critical exponents. Further measurements, in a larger frequency range are thus needed to conclude on the nature of the freezing in GAG. The broad feature is reminiscent of the heterogeneous freezing scenario proposed for Heisenberg kagomé antiferromagnets 47 , and thus suggests that multiple time and lengths scales are involved in this relaxation mechanism.
Our AC susceptibility measurements on GGG and GAG confirm the spin freezing upon cooling below ∼175 mK and ∼300 mK respectively, with a frequency dependent χ and χ . Most interestingly, the temperature dependence of χ for the single crystal of GGG displays an unusual double peaked structure. The frequency depen-dence of each peak follows an Arrhenius law, thus indicating the presence of two distinct thermally activated processes. Single spin magnetization precession is typically on the ps timescale, e.g. Ref. 48, indicating that the first process in GGG, with τ 0 = 3 × 10 −14 s, is related to single spin reversal. The second process (τ 0 = 6×10 −8 s) is too slow to involve single spins, and is thus probing the dynamics of larger structures, which we can reasonably assume to be the ten-ion loops. These results differ from those by Ghosh et al. 8 who probed the ten-ion looped structure using magnetic hole burning and found temperature independent clusters behaving as quantum objects. In our case the signatures remain thermally driven.
In contrast to the single crystal measurements on GGG, the temperature dependence of the powdered GAG AC susceptibility displays a very broad single peaked structure, from which we could not extract a characteristic energy scale. Most probably, a wide distribution of correlation lengths is involved in this dynamical process, which is consistent with the smaller strength of the ten-ion loops correlations in GAG compared with GGG. The larger deviation from the 120 • spin structure in GAG than in GGG obtained in the diffraction measurements may also explain the differences between the observed slow dynamics.

V. SUMMARY AND CONCLUSION
In summary, we have presented an extensive comparative study of GGG and GAG using neutron diffraction, neutron backscattering and time-of-flight spectroscopy and AC susceptibility. We confirm the existence of long range director correlations in GAG, and propose that it will be present in all garnets with antiferromagnetic nearneighbor exchange and relatively strong dipolar interaction or local xy-anisotropy. The reduced D/J 1 value in GAG as well as possible changes in J 2 and J 3 result in weaker director correlations.
On the meV scale, GGG and GAG show similar dynamics with three distinct excitations, INS1-3. In particular, the temperature dependence of the excitation energy and lifetime of the INS3 excitation at 0.58 meV is nearly identical in the two compounds, indicating that it is not driven by near neighbor exchange interactions. The other two excitations are reminiscent of spin waves found in the polarized state of GGG.
On the µeV scale, the spins in GGG fluctuate with a rate that decreases as T 1.5 . The fluctuations are independent of wavevector transfer in the probed range, consistent with the presence of spin diffusion. The slowest dynamics probed by AC susceptibility with low frequencies reveal two distinct fluctuation processes in GGG. One of these is associated with single spin fluctuations, while the other is associated to larger structures, most likely the directors.AC susceptibility on powders of GGG and GAG furthermore give insight into the nature of the freezing of the spins that occurs at low temperature.
We have suggested several directions for further experimental and theoretical work: On the experimental side it would be interesting to ascertain the presence of director correlations in other Gd-based hyperkagome systems, such as Gd 3 Te 2 Li 3 O 12 . In addition, experimental work to determine the further-neighbor exchange constants in GAG would be valuable.
Based on our AC susceptibility data we expect the µeV dynamics of GAG to be different from GGG. This prediction can be straightforwardly tested experimentally on powders of GAG. To determine if the Q-independence of the quasi-elastic broadening is caused by spin diffusion or uncorrelated fluctuations, we propose two experiments: First, similar backscattering experiments as presented here, but reaching lower Q-values, would reach the regime where the width of the signal decreases linearly with decreasing Q in a spin diffusion model. Such an experiment will be difficult due to the small magnetic signal at the required low Q. Second, measurements on an isotope enriched single crystal would reveal any Q-dependence on the signal which is currently lost in powder averaging. Unfortunately, 160 Gd is extremely expensive.
With an isotope enriched single crystal the meV dynamics could also be studied further. We expect INS2 to consist of multiple broad bands of excitations that are similar in nature to the excitations found in the fieldpolarized state.
Finally, it is also clear that more theoretical work is required. The origin of the INS3 excitation remains unclear, as does the exact nature of the excitations associated with the director correlations. Conventional spin wave theory is not applicable to these systems, but another avenue to approach this problem could be numerical Langevin simulations.