Common Glass-Forming Spin-Liquid State in the Pyrochlore Magnets Dy$_2$Ti$_2$O$_7$ and Ho$_2$Ti$_2$O$_7$
Azar B. Eyvazov Ritika Dusad Timothy J. S. Munsie Hanna A. Dabkowska Graeme M. Luke Ethan R. Kassner J. C. Séamus Davis Anna Eyal
cond-mat.str-el




 
equal contribution 
LASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA


equal contribution 
LASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA

 
Brockhouse Inst. for Materials Research, McMaster University, Hamilton, ON, Canada
Department of Physics, McMaster University, Hamilton, Ontario, L8S 4M1, Canada

 
Brockhouse Inst. for Materials Research, McMaster University, Hamilton, ON, Canada

 
Brockhouse Inst. for Materials Research, McMaster University, Hamilton, ON, Canada
Department of Physics, McMaster University, Hamilton, Ontario, L8S 4M1, Canada
Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada

 
LASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA
Current address: Honeywell International Inc., Golden Valley, MN 55422, USA

 
LASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA
CMPMS Department, Brookhaven National Laboratory, Upton, NY 11973, USA
Tyndall National Institute, University College Cork, Cork T12R5C, Ireland


anna.eyal@gmail.com 
LASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA




Despite a well-ordered pyrochlore crystal structure and strong magnetic interactions between the Dy^3+ or Ho^3+ ions, no long range magnetic order has been detected in the pyrochlore titanates  Ho_2Ti_2O_7 and Dy_2Ti_2O_7. To explore the actual magnetic phase formed by cooling these materials, we measure their magnetization dynamics using toroidal, boundary-free magnetization transport techniques. We find that the dynamical magnetic susceptibility of both compounds has the same distinctive phenomenology, that is indistinguishable in form from that of the dielectric permittivity of dipolar glass-forming liquids. Moreover, Ho_2Ti_2O_7 and Dy_2Ti_2O_7 both exhibit microscopic magnetic relaxation times that increase along the super-Arrhenius trajectories analogous to those observed in glass-forming dipolar liquids. Thus, upon cooling below about 2K, Dy_2Ti_2O_7 and Ho_2Ti_2O_7 both appear to enter the same magnetic state exhibiting the characteristics of a glass-forming spin-liquid.



75.50.Lk, 75.47.Lx

Common Glass-Forming Spin-Liquid State 

in the Pyrochlore Magnets Dy_2Ti_2O_7 and Ho_2Ti_2O_7
    Anna Eyal
    December 30, 2023
==============================================================================================



	

§ INTRODUCTION
 

In the pyrochlore lanthanide-oxides with chemical formula A_2B_2O_7, the magnetic rare earth A ions are located at corner sharing tetrahedral sites as shown in Fig. <ref>a <cit.>. These materials support a multitude of exotic magnetic states <cit.>, such as spin-ice<cit.>, spin-slush <cit.>, and various candidates for quantum spin-liquids <cit.>. Dy_2Ti_2O_7 and Ho_2Ti_2O_7 have attracted much interest since the discovery of an entropy deficit in both compounds <cit.>, along with a similarity to the thermodynamic properties of water ice. In both Dy_2Ti_2O_7 and Ho_2Ti_2O_7 the large rare-earth magnetic moments of  μ∼10μ_B <cit.>, at the corners of the corner sharing tetrahedral structure, are subject to strong crystal field interactions <cit.> so that, in the single-ion ground-state, spins are only allowed to point towards (away) from the centers of the tetrahedra that share it (see Fig. <ref>a) <cit.>. Under these circumstances, long-range magnetic dipolar interactions are significant when compared to the nearest-neighbor magnetic exchange coupling. The resulting interaction favors a state in which for each tetrahedron two spins point in and two out, that, by analogy to water ice, was dubbed "spin-ice"<cit.>. One might expect such a constraint to result in an ordered magnetic state that is unique <cit.>, but no such state has ever been observed in these materials at zero magnetic field <cit.>. 




For both Ho_2Ti_2O_7 and Dy_2Ti_2O_7 the strength of the nearest neighbor dipolar interaction is D_nn≈2.35K, while the nearest neighbor exchange interactions are J_nn≈-0.52K for Ho_2Ti_2O_7 and J_nn≈-1.24K for Dy_2Ti_2O_7 <cit.>. Raman spectroscopy reveals that the phonon spectra of these two materials differ only slightly and that the crystal field parameters are very similar <cit.>. Dysprosium ions have 7 f-electrons with J=15/2 for the single ion angular momenta in the ground state, with the crystal field levels all Kramers doublets; holmium ions, on the other hand, have 8 f-electrons with an integer J=8, so its crystal field spectrum has both singlet and doublet energy levels <cit.>. The magnetic heat capacity of both materials has a broad peak at low temperatures (1.2K for Dy_2Ti_2O_7 and 1.9K for Ho_2Ti_2O_7) <cit.>. However, Ho_2Ti_2O_7 has another peak in heat capacity at much lower temperatures, which is believed to be due to its active nuclear magnetism <cit.>. Interpreting the magnetization dynamics of these two systems has proven challenging. Nearest neighbor spin-ice-based models for spin dynamics predict inelastic neutron scattering intensity patterns where ”pinch points” are significantly smeared out compared to the experiments <cit.>, and exchange interactions up to third nearest neighbor must be included in such models to replicate the full complexity of these neutron scattering results <cit.>. Models for the magnetic susceptibility χ(ω,T) based on dipolar-spin-ice <cit.>, or on the dynamics of a dilute gas of mobile monopoles <cit.> representing transitions from two-in-two-out to three-in-one-out configurations<cit.> (see Fig. <ref> b), show significant deviations from the experimental data <cit.>. Moreover, predictions for the temperature dependence of microscopic magnetic relaxation times τ considerably underestimate the divergence of τ at low temperatures <cit.>. Finally, in the absence of magnetic fields, no magnetic order has been detected in either compound <cit.>. 

A recent proposal <cit.> that the magnetic state of Dy_2Ti_2O_7 is the magnetic analog of the diverging viscosity state found in glass-forming dipolar liquids <cit.> provides a different perspective. Classical glass-forming liquids exhibit universally a super-Arrhenius divergence of microscopic dipolar relaxation times τ_0(T) of the Vogel-Tammann-Fulcher (VTF) form τ_0(T)=Aexp(DT_0/(T-T_0 )) <cit.>, a dielectric function ϵ (ω,T) of the Havriliak-Negami (HN) form ϵ (ω,T)= ϵ_∞+  ϵ_0/(1+(iωτ_HN )^α )^γ  <cit.>, and a related time-domain relaxation described by the Kohlrausch-Williams-Watts (KWW) form <cit.> ϵ (t) = ϵ_0 exp [-(t/τ_KWW)^β ]. Observation of this combined VTF/HN/KWW phenomenology provides a strong clear identifier of a supercooled glass-forming dipolar fluid <cit.>. Dy_2Ti_2O_7 was found to exhibit a precise HN form for its magnetic susceptibility χ (ω, T), a general KWW form for the magnetic relaxation, and diverging microscopic magnetic relaxation rates on a VTF trajectory, implying that it is, by analogy, a glass-forming magnetic liquid <cit.>. Here we explore if magnetic fluids with such a phenomenology could be more general in the lanthanide pyrochlore magnetic materials. 

Even if glass-forming spin-liquid phenomenology were common to such materials, the microscopic parameters are still likely to be specific to each compound. In glass-forming dipolar liquids, the measure of the correlated dipole dynamics is called the fragility, D, <cit.> and it characterizes the degree of spatial heterogeneity. D is an indicator of the spread of microscopic relaxation times over different close-by regions in the liquid. The smaller the value of D, the more fragile the liquid and the more spatially heterogeneous its dynamics <cit.>. By analogy, a more fragile glass-forming spin-liquid would mean an enhancement of the super-Arrhenius behavior of its magnetic relaxation times upon cooling. Such a situation could be caused by less efficient tunneling between spin configurations, due perhaps to differences in monopole creation energies and hopping rates. For the pyrochlore magnets discussed, the chemical potential for monopole-pair generation is dependent on nearest neighbor coupling, J_eff <cit.>, which is 1.1K for Dy_2Ti_2O_7 and 1.8K for Ho_2Ti_2O_7. Moreover, the theoretical rate of tunneling of the monopole excitations depends on the off-diagonal components of the dipolar interactions of neighboring spin <cit.>. The reason for this is the strong Ising-like behavior of the magnetic ions <cit.>, with the energy barriers to the first excited crystal field state being Δ∼ 240K for Ho_2Ti_2O_7 and ∼ 380K for Dy_2Ti_2O_7 <cit.>. Additionally, the fact that the effective energy scale for spin-flip dynamics is on the order of several J_eff rather than Δ implies that spin flips occur by quantum tunneling <cit.>. Since the transverse field effects in Ho_2Ti_2O_7 are more pronounced than in Dy_2Ti_2O_7, resulting in a more effective quantum tunneling at low magnetic fields <cit.>, monopole hopping in Ho_2Ti_2O_7 is expected theoretically to be more efficient. In that case, one might anticipate a less fragile glass-forming spin-liquid in Ho_2Ti_2O_7 as compared to Dy_2Ti_2O_7. 


	

§ EXPERIMENTAL SETUP
 

To explore the relationship between the magnetization dynamics of Ho_2Ti_2O_7 and Dy_2Ti_2O_7 we used a boundary-free arrangement to measure the AC susceptibility and time dependent magnetization relaxation characteristics of the two materials at T≤2 K. Single crystals of these materials were grown as boules in O_2 gas under 2 atm pressure in an optical floating zone furnace <cit.>, and were subsequently cut into disks of diameter ∼ 6mm and thickness ∼ 1mm (see Fig. <ref> d). For the boundary-free magnetization measurements, holes of ∼ 2.5mm diameter were drilled through the center of disk shaped samples of both Ho_2Ti_2O_7 and Dy_2Ti_2O_7 crystals. A superconducting toroidal solenoid (STS) was then made by winding a 0.09mm diameter NbTi wire around the toroidal samples (Fig. <ref> c,d). Using a toroidal geometry for both the samples and the magnetization sensors means that the superconducting toroidal solenoid can be used to both drive magnetization flow azimuthally and to simultaneously and directly determine dM/dt throughout. More importantly, it removes any boundaries in the direction of the magnetization transport (Fig. <ref> c). The coil EMF due to changes in both the applied azimuthal field H(t) and sample magnetization M(t,T) is given by


    V_total (t,T)= -μ_0 NA(dH(t)/dt+dM(t,T)/dt)


where N is the number of turns in the solenoid and A is the effective cross-sectional area of the solenoid. Thus, the EMF due to magnetization dynamics in the sample is 


    V(t,T)= -μ_0 NA dM(t,T)/dt


For an applied AC field 


    V(ω,T)= -iμ_0 NA ωM(ω,T)


The definition of the magnetic susceptibility is


    M(ω,T)=χ(ω,T)H(ω)


In a solenoid H_0=nI, where n is the number of turns per unit length, so the EMF is given by


    V=V_x+iV_y=-iμ_0 NAωχ(ω,T)H_0=
     -i Iω L [χ' (ω,T)-iχ” (ω,T)]


where L is the geometric inductance of the STS pickup coil. Currents of up to 200 mA can be applied, using low temperature Nb crimp joints, to the STS coils, yielding azimuthal applied fields of magnitude up to |B|=2.5 mT or |H|=2200 A/m. Such fields are orders of magnitude smaller than those required
to flip spins in these compounds. In addition, this azimuthal field covers a wide range of crystallographic planes in its path.  
The AC susceptibility of the compounds measured was determined typically by applying ∼10 mA currents in a frequency range of 10 Hz – 100 kHz using a 4-probe impedance measurement of the STS. The inductance, L, of the STS was measured at T=50 mK, where neither of the materials show any magnetic activity in the frequency range measured, and then used in equation <ref> to calculate the susceptibility χ (ω,T) data from the voltage readings.


	

§ RESULTS

During transient data acquisition, the voltage over the STS was measured every 20 ms throughout the following excitation protocol: (a) apply magnetic field in a clockwise direction by turning on a current I=50 mA in the STS, (b) set the field to zero by turning off the current, (c) apply a magnetic field in the counter clockwise direction by turning on a current I=-50 mA in the opposite direction and (d) again zero the field. This protocol was repeated 150 times per temperature for each material at each temperature, and the results were averaged to improve data quality and fitting. For both materials, no difference in relaxation characteristics was observed when the magnetic field was turned on or off, as well as when the magnetic field was applied in one azimuthal direction or the opposite. At long times, after the initial sharp change in the field, the EMF that was generated in the STS decayed to zero, indicating that J=dM/dt <cit.> always decays to zero, despite the fact there are no terminating boundaries in our geometry. Figs. <ref> a,b depict the measured magnetization relaxation characteristics dM/dt of Ho_2Ti_2O_7 and Dy_2Ti_2O_7 respectively in the temperature range 0.6 K - 0.95 K. The plots show the measured voltage induced across the STS by the magnetization dynamics of the sample versus time after the application of a DC field. These data sets at each temperature were fitted by a KWW type stretched exponential decay V(t)=V_0 exp(-(t/τ)^β), with fits shown in Fig. <ref> a,b as fine colored curves. Although a simple exponential decay cannot fit any of these data at any temperature, the KWW form provides an excellent fit for all. The insets of Fig.  <ref> a,b show how the stretching parameter, β, is different from unity over the temperature range of the DC measurements for both compounds. More importantly, Figs. <ref> a,b reveal the universal applicability of the KWW form of both samples for the whole temperature range. Here, the normalized EMF V(t)/V_0 is plotted against the modified time parameter x=(t/τ)^β  for each temperature, with the result that all the magnetization transient data from both materials collapse onto a single line with unit slope. This remarkable agreement of magnetization decay dynamics of both Ho_2Ti_2O_7 and Dy_2Ti_2O_7 with a KWW form implies that both these systems are in the same state, a glass-forming spin-liquid.



The AC magnetic susceptibility of the toroidal samples of Ho_2Ti_2O_7  and Dy_2Ti_2O_7  was measured in the temperature range 0.9 K-2 K. For the compounds in this paper, we observed that below 0.5 K the EMF generated in the STS, in the frequency range reported, showed virtually no temperature dependence down to 50 mK, the lowest temperature at which AC measurements were attempted. The AC voltage measured at the lowest temperature was subtracted from the measurement at the temperatures of interest to deduce the susceptibility χ(ω,T)=χ'(ω,T)-iχ”(ω,T). Figs. <ref> c,d present the measured real (χ ') and imaginary (χ”) parts of the susceptibilities for the samples measured versus frequency, in the range 10-10^5 Hz for Ho_2Ti_2O_7 and 10-10^4 Hz for Dy_2Ti_2O_7. The data sets taken at different temperatures are labeled by a color/symbol code as indicated. 



Models of AC susceptibility that assume a single relaxation time of the Debye form χ'-iχ”=χ_0/([1+(iωτ)]), for example those of free monopole motion <cit.>, are not compatible with the measured χ(ω,T) for either Ho_2Ti_2O_7 or Dy_2Ti_2O_7 at any frequency or temperature within these ranges (see Fig. <ref>). By contrast, a Havriliak-Negami form modifies the simple Debye susceptibility with two exponents, α and γ, and corresponds to a system where there is a distribution of relaxation times


    χ'-iχ”=χ_0/[1+(iωτ)^α ]^γ+χ_∞


Figs. <ref> c,d depict our measured data for both Ho_2Ti_2O_7 and Dy_2Ti_2O_7 respectively. The top panels show the real part of the measured magnetic susceptibility versus frequency, as calculated from our measured voltage, using equation <ref>, and the bottom panels present the imaginary part of the susceptibility versus frequency. The different colors/symbols show data from different temperatures in the range 0.9K to 2K. For Dy_2Ti_2O_7, both exponents of the HN fit (equation <ref>) deviate from unity for the majority of the temperature range (inset of Fig. <ref> d), whereas for Ho_2Ti_2O_7 γ is around unity for most temperatures within error. Overall, the susceptibility for both materials shows a very good global agreement with the HN form for all temperatures and frequencies measured, as demonstrated by the fine lines in Fig. <ref> c,d. Fig. <ref> c,d show the collapse of these dynamical susceptibility χ(ω,T) data for all temperatures and both materials onto the single HN form <cit.> as indicated by the fine solid curves. The horizontal axis in the figure is the frequency, scaled by the HN parameters, and the vertical axes are the real and imaginary parts of the scaled HN susceptibility G(γ,χ_0,χ) (a full mathematical derivation can be found in Ref. 
<cit.>).



    G(γ,χ_0,χ)=
    (χ'^2+χ”^2/χ_0^2)^1/2γ(cos(1/γarctanχ”/χ')
    
    -i sin(1/γarctanχ”/χ'))



The scaling parameters, which are the fit parameters of the HN form for each temperature, are plotted in the insets of the figure. The quality of fits, while comprehensively good versus ω and T for both materials (Fig. <ref> c,d), is obviously slightly different between Ho_2Ti_2O_7 and Dy_2Ti_2O_7 (Fig. <ref> c,d). This may not be surprising since the frequency-width and frequency-range of the data from the Ho_2Ti_2O_7 measurements is at least two orders of magnitude wider than that for Dy_2Ti_2O_7 (see Fig. <ref> c,d). In any case, this observation of a universal Havriliak-Negami form for all the χ(ω,T) susceptibility data (Fig. <ref> c,d) constitutes a second robust indication that both these materials are homologous glass-forming spin-liquids.

Finally, to explore the microscopic magnetic relaxation dynamics of these systems, we need a form to relate the relaxation times obtained from the time-domain measurements to those from the frequency-domain. Numerical studies have linked the exponents and relaxation time parameters of the two forms <cit.>, which can be used for a unified analysis of our data. Using the values in Table I of Ref. <cit.> we can generate values for τ_HN/τ_KWW, enabling a conversion of the relaxation times from the time-domain measurements to the frequency-domain <cit.>. 
Fig. <ref> a depicts the combined time- and frequency-domain relaxation-time data for Ho_2Ti_2O_7 (T≥0.8 K) and Dy_2Ti_2O_7 respectively (the relaxation-times obtained from the time-domain measurement, τ_KWW, were converted to τ_HN by the procedure described above) with the horizontal axis being the inverse temperature. Obviously, the relaxation-time data for both materials diverge on a trajectory that is faster than Arrhenius, which would produce a straight line in Fig. <ref> a. Indeed, many groups have previously reported relaxation-time data showing the general behavior of a divergence that is faster than Arrhenius <cit.>, and in particular Ho_2Ti_2O_7 showing a stronger divergence.



However, when the temperature dependence of the relaxation-time is fit to a VTF form,  τ(T)=AexpDT_0/T-T_0, as shown in Fig. <ref> b, our findings indicate that both Dy_2Ti_2O_7 and Ho_2Ti_2O_7 exhibit non-Arrhenius slowing. This form yields A≈8×10^-10s, a fragility parameter D≈60 and a VTF temperature T_0≈191 mK for Ho_2Ti_2O_7, and A≈1.4×10^-4s, D≈14 and T_0≈257mK for Dy_2Ti_2O_7, signifying that Dy_2Ti_2O_7 is a more fragile spin-liquid. These specific parameters, resulting from the best fit to the VTF form give standard errors of 1 to 2 percent, and R^2 of 0.995 and 0.998 respectively.
Systematic errors arising from the fit procedure can be up to several tens of percent, but those have no impact on the resulting function, as can be clearly seen in Fig. <ref>b and by the high R^2. All of the data presented in figures <ref>, <ref> and <ref> are newly acquired for the purposed of comparison between the two compounds: the parameters of Dy_2Ti_2O_7 from this work agree well with previous work <cit.>. The fragility parameters and high-temperature relaxation-times depend strongly on the materials studied, but their T_0, the lowest temperature at which both materials may be expected, by analogy with glass-forming fluids, to enter a magnetic glass phase, are close in value. Overall, we find in the common VTF form for τ(T) (see Fig. <ref>b), a clear indication that both Dy_2Ti_2O_7 and Ho_2Ti_2O_7 are glass-forming spin-liquids.

	

§ DISCUSSION


Previous comparison studies of AC susceptibility of these two compounds <cit.> identified differences in the spread of their microscopic relaxation times. Specifically, the broadness of the absorption spectra, inferred from the width of the imaginary part of the magnetic susceptibility χ”(ω,T) as function of frequency, was found to be greater for Ho_2Ti_2O_7 than for Dy_2Ti_2O_7. The spread of characteristic relaxation times τ, as well as the asymmetry in χ”(ω,T), were also found to be broader in Ho_2Ti_2O_7 <cit.>. The qualitative agreement of these works with the profile of the scaled susceptibilities G(γ,χ_0,χ) shown in Fig. <ref> c,d, alongside the difference in the microscopic energy scales of Ho_2Ti_2O_7 and Dy_2Ti_2O_7 <cit.> indicate that the differences between the two compounds are unlikely due to random disorder or off-stochiometry. In addition, previously reported relaxation times for Ho_2Ti_2O_7 and Dy_2Ti_2O_7, using the inverse of the angular frequency of the peak in χ”(ω,T) (see Fig. 3 in ref. <cit.> for example and Ref. <cit.>), exhibit the distinct characteristics shown in Fig. <ref> - the slope of Ho_2Ti_2O_7 is greater than that of Dy_2Ti_2O_7 and the relaxation times cross around 0.9 K.

The apparent inferiority of the functional HN fits to Ho_2Ti_2O_7 as compared to Dy_2Ti_2O_7 is probably due to a combination of the more than two orders of magnitude wider spread of the relaxation times, and smaller signal sizes. However, these are still the best internally consistent analytic forms for χ(ω,T) in these materials, consistent for both the magnetization and susceptibility measurements and their resulting relaxation time temperature dependence. They yield parameters that agree both with prior works and do not contradict expectations from the different energy scales in these materials. 

The nomenclature of the proposed glass-forming spin-liquid (GFSL) motivates comparisons to existing spin glasses. Even though both have connection to glassy behavior, i.e. magnetic dynamics slow down with decrease in temperature, these two classes of materials are quite physically distinct. The difference between spin glasses and the proposed glass-forming spin-liquid state is both conceptual and in the details. First, quenched disorder is key for spin glasses, whereas there is no intrinsic disorder in the spin-ice compounds we study. Secondly, one of the clear signatures of spin glasses is the presence of a sharp cusp in the real part of the magnetic susceptibility at the transition temperature <cit.>, whereas the magnetic susceptibility of GFSL has a very smooth profile. Lastly, one can distinguish between the two by measuring the magnetic noise spectrum. The noise spectrum of a system carries information about its fluctuations on a microscopic scale. In the spin glass state, for example, one would expect a 1/f noise to  present <cit.>.

The GFSL formalism, underpinning our studies, presents several verifiable predictions for spin-ices: the exact functional forms of the magnetization decay and ac susceptibility, and the divergence of the relaxation times. No other model can account for all of the observed behavior in these compounds in such great detail. In addition, the GFSL suggests a link to the underlying mechanisms through its fragility parameter. The fragility of conventional glass forming liquids reveals the degree of deviation of a system's relaxation time from the Arrhenius profile. The more fragile the glass former, the more curved its relaxation time divergence with respect to temperature. In the VTF formalism, this fragility is manifest as the slope of log(τ) vs 1/T i.e. the temperature dependent energy scale in the Arrhenius form. Therefore, the more fragile a material, the smaller the slope. Examination of the equation that describes the super-Arrhenius form, 
τ(T)=τ_0expDT_0/T-T_0, shows that D is connected to the ac susceptibility characteristics of the glass former through the peak frequency, that is proportional to 1/τ.

In the frustrated pyrochlore magnets we study, fragility appears to indicate the heterogeneous nature of the energy landscape in these materials, similar to what is found in glass-forming dipolar-liquids. Therefore, the difference in fragilities between the two materials may eventually be revealed as due to differences in the underlying correlated and frustrated microscopic behavior of the magnetic monopoles, that is hypothesized to cause the magnetic relaxation in these materials. Here, both monopole creation energies and hopping rates, as well as constraining by Dirac strings would influence the dynamics, and hence the energy landscape. The higher fragility parameter of Ho_2Ti_2O_7, indicating a less fragile GFSL, for example, is consistent with the expected efficient monopole hopping in Ho_2Ti_2O_7 <cit.>. 

Finally, it seems plausible that the observed magnetic dynamics in Ho_2Ti_2O_7 and Dy_2Ti_2O_7 can be reconciled with the microscopic theory of emergent magnetic monopoles in these materials <cit.> by considering correlated transport of these quasiparticles. Here, flips of the real magnetic dipoles are recast as two opposite magnetic charges that, through a sequence of spin flips, may form a fluid of delocalized magnetic monopoles and anti-monopoles <cit.>. At low temperatures, these monopoles should form a dilute neutral gas whose transport characteristics control the magnetization dynamics and the susceptibility <cit.>. However, these monopoles are constrained by a network of Dirac strings, i.e. a trail of flipped spins left behind by a monopole traversing the interconnected tetrahedra. The non-Arrhenius sharp slowing down of spin dynamics may then be attributed to relaxation of Dirac strings <cit.>. 


	

§ CONCLUSION


To summarize: for purposes of comparison, we measured the AC susceptibility χ(ω,T) and time dependent magnetic relaxation behavior in the low-temperature magnetic states of the two materials Dy_2Ti_2O_7 and Ho_2Ti_2O_7. We used identical boundary-free sample geometries within a superconducting toroidal solenoid. 
We find that, for both materials, the DC relaxation follows a stretched exponential, KWW form (Fig. <ref> a,b), the AC susceptibility follows a HN form (Fig. <ref> c,d), and above 0.8 K the relaxation time for both materials diverges along a super-Arrhenius trajectory (Fig. <ref>). These phenomena all indicate that the magnetic state of these two distinct materials is the magnetic analog of a glass-forming dipolar liquid, which seems to be a previously unidentified characteristic of this class of frustrated magnetic materials.They were not anticipated by but appear to be consistent with the DSIM.
The differences between the parameters of this general glass-forming spin-liquid phenomenology for the two materials can offer an insight into the microscopic behavior generating these phenomena. 
Indeed, recent theoretical studies using the spin-ice Hamiltonian extended to include stronger next nearest neighbor interactions, do report the existence of new forms of dynamical magnetic heterogeneity with extremely slow relaxation times for some spins <cit.>. Thus, the type of glass-forming spin-liquid phenomenology that we observe in Dy_2Ti_2O_7 and Ho_2Ti_2O_7, can exist, in theory, in dipolar spin-ice.


	
We acknowledge useful and encouraging discussions with S. Bramwell, C. Castelnovo, J. Chalker, H. Changlani, M. Gingras, E.-A. Kim, M.J. Lawler and J. Sethna. J.C.S.D., R.D. and A. E. acknowledge support from the Moore Foundation's EPiQS Initiative through Grant GBMF4544. 
 






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