---
abstract: 'Extensive work on single molecule magnets has identified a fundamental mode of relaxation arising from the nuclear-spin assisted quantum tunnelling of nearly independent and quasi-classical magnetic dipoles. Here we show that nuclear-spin assisted quantum tunnelling can also control the dynamics of purely emergent excitations: magnetic monopoles in spin ice. Our low temperature experiments were conducted on canonical spin ice materials with a broad range of nuclear spin values. By measuring the magnetic relaxation, or monopole current, we demonstrate strong evidence that dynamical coupling with the hyperfine fields bring the electronic spins associated with magnetic monopoles to resonance, allowing the monopoles to hop and transport magnetic charge. Our result shows how the coupling of electronic spins with nuclear spins may be used to control the monopole current. It broadens the relevance of the assisted quantum tunnelling mechanism from single molecular spins to emergent excitations in a strongly correlated system.'
author:
- 'C. Paulsen'
- 'S. R. Giblin'
- 'E. Lhotel'
- 'D. Prabhakaran'
- 'K. Matsuhira'
- 'G. Balakrishnan'
- 'S. T. Bramwell'
title: Nuclear spin assisted quantum tunnelling of magnetic monopoles in spin ice
---

Introduction {#introduction .unnumbered}
============

In the canonical dipolar spin ice materials ([Dy$_{2}$Ti$_{2}$O$_{7}$]{}, [Ho$_{2}$Ti$_{2}$O$_{7}$]{}) [@Harris1997; @BramwellHarris98; @Ramirez; @BramwellGingras], rare earth ions with total angular momentum $J = 15/2$ (Dy$^{3+}$) and $J = 8$ (Ho$^{3+}$) are densely packed on a cubic pyrochlore lattice of corner-linked tetrahedra. The ions experience a very strong $\langle111\rangle$ crystal field, resulting in two effective spin states ($M_{\rm J} = \pm J$) that define a local Ising-like anisotropy. At the millikelvin temperatures discussed here ($0.08~{\rm K}<T<0.2$ K), a lattice array of such large and closely-spaced spins would normally be ordered by the dipole-dipole interaction [@LuttingerTisza], but the pyrochlore geometry of spin ice frustrates the dipole interaction and suppresses long-range order. Instead, the system is controlled by an ice-rule, that maps to the Pauling model of water ice [@Harris1997; @BramwellHarris98; @Ramirez; @BramwellGingras]. In the effective ground state, the spins describe a flux with closed-loop topology and critical correlations, that may be described by a local gauge symmetry rather than by a traditional broken symmetry [@CMS]. This strongly correlated spin ice state is stabilised by a remarkable self-screening of the dipole interaction [@Isakov; @Gingras]. Excitations out of the spin ice state fractionalise to form effective magnetic monopoles [@CMS; @Ryzhkin], but the excited states are no longer self-screened and this manifests as an effective Coulomb interaction between monopoles. The static properties of spin ice are accurately described by the monopole model [@KaiserDH]. The dynamic properties can also be described by assuming an effective monopole mobility [@JaubertHoldsworth; @Bovo; @Paulsen_Wien], but there have been few studies of the microscopic origin of the monopole motion [@Tomasello].

![image](figure1.pdf){width="10cm"}

The field and energy scales involved in monopole motion are illustrated in Fig. 1, a-e. When a monopole hops to a neighbouring site a spin is flipped (Fig. 1a). For an isolated monopole (far from any others) this spin flip takes place at nominally zero energy cost (Fig. 1b) because contributions from near-neighbour antiferromagnetic superexchange and ferromagnetic dipole–dipole coupling individually cancel. The cancellation of the field contribution relies on the dipolar self-screening [@Isakov] that maps the long-range interacting system [@Gingras] to the degenerate Pauling manifold of the near neighbour spin ice model [@BramwellHarris98]. This surprising cancellation is a key result of the many–body physics of spin ice. In practice, a monopole hop may also involve a finite energy change arising from longitudinal fields at the spin site: the main source of fields is nearby monopoles [@CMS] (Fig. 1b), while further contributions arise from corrections to the mapping, which give a finite energy spread to the Pauling manifold [@Vedmedenko] (here of order $\sim$0.1 K [@Melko]). The mechanism of the hop is believed to be quantum tunnelling and several key signatures of this have been observed in the high temperature regime between 2 K and 10 K [@Ehlers; @Snyder; @JaubertHoldsworth; @Bovo; @Tomasello].

At lower temperatures ($T < 0.6$ K), spin ice starts to freeze [@Snyder]. This is due in part to the rarefaction of the monopole gas whose density $n(T)$ varies as $\sim e^{-|\mu|/T}$ where the chemical potential $|\mu| = 4.35 $ and 5.7 K for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} respectively [@CMS], and also in part to geometrical constraints that create noncontractable, monopole-antimonopole pairs that cannot easily annihilate [@noncontractable]. These factors, which are independent of the monopole hopping mechanism, suggest that the relaxation rate $\nu(T) \propto n(T)$ will fall to exponentially small values at low temperature ($T~<~ 0.35$ K).

Previous thermal quenching experiments have demonstrated monopole populations well below the nominal freezing temperature that are both long lived and able to mediate magnetic relaxation [@PaulsenAQP]. This paradoxical frozen but dynamical character of the system suggests the relevance of resonant magnetic tunnelling, where magnetisation reversal can only occur when the longitudinal field is smaller than the tunnelling matrix element $\Delta E$. The monopolar fields may add a longitudinal component that takes the spin off the resonance condition (Fig. 1c,d) but in addition may add a transverse component that amplifies $\Delta E$: together these lead to a suppression and dispersion of the monopole mobility.

In the following, we will demonstrate experimentally that hyperfine interactions (Fig. 1e) play a significant role in bringing monopoles back to their resonance condition, enabling dynamics at very low temperatures ($T~<~0.35$ K).

Results {#results .unnumbered}
=======

[**Samples**]{}

                      $I$                                                                            
  -- ------------------------------------- ------ -------- -------- --------- ---------------------- --
                      7/2                   4.17    0.3     0.034      5.7     1$\times$10$^{-5}  $  
         $\approx$ 19 $\%$, $^{161}$Dy      5/2    -0.48             -0.0039                         
         $\approx$ 25 $\%$, $^{163}$Dy      5/2     0.67             0.0054                          
      $\approx$ 66 $\%$, $^{\rm other}$Dy    0       0                  0                            
                       0                     0       0        0                                      
                      5/2                   0.67   0.0599   0.0054                                   

To investigate the effect of nuclear spins on the magnetic relaxation in spin ice, we studied four spin ice samples: [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, with $I = 7/2$ and three [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples spanning a range of nuclear spin composition from $I=0$ to $I=5/2$. Details of nuclear spins and hyperfine parameters are given in Table 1. Ho$^{3+}$ is a non-Kramers ion with intrinsically fast dynamics owing to the possibility of transverse terms in the single-ion spin Hamiltonian, while Dy$^{3+}$, being a Kramers ion, has intrinsically much slower dynamics. However, it should be noted that, at low temperature, bulk relaxation is slower in [Ho$_{2}$Ti$_{2}$O$_{7}$]{} than in [Dy$_{2}$Ti$_{2}$O$_{7}$]{}, owing to its larger $|\mu|$ and hence much smaller monopole density (See Supplementary Fig. 1).

[**Thermal protocol**]{}

![Controlled cooling of spin ice below its freezing temperature. How the temperature of the samples varies during and after the AQP: (a) The applied field (black) during an AQP, and the temperatures measured by a small thermometer glued directly on top of the samples (schematically shown in (b)) vs log time for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} (HTO, red) and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} (DTO, blue). The inset shows a zoom of the first 6 seconds vs time. (b) Comparison of the sample cooling rates $dT/dt$ as a function of temperature after the AQP for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  from the data in (a) to the Òequilibrium cooling rateÓ $dT/d \tau$ extracted from ac susceptibility data for the two samples (see Supplementary Fig. 6). The cooling rate for [Ho$_{2}$Ti$_{2}$O$_{7}$]{}  crosses the equilibrium rate at $\sim 0.9$ K, and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  at 0.72 K.[]{data-label="two"}](figure2.pdf){width="7.5cm"}

In previous experiments we have accurately manipulated the monopole density in [Dy$_{2}$Ti$_{2}$O$_{7}$]{} by rapid magnetothermal cooling (Avalanche Quench Protocol, AQP) the sample through the freezing transition, allowing the controlled creation of a non-equilibrium population of monopoles in the frozen regime [@PaulsenAQP]. However it is more problematic to cool samples containing Ho, due to the large Ho nuclear spin which results in a Schottky heat capacity of 7 J mol$^{-1}$ K$^{-1}$ at 300 mK. Indeed this anomaly has been exploited by the Planck telescope where the bolometers are attached to the cold plate by yttrium-holmium feet thus allowing passive filtering with a several hour time constant that was crucial to the operation of the system [@PrivateComm]. For [Ho$_{2}$Ti$_{2}$O$_{7}$]{} this means difficulty in cooling. Therefore during some of the runs the sample temperature was recorded via a thermometer directly mounted on the sample face. Fig. 2a shows the monitoring of the sample temperature as it approaches equilibrium for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} and [Dy$_{2}$Ti$_{2}$O$_{7}$]{}  during and after the AQP. The inset of Fig. 2a shows that only a few seconds are required to cool the samples from 0.9 K to 0.2 K, which is well below the freezing transition. Whereas [Dy$_{2}$Ti$_{2}$O$_{7}$]{} continues to cool, reaching 80 mK after only 10 s, [Ho$_{2}$Ti$_{2}$O$_{7}$]{} takes nearly 2000 s to reach the same temperature. Hence the data shown here were taken at 80 mK for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and 200 mK (and 80 mK when possible) for [Ho$_{2}$Ti$_{2}$O$_{7}$]{}.

[**Monopole density**]{}

We have phenomenologically estimated how the monopole density depends upon the rate of sample cooling, $dT/dt$ and the spin relaxation time $\tau(T) = 1/\nu(T)$, which is derived from the peaks in the imaginary component of the ac susceptibility. Differentiation of $\tau(T)$ to give $d\tau/dT$ and hence $dT/d\tau$, allows definition of an equilibrium cooling rate $dT/d\tau$, that gives the maximum cooling rate that may still maintain equilibrium. Fig. 2b compares $dT/dt$ and $dT/d \tau$ for both [Dy$_{2}$Ti$_{2}$O$_{7}$]{}  and [Ho$_{2}$Ti$_{2}$O$_{7}$]{}. It can be seen that after the AQP, $dT/dt$ for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} crosses the equilibrium curve and goes out of equilibrium at $\approx 0.9$ K, and for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} at $\approx 0.72$ K. The upper limit of the monopole density at low temperature can be estimated by equating it to the theoretical value at the crossing temperature: thus we find one monopole on approximately every $10^3$ tetrahedra for both [Ho$_{2}$Ti$_{2}$O$_{7}$]{} and [Dy$_{2}$Ti$_{2}$O$_{7}$]{}.

[**Spontaneous relaxation**]{}

We studied the effect of wait time $t_{\rm w}$ between the end of the avalanche quench and the application of the field with the aim to determine the effect of nuclear spins on the monopole dynamics. Varying the wait time deep in the frozen regime allowed us to gauge the spontaneous evolution of the zero-field monopole density as a function of time: that is, if monopoles recombine in a time $t_{\rm w}$, then the observed monopole current will be smaller, the longer the wait time. Two separate experiments were designed to study these effects. In the first experiment (Fig. 3) after waiting we applied a constant field and measured the magnetization $M$ as a function of time. In the second experiment (Fig. 4) we investigated the effect of wait time on the magnetothermal avalanches [@Slobinsky; @Jackson; @Krey2012] that occur on ramping the field to high values. Both of these allowed access to the magnetic current density $J_{\rm m} = dM / dt$. Full details of the experimental conditions are given in Supplementary Note 1 and Supplementary Fig. 2.

The monopole current is controlled by multiple factors. In the simplest model [@Ryzhkin] there are three of these: the monopole density $n$, the monopole mobility $u$ (related to the spin tunnelling rate) and the bulk susceptibility $\chi$. Thus $J_{\rm m} = dM/dt = \nu (M_{\rm eq} - M)$ where $M_{\rm eq} = \chi H$ is the equilibrium magnetisation and $\nu \propto u n$. In general it is difficult to deconvolve these various factors. In Ref. it was achieved by independent measurement of $n(T)$ and $\chi(T)$ to reveal $u(T)$. In the present time-dependent experiments we cannot perform such a direct separation, but by studying [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples with different isotopes, it seems reasonable to assume that the susceptibility and starting density are roughly the same, so the variation in mobility (hop rate) will dominate differences between the samples. Inclusion of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} in the comparison gives a further point of reference: the starting monopole densities (see above) and susceptibilities for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} are expected to be comparable to those of [Dy$_{2}$Ti$_{2}$O$_{7}$]{}, while the the tunnel splitting (which controls the intrinsic mobility) is also estimated to be of the same order [@Tomasello] in the appropriate range of internal fields (see Fig. 1 and Ref. , Fig. 5).

![image](figure3.pdf){width="12cm"}

Fig. 3 summarizes results for the relaxation of the magnetization $M(t)$ for the different samples, as well as the value of $M(t = 400~{\rm s})$ and the monopole current $J_{\rm m}(t = 0)$ as a function of wait time, for a constant applied field of 0.08 T. The [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples show a clear progression in wait time effect that correlates strongly with their relative densities of nuclear spin states. Thus the monopoles recombine during the wait period much more effectively the larger the nuclear spin: that is, the larger the nuclear spin the higher the monopole mobility, the faster the recombination, and the fewer the monopoles at the start of the measurement. In Fig. 3e, f, higher mobility means the relaxation curves ($M(t = 400~{\rm s})$ and $J_{\rm m}(t = 0)$) shift both up and to the left, so a crossover in curves is expected – and this is indeed observed at the longer times. Near to equilibrium a second crossover would be expected (i.e. the equilibrium current density is higher for the highest mobility), but this crossover is clearly very far outside our time window. Hence our [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples are always far from equilibrium.

The effects observed for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} are yet more dramatic in [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, consistent with the Ho$^{3+}$ non-Kramers character, large nuclear spin, and large hyperfine coupling. Relaxation at 200 mK covers more than two orders of magnitude but is practically extinguished for long wait times, showing that excess monopoles spontaneously recombine to eliminate themselves from the sample. The plots indicate that the half life for monopole recombination in [Ho$_{2}$Ti$_{2}$O$_{7}$]{} would be approximately 150 seconds (much shorter than the equilibrium relaxation time) and suggests that equilibrium in the monopole density is reached at long times. Using the above estimate for the initial monopole density $n(t = 0) \sim 10^{-3}$, we recover a nominal equilibrium density of $n_{\rm eq} = 10^{-5}$ (per rare earth atom). Although this estimate is an upper limit it is nevertheless far from the expected equilibrium density, $n_{\rm eq}^{\rm 200~mK} \sim 10^{-13}$ (calculated by the method of Ref. , see Supplementary Fig. 1). It continues to evolve with temperature, being lower by a further order of magnitude at $T = 80$ mK (Fig. 3f). Most likely, the actual equilibrium monopole density is amplified by defects and disorder in the sample.

[**Magnetothermal avalanches**]{}

![image](figure4.pdf){width="12cm"}

Fig. 4 illustrates the effect of $t_{\rm w}$ on the magnetothermal avalanches. These occur when the injected power ($\mu_0 H \cdot J_{\rm m}$) overwhelms the extraction of thermal energy from the sample to the heat bath [@Jackson] such that monopoles are excited in great excess as the temperature steeply rises. The faster and more abundant the monopoles, the lower the avalanche field. To obtain the data in Fig. 4, after the AQP and $t_{\rm w}$, the applied field was swept at a constant rate, 0.02 T.s$^{-1}$ up to 0.4 T. If the avalanche field $H_{\rm ava}(t_{\rm w})$, is defined as the field where the magnetization crosses 1 $\mu_{\rm B}$ per rare earth ion, (0.5 $\mu_{\rm B}$ for the $^{163}$Dy sample) then the difference in avalanche field $\Delta H_{\rm ava}= H_{\rm ava}(t_{\rm w}) - H_{\rm ava} (t_{\rm w}={\rm minimum})$ allows us to compare the spread of fields for all samples.

Fig. 4a and b show the experimental results for the isotopically enriched [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples at 80 mK, demonstrating a very clear pattern. In general the spread of $H_{\rm ava}(t_{\rm w})$ becomes larger, the larger the nuclear spin, showing again that the nuclear spins strongly enhance the monopole mobility. Thus, the $^{162}$Dy sample (no nuclear spin) shows negligible evolution of the position of the avalanche field. For $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  (shown in Supplementary Fig. 4) the effect is small, while for the $^{163}$Dy sample (maximum nuclear spin) the effect of $t_{\rm w}$ can be clearly seen as a steady progression of $H_{\rm ava}(t_{\rm w})$ to higher fields for increasing $t_{\rm w}$ due to the smaller initial monopole density at the start of the field ramp. Also shown in the figure are the curves that result from slow conventional zero field cooling (CC) from 900 mK to 80 mK (at 1 mK.s$^{-1}$) followed by a 1000 s wait period. For the $^{162}$Dy and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples the CC avalanche field is offset to higher fields, well outside the distribution of $H_{\rm ava}(t_{\rm w})$. For the $^{163}$Dy sample the CC curves falls within the distribution but near the long wait time curves. Also, we note for $^{163}$Dy, which happens to have better thermal contact, and thus faster cooling during the AQP, the CC curve again falls outside the distribution (shown in Supplementary Fig. 10 b). Thus slow cooling is more efficient at approaching equilibrium in [Dy$_{2}$Ti$_{2}$O$_{7}$]{} than is the AQP cooling followed by a long $t_{\rm w}$, especially for the low nuclear moment samples. This is typical behaviour for frustrated or disordered systems because slow cooling allows the system time to explore all available phase space.

Fig. 4c shows a much greater effect of $t_{\rm w}$ for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} with a larger spread of fields, saturating near 0.32 T for the the longest $t_{\rm w}$. This is again consistent with the conclusion that that the larger the nuclear spin moment, the more effective the spontaneous monopole recombination. The measurements were performed primarily at 200 mK, but the same conclusion follows from measurements at 80 mK. [Ho$_{2}$Ti$_{2}$O$_{7}$]{} also exhibits some unusual behaviour suggesting that the monopole density and magnetization do not approach equilibrium in a simple way. First, the magnetization jumps fall short of the $M$ vs $H$ equilibrium curve taken at 900 mK, even though thermometers placed on the sample indicate that the sample does indeed heat above 900 mK (see Supplementary Note 2 and Supplementary Fig. 3 for more details). Secondly, in contrast to the behaviour of [Dy$_{2}$Ti$_{2}$O$_{7}$]{} discussed above, the CC curve of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} falls in the middle of the distribution of $H_{\rm ava}$($t_{\rm w}$) indicating, unusually, that waiting long enough at low temperature is an equally efficient way of approaching equilibrium as slow cooling.

Discussion {#discussion .unnumbered}
==========

The experimental result demonstrated here is that magnetic monopole dynamics in the frozen regime of spin ice are greatly enhanced by the hyperfine coupling of the electronic and nuclear moments. We now argue that this observation finds a natural – albeit surprising – explanation by analogy with the properties of single-molecule magnets [@review]. These are metal-organic clusters with large composite spins: some of the most studied include the so called Mn$_{12}$ and Fe$_8$ systems, both of which can be thought of as an ensemble of identical, weakly interacting nanomagnets of net spin S = 10 with an Ising-like anisotropy. The degenerate $M_{\rm s} = \pm S$ states are split by the ligand electric field into a series of doublets. At temperatures smaller than the level separation, the spins flip by resonant tunnelling through a quasi-classical barrier. The signature of a resonant tunnelling effect in Fe$_8$ is a peak in the low temperature relaxation rate around $H = 0$ [@Sa]. It quickly became clear that to understand the resonant tunnelling both dipolar and dynamic nuclear spin contributions to the interactions need to be accounted for. The typical dipolar field in such a system is $\approx 0.5$ K, and the relevant tunnel splitting $\Delta E$ of the order 10$^{-8}$ K, meaning that a broad distribution of dipolar field and a static hyperfine contribution would force all the spins off resonance. Prokof’ev and Stamp [@PS] proposed that dynamic nuclear fluctuations can drive the system to resonance, and the gradual adjustment of the dipole fields in the sample caused by tunnelling, brings other clusters into resonance and allows a continuous relaxation. Hence the observation of relaxation in single molecule magnets is fundamentally dependent on the hyperfine coupling with the fields of nuclear spins [@Wernsdorfer00].

The Prokof’ev and Stamp model [@PS] certainly does not apply in detail to spin ice at low temperatures. First, in single molecule magnets the spin of any particular complex in the system is available to be brought to resonance, whereas in spin ice, only those spins that are instantaneously associated with a diffusing monopole are available to tunnel (and this presumes that more extended excitations can be neglected). The remaining spins – the vast majority – are, in contrast, static and instantaneously ordered by the ice rules. The rate of flipping of these quasi-ordered spins, which corresponds to monopole pair creation, is negligible at the temperatures studied and the process is not relevant to our experiments. Thus, even at equilibrium, spin ice has an effective number of flippable spins that depends on temperature (see Supplementary Fig. 1). Away from equilibrium, where our experiments are performed, the number of flippable spins in spin ice further depends on time, with monopole recombination depleting their number. In addition, it seems reasonable to assume that the reduction of the density of monopoles is even more important during the relaxation process; as monopoles move through the matrix magnetizing the sample they will annihilate when they encounter a monopole of opposite charge, or become trapped on a defect or on the sample surface. This feature of spin ice is a second important difference with single molecule magnets, as modelled in Ref. 27.

A third difference relates to the distribution of internal fields in the system. In spin ice only, the actual field associated with a flippable spin, both before and after a flip, is a monopolar field. Flipping a spin transfers a monopole from site to site (Fig. 1a), dragging the monopolar field with it: a field that is much stronger and of longer range than any conventional dipole field. However, the change in field on a spin flip is dipolar, as in single molecule magnets.

In short, the flippable spins in spin ice are really an aspect of the emergent monopole excitation rather than a perturbed version of an isolated (composite) spin as assumed for the single molecule magnets in Ref. . Yet despite this difference, it seems reasonable to suggest that the basic idea of Ref. does apply to spin ice. The longitudinal monopolar fields will take flippable spins off resonance (Fig. 1c-e), while the transverse ones will tend to broaden the resonance well beyond the tunnel splitting calculated for an isolated spin. i.e. $\Delta E= 10^{-5}$ K [@Tomasello]. An applied field can also take flippable spins on or off resonance or broaden the resonance, depending on its direction. Nevertheless, in zero applied field, at very low temperatures we would expect all flippable spins associated with isolated monopoles to be off resonance and hence unable to relax, unless they are brought back to resonance by a combination of the monopole fields and the fluctuating nuclear spins: nuclear assisted flipping of spins will then bring further spins to resonance via the change in dipolar fields, as in the Prokof’ev-Stamp picture [@PS]. Our experimental results for the wait time dependence of various properties clearly support this proposition: in zero field (during $t_{\rm w}$) the sample with no nuclear spin is scarcely able to relax its monopole density, while the larger the nuclear spin, the quicker the relaxation. For flippable spins associated with closely–spaced monopole-antimonopole pairs the situation is slightly different. Although they are strongly off-resonance (Fig. 1b), the decreasing transition matrix elements will be compensated by the increasing Boltzmann factors required for detailed balance. Also, for the final recombination, a favourable change in exchange energy will reduce the field required to bring spins to resonance (see caption, Fig. 1).

We note in passing that the differences between single molecule magnets and spin ice are also evident in our data. Specifically, a $t^{1/2}$ initial relaxation of the magnetisation is a property of single molecule magnets, with the $t^{1/2}$ form arising from the dipole interactions  [@PS; @Pauling-t-half]. Given the very unusual field distribution in spin ice, and the complicating factor of monopole recombination, as described above, it is hardly likely that this functional form will apply. We test for a $t^{1/2}$ decay in the Supplementary Fig. 5 and confirm that it can only be fitted over a narrow time range: to calculate the true time dependence in spin ice poses a theoretical problem.

Our main result has implications for both the theory of spin ice and the theory of nuclear spin assisted quantum tunnelling. First, in previous work [@Paulsen_Wien] we have shown how the low-temperature quenched monopole populations of [Dy$_{2}$Ti$_{2}$O$_{7}$]{} obey the nonlinear and non-equlibrium response of monopole theory [@Kaiser2] that was developed assuming a single hop rate. In view of our findings, the theory should apply most accurately to the [Dy$_{2}$Ti$_{2}$O$_{7}$]{}  sample with no nuclear spins and least accurately to [Ho$_{2}$Ti$_{2}$O$_{7}$]{}  where the hyperfine splitting energies are of a similar order to the Coulomb energies. In other measurements, presented in Supplementary Figs. 7 and 8, we confirm that this is the case; hence a generalisation of the theory of Ref. to include the effect of nuclear spins seems an attainable goal. We also note that [Ho$_{2}$Ti$_{2}$O$_{7}$]{} offers the unusual situation that, at low temperatures ($<$ 0.35 K) and sufficient wait times, the nuclear spins are ice-rule ordering antiparallel to their electronic counterparts; hence spin ice offers a rare chance to investigate the effect of correlation on nuclear spin assisted quantum tunnelling in a controlled environment. Perhaps this will shed light on some of the unusual properties particular to [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, as noted above.

Spin ice thus exemplifies a remarkable extension of the concept of nuclear spin assisted quantum tunnelling [@PS] to the motion of fractionalised topological excitations [@CMS]. This is made possible by the fact that the emergent excitations of the system – the monopoles – are objects localised in direct space that move through flipping spins. As well as illustrating this generic point, our result may also have practical consequences. We have established how coupling with nuclear spins controls the magnetic monopole current and the spectacular magnetothermal avalanches: hence any experimental handle on the nuclear spins of the system would also be a rare experimental handle on the monopole current. Any future application of magnetic monopoles in spin ice will surely rely on the existence of such experimental handles.

Methods {#methods .unnumbered}
=======

[*Samples.* ]{}Single crystals were grown by the floating zone method for all samples, the natural $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{}  samples (DTO, HTO) were prepared at the Institute of Solid State Physics, University of Tokyo, Japan, and $^{162}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}, $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} at Warwick University and Oxford University respectively.

[*Measurements.* ]{} Measurements were made using a low temperature SQUID magnetometer developed at the Institut Néel in Grenoble. The magnetometer is equipped with a miniature dilution refrigerator with a base temperature of 65 mK. The fast dynamics after a field change were measured in a relative mode, the slower measurements were made by the extraction method, and the initial relative measurements were adjusted to the absolute value extraction points. The field could be rapidly changed at a rate up to 2.2 T s$^{-1}$.

For all the data shown here the field was applied along the \[111\] crystallographic direction. Measurements were also performed perpendicular to the \[111\] direction, as well as along the \[001\] and \[011\] directions and on a polycrystalline sample, examples of which are discussed in Supplementary Note 3. In total ten different samples were studied. The direction of the applied field as well as differences in the sample shapes and thermal contact with the sample holder can effect some of the details of the measurements. However this does not change the main conclusion of the paper: the demonstration of the importance of nuclear assisted quantum tunnelling to the relaxation.

The measurements of temperature vs time shown in Fig. 2, a bare-chip Cernox 1010-BC resistance thermometer from LakeShore Cryogenics was wrapped in Cu foil and glued on top of the sample as shown in the inset of Fig. 2b.

Cooling [Ho$_{2}$Ti$_{2}$O$_{7}$]{} was difficult and warming was also tricky using the AQP, depending on the initial temperatures and wait times. Therefore to ensure the sample was heated above 900 mK, two AQP were used, separated by 300 seconds, which explains why the starting temperature for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} was higher in Fig. 2 (see Supplementary Note 2 and Supplementary Fig. 3 for further discussions).

A schematic of the AQP used for the preparation of the samples is shown in Fig. 3d. First a field of $-0.3$ T was applied and the sample was allowed to cool to base temperature for 20 minutes. The field was then reversed at 2.2 T s$^{-1}$ to $+ 0.3$ T for 4 s then reduced to zero. After a wait period ranging from 10 to 50,000 s, a field of 0.08 T was applied and the relaxation of the magnetization was recorded. The field $B= 0.08$ T was chosen because it is large enough to get sizeable relaxation, but small compared to the avalanche fields shown in Fig. 4. In this way, when applying the magnetic fields, the relaxation is well behaved and the sample does not heat.

The AQP used for the data of Fig. 4 was similar to the above, except the avalanche field was $\pm 0.4$ T. After the wait period the field was ramped at 0.02 T s$^{-1}$ while the magnetization and temperature of the sample were continuously recorded. For the slow CC protocol measurements shown in Fig. 3, the samples were first heated to 900 mK for 10 s, then cooled at a rate of approximately 0.01 K s$^{-1}$, followed by a waiting period of 1000 s.

Data availability {#data-availability .unnumbered}
=================

Information on the data underpinning the results presented here, including how to access them, can be found in the Cardiff University data catalogue at http://doi.org/10.17035/d.2019.0069144874.\
The datasets obtained and/or analyzed in this study are also available from the corresponding author on reasonable request.

[99]{} Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske T. & Godfrey, K. W. Geometrical frustration in the ferromagnetic pyrochlore Ho$_2$Ti$_2$O$_7$. [*Phys. Rev. Lett.*]{} [**79**]{}, 2554 – 2557 (1997). Bramwell, S. T. & Harris, M. J. Frustration in Ising-type spin models on the pyrochlore lattice. [*J. Phys.: Condens. Matter*]{} [**10**]{}, L215 - L220 (1998). Ramirez, A. P., Hayashi, A., Cava, R. J., Siddharthan, R. B. & Shastry, S. Zero-point entropy in Ôspin iceÕ. [*Nature*]{} [**399**]{}, 333 – 335 (1999). Bramwell, S. T. & Gingras, M. J. P. Spin ice state in frustrated magnetic pyrochlore materials. [*Science*]{} [**294**]{}, 1495 – 1501 (2001). Luttinger, J. M & Tisza, L. Theory of dipole interactions in crystals. [*Phys. Rev.*]{} [**70**]{}, 954 (1946). Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. [*Nature*]{} [**451**]{}, 42–45 (2008). den Hertog, B. C. & Gingras, M. J. P. Dipolar interactions and origin of spin ice in Ising pyrochlore magnets. [*Phys. Rev. Lett.*]{} [**84**]{}, 3430 (2000). Isakov, S. V., Moessner, R. & Sondhi, S. L. Why spin ice obeys the ice rules [*Phys. Rev. Lett.*]{} [**95**]{}, 217201 (2005). Ryzhkin, I. A. Magnetic relaxation in rare-earth pyrochlores. [*J. Exp. and Theor. Phys.*]{} [**101**]{}, 481–486 (2005).

Kaiser, V., Bloxsom, J. A., Bovo, L., Bramwell, S. T., Holdsworth, P. C. W. & Moessner, R. Emergent Electrochemistry in Spin Ice: Debye–H[ü]{}ckel Theory and Beyond. [*Phys. Rev. B*]{} [**98**]{}, 144413 (2018). Paulsen, C., Giblin, S. R., Lhotel, E., Prabhakaran, D., Balakrishnan, G., Matsuhira, K. & Bramwell, S. T. Experimental signature of the attractive Coulomb force between positive and negative magnetic monopoles in spin ice. [*Nature Phys.*]{} [**12**]{}, 661 (2016). Jaubert, L. D. C. & Holdsworth, P. C. W. Signature of magnetic monopole and Dirac string dynamics in spin ice. [*Nature Phys.*]{} [**5**]{}, 258 - 261 (2009). Bovo, L., Bloxsom, J. A., Prabhakaran, D., Aeppli, G. & Bramwell, S. T. Brownian motion and quantum dynamics of magnetic monopoles in spin ice. [*Nature Comms.*]{} [**4**]{}, 1535 (2013). Tomasello, B., Castelnovo, C., Messier, R. & Quintanilla, J. Single-ion anisotropy and magnetic field response in the spin-ice materials Ho$_{2}$Ti$_{2}$O$_{7}$ and Dy$_{2}$Ti$_{2}$O$_{7}$. [*Phys. Rev. B*]{} [**92**]{}, 155120 (2015). Vedmedenko, E. Y. Dynamics of bound monopoles in artificial spin ice: How to store energy in Dirac strings. [*Phys. Rev. Lett.*]{} [**116**]{}, 077202 (2016). Melko, R. G., den Hertog, B. C. & Gingras, M. J. P. Long range order at low temperatures in dipolar spin ice. [*Phys. Rev. Lett.*]{} [**87**]{}, 067203 (2001). Ehlers, G., Cornelius, A. L., Fennell, T. Koza, M., Bramwell, S. T. & Gardner, J. S. Evidence for two distinct spin relaxation mechanisms in ‘hot’ spin ice Ho$_{2}$Ti$_{2}$O$_{7}$. [*J. Phys.: Condens. Matter*]{} [**16**]{}, S635–S642 (2004). Snyder, J., Ueland, B. G., Slusky, J. S., Karunadasa, H. Cava, R. J. & Schiffer, P. Low-temperature spin freezing in the Dy$_2$Ti$_2$O$_7$ spin ice. [*Phys. Rev. B*]{} [**69**]{}, 064414 (2004). Castelnovo, C., Moessner, R. & Sondhi, S. L. Thermal quenches in spin ice. [*Phys. Rev. Lett.*]{} [**104**]{}, 107201 (2010). Paulsen, C., Jackson, M. J., Lhotel, E., Canals, B., Prabhakaran, D., Matsuhira, K., Giblin, S. R. & Bramwell, S. T. Far-from-equilibrium monopole dynamics in spin ice. [*Nature Physics*]{} [**10**]{}, 135–139 (2014).

Planck Collaboration. Planck Early Results. II. The thermal performance of Planck, [*Astronomy & Astrophysics*]{} [**536**]{} (2011). doi: 10.1051/0004-6361/201116486. Slobinsky, D., Castelnovo, C., Borzi, R. A., Gibbs, A. S., Mackenzie, A. P., Moessner, R. & Grigera, S. A. Unconventional Magnetization Processes and Thermal Runaway in Spin-Ice Dy$_2$Ti$_2$O$_7$. [*Phys. Rev. Lett.*]{} [**105**]{}, 267205 (2010). Jackson, M. J., Lhotel, E., Giblin, S. R., Bramwell, S. T., Prabhakaran, D., Matsuhira, K., Hiroi, Z., Yu, Q. & Paulsen, C. Dynamic behavior of magnetic avalanches in the spin-ice compound Dy$_{2}$Ti$_{2}$O$_{7}$. [*Phys. Rev. B*]{} [**90**]{}, 064427 (2014). Krey, C., Legl, S., Dunsiger, S. R., Meven, M., Gardner, J. S., Roper, S. R. & Pfleiderer, C. First Order Metamagnetic Transition in Ho$_2$Ti$_2$O$_7$ Observed by Vibrating Coil Magnetometry at Milli-Kelvin Temperatures. [*Phys. Rev. Lett.*]{} [**108**]{} 257204.

Gatteschi, D. & Sessoli, R. Quantum Tunneling of Magnetization and Related Phenomena in Molecular Materials. [*Angewandte Chemie*]{} [**42**]{}, 268 – 297 (2003).

Sangregorio, C., Ohm, T., Paulsen, C., Sessoli, R. and Gatteschi, D. Quantum tunneling of the magnetization in an Iron cluster nanomagnet. [*Phys. Rev. Lett.*]{} [**78**]{}, 4645 – 4648 (1997). Prokof’ev, N. V. & Stamp, P. C. E. Low-temperature quantum relaxation in a system of magnetic nano molecules. [*Phys. Rev. Lett.*]{} [**80**]{}, 5794 (1998).

Wernsdorfer, W., Caneschi, A., Sessoli, R., Gatteschi, D., Cornia, A., Villar, V., & Paulsen, C. Effects of Nuclear Spins on the Quantum Relaxation of the Magnetization for the Molecular Nanomagnet Fe$_8$. [*Phys. Rev. Lett.*]{} [**84**]{}, 2965 – 2968 (2000).

Ohm, T., Sangregorio, C. & Paulsen, C. [*J. Low Temp. Physics*]{} [**113**]{}, 1141 – 1146 (1998). Kaiser, V., Bramwell, S. T., Holdsworth, P. C. W. & Moessner, R. ac Wien Effect in Spin Ice, Manifest in Nonlinear, Nonequilibrium Susceptibility. [*Phys. Rev. Lett.*]{} [**115**]{}, 037201 (2015).

[**Acknowledgements:**]{} S.R.G. thanks Cardiff University for ‘seedcorn’ funding, and acknowledges the EPSRC for EP/L019760/1. S.T.B. thanks Patrik Henelius for communicating his independent ideas on nuclear assisted quantum tunnelling in spin ice, and acknowledges the EPSRC for EP/S016465/1. E.L. and C.P. acknowledge financial support from ANR, France, Grant No. ANR-15-CE30-0004. G.B. wishes to thank financial support from EPSRC, UK, through grant EP/M028771/1.\
[**Author Contributions:**]{} The experiments were designed and performed by C.P. with inputs and discussions from E.L. and S.R.G. The data were analyzed by C.P., E.L., S.R.G., and S.T.B. Contributed materials were fabricated by K.M., D.P. and G.B. The paper was written by C.P., E.L., S.R.G., and S.T.B.

[**Nuclear spin assisted quantum tunnelling of magnetic monopoles in spin ice\
Supplementary Information**]{}

![\[fig1SI\] [**Density of single-charge monopoles (equilibrium number per diamond lattice site in zero field) versus temperature**]{}, calculated by the Debye-Hückel theory of Kaiser et al. [@KaiserDH]. The analytic calculation is very accurate for the monopole model of spin ice: it includes both single and double-charge monopoles, but only the single-charge monopoles are relevant at the temperatures we study. The density of ‘flippable’ spins per spin site is $3/2$ times the monopole density.](Plot_SI_nvsT.pdf){width="8cm"}

Experimental details
--------------------

[**Creating a large density of monopoles using the avalanche quench protocol in [Ho$_{2}$Ti$_{2}$O$_{7}$]{}.**]{} We have previously described the avalanche quench protocol (AQP) in detail for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} (see Ref.  and its Supplementary Information). From magnetisation measurements recorded during and after the AQP, we inferred that samples of [Dy$_{2}$Ti$_{2}$O$_{7}$]{} heat systematically to temperatures above 900 mK, even though the reference thermometer on the sample holder only registered a small jump [@Jackson14].

However from the onset, it was clear that [Ho$_{2}$Ti$_{2}$O$_{7}$]{} was different. For example, when performing measurements where the field is ramped at a steady rate, the magnetic avalanches of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} never reach the 900 mK equilibrium value as seen in Fig. 4c. Sometimes depending on previous measurements, the AQP worked very poorly, or did not seem to work at all.

This was the motivation for measuring the sample temperature directly by mounting a thermometer on the samples during some of the runs.

[**Temperature measurements during the AQP.**]{} We attempted direct temperature measurements with 4 different thermometers; 2 homemade RuO$_2$ resistance thermometers (filed down to reduce mass with wires attached with silver epoxy), a Cernox 1010-SD thermometer with platinum leads, and a bare-chip Cernox 1010-BC, both from LakeShore Cryogenics. All had short comings but in the end most of the measurements shown here were made using the bare chip thermometer. This was the lightest of the four, but had a small but noticeable magnetoresistance that we have corrected for. A constant current source delivered 10 nA, and the voltage was measured with a Stanford Instruments model 830 lock-in amplifier running at 1100 Hz. This setup was a compromise, the measurements of the temperature were fast, but prone to some drift and noise.

During normal measurements samples are sandwiched between two long narrow pieces of Cu that are anchored to the mixing chamber of a miniature dilution refrigerator. The samples are glued in place, then teflon tape is tightly wrapped around the Cu strips, clamping the samples to the Cu. For measurements with the thermometer glued to the sample, only one Cu strip was used, the second was suspended away from the thermometer, as shown in Supplementary Figure \[fig1SI\].

![\[fig1SI\] [**Pictures of the sample mounting**]{}. Upper left: Sample of [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and Cernox bare chip resistor with leads protected by kapton tape. Upper right: [Ho$_{2}$Ti$_{2}$O$_{7}$]{} glued onto the bare chip resistor with thin Cu foil protruding. Lower left: the sample + thermometer have been turned over, and the sample surface has been glued onto the Cu sample holder using GE varnish. The Cu foil has been folded back to cover the upper half of the sample and bare chip resistor. Lower right: standoffs hold upper part of sample holder away, teflon tape has been wrapped around the sample and thermometer. The thermometer is isolated from the Cu sample holder by the sample.](figure1_SI.pdf){width="8.5cm"}

Results
-------

As already discussed, cooling samples that contain Ho can be problematic, and this is also true for heating samples with the AQP when the starting temperature was well below 200 mK. This is shown in Supplementary Figure \[fig2SI\] for a series of AQP taken on [Ho$_{2}$Ti$_{2}$O$_{7}$]{} when the sample was first cooled to 65 mK after waiting 4 hours. Point (a) is the beginning of sequence when we applied a field of $-0.3$ T on the sample followed by a wait period of 180 s. For [Dy$_{2}$Ti$_{2}$O$_{7}$]{}, already a large $>1$ K spike in temperature would be seen at (a) as the sample rapidly magnetizes in the field, but for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} only a small jump of about 0.3 K was recorded. The jump in temperature at the first AQP is also small, only reaching about 0.6 K, compared to $>1.4$ K for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} under similar conditions. At (c) we begin a second AQP, again setting $\mu_0 H=-0.3$ T, and waiting 180 s. But this time the jump in temperature reaches nearly 0.8 K, and the second AQP at (d) shows that now, the sample has warmed $>1.3$ K.

During the AQP, heat from the flipping of the electronic spins is absorbed by the sample. But because of the large heat capacity of the Ho nuclei, much of the energy is absorbed by the nuclear spin bath, raising its temperature but resulting in a small overall jump in sample temperature. However for the second AQP, the starting sample temperature is now greater, nearly 0.16 K, and this is enough to heat the spins above one Kelvin.

Thus [Ho$_{2}$Ti$_{2}$O$_{7}$]{} measurements were systematically made with 2 or 3 AQP in succession in order to ensure the sample is warmed above 1 K.

Note that the need for several AQP also suggests that at low temperature, well below 300 mK, the nuclear spins begin to freeze out, and anti-align with their respective electronic spin, thus 2 in – 2 out for the electronic spin becomes 2 out – 2 in for its nuclear counterpart.

![\[fig2SI\] [**The applied field and the sample temperature for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} as a function of running time for a double AQP.**]{} The sample was cooled for 4 hours to base temperature of approximately 65 mK. (a) At $t=0$, the field was changed from 0 to $-0.3$ T. A relatively weak jump in the temperature can be seen. (b) At $t=180$ s, the first AQP is performed: the field goes from $-0.3$ to $+0.3$ T, then after 4 s from $+0.3$ T to zero. The temperature on the sample reaches about 0.55 K, not sufficient to randomize the spins. (c) At $t=200$ s the field is again put at $-0.3$ T in preparation for the next AQP. This time the jump in sample temperature is larger, but still less than required. (d) At $t=380$ s the second AQP takes place, warming the sample above 1 K. ](figure2_SI.pdf){width="8cm"}

Another nagging problem was that the samples of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} did not reach $M=0$ after the AQP. The origin of this is not clear, but our data suggests that [Ho$_{2}$Ti$_{2}$O$_{7}$]{} cools too fast. When the field is switched off, the applied field $H$ goes to zero before the sample even starts to change its magnetisation. The sample then feels the internal field $H_{\rm internal}=-D\cdot M$ where D is the demagnetisation factor and avalanches against this. As the magnetisation decreases, $H_{\rm internal}$ also decreases, the sample heats, but then cools so rapidly that the magnetisation gets ‘stuck’ at a small positive value of the order 1 emu/g. For convenience, the solution was to add a small overshoot for the field of about $-0.004$ T for 1 s, then switch back to $H=0$. This resulted in a starting $M$ closest to zero. Note that measurements without the overshoot gave the same results, but with an offset. This ultra rapid cooling may also explain why the avalanches shown in Fig. 4c for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} (while ramping of the field) fall below the equilibrium value expected for 900 mK, in contrast to [Dy$_{2}$Ti$_{2}$O$_{7}$]{}.

![\[fig3SI\] [**Plots for natural [Dy$_{2}$Ti$_{2}$O$_{7}$]{} corresponding to Figs. 3 and 4 of the main manuscript.**]{} (a) Avalanches of the magnetisation recorded while the field was ramped at 0.02 T/s for $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} (DTO). The samples were first prepared using the AQP and then followed by various wait times except for the curve marked ‘ZFC’, where the sample was first prepared using the conventional zero field cooled (CC) protocol (red circles). Also shown is the equilibrium $M$ vs $\mu_0 H$ taken at 900 mK (solid black dots). (b) Magnification, showing the spread in avalanche fields as a function of the wait time, and the ZFC far outside the pack. (c) The effect of wait time on the relaxation of the magnetisation $M$ vs time for $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} measured at 80 mK. The samples were again first prepared using the AQP. After the specified wait periods, a field of 0.08 T was applied and the magnetisation as a function of time was recorded.](figure3_SI.pdf){width="\textwidth"}

![\[fig5SI\] [**Time dependence of the magnetisation at short times**]{}. The same data of Fig. 3 in the main text as well as data for the $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} sample plotted against $\sqrt{time}$. An important prediction from ProkofÕev and Stamp [@PS] was that the initial relaxation of the magnetisation should follow a square root time dependence. This worked well for the SMM Fe$_8$ up to 1000 s or more. The situation for spin ice is quite different, the right hand side of the figure shows the data can at best be fitted over a very restricted range in time only up to 1 s. ](figure5_SI.pdf){width="13cm"}

![\[fig6SI\] [**Experimental values of relaxation time $\tau$ from susceptibility and magnetisation measurements.**]{} Relaxation time $\tau$ vs temperature for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}. The green and blue data points ($\tau$ less than 100 s) were taken from the peaks in the imaginary susceptibility. The red points for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} come from analyzing dc relaxation (all raw data was first corrected for demagnetisation effects). The slope $dT/d\tau$ defining the equilibrium cooling rate shown in Fig. 2 are taken from fits to these curves. ](figure6_SI.pdf){width="7.2cm"}

![\[fig7SI\] [**Effect of monopole current $J_m=dM/dt$ on wait time for [Ho$_{2}$Ti$_{2}$O$_{7}$]{}**]{} left: Monopole current $J_m=dM/dt$ obtained from the relaxation of the magnetisation of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} measured after different waiting times, and measured at 800 Oe. $J_m$ is obtained by extrapolating the derivative of the magnetisation with respect to time at $t=0$ (See Ref. for the detailed procedure). right: $J_m$ vs $1/T$ obtained from the saturation value of the left figure, i.e. when the current value does not depend anymore on the waiting time. ](i_800_Ho.pdf "fig:"){width="7.2cm"} ![\[fig7SI\] [**Effect of monopole current $J_m=dM/dt$ on wait time for [Ho$_{2}$Ti$_{2}$O$_{7}$]{}**]{} left: Monopole current $J_m=dM/dt$ obtained from the relaxation of the magnetisation of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} measured after different waiting times, and measured at 800 Oe. $J_m$ is obtained by extrapolating the derivative of the magnetisation with respect to time at $t=0$ (See Ref. for the detailed procedure). right: $J_m$ vs $1/T$ obtained from the saturation value of the left figure, i.e. when the current value does not depend anymore on the waiting time. ](i_800sat_vs1-T.pdf "fig:"){width="7.5cm"}

![\[fig8SI\] [**Effect of the nuclear spins on the monopole current**]{}. Monopole current $J_m=dM/dt$ vs $\sqrt{H}$ determined for $^{\rm nat}$Dy and $^{162}$ [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples (left), $^{163}$ [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} samples (middle), and in different cooling conditions for the [Ho$_{2}$Ti$_{2}$O$_{7}$]{} sample. $J_m$ is obtained by extrapolating the derivative of the magnetisation with respect to time at $t=0$ (See Ref. for the detailed procedure). Natural and $^{162}$Dy (no nuclear spin) [Dy$_{2}$Ti$_{2}$O$_{7}$]{} follows the $\sqrt{H}$ behavior expected for magnetic monopoles interacting through the Coulomb force, $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} samples do not, whatever the cooling process and so the initial density of monopoles. This result shows, as suggested in the main text, that the idealised emergent chemical kinetics of monopole theory does not apply in $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, where the dynamics is strongly affected by the nuclear spin effects, because the hyperfine splitting energies are of a similar order to the Coulomb energies. ](ivsh_DTO_no_N.pdf "fig:"){width="5.5cm"} ![\[fig8SI\] [**Effect of the nuclear spins on the monopole current**]{}. Monopole current $J_m=dM/dt$ vs $\sqrt{H}$ determined for $^{\rm nat}$Dy and $^{162}$ [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples (left), $^{163}$ [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} samples (middle), and in different cooling conditions for the [Ho$_{2}$Ti$_{2}$O$_{7}$]{} sample. $J_m$ is obtained by extrapolating the derivative of the magnetisation with respect to time at $t=0$ (See Ref. for the detailed procedure). Natural and $^{162}$Dy (no nuclear spin) [Dy$_{2}$Ti$_{2}$O$_{7}$]{} follows the $\sqrt{H}$ behavior expected for magnetic monopoles interacting through the Coulomb force, $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} samples do not, whatever the cooling process and so the initial density of monopoles. This result shows, as suggested in the main text, that the idealised emergent chemical kinetics of monopole theory does not apply in $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, where the dynamics is strongly affected by the nuclear spin effects, because the hyperfine splitting energies are of a similar order to the Coulomb energies. ](ivsh_nuk_Hob4.pdf "fig:"){width="5.5cm"} ![\[fig8SI\] [**Effect of the nuclear spins on the monopole current**]{}. Monopole current $J_m=dM/dt$ vs $\sqrt{H}$ determined for $^{\rm nat}$Dy and $^{162}$ [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples (left), $^{163}$ [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} samples (middle), and in different cooling conditions for the [Ho$_{2}$Ti$_{2}$O$_{7}$]{} sample. $J_m$ is obtained by extrapolating the derivative of the magnetisation with respect to time at $t=0$ (See Ref. for the detailed procedure). Natural and $^{162}$Dy (no nuclear spin) [Dy$_{2}$Ti$_{2}$O$_{7}$]{} follows the $\sqrt{H}$ behavior expected for magnetic monopoles interacting through the Coulomb force, $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} samples do not, whatever the cooling process and so the initial density of monopoles. This result shows, as suggested in the main text, that the idealised emergent chemical kinetics of monopole theory does not apply in $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, where the dynamics is strongly affected by the nuclear spin effects, because the hyperfine splitting energies are of a similar order to the Coulomb energies. ](ivsh_Hob4.pdf "fig:"){width="5.5cm"}

![\[fig9SI\] [**The effect of wait time $t_{\rm w}$ on the magneto-thermal avalanches for two different samples of natural DTO, and for two different runs**]{} (a) shows the value of the avalanche field $H_{\rm ava}$($t_{\rm w}$), defined as the field where the magnetization crosses 1 $\mu_{\rm B}$ per rare earth ion. (b) is a plot of the difference in avalanche field $\Delta H_{\rm ava}= H_{\rm ava}(t_{\rm w}) - H_{\rm ava} (t_{\rm w}=minimum)$. ](figure9_SI.pdf){width="16cm"}

Different Samples and Measuring Directions
------------------------------------------

During the course of this study, 10 different samples were measured with some samples measured along multiple axis. The mass of the samples ranged between 3 to 40mg, and they had various shapes. No corrections for demagnetization effects have been taken into account for the results presented in the main text. This is because for most of the measurements shown in the main text, the sample was far from equilibrium, and the magnetization was very small, and thus the demagnetizing field -NM was small. Nevertheless the sample shape and field direction do effect the observed relaxation curves and the avalanche fields.

In this section we show that although there are some variations between samples, between cooling runs, and for different directions, these differences do not change the main conclusion of the paper; the demonstration that nuclear assisted quantum tunneling is operative regardless of field direction.

Supplementary Figure 9(a) shows the effect of $t_{\rm w}$ on the magneto-thermal avalanches for two different samples of natural DTO, and for two different runs. Sample 1 (also shown in the main text) was rectangular shaped parallelepiped and sample 2 was a square thin platelet shaped sample. The measurements shown in the figures were taken with the field along the \[111\] axis for both samples. The left panel shows the value of the avalanche field $H_{\rm ava}$($t_{\rm w}$), defined as the field where the magnetization crosses 1 $\mu_{\rm B}$ per rare earth ion. The curves are clearly offset from one another, even the two curves taken on the same sample, but during different runs. The initial position of the avalanche field is very sensitive to thermal contact with the sample holder. For sample 2 run 1, the thermal contact was made using two Cu bands with the sample sandwiched between the two. For sample 2 run 2 only one Cu band was used, thus the thermal contact was worse. The better thermalized sample has a higher avalanche field, because leading up to the avalanche heat could be more efficiently evacuated from the sample. Supplementary Figure 9(b) is a plot of the difference in avalanche field $\Delta H_{\rm ava}= H_{\rm ava}(t_{\rm w}) - H_{\rm ava} (t_{\rm w}=minimum)$. As can be seen, for sample 2, the two runs collapse onto one another, but the shape of the curve for sample 1 is slightly different. A more systematic study needs to be made to understand if this is a shape dependent effect, or sample dependent.

![\[fig10SI\] [**Measurements made on a polycrystalline sample of [$^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}]{} (sample 2)**]{} The sample was first prepared using the same avalanche quench protocols (AQP) outlined in the main text and methods section, followed by various waiting times. (a) shows the effects of wait time on the relaxation of the magnetization in a field of 0.8 T at 80mK. (b) shows the effects of wait time on the position of the avalanche field when the field is ramped from 0 to 0.4 T at a constant rate of 0.02 T/s at 80mK. Also shown in the figure is the conventional zero field cooling (CC) curve were the sample was slowly cooled from 900 mK to 80 mK (at 1 mK/s) followed by a 1000 s wait period. For this sample the CC avalanche field is offset to higher fields, and is well outside the distribution of $H_{\rm ava}$($t_{\rm w}$). ](figure10_SI.pdf){width="16cm"}

For $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}, two samples were studied. Sample 1 was an odd shaped disk. The \[111\] direction was perpendicular to the surface of the disk, and resulted in a very large demagnetization factor for this direction. Importantly this resulted in difficulty thermalizing the sample along this direction to our Cu sample holder. This resulted in a much less efficient AQP cooling. We estimate that the sample took about 5 seconds to cool below 500mK, and about 40 seconds to cool below 100mK. This is much slower than the usual AQP as described in Fig. 2 of the main text, but still faster than the CC method. Sample 1 was measured along the \[111\] direction (shown in the main text) and perpendicular to the \[111\] direction. Sample 2 was a polycrystalline sample and the effects of wait time on the relaxation of the magnetization and position of the avalanche field are shown in Supplementary Figure 10 (a) and (b). These data sets are very similar to those presented in the main text in terms of the strength of the effect of wait time for $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}. (see Figures 3 and 4)

However there are two interesting differences.

Firstly, Supplementary Figure 11 (a) shows the monopole current $J_{\rm m}=dM/dt$ at $t=0$ vs log wait time for sample 1 \[111\] and perpendicular to \[111\] as well as for polycrystalline sample 2. As can be seen in the figure, although the slopes of the three curves are roughly the same, the \[111\] data fall significantly below the two perpendicular curves. Most likely this is not an intrinsic effect, but comes from the poor thermalization for the \[111\] sample run: as mentioned above, for this direction after the AQP the sample cooled much slower, therefore the initial monopole density at the beginning of the wait period was much reduced, so the initial monopole current was less, shifting the \[111\] curve down in the plot. A second difference that can be seen in Supplementary Figure 10 (b) is that the avalanche field for the CC method occurs at much higher fields and is well outside the distribution of curves obtained by the AQP method. The same result was found for sample 1 perpendicular to \[111\]. This can be contrast to the data shown for the \[111\] sample in the main text, and again can be explain by the slower cooling for the \[111\] sample run.

![\[fig11SI\] [**comparison of different samples and different measuring directions for HTO, [$^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}]{} and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  (DTO)**]{} (a) the monopole current $J_{\rm m}=dM/dt$ at $t=0$ vs log wait time for two samples of [$^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}]{} and three samples of HTO. (b) Plot of difference in avalanche field $\Delta H_{\rm ava}= H_{\rm ava}(t_{\rm w}) - H_{\rm ava}(t_{\rm w}=minimum)$ against wait time for various directions and various samples of HTO, [$^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}]{} and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  (DTO). The top 4 curves in the figure are measurements for 3 different samples of HTO (squares). Sample 1 was a needle shaped sample measured along the \[111\] (long) direction. Sample 2 was also needle shaped and measured along the \[001\] direction. Sample 3 was a square platelet, and was measure along the \[001\] and \[110\] axis. The middle 3 curves are for two different samples of $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}(solid dots) measured at 80mK. Sample 1 was measured along the \[111\] direction and perpendicular to the \[111\] direction, and sample 2 was a poly-crystal. The bottom two curves are for natural DTO (triangles) taken on two different samples along the \[111\] and \[001\] directions. (same data as shown in Supplementary Figure 9 (b)) ](figure11_SI.pdf){width="16cm"}

The results for measurements on 3 different samples of HTO are also shown in Supplementary Figure 11. Sample 1 was a needle shaped sample measured along the \[111\] (long) direction. Sample 2 was also needle shaped and measured along the \[001\] direction. Sample 3 was a square platelet, and was measure along the \[001\] and \[110\] axis. Supplementary Figure 11 (a) shows the monopole current $J_{\rm m}=dM/dt$ at $t=0$ vs log wait time for sample 1 \[111\] (also in the main text) compared to sample 2 \[110\]. The effect of wait time on the currents for these two samples are very similar; the rate at which monopoles recombine is seen to be much faster than that of $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}, and both seem to saturate at very long wait times. Supplementary Figure 11 (b) are plots of difference in avalanche field $\Delta H_{\rm ava}= H_{\rm ava}(t_{\rm w}) - H_{\rm ava}(t_{\rm w}=minimum)$ against log wait time for the three samples of HTO, as well as two samples of [$^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}]{} and for two samples of $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  (DTO) (same data as shown in Supplementary Figure 9 (b)).

[99]{} Kaiser, V., Bloxsom, J. A., Bovo, L., Bramwell, S. T., Holdsworth, P. C. W. & Moessner, R. Emergent Electrochemistry in Spin Ice: Debye–H[ü]{}ckel Theory and Beyond. [*Phys. Rev. B*]{} [**98**]{}, 144413 (2018).

Paulsen, C., Giblin, S. R., Lhotel, E., Prabhakaran, D., Balakrishnan, G., Matsuhira, K., and Bramwell S. T. Experimental signature of the attractive Coulomb force between positive and negative magnetic monopoles in spin ice. [*Nature Physics*]{} [**12**]{}, 661 (2016).

Jackson, M. J., Lhotel, E., Giblin, S. R., Bramwell, S. T., Prabhakaran, D., Matsuhira, K., Hiroi, Z., Yu, Q., and Paulsen, C. Dynamic behavior of magnetic avalanches in the spin-ice compound [Dy$_{2}$Ti$_{2}$O$_{7}$]{}. [*Phys. Rev. B*]{} [**90**]{}, 064427 (2014).

Prokof’ev, N. V. & Stamp, P. C. E. Low-temperature quantum relaxation in a system of magnetic nano molecules. [*Phys. Rev. Lett.*]{} [**80**]{}, 5794 (1998).
