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# Supercooled spin liquid state in the frustrated pyrochlore Dy_{2}Ti_{2}O_{7}

Contributed by J. C. Séamus Davis, June 8, 2015 (sent for review April 10, 2015; reviewed by Zohar Nussinov and Peter Schiffer)

## Significance

Frustrated magnetic pyrochlore systems, in which there are many possible favored spin configurations, may host a variety of exotic magnetic phases. For example, some models predict that Dy_{2}Ti_{2}O_{7} hosts a fluid of mobile magnetic “monopoles” that interact via a magnetic Coulomb interaction. Here, we introduce a novel measurement technique that realizes periodic boundary conditions, and use it to examine the magnetization transport dynamics of tori of Dy_{2}Ti_{2}O_{7}. We identify multiple phenomena in the dynamics of Dy_{2}Ti_{2}O_{7} that are characteristic of a supercooled magnetic liquid approaching a glass transition. This highly unusual classical spin liquid forms in a structurally ordered crystal and therefore it may constitute the approach to a novel magnetic glass state.

## Abstract

A “supercooled” liquid develops when a fluid does not crystallize upon cooling below its ordering temperature. Instead, the microscopic relaxation times diverge so rapidly that, upon further cooling, equilibration eventually becomes impossible and glass formation occurs. Classic supercooled liquids exhibit specific identifiers including microscopic relaxation times diverging on a Vogel–Tammann–Fulcher (VTF) trajectory, a Havriliak–Negami (HN) form for the dielectric function _{2}Ti_{2}O_{7} has become of interest because its frustrated magnetic interactions may, in theory, lead to highly exotic magnetic fluids. However, its true magnetic state at low temperatures has proven very difficult to identify unambiguously. Here, we introduce high-precision, boundary-free magnetization transport techniques based upon toroidal geometries and gain an improved understanding of the time- and frequency-dependent magnetization dynamics of Dy_{2}Ti_{2}O_{7}. We demonstrate a virtually universal HN form for the magnetic susceptibility _{2}Ti_{2}O_{7} therefore exhibits the characteristics of a supercooled magnetic liquid. One implication is that this translationally invariant lattice of strongly correlated spins may be evolving toward an unprecedented magnetic glass state, perhaps due to many-body localization of spin.

Cooling a pure liquid usually results in crystallization via a first-order phase transition. However, in glass-forming liquids when the cooling is sufficiently rapid, a metastable “supercooled” state is achieved instead (1⇓–3). Here, the microscopic relaxation times diverge until equilibration of the system is no longer possible at a given cooling rate. At this juncture there is generally a broad peak in the specific heat preceding the glass transition, at which no symmetry-breaking phase transition occurs (Fig. 1*A*). The antecedent fluid exhibits a set of phenomena characteristic of the supercooled liquid state (1⇓–3). For example, the divergence of microscopic relaxation times *D* characterizes the extent of the super-Arrhenius behavior (Fig. 1*B*). One way to establish *α* and *γ* describe, respectively, the broadening and asymmetry of the relaxation in frequency compared with a simple Debye form (*α* = γ = 1), *C*. In the time domain, this relaxation is described by the classic Kohlrausch–Williams–Watts (KWW) form (7)** P**(t),

*D*). Debye relaxation corresponds to

## Magnetization Dynamics Studies of Dy_{2}Ti_{2}O_{7}

Frustrated magnetic pyrochlores are now the focus of widespread interest because of the possibility that they can support different exotic magnetic phases (9⇓⇓⇓⇓–14). The pyrochlore Dy_{2}Ti_{2}O_{7} is one of the most widely studied; it consists of highly magnetic Dy^{3+} ions in a sublattice comprising corner-sharing tetrahedra (Fig. 2*A*), and an interpenetrating octahedral lattice of Ti^{4+} and O^{2−} ions playing no magnetic role. Crystal fields break the angular-momentum-state degeneracy and cause the Dy^{3+} moments (_{2}Ti_{2}O_{7} down to *T* ∼ 50 mK. Additionally, a broad peak in the specific heat (18⇓⇓⇓–22) centered around _{2}Ti_{2}O_{7}. Instead, dipole and exchange interactions combine to create an effective nearest-neighbor Ising interaction of the form _{2}Ti_{2}O_{7} tetrahedron; these can be mapped to the Bernal–Fowler (“2-in, 2-out”) rules that govern hydrogen bond configurations in water ice but now it is a 2-in, 2-out arrangement of Dy^{3+} moments (Fig. 2*B*). This elegant “spin-ice” configuration has been firmly established (10, 14, 21, 22, 24). Theoretically, the long-range dipole interactions can also generate magnetic ordering (10, 14, 25) but, significantly, this has not been observed down to temperatures below 50 mK.

The magnetic excited states of Dy_{2}Ti_{2}O_{7} are then of great interest. One conjecture is that the magnetization dynamics may be described as a fluid of emergent delocalized magnetic monopoles (26). The widely used dipolar spin-ice model (DSIM) (23) can be used to derive this magnetic monopoles in spin-ice (MMSI) picture. DSIM incorporates nearest-neighbor exchange interactions and long-range dipole interactions:*a* is the nearest-neighbor distance between moments, *J* is an exchange strength, and ^{3+} dipoles are recast as two opposite magnetic charges which, through a sequence of spin flips, are hypothesized to form a fluid of delocalized magnetic monopoles (red and green in Fig. 2*C*) (26). At low temperatures these monopoles might then form a dilute neutral gas analogous to an electrolyte, so that a Debye–Hückel electrolyte model (27) suitably modified for magnetic monopoles may be used to describe a “magnetolyte” state (28). Such a fluid of delocalized magnetic monopole excitations, if extant, would constitute a highly novel magnetic state.

However, many observed properties of Dy_{2}Ti_{2}O_{7} remain unexplained when they are analyzed using the DSIM/MMSI models. Although DSIM captures some of the diffuse neutron-scattering features of Dy_{2}Ti_{2}O_{7}, simulations based on Eq. **5** give an incomplete description of the data; additional exchange contributions from next-nearest and third-nearest neighbors are required to actually fit the measured scattering intensities precisely (29). Similarly, whereas the magnetic susceptibility _{2}Ti_{2}O_{7} cannot be described by the simple Debye form (e.g., Eq. **2** with *α* = *γ* = 1) that would be expected of a typical paramagnet (33). Microscopic approaches do not capture precisely the Dy_{2}Ti_{2}O_{7} magnetization dynamics. For example, DSIM simulations do not accurately reproduce the measured _{2}Ti_{2}O_{7} become ultraslow (20, 21, 30, 33, 35⇓–37); this reflects a divergence of microscopic relaxation times (30, 33) that is unexplained quantitatively within the present DSIM/MMSI models. Thus, although Debye–Hückel calculations (28) and DSIM/MMSI simulations (38, 39) offer clear improvements over a simple Arrhenius form for _{2}Ti_{2}O_{7} have yielded direct and unambiguous evidence of a fluid of delocalized magnetic monopoles. This motivates the search for a more accurate identification and understanding of the low-temperature magnetic state of this important compound.

## Experimental Methods

To explore these issues, we introduce a novel boundary-free technique for studying magnetization transport in Dy_{2}Ti_{2}O_{7}. The innovation consists primarily of using a toroidal geometry for both the Dy_{2}Ti_{2}O_{7} samples and the magnetization sensors, an arrangement with several important benefits. The first is that the superconducting toroidal solenoid (STS) can be used to both drive any magnetization flows azimuthally and to simultaneously and directly detect d** M**/dt throughout the whole torus. More significantly, this topology removes any boundaries in the direction of the magnetization transport. To achieve this sample topology, holes were pierced through disks of single-crystal Dy

_{2}Ti

_{2}O

_{7}(

*SI Text*, section S1). A superconducting toroidal solenoid of CuNi-clad NbTi wire (diameter ∼0.1 mm) is then wound around each Dy

_{2}Ti

_{2}O

_{7}torus (

*SI Text*, section S1). Typical samples had an inner diameter ∼2.5 mm, outer diameter ∼6 mm, and thickness ∼1 mm. The superconducting circuitry used to drive and measure d

**/dt is shown schematically in Fig. 2**

*M**D*, along with indicators of the azimuthal

*Bϕ*generated by the STS (blue arrows) and the putative flow of a mixed-sign monopole fluid (red/green arrows). The all-superconducting four-point I–V measurement circuit used to generate the

*Bϕ*-field and simultaneously measure the electromotive force (EMF) induced by d

**/dt is shown schematically in Fig. 2**

*M**D*. These toroidal sample/coil assemblies were mounted on a dilution refrigerator and studied at temperatures 30 mK <

*T*<3 K, and using currents not exceeding 30 mA in the STS (

*Bϕ*< ∼1 G, so always in the low-field limit

### Time-Domain Measurements.

Elementary magnetization dynamics experiments in the time domain are then carried out using the following repeated measurement cycle. First, a current (and thus *Bϕ*) is switched on in the STS, and the EMF across this same coil *V*(t) is measured from when the switch-on transient ends until *V*(t) drops below our noise level. Next, the current (and *Bϕ*) is turned off and we again measure the *V*(*t*) response. The current is then turned on in the reverse direction (−*Bϕ*) for the same time interval and turned off for the same time interval, and the procedure is repeated. Complete data sets for this sequence as a function of temperature are shown in *SI Text*, section S3.

Fig. 3*A* shows the time-dependent magnetization of Dy_{2}Ti_{2}O_{7} determined from *V**A*). To study these measurements in the context of KWW, we fit each transient **3** and determine *B*; for these temperatures *β* ∼ 0.75. There we see that, for all measured transients over a range 600 mK < *T* < 900 mK, the _{2}Ti_{2}O_{7} is very well represented by a KWW time dependence. A complete set of all measured transients and KWW fits between 575 mK and 900 mK is shown in the *SI Text*. This puts the ultraslow magnetization dynamics in the same empirical class as classic supercooled liquids (1⇓⇓–4). Stretched-exponential relaxation has been previously seen in studies of single-crystal rods of Dy_{2}Ti_{2}O_{7} (37) but there it was proposed that this relaxation occurred due to the open boundary conditions. However, our sample topology is a physical realization of periodic boundary conditions implying that our observed KWW relaxation is actually a fundamental property of the system.

### Frequency-Domain Measurements.

If KWW magnetization transient dynamics as in Fig. 3 are evidence for supercooling of a correlated magnetic fluid, the equivalent phenomena should also be observed as a universal HN form for the frequency-dependent complex magnetic susceptibility *Bϕ* *SI Text*, section S2). Directly from the experimental setup (*SI Text*, section S2) one can write *L* is the effective geometrical inductance of the STS. A complete set of *A* and *B* and described in *SI Text*, section S2. Equivalent phenomena were measured in three different Dy_{2}Ti_{2}O_{7} tori ruling out any specific sample preparation or torus/coil geometry effects as the cause of reported phenomena. Empirically, the temperature and frequency dependence of the _{2}Ti_{2}O_{7} has been identified. Here, we show that an HN form for a magnetic susceptibility

provides a comprehensive accurate internally consistent description for both _{2}Ti_{2}O_{7}. To achieve this, we need to demonstrate that all our disparate *A* and *B*) have the same HN functional form. We define a scaled susceptibility *γ*,

where we have neglected *SI Text*, section S2). As shown analytically in *SI Text*, section S2, if plotting *C* and *D* shows such a data collapse for **7** using the susceptibility of Eq. **6** with χ_{0} = γ = 1. It is clear that _{2}Ti_{2}O_{7} data precisely and comprehensively. To our knowledge, this is the first time that both _{2}Ti_{2}O_{7} have been quantitatively and simultaneously described by a single internally consistent function, across very wide frequency–temperature ranges and with Kramers–Kronig consistency. More importantly, because this HN form for magnetic susceptibility is functionally indistinguishable from the

### Diverging Microscopic Relaxation Rates.

A final test of the supercooled magnetic liquid hypothesis would be to show that the magnetic relaxation times diverge on a VTF trajectory for Dy_{2}Ti_{2}O_{7}. To determine microscopic relaxation times *T*) spanning the whole temperature range, we used *T*) from fitting Eq. **6** simultaneously to *T* < 3 K, and *T*) from the time-domain V(*t,T*) fitted by Eq. **3** and then converted to the equivalent *T*) for 0.58 K < *T*<0.85 K using Eq. **4**. As shown in Fig. 4*E*, the resulting *T*) of Dy_{2}Ti_{2}O_{7} can, indeed, be represented by a VTF function (Eq. **1**) with high precision over many orders of magnitude (_{2}Ti_{2}O_{7} for 0.5 K < *T* < 3 K is a supercooled magnetic liquid.

## SI Text

## S1. Toroidal Dy_{2}Ti_{2}O_{7} Sample and Toroidal Superconducting Sensor Coil

The Dy_{2}Ti_{2}O_{7} samples used in this project were synthesized by the G.M.L. Group at McMaster University using an optical floating zone furnace. The original sample boule was grown at a rate of ∼7 mm/h in O_{2} gas under 2 atm of pressure, and was subsequently cut into disks of diameter ∼6 mm and thickness ∼1 mm. X-ray diffraction on the resultant crystal was sharp and showed no signs of twinning or the presence of multiple grains. Performance of Rietveld refinement on the diffraction data yields a unit cell lattice constant of 10.129 Å; this implies a maximum possible level of “spin stuffing” (substitution of Dy^{3+} ions on Ti^{4+} sites) of ∼2.9% and a most likely spin-stuffing fraction <1%.

To create boundary-free conditions for our measurements of magnetization dynamics, we pierced holes of diameter ∼2.5 mm through each Dy_{2}Ti_{2}O_{7} disk using diamond-tipped drill bits. Fig. S1*A* shows two typical samples after the completion of the drilling process. We also manufactured equivalent tori from Stycast 1266 for control measurements; these were cut and drilled to match the geometrical details of our Dy_{2}Ti_{2}O_{7} samples. Because Stycast 1266 has negligible magnetic activity in our parameter space, it functioned as a good material for control tests; we observed no significant temperature-dependent magnetic signals in this epoxy.

CuNi-clad NbTi wire (thickness 0.1 mm) was wound around each toroidal Dy_{2}Ti_{2}O_{7} sample, creating a superconducting toroidal solenoid (STS) (Fig. S1*B*). Lakeshore varnish (VGE-7031) was used to mount our samples on a stage connected to the mixing chamber of a dilution refrigerator, and we performed measurements at temperatures from 30 mK up to 3 K. Sample temperature was measured with a Lakeshore RuO_{2} (Rox) thermometer, and temperature was controlled using a Lakeshore 370 ac resistance bridge.

## S2. ac Magnetization Dynamics Experiments

For ac measurements four-probe I–V circuit configuration (Fig. 2*D* in the main text) was used. A lock-in amplifier was used to apply ac currents (of up to 30 mA) to the STS and to simultaneously measure the STS EMF. We begin our ac analysis by writing the STS EMF in terms of the changing magnetic flux:*N* is the number of coil turns in the STS and *A* is the effective coil cross-sectional area. Using the standard definition of the frequency-dependent magnetic volume susceptibility, *N* is total number of STS turns, *n* is the turns per length in the STS, *A* is the effective cross-sectional area of the coil, *I* is the applied current, and *L* is the effective geometric inductance of the STS. We specifically measured the EMF components that were in phase (“X”) and 90° out of phase (“Y”) with the applied current: _{2}Ti_{2}O_{7} measurements, showed no temperature dependence; all signals in the Stycast coils were due simply to standard circuit responses to a changing magnetic field. These observations confirmed that the circuit geometry and conductance in the area of the experiment had no significant variation in our measured temperature range. Any temperature-dependent changes in the frequency-dependent STS EMF are due to changes in the magnetization dynamics of the Dy_{2}Ti_{2}O_{7} samples.

Because the microscopic relaxation times of Dy_{2}Ti_{2}O_{7} are sensitive to its temperature (30), it was essential to verify that our samples and measurements stabilized at each set temperature. We determined thermalization times by performing tests in which we took lock-in amplifier readings for several hours after temperature and frequency changes; Fig. S2 shows typical results of these measurements. After frequency changes the EMF readings settled almost immediately (within seconds) to stable long-time values, whereas the readings reached stable values after less than 10 min following temperature changes. To accommodate this thermalization time, we waited at least 15 min after temperature changes before recording data in both the ac and dc experiments.

We measured the geometric inductance *L* of our STS systems by measuring *T* = 30 mK; these frequencies are well above the frequency range in which we observed significant Dy_{2}Ti_{2}O_{7} dynamics, so for these measurements the STS EMF is dominated by the vacuum inductor signal

Fig. S3 *A* and *B* shows the real and imaginary parts, respectively, of typical STS EMF data at 500 and 900 mK, with 50-mK data treated as background and subtracted. There is a clear difference in the EMFs measured at 900 and 50 mK; we use this temperature dependence of the STS signals to study the frequency-dependent magnetization dynamics of Dy_{2}Ti_{2}O_{7} (see below). At 500 mK, however, the STS EMF is indistinguishable from the EMF at 50 mK. Previous susceptibility measurements (33) indicate that Dy_{2}Ti_{2}O_{7} has negligible magnetic activity in our frequency range (2–10,000 Hz) at temperatures as low as several hundred mK; we therefore assume that the data at 50 mK are due completely to non-Dy_{2}Ti_{2}O_{7} sources, and we conclude that all data in our parameter space taken at *T* ≤ 500 mK are due to non-Dy_{2}Ti_{2}O_{7} background sources.

For experiments at higher temperatures, we subtracted the 500-mK data to isolate Dy_{2}Ti_{2}O_{7} contributions to the STS EMF:*α* and *γ* describe the spread and asymmetry, respectively, of the relaxation in frequency space. *A* and *B* shows the quadrature susceptibility components at two representative temperatures, along with fits to Debye and HN forms. Debye relaxation is clearly an inadequate description of our data; one needs the full set of HN parameters to reproduce the observed magnetization dynamics.

Fig. S5 *A* and *B* shows the full set of real and imaginary parts of the susceptibility calculated from the observed EMFs using Eq. **S2**. These experiments were performed between 0.8 and 3 K. We performed simultaneous fits of *A* and *B* show the results of these fits; our observations are clearly described very well by HN relaxation. The small residuals [*A*, *Inset*; and *B*, *Inset*] further illustrate the quality of these fits; throughout all of our parameter space the residuals are a few percent or less of the signal size. HN spectra are generally found in supercooled liquids. The temperature dependencies of the HN parameters *A*), *B*), *α* (Fig. S6*C*), and *γ* (Fig. S6*D*) are shown in Fig. S6; *E* of the main text.

We can further show the broad applicability of HN relaxation to Dy_{2}Ti_{2}O_{7} dynamics by finding scaled HN variables that collapse the *T*-dependent *α* and *γ*:**S4** as*G* give us effective scaled variables for the real and imaginary susceptibility components, respectively:**S6** as*α* varies weakly with temperature (Fig. S6*C*), so with these definitions we achieve excellent universal collapse of our ac data (Fig. 4 *C* and *D* in the main text). The characteristic curve of this collapse is a Cole–Cole curve with

## S3. Time-Domain Experiments

During time-domain experiments we used a four-probe superconducting I–V circuit configuration to apply current to and measure the EMF across our STSs (Fig. 2*D* in the main text). For these studies we applied dc currents as high as 25 mA using an Agilent 33210A waveform generator to create a current supply; we simultaneously measured the STS EMF, *V*(*t*), with a Keithley 2182A nanovoltmeter. As shown in Fig. S7*A*, the current application protocol is as follows (green dashed lines): (*i*) apply current in one direction for a time interval *t*, (*ii*) turn off the current for an identical interval *t*, (*iii*) apply current in the opposite direction for *t*, (*iv*) turn off the current for *t* and then return to step (*i*). We typically repeated this sequence hundreds of times to improve our signal-to-noise ratio and achieve robust fits to the data. Fig. S7*A* depicts a typical complete measurement sequence at three experimental temperatures. The STS *V*(*t*) was measured every 20 ms throughout the experiment, and we observed that 20 s is sufficient for the induced EMF to drop below our noise level ∼1 nV at all temperatures above 550 mK. Our ability to resolve decay curves at higher temperatures was limited by the time resolution of the nanovoltmeter.

Our *V*(*t*) are direct measurements of changes in the sample magnetization density *M* over time:*N* is the number of coil turns in the STS and *A* is the effective STS cross-sectional area. The second term, which describes the STS response to changes in the applied field itself, is present even when the STS encloses vacuum. Because the only change in *H* occurs when we turn the applied current on or off, the contributions of this term are limited to very short times (<∼100 ms) after current changes. For our fits we examined data taken ≥200 ms after current changes, when only the first term in Eq. **S8** contributes.

We can relate the measured EMF to the time-dependent magnetic volume susceptibility *χ*(*t*) by considering the STS as having inductance *L* with *I* as the field-generating current. Taking the time derivative of *M*(*t*) and inserting it into Eq. **S8** gives the final expression for the long-time STS EMF:

Fig. S7*B* depicts the typical measured EMF (symbols) generated for *t* ≥ 200 ms after turning off the STS current at temperatures from 575 mK up to 900 mK; at these times the EMF is given by Eq. **S9**. The STS EMF shows slower-than-exponential decay, and the decay times increase dramatically with decreasing temperature. The lines in Fig. S7*B* are fits to the KWW form*β* is a stretching exponent. Here, *β* = 1 corresponds to standard Debye relaxation, whereas *β* < 1 indicates the presence of a more complex landscape of energy barriers and dynamics. The KWW fits to our measured *V*(*t*) were performed using a least-squares method.

Fig. S7*B* shows that the KWW form describes the time-domain magnetization dynamics very well. If there is truly universal applicability of the KWW function to our data, we should be able to define scaled variables such that data from all temperatures collapse onto a single function. We achieve this by defining a scaled time **S10**, relaxation with a KWW form should collapse onto a simple exponential given by *B* in the main text).

Fig. S7*B* (*Inset*) shows residuals for the KWW fits, i.e., *V*(*t*) data. Fig. S8 shows the temperature dependence of the exponent *β*; this exponent shows a weak increasing trend, but in most of the temperature range we find _{2}Ti_{2}O_{7}. The temperature dependence of *E* of the main text, as is the behavior of the relaxation time obtained from our ac experiment (see below).

## S4. Super-Arrhenius Diverging Microscopic Relaxation Times

The temperature dependence of the observed relaxation times differs substantially from the standard Arrhenius form *D* characterizes the extent of departure from an Arrhenius form (often referred to as the “fragility” of the liquid), and

To apply this formalism to our Dy_{2}Ti_{2}O_{7} measurements, we must present the results from our separate dc and ac measurements in a unified manner. We have done this by converting **4** of the main text; the complete temperature dependence of the relaxation times is plotted in Fig. 4*E* in the main text and in Fig. S9. We performed ac and dc experiments at overlapping temperatures of 800–850 mK to make sure that the relaxation times obtained by these two independent techniques give a consistent description of Dy_{2}Ti_{2}O_{7} magnetization dynamics. Fig. S9 shows that the microscopic relaxation time in Dy_{2}Ti_{2}O_{7} varies smoothly throughout our entire temperature range, even when we cross over from our ac results to dc results. Results for _{2}Ti_{2}O_{7} dynamics from 575 mK up to 3 K. **S11**.

## Conclusions and Discussion

One may now reconsider the anomalous phenomena of Dy_{2}Ti_{2}O_{7} in this new context. Empirically, our measured **2**; Fig. 4 *A* and *B*). Also, whereas the Curie–Weiss temperature *T*_{CW} ∼ 1.2 K (16) implies a tendency toward ferromagnetic order, no ordering is observed and, instead, a broad peak in specific heat *C(T)* appears just below *T*_{CW} (18⇓⇓⇓–22); this is as expected for a supercooled liquid (Fig. 1*A* and refs. 1⇓–3). The location of Dy^{3+} moments in the highly anisotropic environment of the Dy_{2}Ti_{2}O_{7} prevents ferromagnetic ordering at a temperature that might be expected from the nearest-neighbor interaction energy scale; this may be analogous to preventing the onset of a crystalline phase in glass-forming liquids due to anisotropic interactions between molecules. Moreover, ultraslow macroscopic equilibration is widely reported at lower temperatures (20, 35, 37) and is also just what one expects in a supercooled liquid approaching the glass transition (1⇓–3). Our observed stretched-exponential form for ultraslow magnetization relaxation agrees well with previous studies (37). Although actually at odds with the predictions of DSIM/MMSI simulations for periodic geometries (39), this phenomenon is characteristic of a supercooled fluid (1⇓–3). Finally, the published data on divergences of microscopic relaxation times (30, 33) are in good empirical agreement with ours, implying that the VTF form for *τ*(*T*) is general for Dy_{2}Ti_{2}O_{7} (Eq. **1**; Fig. 4*E*). Thus, we conjecture that overall magnetization dynamics of Dy_{2}Ti_{2}O_{7} are best explained if the system is a classical correlated-spin liquid that is supercooled and approaching a glass transition. Within this picture, the divergence temperature *E*) provides an estimate of the lowest temperature at which a metastable magnetic liquid state can survive under arbitrary cooling protocols; below this temperature we expect that Dy_{2}Ti_{2}O_{7} must transition into either a heterogeneous glass phase or (with extremely slow cooling) a phase with global magnetic order.

To recapitulate: For the magnetic pyrochlore system Dy_{2}Ti_{2}O_{7} we discover that the magnetic susceptibility *T*) occur on the VTF trajectory. When, in combination with a broad specific heat peak, this phenomenology is observed for the _{2}Ti_{2}O_{7} is a supercooled classical spin liquid, approaching a glass transition. However, we emphasize that one should not expect any consequent magnetic glass to be a classic spin glass, because all of the Dy_{2}Ti_{2}O_{7} spins are at ordered crystal lattice sites with locally identical spin Hamiltonians. And, indeed, Dy_{2}Ti_{2}O_{7} is known to exhibit a very different field dependence from what is seen in classical spin glasses (30). The supercooled liquid characteristics of magnetization dynamics in Dy_{2}Ti_{2}O_{7} (Figs. 3 and 4) more likely imply some form of persistent heterogeneous freezing in the microscopic configurations of strongly correlated spins (40, 41). Such a situation could also be described (somewhat redundantly) as freezing of monopole configurations. However, in terms of actual magnetization transport, the observed stretched-exponential time dependence of magnetization (Fig. 3; ref. 37) contradicts the predicted dynamics of both DSIM and MMSI models (39). Instead, one intriguing possibility is that the state of Dy_{2}Ti_{2}O_{7} actually represents translationally invariant many-body localization of the spins (42⇓–44). It will be fascinating, in this context, to reconsider the absence of magnetic ordering in other frustrated pyrochlores so as to determine if supercooled classical spin liquids occur therein.

## Acknowledgments

We are grateful to E. Fradkin, B. Gaulin, M. Gingras, S. Grigera, D. Hawthorn, R. Hill, E.-A. Kim, J. Kycia, M. J. Lawler, A. P. Mackenzie, R. Melko, and J. Sethna for very helpful discussions and communications. This research is funded by the Gordon and Betty Moore Foundation’s Emergent Phenomena in Quantum Systems Initiative through Grant GBMF4544 and by the Engineering and Physical Sciences Research Council Programme Grant “Topological Protection and Non-Equilibrium States in Correlated Electron Systems.”

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: jcseamusdavis{at}gmail.com.

Author contributions: E.R.K. and J.C.S.D. designed research; E.R.K., A.B.E., and B.P. performed research; E.R.K. and A.B.E. analyzed data; T.J.S.M. and H.A.D. synthesized the samples; G.M.L. supervised the research and synthesized samples; J.C.S.D. supervised the research; and E.R.K., A.B.E., G.M.L., and J.C.S.D. wrote the paper.

Reviewers: Z.N., Washington University in St. Louis; and P.S., University of Illinois at Urbana–Champaign.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1511006112/-/DCSupplemental.

Freely available online through the PNAS open access option.

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- Abstract
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_{2}Ti_{2}O_{7} - Experimental Methods
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_{2}Ti_{2}O_{7}Sample and Toroidal Superconducting Sensor Coil - S2. ac Magnetization Dynamics Experiments
- S3. Time-Domain Experiments
- S4. Super-Arrhenius Diverging Microscopic Relaxation Times
- Conclusions and Discussion
- Acknowledgments
- Footnotes
- References

- Figures & SI
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