Intermediate magnetization state and competing orders in Dy2Ti2O7 and Ho2Ti2O7

Abstract

Among the frustrated magnetic materials, spin-ice stands out as a particularly interesting system. Residual entropy, freezing and glassiness, Kasteleyn transitions and fractionalization of excitations in three dimensions all stem from a simple classical Hamiltonian. But is the usual spin-ice Hamiltonian a correct description of the experimental systems? Here we address this issue by measuring magnetic susceptibility in the two most studied spin-ice compounds, Dy₂ Ti₂ O₇ and Ho₂ Ti₂ O₇ , using a vector magnet. Using these results, and guided by a theoretical analysis of possible distortions to the pyrochlore lattice, we construct an effective Hamiltonian and explore it using Monte Carlo simulations. We show how this Hamiltonian reproduces the experimental results, including the formation of a phase of intermediate polarization, and gives important information about the possible ground state of real spin-ice systems. Our work suggests an unusual situation in which distortions might contribute to the preservation rather than relief of the effects of frustration.

ARTICLE

Received 18 Dec 2015 | Accepted 15 Jul 2016 | Published 25 Aug 2016

R.A. Borzi1,2, F.A. Gmez Albarracn2,3, H.D. Rosales2,3, G.L. Rossini2,3, A. Steppke4,5, D. Prabhakaran6, A.P. Mackenzie4,5, D.C. Cabra2,3 & S.A. Grigera1,2,4

Among the frustrated magnetic materials, spin-ice stands out as a particularly interesting system. Residual entropy, freezing and glassiness, Kasteleyn transitions and fractionalization of excitations in three dimensions all stem from a simple classical Hamiltonian. But is the usual spin-ice Hamiltonian a correct description of the experimental systems? Here we address this issue by measuring magnetic susceptibility in the two most studied spin-ice compounds, Dy2Ti2O7 and Ho2Ti2O7, using a vector magnet. Using these results, and guided by a theoretical analysis of possible distortions to the pyrochlore lattice, we construct an effective Hamiltonian and explore it using Monte Carlo simulations. We show how this Hamiltonian reproduces the experimental results, including the formation of a phase of intermediate polarization, and gives important information about the possible ground state of real spin-ice systems. Our work suggests an unusual situation in which distortions might contribute to the preservation rather than relief of the effects of frustration.

DOI: 10.1038/ncomms12592 OPEN

Intermediate magnetization state and competing orders in Dy2Ti2O7 and Ho2Ti2O7

1 Instituto de Fsica de Lquidos y Sistemas Biolgicos (IFLYSIB), UNLP-CONICET, La Plata 1900, Argentina. 2 Departamento de Fsica, Facultad de Ciencias Exactas,Universidad Nacional de La Plata, 1900 La Plata, Argentina. 3 Instituto de Fsica de La Plata, UNLP-CONICET, 1900 La Plata, Argentina. 4 School of Physics and Astronomy, University of St. Andrews, St. Andrews KY16 9SS, UK. 5 Max Planck Institute for Chemical Physics of Solids, Nthnitzer Str. 40, Dresden, Germany. 6 Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK. Correspondence and requests for materials should be addressed to S.A.G. (email: mailto:[email protected]

Web End [email protected] ).

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Spin-ice systems owe their name to an analogy of their ground-state construction rules to those in Paulings model for proton disorder in water ice1,2. In their simplest variant,

they can be described in terms of centre pointing classical Ising spins at the binding points of a network of corner-shared tetrahedra forming a pyrochlore lattice (see Fig. 1a). In real materials, such as Dy2Ti2O7 (DTO) and Ho2Ti2O7 (HTO), this is an accurate description for the low temperature behaviour. It was recognized early that, given the large magnetic moments of the rare-earth ions (of the order of 10 mB) in addition to a ferromagnetic nearest neighbour coupling J (of the order of 1 K), a dipolar term was necessary in the Hamiltonian to describe the experimental results35. The minimal model that became the norm in the description of spin-ice materials is the standard dipolar spin-ice model (s-DSM):

H JXi;jh iSi Sj Dr31

Xi4jSi Sj rij

3

" #

3 Si rij

Sj rij

rij

5

1

where r1 is the nearest-neighbour distance, rij is the distance between spins i and j, Si is a classical spin of unit length, and the

dipolar constant, D, is of the same order than J. Spin-ice systems belong to the class of geometrically frustrated magnets. Frustration is possible in other ways, but the class of geometrically frustrated materials has the advantage of better experimental control and theoretical description than the case where disorder is needed. An inherent weakness of geometrical frustration is that it might be relieved by distortions to the lattice68. The spin-ice materials DTO and HTO are fairly robust to distortions; experimentally, the properties of the frustrated state and its low-lying excitations are seen to dominate the intermediate temperature regime9. However, as we shall see, the effects of small distortions in this system might account for some specic experimental features and determine the presence or absence of an ordered state at low temperatures.

The different limits of the classical s-DSM already contain the most singular and attractive features of spin-ice materials, notably residual entropy, emergent gauge structure, fractionalized monopolar excitations1012, and two-dimensional and three-dimensional Kasteleyn transitions1315. The s-DSM also predicts an ordered state at very low temperatures16. As the eld developed, it was soon recognized that additional interaction

[111]

a

b

c d

8

8

6

6

(a.u.)

4

(a.u.)

4

2

2

0

0

1 2 1 2 3

3

Field (T)

Field (T)

Figure 1 | Magnetic susceptibility at the polarization transition. The data shown are from HTO, but are representative of both materials. (a) A schematic view of the pyrochlore lattice: spins are represented as black arrows, and tetrahedra are coloured light blue. The coloured lines show, starting from the white circle, a rst (light blue), second (red) and two types of third (green and light green) nearest neighbour. (b) Sign convention for the angle shown in a single tetrahedron: rotations towards [112] are negative while those towards [110] are positive. (c) Magnetic susceptibility: when the eld is rotated towards [110] the peak remains unique. (d) As the eld is rotated towards [112], the peak splits into two, the upper one moves faster with eld as the angle is increased. The temperature for all curves is xed at 0.18 K and in all cases a smoothly varying background has been subtracted for clarity.

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a

terms are needed to properly account for the experiments and an effort was made to determine an empirical Hamiltonian from the experimental data17,18. In particular, the most complete work of this type is that of Yavorskii et al.19. In their approach, Monte Carlo simulations of a model with dipolar interactions, and isotropic rst (J1), second (J2) and third (J3) nearest-neighbour exchange interactions are compared with DTO experimental susceptibility, specic heat and magnetization data for zero magnetic eld H2022, with H applied in the 112 direction23,24, and with the empirical dependence of its polarization transition with H//11125. By means of this comparison they put bounds to the values of J1, J2 and J3 and arrive at an empirical Hamiltonian, the generalized dipolar spin-ice model (g-DSM). The g-DSM gives a very good phenomenological description of several of the spin-ice features and reproduces the low temperature neutron scattering pattern of DTO noticeably better than the s-DSM19.

Despite its successes, the g-DSM model does not fully reproduce the low temperature behaviour of the spin-ice materials. In this work we address two instances of these, which were unknown at the time the model was developed: the doubling of the polarization transition with elds in the neighbourhood of 111 (refs 26,27) and the recent report of an upturn in the specic heat below 0.6 K28. Based on our experimental results and those in the literature, we construct a new spin-ice model and show that in addition to reproducing the experimental features, it predicts an intermediate magnetization state for elds near [111] and the existence of different types of possible order at low temperatures when no eld is applied.

ResultsMagnetic susceptibility. We start by discussing the magnetic susceptibility at low temperatures when an external constant eld is applied in the neighbourhood of the crystallographic [111] direction. For our experiments, single crystalline samples of DTO and HTO were used. The a.c. susceptibility was measured in a dilution refrigerator with an external eld applied using a triple-axis vector (see Methods section).

The change in a.c. susceptibility at the polarization transition is plotted in Fig. 1 for the HTO single crystal (analogous behaviour is seen in the DTO sample) as a function of eld at a xed temperature of T 0.18 K for different angles from the [111]

direction (see Fig. 1b). Figure 1c shows eld rotations from [111] towards [110], while Fig. 1d shows rotations towards [112]. The usual spin-ice models predict a single transition between the partially frustrated kagome-ice state at low elds and the polarized state at high elds. In the experiments, this is true when the eld is perfectly aligned with [111], or when it is rotated towards [110], but the transition splits into two peaks, with an intermediate polarization region, when the eld is rotated towards [112]. This behaviour is seen more clearly in Fig. 2, where the change in susceptibility is shown for both materials as an interpolated contour plot. In this gure, it is also quite noticeable that the angular dependence of the lower transition eld is quite small when the eld is rotated towards [112]. Similar behaviour had been previously observed in magnetization experiments in DTO26.

The failure of the usual spin-ice Hamiltonians to reproduce an experimental feature common to both spin-ice materials suggests that there is an additional physical mechanism that is being neglected in the theoretical treatment. In the following, we argue that a missing ingredient can be the effect of distortions in the magnetic couplings of the pyrochlore lattice.

The role of distortions. We analyse how the s-DSM Hamiltonian (equation (1)) is modied in the presence of distortions. We

consider the simplest case: classical Einstein phonon modes in the pyrochlore lattice, parameterized as uiEri ri0, with a

linear dependence of the exchange couplings on distances: J(rij)EJ0[1 a rij0 (uj ui)], where J0 is the undistorted

exchange constant and a is the spin-phonon coupling constant, and a simple quadratic term for the energy cost of distortions. Expanding the variation of H up to linear order in ui, one obtains

a magnetoelastic Hamiltonian that can be written compactly as:

Hme

2.8

DTO

2.4

6

Field (T)

2.0

1.6

0

1.2

0.8

10

0 10

Angle (deg)

[112] [110]

b

HTO

3.5

3.0

6

Field (T)

2.5

0

2.0

1.5

10

0 10

Angle (deg)

Figure 2 | Contour plots of the magnetic susceptibility for DTO and HTO. (a,b) Interpolated contour plot of the magnetic susceptibility, for DTO and HTO respectively, at T 0.18 K. Each graph is constructed based on

approximately 40 traces of w versus angle at xed temperature and eld. Positive (negative) angles correspond to a rotation towards [110] ([112]).

For both materials the transition widenseventually resolving into two peaksas the eld is rotated towards [112]. The effect is more pronounced for HTO. The origin of this difference could lie on the non-Kramers nature of Ho3 which makes it more susceptible to its local environment.

Fij 2

where K is the elastic constant for classical Einstein phonons, N(i) stands for the neighbours of site i and Fij encodes the quadratic spin interactions between a site i and a neighbour j.

An effective Hamiltonian can be obtained from Hme by

integrating out the phononic d.f.29 (see Methods section). This effective Hamiltonian can be cast into a simple form where the

H

XiK 2ui2

Xiui

Xj

Ni

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effect of distortions to the lattice is translated into a change in the nearest-neighbour interaction and the emergence of further neighbour interactions. These interactions have their origin in the local changes in the magnetic interaction due to changes in the relative positions of the lattice sites, and are thus strongly dependent on the lattice connectivity between neighbours. Further neighbours of equal distance but connected to the original site by a different number of lattice sites develop different interaction terms. In the case of the pyrochlore lattice (see Fig. 1a), this mechanism provides two kinds of third nearest neighbour interactions, J3 and J30.

The model. While it cannot be expected that this simple analysis based on classical phonons will give accurate values for the effective exchange constants of the real system, which consists of a far more complicated lattice once all the non-magnetic atoms are taken into account, it provides the key observation that, regardless the specic material, further neighbour interactions should be considered and distinction made between differently connected neighbours of the same order, even if these terms or differences were negligible from the exchange integrals. With this information, we constructed a model to third nearest neighbours that incorporates these distinctions, the distorted dipolar spin-ice model (d-DSM), and explored it using Monte Carlo simulations (see Methods section and Supplementary Note 1). As additional constraints, we keep J2 and the mean value (J3 J30)/2, within the

rst set of intervals determined by Yavorskii et al. for the g-DSM19 to retain compatibility with the experimental results this model was tted to (see also Supplementary Note 2 and Supplementary Fig. 1).

Figure 3 shows the magnetic susceptibility for the d-DSM, with J1 3.41 K, J2 0, J3 0.02 K, J30 0.07 K and D 1.3224 K

(Fig. 3a), and for the g-DSM model (Fig. 3b), both as interpolated contour plots. Unlike the s-DSM or g-DSM, and in keeping with the experiments, the susceptibility calculated in the d-DSM has a atter angular dependence of the critical eld for rotations towards [112], showing a broad peak that eventually resolves into two peaks, and remains unique when the eld is rotated towards [110], all of which is in correspondence with the behaviour shown in Fig. 2 for the experimental results in the neighbourhood of [111]. Figure 3b shows the best possible t to the data using the g-DSM: by forcing the parameters towards the higher end of the intervals given in ref. 19, it is possible to eventually achieve a small doubling of the transition when rotating towards [112], but it completely fails to reproduce the angular dependence of the critical eld. The addition in the d-DSM of different types of J3 is

therefore necessary to be able to reproduce the susceptibility experiments close to [111], even at a qualitative level. Further terms and analysis are required to reproduce in detail the full angular dependence.

Intermediate phase. A known consequence of the coupling between spin d.f. and distortions is the stabilization of plateaux in the magnetization (see for example, refs 30,31). It therefore comes as little surprise that the doubling of the polarization transition corresponds, in the limit of low temperature and perfectly homogeneous tensions, to the presence of a plateau between the kagome-ice and the fully polarized state (see Fig. 4a). This plateau marks an intermediate ordered state where the ice rule is broken in only half of the tetrahedra. To visualize these states, it is helpful to mark by a red or blue dot the defect or excess of inward pointing spins. Defects of different colour interact approximately as charges of opposite sign11. Full polarization corresponds in this picture to a complete checkerboard pattern of blue and red dots. The intermediate states of Fig. 4 are easily pictured as alternate

arrangements of stripes of the checkerboard pattern separated by empty stripes (see Fig. 4).

The ground state. One of the remarkable predictions of all dipolar spin models is that, contrary to initial expectations, spin-ice materials may have no extensive residual entropy even in the absence of an applied magnetic eld. As shown by Melko et al.16, the introduction of dipolar interactions leads to a sharp transition at B0.18 K into an ordered ground state. At present there is no direct experimental determination of the ground state of spin-ice systems, since below 0.6 K the characteristic relaxation time of the system seems to grow faster than exponentially32,33 and most measurements probe properties of the system out of equilibrium34. A recent experiment28, in which the zero eld low temperature specic heat was measured over much longer timescales, reports the possible observation of a peak in the specic heat that might correspond to the onset of order in DTO. However, the temperature at which this onset is seen (B0.6 K) is not compatible with the prediction of Melko et al. for the s-DSM.

a

1.4

6

1.2

Field (T)

1.0

0

0.8

5

0 5

Angle (deg)

b

1.4

6

Field (T)

1.2

1.0

0

0.8

5

0 5

Angle (deg)

Figure 3 | Magnetic susceptibility from Monte Carlo simulations.(a) Contour plot of the susceptibility as extracted from the d-DSM, which shows a doubling of the transition and a marked anisotropy similar to that seen in the experiments (compare with previous data and ref. 26). (b) The best possible t to the data achievable with the g-DSM. In both cases the colour scale is non-linear to avoid saturation from the high intensity peaks.

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a b

5.0

M ( Bper Dy ion)

4.5

4.0

1.0 1.1

Field (T)

Figure 4 | Plateu in the magnetization and intermediate state of the model with distortions (d-DSM). (a) The simulated magnetization curve in a homogeneous system as a function of eld for T 0.1 K with the eld at an angle of 5 from [111]. A mid-plateau is stabilised between the kagome-ice

and the fully polarized state. (b) A sketch of this intermediate state projected along [100]. The coloured circles mark tetrahedra where the ice-rule is

broken either by excess (blue) or defect (red) of inward pointing spins.

The situation is very similar for the g-DSM: the additional interaction terms of this model leave the ordered state unaltered, and the specic heat (by construction), shows no upturn down to 0.2 K.

The model presented in this work (the d-DSM) is a different scenario altogether. In the simple homogeneous case, and within the constraints on the parameters imposed by the experiments (crucially our susceptibility near [111]) two different ordered ground states are possible depending on the relation between the exchange constants.

Figure 5a,b shows cuts along [100] of the two ordered states obtained at low temperatures for different values of the exchange constants. For small values of J2, the relevant parameter that switches between the two types of order is the difference,

DJ3 J30 J3. The scale in the middle (Fig. 5c) indicates, within

the region compatible with the experiments (shaded in blue), the values of DJ3 (for xed average J3, and J2 0) where each type of

order is seen. As expected, for low values of DJ3, the ordered state is identical to the one originally found in the s-DSM. A simple description of this state (state I) is to consider it an antiferromagnetic arrangement of ferromagnetic chains of spins (indicated with the same colour in the gure) at axes rotated p/4 with respect to the original lattice. The second state (state II) can be thought as a simple extension of state I where the unit blocks are now pairs of ferromagnetic spin chains (also indicated with the same colour in the right hand side gure) ordered antiferromagnetically. In both cases, the non-shaded region in the gure corresponds to the unit cell.

These are the two states of lower energy that have been identied depending on the degree of asymmetry in the two J3. It is straightforward to conjecture that there is a wealth of possible intermediate states with energies differing by a few mK (in 1/k), constructed from all the possible periodic intercalations of different numbers of alternating single and double chains in each direction, or even of the alternation of bundles of a higher number of chains. This by itself has already the consequence that it will be extremely difcult for any experiment to probe equilibrium properties, but the situation in a real sample is even more complicated: the most likely experimental state is that the sample will have a distribution of tensions and distortions within its volume. This will lead to local changes in DJ3, coming from the distortion-induced contribution to the two J3, which in turn will result into local energetic rearrangements of the possible states. No long-range ordered state will be favourable in the whole

sample. The effect of tensions and distortions could also be intimately related with the experimental observation of freezing or glassy behaviour below temperatures of about 0.6 K3538.

Specic heat. Figure 5d shows the zero-eld specic heat divided by temperature, calculated for the d-DSM with the same values used for Fig. 3, compared with the experimental data for DTO from refs 12,28 (a nite size analysis of the simulated specic heat is shown in Supplementary Note 3 and Supplementary Fig. 2). Both experimental sets of data coincide at high temperatures, down to about 0.6 K below which the characteristic time of the system becomes extremely long and the results from ref. 28, taken over longer times, are believed to be closer to equilibrium. The upturn seen in this data has been interpreted as a possible sign of onset of long-range order. The specic heat calculated from the d-DSM matches the high temperature behaviour of the experiments and shows a sharp peak at a temperature close to0.2 K, which corresponds to the ordering transition into state I or II (depending on the value of DJ3). It would be tempting to associate this peak with the upturn of ref. 28, and match the two sets of data by making small adjustments to the exchange constants of the d-DSM. However, as discussed before, it might not be realistic to expect a homogeneous long-range ordered state in a real sample, but rather a wide range of local congurations and a multitude of energetically close excitations. In this case, features matching those seen in the specic heat data of ref. 28 have their origin in a Schottky-type peak consequence of this proliferation of states of very similar energies in this range of temperatures. A more denitive answer to this question would come from neutron scattering experiments performed under the same relaxation conditions as the specic-heat experiment.

DiscussionIn summary, in this work we present experiments performed in the two paradigmatic examples of spin-ice materials, DTO and HTO, where we have measured the angular dependence of susceptibility using precise alignment with the crystallographic axes, concentrating in particular in the polarization transition with eld in the neighbourhood of [111]. We show that a generic feature of these two materials, absent from any theoretical spin-ice model in the literature, is a marked anisotropy in w, which shows a doubled transition when the eld is rotated towards [112]. This doubled transition had been previously observed in

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a b

c

Ordered state I

Ordered state II

0.05

0.1 0.12

J 3

Compatible with experiments

d

3

Cv/T(J per K2 Dy mol)

2

1

0 0 1 2 3

T (K)

Figure 5 | Ordered states of the model with distortions (d-DSM). Within the range of parameters xed by the experimental susceptibility data, and depending on the difference DJ3 between the third-nearest-neighbours exchange constant, the system orders into two possible states. (a,b) Cut along [100] of the two ordered states. The non-shaded regions correspond to the unit cells. (c) Ranges in DJ3 where each state is observed. (d) The signature of this transition in the specic heat divided by temperature, compared with the results obtained for DTO in refs 12,28.

specic heat and magnetization experiments in DTO23,26, but not in HTO. We considered the possibility of distortions, a missing ingredient from the usual spin-ice models, and by performing the simplest possible analysis on their effect on the s-DSM we found that they provide a justication for the addition of interactions between second and two kinds of third nearest neighbours. This is a generic feature of distorted spin-ice materials, and applies even if the exchange terms for the specic material were negligible. There is a crucial difference in the current treatment of thermal vibrations to what already exists in the literature, which is a consequence of the presence of long-range interactions in this class of materials. Thermal vibrations usually just renormalize constants, but in the present case dipolar long-range interactions lead to the appearance of couplings that would not exist in the absence of vibrations. Thus, our treatment of the simple d-DSM leads to a more complex Hamiltonian, that resembles that of for example, ref. 19, but is not obtained in a purely empirical manner.

Provided with these ingredients we constructed a new model (the d-DSM) and explored its characteristics using Monte Carlo simulations. The new model reproduces the anisotropy found in the experiments with elds in the neighbourhood of [111], and, despite the fact that it was not built based on neutron scattering data or Cv, the S(q) calculated for the d-DSM reproduces the experimental results (see Supplementary Note 4 and Supplementary Fig. 3) and we have seen that the same is true for Cv (Fig. 5). More importantly, it predicts an intermediate magnetization state, thus explaining the hitherto unresolved issue of a doubled polarization transition with eld close to the [111] direction, and also predicts the existence of different types of possible order at low temperatures when no eld is applied. These results from the d-DSM are valid regardless the ultimate origin of the different interaction terms. The possibility that these further order terms could be a consequence of distortion would mean that the small tensions present in real samples could result in the absence of long-range order at low temperatures. This is probably

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the most surprising conclusion of this work, that distortions, which usually result in a relief of frustration, might be the reason why the system would remain disordered at very low temperatures.

Very recently, Henelius et al.39 have presented an empirical spin-ice Hamiltonian constructed to describe three main sets of experiments in DTO (the structure factor S(q), the specic heat Cv and the magnetization for H near [112]). The model Hamiltonians presented in both studies have several points of contact. In particular, both models predict the same possible ordered ground states for H 0 (those described in Fig. 5) and,

crucially, the likelihood of a disordered ground state as a consequence of internal stress (in our case given by external strain, in theirs by impurities) and quasi-degeneracy. The similarity of these ndings is remarkable taking into account that the core experiments used in their work and in ours are very different.

Methods

Experiments. For our experiments, single crystalline samples of DTO and HTO where grown using an optical oating zone in St. Andrews and Oxford, and they were oriented and cut into prisms of approximate dimensions (0.7 0.7 3) mm3,

with the long axis oriented along [111]. The a.c. susceptibility was measured in a dilution refrigerator using two sets of detection coils, positioned so as to avoid cross-talk effects, and a drive coil. Each detection coil set comprises of two coils, connected antiparallel to each other. The drive coil is concentrically wound around the pick-up coils. The samples were placed in each of the upper pick-up coils and thermally grounded to the mixing chamber through silver and copper wires.

A low-frequency and very low-amplitude excitation eld of about 3 10 5 T

r.m.s. was generated by the ac current in the drive coil, and the response detected by a lock-in amplier. For accurate control of the external magnetic eld, we used a bespoke triple-axis vector magnet capable of producing 9/1/1 T along z, x and y. Careful centring using the vector eld capability allowed y to be determined to an accuracy of 0.1 relative to the crystallographic axis [111].

Theoretical analysis of the effect of distortions. A simpler effective Hamiltonian can be obtained from Hme (equation (2)) by integrating out the

phononic d.f.26. In the spin-ice model Si si ei, with s 1, which leads to

important simplications: since si2 1, the usual quartic terms become quadratic.

Still, due to the long-range nature of dipolar couplings one encounters terms Fij of

any range, and the analysis of the effective magnetic model is rather complex. To compare the predictions of our model with those in the literature we have considered an expansion of the corrections to the full effective magnetic Hamiltonian up to third neighbours.

The effective magnetic model obtained reads

Heff =J Jeff1

Xoi;j41

sisj Jeff2

Monte Carlo simulations. For the d-DSM model we used the following Hamiltonian:

Hd

J1

Xoi;j41

sisj J2

Xoi;j42

sisj J3

Xoi;j43

sisj J03

Xoi;j430

sisj

Dr31

Xi;jSi Sj rij

3

" #

m

3 Si rij

Sj rij

rij

5

5

XiSi B

where the oi,j43 and oi,j43 distinguish between third nearest neighbours connected through the lattice by a minimum of one or two other lattice sites respectively (see Fig. 1a).

We performed Monte Carlo simulations, using Ewald summations to take into account long-range dipolar interactions42,43 in a conventional cubic cell for the pyrochlore lattice. We simulated systems with L L L cells with periodic

boundary conditions. Basic parameters were taken from ref. 19, including cell size, lattice parameter and magnitude of magnetic moments. Thermodynamic data as a function of eld were collected using a variation of single spin-ip Metropolis algorithm. For Fig. 3 we chose a temperature of 0.4 K to avoid problems associated with hysteresis. For the specic heat data below 0.6 K we mixed the usual Metropolis with a dynamics that conserves defects for a certain number of Monte Carlo steps before letting the system relax (the algorithm is a variation of the one used in ref. 43 and is described in Supplementary Note 1). In the case of magnetization measurements at 0.1 K (Fig. 4), data at each eld were collected after a slow annealing, averaging over ve independent runs.

The average value of the third neighbour interactions was kept within the range specied in ref. 19 to retain compatibility with previous experiments in DTO. As an example, with the values of J used in Figs 35, we studied the transition to ferromagnetic ordering with eld nearly along the [112] direction19,21. We obtained a transition temperature of E0.35 K, which could be justied considering angular deviations of o0.5 off perfect alignment.

Regarding the double peak near [111], we contrasted our simulations with the static magnetization data obtained in ref. 26, which shows narrower features than the a.c. susceptibility. The range xed in Fig. 5 was determined looking for special key details: (i) a magnetization increase of only B0.025m when the eld is tilted from 0 to 5 (towards [112]) at a xed eld of 0.9 T, (ii) an increment for the

rst critical eld of o0.06 T when tilting the eld from 0 to 5 and (iii) a

difference between the two critical elds at 5of E0.1 T.

Data availability. All relevant data are available from the authors.

References

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Xoi;j42

sisj Jeff3

Xoi;j43

sisj

J0eff3

Xoi;j43

sisj further dipolar terms

3

where oi,j4r indicates r-th range neighbours. The couplings up to third neighbours are

Jeff1

13

5 3

D

J 2d; Jeff2

D 3

3

p J d; Jeff3

D8J 2d; J0eff3

D8J 4

where d is the contribution arising from the phonons, d J=4ka2a a=3 5D=J2.

Further range terms are the unmodied dipolar interactions.

The rst observation is that the crystal environment distinguishes two different third neighbour couplings in Heff: J3eff and J30eff distinguish between third nearest neighbours connected through the lattice by a minimum of one or two other lattice sites respectively. One of them is sensitive to distortions, while the other is not. A second observation is that the relative strengths of all couplings in Heff depend

only on two microscopic parameters: D/J and a/J. Corrections of longer-range magnetic couplings can be done systematically, depending on the same microscopic parameters. We have checked that fourth-order corrections do not change the main results of this work.

It is not realistic to expect that a simple classical model such as the one presented here will give an accurate description of the real system at low temperatures. The presence of soft modes (speculated for spin-ice materials in ref. 40 or seen in Tb2Ti2O7 (ref. 41) could be underlying the relative success of such a treatment, and is worthy of being investigated.

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Acknowledgements

We thank R. Moessner, C. Castelnovo and M. Gingras for helpful discussions, and the nancial support of ANPCYT through PICT 2013-2004 and PICT 2014-2618 and CONICET (Argentina), the EPSRC and the Royal Society (UK).

Author contributions

D.P. prepared the HTO crystal samples. R.A.B., S.A.G. and A.S. designed and performed the susceptibility measurements. F.A.G.A., H.D.R., D.C.C. and G.L.R. did the theoretical analysis, R.A.B. performed the Monte Carlo simulations The manuscript was initially drafted by S.A.G. and optimized by S.A.G., R.A.B., D.C.C. and A.P.M. with input and discussion from all authors.

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How to cite this article: Borzi, R.A. et al. Intermediate magnetization state and competing orders in Dy2Ti2O7 and Ho2Ti2O7. Nat. Commun. 7:12592doi: 10.1038/ncomms12592 (2016).

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