Brownian motion and quantum dynamics of magnetic monopoles in spin ice

Abstract

Spin ice illustrates many unusual magnetic properties, including zero point entropy, emergent monopoles and a quasi liquid-gas transition. To reveal the quantum spin dynamics that underpin these phenomena is an experimental challenge. Here we show how crucial information is contained in the frequency dependence of the magnetic susceptibility and in its high frequency or adiabatic limit. The typical response of Dy₂ Ti₂ O₇ spin ice indicates that monopole diffusion is Brownian but is underpinned by spin tunnelling and is influenced by collective monopole interactions. The adiabatic response reveals evidence of driven monopole plasma oscillations in weak applied field, and unconventional critical behaviour in strong applied field. Our results clarify the origin of the relatively high frequency response in spin ice. They disclose unexpected physics and establish adiabatic susceptibility as a revealing characteristic of exotic spin systems.

ARTICLE

Received 14 Sep 2012 | Accepted 25 Jan 2013 | Published 26 Feb 2013

Spin ice illustrates many unusual magnetic properties, including zero point entropy, emergent monopoles and a quasi liquidgas transition. To reveal the quantum spin dynamics that underpin these phenomena is an experimental challenge. Here we show how crucial information is contained in the frequency dependence of the magnetic susceptibility and in its high frequency or adiabatic limit. The typical response of Dy2Ti2O7 spin ice indicates that monopole diffusion is Brownian but is underpinned by spin tunnelling and is inuenced by collective monopole interactions. The adiabatic response reveals evidence of driven mono-pole plasma oscillations in weak applied eld, and unconventional critical behaviour in strong applied eld. Our results clarify the origin of the relatively high frequency response in spin ice. They disclose unexpected physics and establish adiabatic susceptibility as a revealing characteristic of exotic spin systems.

DOI: 10.1038/ncomms2551 OPEN

Brownian motion and quantum dynamics of magnetic monopoles in spin ice

L. Bovo1, J.A. Bloxsom1, D. Prabhakaran2, G. Aeppli1 & S.T. Bramwell1

1 Department of Physics and Astronomy, London Centre for Nanotechnology, University College London, 1719 Gordon Street, London WC1H OAH, UK.

2 Department of Physics, Clarendon Laboratory, University of Oxford, Park Road, Oxford OX1 3PU, UK. Correspondence and requests for materials should be addressed to L.B. (email: mailto:[email protected]

Web End [email protected] ).

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In spin ice materials like Ho2Ti2O7 or Dy2Ti2O7 (refs 17) magnetic rare earth ions (for example, Ho, Dy) occupy a lattice of corner-linked tetrahedra. In the low temperature spin

ice state two atomic magnetic moments or spins point into, and two point out of each tetrahedron. This is equivalent to the ice rule that determines proton congurations in water ice1,2, and hence spin ice has a residual entropy equal to the Pauling entropy of water ice3. The thermodynamic properties of spin ice are well described by a classical spin Hamiltonian with a dominant dipoledipole interaction4,8. The self-screening of the latter establishes the ice rule ground state4, but this property does not extend to excited states5. A spin ip out of the ice rule manifold creates a dipolar magnetic excitation that may fractionalize to produce free defects. These inhabit the diamond lattice formed by tetrahedron centres and behave as magnetic monopoles on account of the integrated dipoledipole interaction5,6.

The spin ices are part of the family of rare earth pyrochlores, a series of frustrated magnets for which collective quantum effects have been widely discussed915. Recent theoretical work16,17 does not rule out the possibility that such effects may be relevant to Ho2Ti2O7 and Dy2Ti2O7 but, to a good approximation, the monopoles may be treated as classical objects, with local quantum mechanics setting local parameters such as attempt frequencies.

The magnetic monopole current density in spin ice is dened as the rate of change of magnetization: J qM/qt, with the

conductivity proportional to the monopole density6. However, even in an innite system, magnetic monopoles in spin ice cannot sustain a direct current (dc), on account of the destruction of the spin ice entropy by magnetization of the system6. This means that dc magnetricity in spin ice18,19 is necessarily transient20,21. Alternating current (ac) magnetricity does not suffer from this limitation as monopoles can in principle be driven indenitely back and forth by an oscillating magnetic eld. The theory of ac-current6 has not yet been tested as existing ac-magnetization studies either precede the theory22,23 or focus on the low temperature regime24,25 where complicating factors are expected1921.

Here, we present the rst experimental test of the theory of Ryzhkin6, where we add experimental support to a number of ideas and arguments about monopole diffusion20,21,26 and spin

tunnelling22,27,28, and derive new information on the microscopic processes involved. Our dynamical magnetization measurements also estimate the isothermal susceptibility wT and the adiabatic susceptibility wS. Although the former is a much discussed magnetic response function, the latter is typically neglected. We also report a striking contrast between the temperature dependence of wS and wT in weak applied eld, showing that wT is best interpreted as a spin response, while wS is best interpreted as a monopole response. This contrast has its root in the fact that congurational entropy ultimately connes the monopoles when they are driven by a magnetic eld6. Finally, in strong applied eld along the cubic [111] direction, spin ice exhibits a liquidgas type phase transition with a critical point29 at m0HC 0.929 T,

TC 0.36 K. This transition has been interpreted as a monopole

condensation5 and has been treated in renormalization group theory30. We extend our comparison of wT and wS to the supercritical regime at T4TC, where we observe strong signatures of critical behaviour and nd that monopoles behave increasingly like dipole pairs, in agreement with the comments of Shtyk and Feigelman30.

ResultsGeneral features of the magnetic response in spin ice. The temperature and eld regimes probed in this paper are illustrated in Fig. 1. In the gure we show the relationship between mono-poles and dipoles in spin ice and we broadly dene regimes of monopolar response and dipolar response. It should be emphasized that, as in other cases where novel quasiparticles accurately account for the low-energy physics, monopole and spin descriptions are never in conict. Instead, certain properties are best discussed in terms of spins and others are best discussed in terms of monopoles. Our study claries how this division should be made.

One of our key results is that in spin ice, the adiabatic susceptibility wS is nite, with the ratio wS/wT increasing with increasing applied static magnetic eld. This basic observation is emphasized in Fig. 2, which displays representative experimental data at T 1.95 K, in different applied static magnetic elds.

The gure clearly shows the presence of a nite offset, wS, in the

limit of innite frequency.

Hlocal

100 111

1.2

Critical point

100 111

110

1.0

0.8

Happlied

Hlocal

[afii9839] 0H(T)

Dipole regime

0.6

110

0.4

Happlied

0.2

Happlied

Monopole regime

Hlocal

0.0

0 1 2 3 4 5 6 7 8

T (K)

Figure 1 | Field-temperature phase diagram of spin ice. (a) Dy2Ti2O7 with H k 111 . The full line is a line of rst order phase transitions, terminating in a

classical critical point, that has been interpreted as a monopole condensation5. Monopoles are deconned in zero eld but become conned in an applied eld. The right hand diagrams show how the monopoles reform ippable spins or dipole pairs near the critical eld. Dotted lines are guides to the eye. Experimental points with error bars (s.d.) show the applied eld of the maximum in the adiabatic susceptibility measured here. Hopping of emergent magnetic monopoles: fragment of spin ices cubic pyrochlore lattice, which consists of corner-linked tetrahedra, showing spin congurations (arrows). (b) Illustrates crystallographic axes, the applied eld direction and how internal elds may be transverse to the local spin direction. Blue (red) circles represent negative (positive) monopoles. (c) Illustrates how a monopole hop can be associated with a spin ipped by a transverse eld or a tunnelling event through a potential barrier.

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28

0 T

1

24

~[afii9851]T [afii9851]s

[afii9851]

0.1

20

16

[afii9851] (102 )

0.01

12

0.1 1 10

0.86 T

[afii9853] (103 Hz)

[afii9851]S

8

0 T

4

10 T

0 0 2 4 6 8 10

[afii9853] (103 Hz)

Figure 2 | Finite adiabatic susceptibility in spin ice. The adiabatic (wS) and

isothermal (wT) susceptibilities are estimated as the real part of the frequency dependent susceptibility wo in the limits o ! 1 and o ! 0,

respectively, as indicated in the main plot. Experimental data are atT 1:95 K at applied static magnetic eld m0 jHj 0 (black circles), 0:86

(grey circles) and 10 T (light grey circles). At 10 T both susceptibilities are nearly zero. At 0.86 T, the two are of similar magnitude. At zero applied eld wS wT, but still nite. The respective lines are the t to a ColeCole

model (see text). The inset shows the m0H 0 data on loglog scales. The

clear deviation from a linear curve at large o conrms the presence of a nite offset, wS. Here the blue line is the t to the ColeCole function using nite wS and the red line is the same t with wS constrained to be zero.

; 2

where a is the ColeCole parameter that determines the width of the log-relaxation time distribution (see Methods). Representative experimental data and results, along with characteristic ts, are shown in Fig. 3. The model ts the experimental data well at T43.5 K but describes only the high frequency part at the lowest temperatures (for details of ts in this range, Supplementary Fig. S1).

Figure 3d shows the temperature evolution of the monopole mobility, u(T). At T410 K the apparent mobility diverges in accord with an expected Orbach type spin ip process22,27,28 that is not considered further here. At lower temperatures u(T) becomes accurately proportional to 1/T, which is consistent with the NernstEinstein equation for the Brownian diffusion of magnetic monopoles:

u

DQkT : 3

Here, our transformed data indicates that the diffusion constant D is temperature-independent, as shown in the inset of Fig. 3d. The athermal diffusion constant shows that the observed temperature dependence of the magnetic relaxation22, in this temperature range, is completely accounted for by the temperature evolution of the monopole density, the isothermal susceptibility and the temperature factor characteristic of Brownian diffusion. This general behaviour is insensitive to small applied eld (typically m0Ho50 mT, Fig. 3d). Writing

D n0a2/6 where a is the diamond lattice constant and v0 the

monopole hop rate26,32, we nd a temperature-independent hop rate of n0 2.43(6) 103 s 1. This athermal hop rate may be

treated as evidence of quantum tunnelling of the spin involved in the monopole hop (Fig. 1c). Note that, owing to the several non trivial temperature-dependent terms in the expression for t(T), it could be quite misleading to plot log(t) versus 1/T (Supplementary Fig. S2) as is commonly done when considering thermally activated relaxation processes: however, a treatment of this sort is given in Supplementary Note 3.

Our results are fully consistent with the theory of Ryzhkin6 and the numerical analysis of Jaubert and Holdsworth20, which assumed an athermal hop rate, but also indicate an essential renement that must be made to both these approaches. That is, we nd a nite a, suggesting a signicant dispersion of relaxation times, as previously observed23, rather than the single relaxation time assumed in the theory. However, the theory neglects monopole interactions (except insofar as they determine x(T)), which might be expected to inuence the hopping rate of individual monopoles. A detailed theory of such effects is far beyond the scope of this contribution, but we may derive a

Magnetic relaxation in zero DC eld. The theory of Ryzhkin6 applies the thermodynamics of irreversible processes to the problem of magnetic monopole transport in spin ice, specically exploiting the spin ice-water ice analogy and the Jaccard theory of defect motion in water ice31. It is shown that the usual entropy gain of driving a current (in this case a monopole current) is opposed by the entropy loss in magnetization of the system. A current is only possible when the dynamical magnetization M(t) is less than its equilibrium value wTH. The expression for the frequency dependent susceptibility of spin ice, w(o) calculated in Ryzhkins theory6 therefore depends on both wT and the monopole transport coefcients. This result is discussed at length in Ref. 32 where it is suggested, following the classical thermodynamic arguments of Casimir and Du-Pr33, that for practical purposes, the expression for the susceptibility calculated in Ryzhkins theory6 should be modied to include a nite adiabatic susceptibility wS. With this modication, the result of

Ryzhkins theory6 becomes:

wo wS

wT wS

11 iot

; 1

where t is a relaxation time. Although this expression looks like that for ordinary paramagnetic relaxation, there is an important difference, in that t is a function of monopole rather than spin parameters. It may be written t 1 m0uQx/V0wT where u is

the monopole mobility, Q 4.266 10 13 J T 1 m 1 the

monopole charge5, x the total monopole density per diamond lattice site, and V0 1.29 10 28 m3 the volume per diamond

lattice site32. The relaxation time t therefore depends on three temperature-dependent parameters: u(T), x(T) and wT(T). Of

these, only the isothermal susceptibility wT(T) can be directly measured. The density x(T) evolves with temperature in a way that cannot be expressed in closed form26, but we have estimated it with sufcient accuracy by tting specic heat data to Debye Hckel theory (refs 26,34), Methods and Supplementary Note 1).

Knowledge of the experimental x(T) and wT(T) allows us to determine the monopole mobility u(T) by dividing our measured t 1(T) by x(T)/wT(T).

In practice we found the assumption of a single relaxation time to be too restrictive, and therefore considered a modied model with one extra parameter introduced to describe a distribution of relaxation times. The ColeCole model35, which describes a roughly Gaussian distribution of log-relaxation times, was chosen because it was found to t the experimental data and has some physical justication in that its approximately Gaussian cutoff at high frequency is consistent with a monopole hopping model: certain other commonly-used models are less suitable in this regard (Supplementary Fig. S1 and Supplementary Note 2, for a discussion). The expression used to t the data therefore took the form:

wo wS

wT wS

11 iot1 a

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1.2

0.8

[afii9851]

1.2

0.4

1.0

0.0

0.1[afii9840] (103 Hz)

[afii9840] (103 Hz)

1 10

0.8

[afii9839](106 m s1 T1)

0.3

0.0 0.0 0.2 0.4 0.6 0.8 1.0

0.2

[afii9851]

[afii9851]

0.6

40

0.1

D(1017 m2 s1)

0.0

30

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0.4

20

0.3

10

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0 0 2 4 6 8 10 12 14

0.1

T (K)

0.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6

[afii9851]

1/T (K1)

Figure 3 | Ac-magnetricity and monopole diffusion in spin ice, Dy2Ti2O7. A representative t to the ac-magnetization data at Hdc 0: (a) real and

(b) imaginary susceptibilities and (c) ColeCole plot (an argand diagram). The lines (red, T 4.5 K; blue, T 8 K; green, T 14 K) t all data with the same

set of four parameters at a given temperature: wT, wS, a and t, as dened in the text. (d) Monopole mobility measured at applied elds m0 j H j 0

(full black circles), 3 (full dark grey), 10 (full light grey), 18.5 (open black) and 38.5 (open grey) mT. Error bars represent the s.d. The red line is u A=T

where A 2:272 10 6 mKs 1T 1. This is characteristic of Brownian diffusion of monopoles with a temperature-independent diffusion constant (inset).

The blue line is u Be C=T where B 391 ms 1T 1, C 2501 K, characteristic of a previously identied Orbach-like spin ip process arising from the

coupling to excited crystal eld states. This process is extinct below 10 K, giving way to monopole diffusion as the cause of magnetic relaxation in spin ice.

3.2

2.8

2.4

2.0

[afii9846]2 ln [afii9848]

1.6

p , for small h1, h2. Then using the additivity property of cumulants (here the variance) of uncorrelated random variables, we nd

s2lnt s21 xs22: 5

Our measured a(T) may be transformed37 to give the quantity on the left (Methods) and hence we can test the above expression. Figure 4 conrms a very satisfactory agreement between theory and experiment in zero and weak applied eld, with the tted s2 increasing rapidly in an applied dc eld of 1mT, but thereafter more slowly. We may rmly conclude that the observed dispersion of rates is in large part a monopole property that obeys Equation 5. Given our derivation of Equation 5, our result suggests that monopole hopping is assisted by mutual monopole interactions, an inference that we examine further in the Supplementary Note 4 and Supplementary Fig. S3.

Isothermal and adiabatic susceptibilities. The isothermal susceptibility wT extracted from the ts to theory is in close agreement with the directly measured wT (Supplementary Fig. S4). At low temperature the isothermal susceptibility is predicted to be twice the Curie susceptibility6, wT 2C/T, but recent work38 has

established that in spin ice there is a gradual crossover from a Curie constant C at very high temperature to the expected 2C at low temperature. Our results are consistent with a gradual evolution from wTE1.8C/T at T 2 K to wTE1.2C/T at T 10 K,

consistent with this crossover, assuming a Curie constant of3.95 K (Supplementary Note 5). However a much more detailed experimental and theoretical study of the Curie Law crossover in

1.2

0.8

0.4

0.0 0 2 4 6 8 10 12 14

T (K)

Figure 4 | Monopole signatures. The experimental variance in logarithmic relaxation time s2lnt (circles, errors bars represent s.d.) compared with the predicted form for monopolar-eld assisted tunnelling. Red and green indicate, respectively, applied elds of m0H 0 and the set of nite elds

listed in the caption of Figure 3. For each curve there are two tted parameters s21 and s22, which describe the mean square static eld and mean square monopole eld, respectively. The line is the function s21 xs22 where x(T) is the monopole density. The shading indicates the maximum possible systematic error (absolute minima and maxima) in the monopole density. We nd s1,2 0.84(1),3.40(5), respectively, in zero eld and

1.15(1),3.40(5) in nite eld. The deviation at T410 K is related to a change in relaxation mechanism (see text).

working model using the results of Bramwell32, where it was found that the mean square monopole eld at a point is proportional to the monopole density. A heuristic argument for this is that the squared Coulomb eld per monopole scales as 1/r4 (where r is distance) while the number of monopoles scales as xr2dr; hence, neglecting monopole correlations, the mean square eld scales as x times a nite denite integral over r. We assume that in zero applied eld, spins are ipped by transverse elds36 arising from the dense ensemble of atomic dipoles, and we

decompose the instantaneous local transverse dipolar eld as follows:

H H01 h1 h2 x

p ; 4 where H0 is an effective eld that causes ipping at rate n0. Here, hi (i 1,2) are assumed to be uncorrelated random variables

with zero mean and variance s2i. Assuming t 1pH, it follows that ln t ln H0 h1 h2 x

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20

[afii9853]

0

18

16

14

12

[afii9851]T(HC)

[afii9851] (102 )

10

0.2[afii9851]T(0)

[afii9851]S(0)

[afii9853]

8

6

4

[afii9851]S(HC)

2

0

0 2 4 6 8 10 12 14

T (K)

2.0

1.5

2 )

[afii9851] S (10

1.0

0.5

T (K)

Figure 5 | Monopole and dipole signatures revealed by comparing the measured adiabatic (vS) and isothermal (vT) susceptibilities. (a) the susceptibilities are shown as a function of applied dc eld at zero eld (black circles) and near the crossover eld (grey circles) shown as a red dotted line in Fig. 1. Error bars represent s.d. The full red line is the experimentally measured monopole density tted to wS0 by the

adjustment of a scale factor (see text). The blue line is a CurieWeiss law tted to wT0 (Supplementary Fig. S4). The light blue line is the theoretical

prediction wSHC C0=T TC (see text). (b) Measured wS as a function

of weak applied eld 0 m0H 0:05 T (circles, same colour code is

maintained as in Fig. 3). The full red line is the experimentally measured monopole density tted to wS0 by the adjustment of a scale factor (see

text), also showing the maximum systematic uncertainty (absolute minima maxima) in the measured monopole density (red shading).

T qM=qT

2

CH ; 6

where CH Z0 is the specic heat at xed applied eld H. For an ideally paramagnetic rare earth salt a typical behaviour of wS would be to roughly track the increase of wT as T-0 according to the Curie Law wT C/T. The striking difference we observe

between wT(T) and wS(T) (Fig. 5a) reects a transition from spins to monopoles as the natural variables by which to describe the magnetic response, monopoles being more appropriate at high frequency.

Nevertheless, we can explore the origin of wS in spin language if we consider a monopole as a label for a set of ippable spins (Fig. 1). We assume that the adiabatic susceptibility is equal to the isolated susceptibility, which in a semi-classical approximation is given by39:

wS Z 1 X

n

0.0 0 2 4 6 8 10 12 14

qMn @H

Dy2Ti2O7 would be worthwhile. The Curie like wT is of course characteristic of a spin system: indeed there is no direct monopole signature in this quantity. This may be traced to the congurational entropy in the problem, which in applied eld connes the monopoles6,32, making the magnetic response spin-like at long time.

The thermodynamic adiabatic susceptibility wS is the acsusceptibility extrapolated to innite frequency (or more strictly to a frequency where spin-spin relaxation is active but where spin-lattice relaxation is not33). Figure 5b illustrates a striking correlation between our measured adiabatic susceptibility wS(T)

and the measured monopole density x(T) (detailed checks on this result are given in Supplementary Note 6 and Supplementary Figs S5, S6). Thus, we nd wS w0x(T) with w0 0.030(1),

a temperature-independent constant. The athermal prefactor suggests that monopole diffusion is not involved in this magnetization process. Instead, in a simple classical interpretation, we may imagine a frictionless, and hence reversible, displacement of magnetic monopoles by distance r in the applied eldlike a driven plasma oscillation. We write the force on a positive monopole as m0H(o)Q Kr(o), where K is the force

constant, and use the fact that the magnetization is the magnetic moment per unit volume, or M(o) (x/V0)Qr(o). From these

relations we nd wS xm0Q2/KV0, which is of the observed form,

wSpx. From the value of wS, we nd KE0.06 N m 1, implying an energy barrier between lattice sites at a distance r a/2 of

order 100 K. The latter seems too large to be a Coulombic barrier, and is more likely connected with the crystal eld energy scale of several hundred kelvin. The amplitude of motion implied in our eld of m0H 5 10 5 T is of the order of femtometres, which is

much less than one lattice spacing. This frictionless oscillation of the monopole ensemble is reminiscent of a plasma oscillation in an electrical plasma, though the absence of an accelerative term in the equation of motion means that the monopole plasma oscillation cannot occur in the absence of a driving eld. Of course a nite wS in a magnetic system can always be formally represented as an oscillation of magnetic charge, but in this case the proportionality of wS with x shows that it is associated with the displacement of recognizable positive and negative magnetic monopoles.

In a magnetic system wS is always less than the isothermal susceptibility wT, as it obeys the thermodynamic relation:

wS wT

e En=kT: 7

Here, Z is the partition function, n labels the energy states of the system, and Mn is the magnetic moment per unit volume of the state n. Our experimental observation that wS w0x(T) is

obtained if the ground state is assigned null adiabatic susceptibility and the monopole excited state is assigned qMn/qH w0, where w0 is temperature independent. As

Mn V 1qEn/q(m0H) (where V is volume) our result reveals a

quadratic term in the energy per monopole: E0n Vm0w0=2H2.

A quadratic energy expression generally indicates stretchable magnetic moments. A small quadratic (Van Vleck) term is expected for a free Dy3 ion through mixing of the ground state with states of higher total angular momentum J. However, in our case, the observation that only ippable spins contribute to w0 and that ippable spins and non-ippable ones are distinguished only by a thermal energy scale at these temperatures, appears to rule out any single spin mechanism. It is interesting to note that the monopole spin texture is predicted to produce an electric dipole40

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14

2.5

12

2.0

Width (K)

10

2.5

Width (K)

1.5

2.0

1.5

[afii9851] s (102 )

8

1.0

1.0

0.5

6

0.5

0.0 0.0 0.5 1.0 1.5 2.0 2.5

t1/2 (K1/2)

4

0.0 0 1 2 3 4 5 6 7 8

t (K)

2

0 0.0 1.0 2.0 3.0 4.0 5.0 6.0

[afii9839]0H (T)

Figure 6 | Evidence of unconventional critical behaviour. Adiabatic susceptibility versus applied magnetic eld, showing an unusual Lorentzian eld dependence, suggestive of anomalous critical behaviour. The gure illustrates zero response at strong eld and a peak at the eld where the ice rule breaks. Error bars represent s.d. The lines are ts to a Lorentzian function at the following temperatures: blue, T 1.95 K; light blue, T 2.5 K; purple,

T 3.0 K; lilac, T 4.0 K; light green, T 6.0 K; green, T 8.0 K. Scaling of Lorentzian width (D) (inset): the red line is the function D / t

p . Experimental

data appear to rule out the possibility of a different scalingfor example, linear scale h/t.

and it appears from our result that it is associated with a magnetic polarizability as well.

These ndings have very important ramications for the monopole description of spin ice. Previous work by neutron spin echo27,28 on Ho2Ti2O7 and mSR (ref. 41) on Dy2Ti2O7 has suggested a high frequency response that thermally evolves to low temperature, and that at rst sight seems disconnected from the monopole picture. However, our results indicate that the dynamical spectrum in the approach to the high frequency limit is fully accounted for by magnetic monopoles, and they clearly explain the thermal evolution observed in the previous work. A very recent thermal conductivity study42 indirectly estimates a diffusion constant for magnetic monopoles that is much faster than ours, but our adiabatic susceptibility results show that there is no necessary contradiction, as monopoles mediate a dynamical response over a very broad frequency range. Finally, our results rule out any signicant spectral weight beyond that associated with monopoles, contrary to a recent proposal43.

Adiabatic susceptibility in applied eld. Figure 1 shows the spin ice phase diagram for a dc-magnetic eld applied along the cubic [111] direction. A small applied eld orders one spin per tetrahedron in the pyrochlore structure, but maintains the ice rule of two spins in and two out per tetrahedron, thus creating the so called kagome ice phase of two-dimensional disordered sheets, which still possess residual entropy29,44,45. With increasing eld at T 0 there is a breaking of the ice rules, pictured as the

ipping of one spin per tetrahedron, to create an ordered three in, one out state. Extending from this point there is a line of rst order phase transitions that terminates in a critical end point. The positive slope of this line reects the destruction of the spin ice entropy by the applied eld, according to the Clapeyron equation. In the monopole representation, the applied eld tunes the chemical potential of monopole-antimonopole pairs such that the increased monopole density drives a rst order condensation from a sparse monopole uid to a dense liquid (or perhaps better, ionic crystal) of alternating positive and negative monopoles5. The detailed theory of magnetic relaxation near

the critical point30 predicts mean eld critical exponents modied by logarithmic corrections. Here, we are interested in the supercritical region at temperatures well above the critical point, where the system may be described as a dense monopole plasma. Recently, a peak in the ac-susceptibility at nite frequency was observed in this region46. We examined the behaviour of the adiabatic susceptibility as a function of eld in this regime, to compare it with our zero-eld measurement.

In weak elds (m0Ht0:3 T) the thermal evolution of wS(T) shows a slow increase with eld, including a noticeable peak at higher temperature (Fig. 6). In much stronger elds (m0H 1 T)

the adiabatic response is completely suppressed as would be expected (Fig. 6), but at an intermediate eld (m0HE0.920(8) T), wS(H) exhibits a striking peak very near to the (internal) eld of the zero temperature phase transition. At this eld the ice rule is locally broken44 and 1/4 of the spins in the sample may then be ipped at zero energy cost. However, in contrast to the zero-eld result, wS(T) measured near this crossover eld (0.86 T)

exhibits a simple Curie law, wS C0/T (Fig. 5a), indicating a

different type of magnetic current to that observed in the weak eld limit, as anticipated in Shtyk and Feigelman30. We may regard the magnetic response in this regime as characteristic of switching magnetic dipoles, rather than magnetic monopoles. Note that the temperature evolution was measured at this point just off the peak maximum as it was found that systematic errors in tting to the ColeCole function are minimized at this point (Supplementary Note 6). The Curie constant C0 may be calculated under the assumption that 1/4 of the spins are thermally active and that these have a projection of 1/3 of their full classical value on the eld direction. Thus, we predict C0 C/36 0.1097 where CE3.95 is the high temperature Curie

constant. A t of the experimental data to the expression wS(m0H 0.86 T) a/(T TC) gave a 0.090(5), TC 0.4(2) in

close agreement with our prediction (Fig. 5a).

The striking 1/T divergence and location of the peak position in the H T plane (Fig. 6) suggests that the adiabatic

susceptibility is dominated by the classical critical point for monopole condensation (Fig. 1), for which the isothermal

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susceptibility wT is predicted30 to diverge as 1/|T TC|g, with

g 1 and TC T here. Usually the ratio wT/wS, analogous to the

LandauPlaczek ratio in a uid, should diverge towards the critical point. However, our data for wT and wS (Fig. 5a) illustrate a fairly typical paramagnetic response, as discussed above, with wS wT. Hence, both wS and wT diverge, but the latter is always

larger, as required by thermodynamics (Equation 6).

According to Shtyk and Feigelman30, the eld dependence of the susceptibility should be mean eld like, with logarithmic corrections: hence, we would expect w j H HC j1=d 1 with

d 3. However, we observe an exponent of 2 rather than 2/3

(Supplementary Fig. S7). Thus, dening reduced variables t T TC (with TC 0.36 K here), and h H HC, we nd to

a good approximation (suppressing dimensional constants):

wS

1t h2

). However, one may ask the question, how are these expectations consistent with the diffusive motion of monopoles that our experiments have revealed? An answer might be found in the theory of Chen et al.55 on the hopping of ionic defects in water ice, where it is shown how diffusive motion of defects can arise in a tight binding model that is constrained by the ice rules.

Our second result is that the adiabatic susceptibility gives a clear perspective on the magnetic properties of spin ice, revealing a direct measure of the magnetic monopole concentration and critical behaviour in applied eld. Our analysis of the adiabatic susceptibility has revealed a new property of magnetic mono-poles: their partial magnetic polarization by an applied eld. More generally we may conclude that the adiabatic susceptibility, often ignored as an uninteresting by-product of ac-susceptibility analysis, may contain a wealth of information about strongly correlated spin systems at low temperature.

If we combine these results with the remarkable prediction that a monopole will carry an electric dipole moment (the equivalent of its spin, if we reverse the roles of electricity and magnetism)40 a fascinating picture of the local properties of the monopole is starting to emerge. In general, the properties that we have discovered will have an important inuence on any future application of magnetic monopoles in spin ice that seek to exploit their magnetic and thermal response.

Methods

The dynamical magnetization of a 0.0326(1) g cubic crystal of Dy2Ti2O7 was measured with the ACMS (AC-Measurement System) option of a PPMS (Physical Property Measurement System, Quantum Design). Alternating and dc magnetic elds (Hac and Hdc, respectively) were applied parallel to the cubic [111] axis of the sample. Data were collected at different temperatures between 1.9 K rTr14 K in the ac-frequency range of 10 Hz to 10 kHz. A variable dc eld of m0H 0 10 T

was applied (at low eld the absolute eld was calibrated in dc-sweep measurement). Scans were taken at different ac elds in the range m0Hac 0.05

3 10 4 T to dispel the possibility of non-linear response of the system.

The results presented here were taken at m0Hac 5 10 5 T. Data were

corrected taking into account a demagnetizing factor D 1=3 to give

wac Mac= Hac DMac

. The calibrated response function of the instrument

was checked by measurement of a very dilute paramagnetic salt (Supplementary Fig. S5, panel b and c).

The data were tted to the phenomenological model for the frequency dependent susceptibility described in (refs. 33,35). By separating Equation 2 it is possible to derive analytical expressions for the real and imaginary parts and Argand diagram (ColeCole plot), which were each tted to the experimental data at a given temperature using a single set of parameters wS, wT, t, a.

The ColeCole formalism assumes a symmetric unimodal distribution of logarithmic relaxation times ln t0 with mean lnt. It can be shown that37:

s2lnt

p2

: 8

This implies h=

p scaling (Fig. 6, inset), which is formally characteristic of a zero dimensional phase transition (Supplementary Note 7). An alternative interpretation of the eld dependence is in terms of a classical single spin ip process, associated with the free moments in the eventual ordered structure, which would be characterized by a response of the type wS t=t2 h2 and hence h/t scaling, but our data appear to

distinctly rule against this possibility (Fig. 6, inset). Thus, the behaviour of wS(H) seems inconsistent with both the monopole

and T N xed points. One possibility, as discussed further in

Supplementary Note 7, is that the susceptibility is dominated by a zero temperature quantum critical point, but again with anomalous exponents. It is noteworthy several other examples of anomalous exponents have been reported for the quantum critical behaviour of rare earth magnets47,48.

DiscussionAlthough the concept of magnetic monopoles in spin ice is supported by much experimental evidence4951, the microscopic mechanism of monopole motion has yet to be identied. Our investigation of Dy2Ti2O7 has isolated the characteristics of this mechanism to which any future theory must conform.

Our rst result is that, over the temperature ranged probed, monopoles obey the NernstEinstein equation with temperature independent diffusion constant, a strong signature of Brownian diffusion. It should be emphasized that this is an experimental result and not a theoretical input. We envisage Brownian diffusion in the sense of an electrolyte, where the motion of oppositely charged ions is strongly correlated, yet the Nernst Einstein equation is obeyed if the Debye length is sufciently small (as it is here)32.

It would be useful to apply our methods to investigate monopole diffusion in the low temperature regime, as zero-eld measurements in that regime await an unambiguous interpretation in the monopole picture24,25, and the theory of Shtyk and Feigelman30 has yet to be comprehensively tested. In this context we emphasize that the characteristic relaxation time t(T) depends on at least three temperature-dependent factors, which shows that it cannot be treated in terms of an effective activation energy. However, our method disentangles all three factors and allows direct experimental estimation of the monopole mobility, which may be used to infer the monopole hop rate. We note a result52,53 published while this paper was in nal revision, that presents signicant evidence of a low temperature (o1.5 K) crossover to a regime where the monopole hop rate is roughly proportional to monopole density (see also Castelnovo et al.26). The precise temperature dependence of the hop rate can be experimentally determined by the general method presented here.

It is interesting to discuss our result of Brownian motion in the context of band theory. Just as water ice can be thought of as an intrinsic protonic semiconductor54, so spin ice may be expected to be an intrinsic semiconductor for magnetic monopoles. These are produced by the thermal unbinding (or fractionalisation5) of conventional magnetic excitons. They tunnel from site to site and have an effective mass determined by the inverse Debye length (proportional32 to

x=T p

t

: 9

In the text we label s2lnt simply as s2lnt for ease of reading, although strictly t is a xed parameter at a given temperature. It should be noted that there is no general way to derive the true mean relaxation time ht0i from ac-susceptibility data: here

we approximate it to the ColeCole parameter t.

The dimensionless monopole density xT was estimated by tting experimental

specic heat data to DebyeHckel theory26,34. The specic heat was represented as

the temperature derivative of the energy per diamond lattice site:

u m mDHx uDCM; 10 where m is the monopole chemical potential, mDHT is the Debye-Hckel

correction, calculated self consistently with the dimensionless monopole density xT, and uDCMT is a correction for double charge monopoles. The experimental

specic heat data, taken between 0:4 K T 10 K, were tted by adjusting m, with

the best t value m=k 4:33 K. The theory is not exact in the temperature range

of interest and Figs 4,5 report an approximate envelope of systematic error, found

1

3 1 a2 1

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by extrapolating the theory between low and high temperature according to different schemes (Supplementary Note 1).

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Acknowledgements

It is a pleasure to thank A. Fisher, C. Castelnovo and P. Holdsworth for valuable discussions; R. Aldus and M. Ellerby for their involvement in the specic heat work. The authors are also grateful to EPRSC for its support of the project.

Author contributions

L.B. did the experiment and analysed the data. S.T.B. conceived the idea of focusing on wS and derived the theory. S.T.B., L.B. and G.A. planned the experiments and interpreted the results. S.T.B. and L.B. drafted the paper with input from G.A. J.A.B. derived the density versus temperature from specic heat data. D.P. grew the crystal. All authors commented on the nal manuscript.

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How to cite this article: Bovo, L. et al. Brownian motion and quantum dynamics of magnetic monopoles in spin ice. Nat. Commun. 4:1535 doi: 10.1038/ncomms2551 (2013).

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