Spin liquid and ferroelectricity close to a quantum critical point in PbCuTe$_2$O$_6$

Geometrical frustration among interacting spins combined with strong quantum fluctuations destabilize long-range magnetic order in favour of more exotic states such as spin liquids. By following this guiding principle, a number of spin liquid candidate systems were identified in quasi-two-dimensional (quasi-2D) systems. For 3D, however, the situation is less favourable as quantum fluctuations are reduced and competing states become more relevant. Here we report a comprehensive study of thermodynamic, magnetic and dielectric properties on single crystalline and pressed-powder samples of PbCuTe$_2$O$_6$, a candidate material for a 3D frustrated quantum spin liquid featuring a hyperkagome lattice. Whereas the low-temperature properties of the powder samples are consistent with the recently proposed quantum spin liquid state, an even more exotic behaviour is revealed for the single crystals. These crystals show ferroelectric order at $T_{\text{FE}} \approx 1\,\text{K}$, accompanied by strong lattice distortions, and a modified magnetic response -- still consistent with a quantum spin liquid -- but with clear indications for quantum critical behaviour.


Introduction
Competing interactions combined with strong quantum fluctuations are considered a major guiding principle for the realization of a quantum spin liquid. This long sought state of matter is characterized by exciting properties such as macroscopic entanglement and fractionalized excitations, see refs. [1][2][3][4][5] for recent reviews. As quantum fluctuations are enhanced for low spin values and low lattice coordination, most efforts have been devoted to low-dimensional (low-D) quantum spin (S = 1/2) antiferromagnets, with prominent examples including the 2D kagome system Herbertsmithite 6,7 and some layered triangular-lattice charge-transfer salts [8][9][10] .
For 3D lattices, the search has focused on materials where the spins reside on a pyrochlore 11,12 or hyperkagome lattice 13 , a 3D network of corner-sharing triangles. As in 3D competing ground states are expected to become more relevant, perturbations from purely Heisenberg spin scenarios due to, e.g., Dzyaloshinskii-Moriya interaction 14,15 or spin-orbit coupling 16 gain in importance and may eventually govern the materials' ground state 1 . This raises the general question about the stability of a quantum spin liquid phase in 3D. In fact, for Na4Ir3O8, a 3D effective S = 1/2 spin liquid candidate system with a hyperkagome lattice 13 , evidence was reported for a nearby quantum critical point 17a T = 0 instability to some nearby (possibly magnetically) ordered state. More recently, PbCuTe2O6 has attracted considerable interest as another 3D quantum spin liquid candidate system featuring a highlyconnected hyperkagome lattice [18][19][20] , where S = 1/2 Cu 2+ spins are coupled by isotropic antiferromagnetic interactions. According to magnetic studies, mainly on pressed-powder samples [18][19][20] , this system lacks long-range magnetic order down to 0.02 K 19 and shows diffuse continua in the magnetic spectrum 20 consistent with fractional spinon excitations. Puzzling issues relate to the appearance of small anomalies in the powder samples around 1 K of unknown origin [18][19][20] and signs of a phase transition around this temperature in first-generation single crystals 20 .
In the present work, we present a comprehensive study of thermodynamic, magnetic and dielectric properties of PbCuTe2O6, with special focus lying on its low-temperature state. To this end, we have investigated several single crystals and compare the results with data on pressed-powder samples. The salient results of our study are (i) the observation of a phase transition in single crystalline material around 1 K into a ferroelectrically-ordered state, which is accompanied by strong lattice distortions. (ii) This phase transition and the state at T  1 K are characterized by a finite magnetic susceptibility without any indication for long-range magnetic order consistent with a gapless quantum spin liquid state. (iii) The bulk ferroelectric transition and its accompanying lattice distortions are absent in the powder samples.
(iv) For these powder samples, the ratio /Ce, with α the thermal expansion coefficient and Ce the specific heat, probing the thermal Grüneisen parameter p, is practically independent of temperature for T  1.6 K. In contrast, a corresponding uniaxial Grüneisen parameter for the single crystals shows a strong increase for T  0, potentially indicating quantum critical behaviour. In fact, clear evidence for a nearby field-sensitive quantum critical point is observed in the electronic magnetic Grüneisen parameter B,e, yielding a divergence for T  0.

Material, crystal structure and magnetic interactions
PbCuTe2O6 crystallizes in a non-centrosymmetric cubic structure with space group P4132 (No. 213), see Supplementary Fig. 1 and Note 1. According to density functional theory calculations 20 based on room-temperature structural data, the magnetic lattice can be described by isolated equilateral S = 1/2 triangles with nearest-neighbour interaction J1 = 1.13 meV, which are coupled via the secondnearest neighbour interaction J2 = 1.07 meV into a hyperkagome lattice. The weaker third-and fourth-nearest neighbour interactions J3 = 0.59 meV and J4 = 0.12 meV couple the spins into chains. Another special feature of PbCuTe2O6, which has been largely ignored until now, relates to its dielectric degrees of freedom 21 . The material contains polar building blocks originating from the free electron pairs (lone pairs) of the Te 4+ ions in the oxotellurate tetrahedrons and the asymmetrically coordinated Pb 2+ ions. These characteristics together with the non-centrosymmetric structure imply the possibility of ferroelectric order interacting with the strongly frustrated quantum spin system. In fact, a magneto-dielectric effect was observed in SrCuTe2O6 22 which is isostructural to PbCuTe2O6 but features a magnetic network with predominantly 1D character.
Measurements were performed on single crystals (sc) from 4 different batches grown by utilizing two different techniques (see Methods and ref. 23), and pressedpowder (pd) samples (see Methods) from the same batch prepared as described in ref. 20. In what follows, the samples will be specified by giving their batch number followed by an alphabetic character to distinguish different samples from the same batch. Figure 1 shows the results of the electronic specific heat, Ce, measured on single crystal sc #5(b). The data reveal a pronounced -shape phase transition anomaly around 1 K, signalling a second-order phase transition, on top of a broad maximum.

Thermodynamic properties
The figure also includes data for a pressed-powder sample pd #(b) showing a distinctly different behaviour. Here we find the broad maximum in Ce(T) centred around 1.4 K, followed by a smooth reduction upon further cooling similar to previous reports on powder material [18][19][20] . As also revealed in these studies, we observe signatures for a tiny feature around 1 K and a Ce(T)  T n dependence with n  2 at low temperatures T  0.4 K More insight into the nature of the phase transition in the single crystalline material can be obtained by studying the elastic properties via thermal expansion measurements, cf. and [1][2][3][4][5][6][7][8][9][10]. For the data at higher temperatures down to about 1.7 K, we find a smooth isotropic reduction of Li(T)/Li with decreasing temperature, consistent with a cubic structure. However, on further cooling the thermal contraction becomes increasingly stronger and develops a pronounced anisotropy, indicating deviations from cubic symmetry at low temperatures. At around 1 K we find a sharp break in the slope in Li(T)/Li along both directions, consistent with a second-order phase transition. We stress that the evolution of a non-isotropic lattice strain from a cubic high-temperature state implies the formation of structural domains. As a result, the Li(T)/Li data for T  1.7 K in Fig. 2 could potentially be affected by the material's domain structure. In general, the formation of domains can be influenced by the application of uniaxial pressure to the crystal in its high-temperature phase. This is actually the case in our thermal expansion measurements along the measuring direction, where uniaxial pressure of typically 0.01 -5 MPa is applied, depending on the sample geometry and the chosen starting capacitance 24 , see also Methods and Supplementary Note 3 . In the main panel of Fig. 2 we show the coefficient of thermal expansion i(T) = 1/Lid(Li(T))/dT for single crystal sc #5(c) along the [110] direction.
Since in this experiment a rather high uniaxial pressure of about (6.5  1.3) MPa was realized, we believe that these [110] data represent a preferential domain orientation coming close to a mono-domain structure, see Fig. 6 in the Supplementary. Therefore, we confine the discussion of the temperature dependence to these  [110] data. Upon cooling, [110] shows an extraordinarily strong increase followed by a sharp negative phase transition anomaly slightly below 1 K, and a second maximum

Dielectric constant and polarization
Given the material's non-centrosymmetric structure and its polar building blocks, comprising two subsystems of stereochemically active lone pairs associated with the Te 4+ and Pb 2+ ions, the observation of a lattice distortion suggests an involvement of the electric degrees of freedom. To probe the dielectric response, measurements of the dielectric constant were performed on single crystals #1(a), #4 and #5(b) for temperatures T  1.3 K. In Fig. 3 we show exemplarily the results of the normalized dielectric constant ' for sc #5(b). The data reveal an enhanced background dielectric constant of 'b  18 in the temperature range investigated and a well-pronounced peak centred at 0.97 K, signalling a ferroelectric transition at TFE = 0.97 K. This assignment is further corroborated by a Curie-Weiss-like behaviour on approaching the maximum in ' from both sides, see inset (a) of Fig. 3. In the inset (b) of Fig. 3 we show the frequency dependence of the dielectric anomaly around 1 K. The measurements, covering approximately two decades in frequency, reveal a distinct suppression of the peak with increasing frequency whereas the position of the peak remains essentially unchanged. This behaviour is typical for an order-disorder-type ferroelectric transition 26,28 where electric dipoles that are disordered at high temperature order with a net overall polarization below TFE. In contrast, for the pressed-powder sample pd #(b) we find only a tiny anomaly in ' around 1 K, see It should be noted that the detected saturation polarization Ps of PbCuTe2O6 of about 5 C/m 2 (= 0.5 nC/cm 2 ) is rather small compared, e.g., to the well-established lonepair ferroelectric BiFeO3 29 or to classical displacive ferroelectrics like BaTiO3 30 , revealing Ps values of several tens of C/cm 2 . This is consistent with the rather small amplitude of the anomaly observed in the dielectric constant at the transition (Fig. 3).
Small values of '(T) and Ps are, e.g., also found in some improper ferroelectrics like TbMnO3 31 . However, it is conceivable that Ps is higher for field directions other than the presently used E || [110] geometry. To clarify this question, systematic investigations with different contact geometries and/or crystal cuts have to be performed, which is out of the scope of the present work.
PbCuTe2O6 has a cubic chiral but non-polar crystal structure at high temperatures (space group P4132). Consequently, the onset of ferroelectricity is necessarily accompanied by the lowering of the crystal symmetry. The highest-symmetry polar subgroups of the parent cubic group are the tetragonal P41 group and the rhombohedral R3 group. Though both space groups are compatible with the presence of ferroelectricity, we expect the rhombohedral one being realized in this compound for the following reason: The cubic to rhombohedral distortion maintains the three-fold symmetry of the kagome lattice, while in case of the cubic to tetragonal distortion the kagome lattice would not be regular anymore. Thus, the former would preserve a high degree of frustration of the spin-spin interactions, while the latter would reduce it and by this likely promotes the onset of magnetic ordering, not observed experimentally. Again, future polarization measurements along different crystallographic directions should help clarifying this issue.

Discussion
The different dielectric and lattice properties observed here for the pressed-powder material as opposed to the single crystals, can be rationalized by considering that the powder samples studied here and in the literature [18][19][20] were all prepared by a solidstate reaction method where the material is subject to multiple grinding processes interrupted by special heat treatments. Correspondingly, these samples constitute a more or less homogeneous collection of grains with some distribution of grain sizes, which may vary from sample to sample and the heat treatment applied. Our finding of a ferroelectric transition in the single crystalline material and the suppression of this transition in the pressed-powder samples is consistent with results on grain-size effects in ferroelectric ceramics (see, e.g., refs. 32-34 and Supplementary Note 7), yielding a critical grain size below which the transition disappears. Triggered by the results of the present work, the influence of the grain size on the 1 K phase transition in PbCuTe2O6 was systematically investigated in ref. 23. In their studies, the lowtemperature specific heat on single-and polycrystalline samples was measured both after crushing the samples (thereby reducing the size of the crystallites) and after annealing them (thereby increasing the size of the crystallites and reducing dislocations and grain boundaries). According to ref. 23, the phase transition anomaly around 1 K is drastically reduced for crystallites of diameter 30 m, and completely vanishes for diameters below 10 m.
The drastic reduction of the anomaly in ' around 1 K for the powder sample, together with the smooth behaviour in the specific heat and thermal expansion ( Fig. 1 and Fig. 2) are fully consistent with the absence of a bulk ferroelectric transition and the accompanying lattice distortion in the powder sample. At the same time, as demonstrated by various advanced magnetic measurements 19,20 , these powder samples lack long-range magnetic order down to temperatures as low as 0.02 K, indicating that in the non-distorted cubic low-temperature state of PbCuTe2O6, the frustration of the magnetic network is strong enough to suppress long-range ordering consistent with the formation of a quantum spin liquid state. This raises the question to what extent the magnetic lattice and thus the degree of frustration is altered by the lattice distortions revealed in single crystalline material. In this context it is interesting to note that the broad maximum in the low-temperature specific heat, a feature which is considered a hallmark of strongly frustrated spin systems 13,17,18,35,36 , shows a considerable shift from Tmax  1.4 K for the powder sample to below 1 K for the single crystalline material. It is tempting to assign this shift to alterations of the exchange coupling constants in the frustrated spin system in the single crystals due to ferroelectric ordering. This is consistent with results for the magnetization M(T) Another even more pronounced difference in the materials' low-temperature magnetic/electronic state becomes apparent by studying the ratio [110]/Ce for sc #5, a quantity which is proportional to the uniaxial thermal Grüneisen parameter p [110] (see ref. 37) and compare this to /Ce for the pressed-powder sample pd #, cf. inset of Fig. 5. Note, since for T  1 K the lattice contribution to  is very small, its influence can be neglected in discussing the critical behaviour of the Grüneisen ratio. Figure 5 reveals a practically temperature-independent ratio for the pressed-powder sample in the temperature range investigated. This is consistent with the notion that the lowenergy excitation spectrum in the powder material is governed by a single energy scale, the pressure dependence of which is probed by p. We link this energy scale to the maximum in  and Ce at Tmax around 1.4 K which reflects magnetic correlations in the strongly frustrated hyperkagome lattice. In contrast, for the single crystal, we find a distinctly different behaviour. Whereas the ratio at higher temperatures lies close to the value for the powder material, it shows a distinctly different temperature dependence upon cooling. Of particular interest is the behaviour at lowest temperatures where the influence of the phase transition anomalies in [110] and Ce around TFE  1 K is expected to be small, suggesting it is related to the magnetic sector. In fact, the data reveal a growing ratio [110]/Ce upon cooling for low temperatures. However, in light of the substantial error bars involved, the data do not allow to make a definite statement on the asymptotic behaviour, especially whether or not this ratio diverges for T  0. In general, a diverging Grüneisen ratio p directions. Remarkably, an anisotropic thermal expansion response with quantum critical behaviour only along one direction was observed for the heavy fermion metal CeRhSn 39 featuring a frustrated 2D kagome lattice.
In conclusion, comprehensive investigations on PbCuTe2O6 including thermodynamic, magnetic and dielectric probes, reveal markedly different behaviour for single crystalline material as compared to pressed-powder samples: whereas the low-temperature properties of the powder material are consistent with the recently proposed gapless quantum spin liquid state, an even more exotic behaviour is observed for the single crystals. Here, we find a ferroelectric transition at TFE  1 K, accompanied by pronounced lattice distortions, and somewhat modified magnetic signatures -still consistent with a quantum spin liquid -but with clear indications for quantum critical behaviour. These findings call for low-temperature structural investigations on single crystals as key input for determining their T-dependent exchange interactions. It would be interesting to extend these low-temperature structural investigations also to the pressed-powder samples, as the non-critical behaviour observed there is assigned to the lack of those lattice distortions.
In the absence of such low-T structural information, we speculate that the lattice distortions in single crystalline material and the associated changes in the electronic structure are likely to alter the magnetic network in a way that drives the system close to a quantum critical point. Based on the trend revealed by comparing the maximum in the specific heat at Tmax for various frustrated antiferromagnets 13 , we are inclined to assign the reduction of Tmax from 1.4 K for the powder to below 1 K for the single crystals to an increase in the degree of frustration, thereby driving the system quantum critical. It is interesting to note that quantum critical behaviour, assigned to geometrical frustration, has been observed also in other materials where a 3D frustrated spin system interacts with charge degrees of freedom such as in the Kondo lattice Pr2Ir2O7 40 or the valence fluctuator -YbAlB4 41 .
In contrast to these systems, PbCuTe2O6 features a spin system which is devoid of effects related to strong spin-orbit interactions, crystalline electric fields or Kondo-type interaction with delocalized charges. Thus, single crystalline PbCuTe2O6 offers the possibility for exploring quantum criticality resulting from a strongly frustrated hyperkagome spin system interacting with ferroelectricitya scenario holding promise for fascinating physics. Especially, through comparison with the noncritical behaviour of the undistorted low-T state realized in the pressed-powder samples of PbCuTe2O6, one may expect to find interesting magneto-electric coupling effects mediated by the lattice deformations.

Samples investigated.
Single crystals of PbCuTe2O6 were grown by utilizing two different techniques: a travelling solvent floating zone (TSFZ) method for sc #1, sc #2, and sc #4, and a top-seeded solution growth (TSSG) technique for sc #5. Details of the growth conditions and the sample characterization, including polarized optical microscopy, X-ray Laue and X-ray powder diffraction, are given in ref. 23. According to these studies, the room-temperature structure of the resulting single crystals can be best refined in a cubic structure with space group P4132 and a lattice parameter a = 12.4967 Å (TSSG) and 12.502 Å (TSFZ), in agreement with previous results [18]. Whereas the single crystals grown by the TSFZ technique may contain non-magnetic foreign phases with a volume fraction of 8 % at maximum, no foreign phase could be detected in these measurements for the crystals grown by using the TSSG method.
The pressed-powder samples were obtained by using "polycrystalline ceramic powder" from the solid-state reaction of the precursor oxides, as described in  (2)

Dielectric measurements.
For measurements of the dielectric constant, a plate capacitor arrangement was realized by attaching two electrodes (silver paste) to opposite parallel surfaces of the samples. The dielectric constant was derived from the capacitance, read out by using an LCR meter (Agilent E4980), and the geometrical dimensions of the plate capacitor. This procedure implies an uncertainty in the experimental data of about  5 %. Measurements were performed by using a top-loading 3 He-4 He dilution refrigerator.

Polarization.
The polarization measurements were performed using the same capacitor-contact Typical fields applied were of the order of several kV/cm.

Ac-susceptibility.
For measurements of the ac-susceptibility, a homemade susceptometer adapted to a top-loading 3 He-4 He dilution refrigerator was employed. The ac-susceptometer was calibrated via magnetization measurements up to 5 T by comparing the results with data obtained by using a SQUID magnetometer (Quantum Design MPMS).

Data availability
All the raw and derived data that support the findings of this study are available from the authors upon reasonable request.
[29] Lu, J., et al.      Representative error bar, denoting standard deviation due to statistical treatment of raw data.

Crystal structure, structural motives and magnetic as well as dielectric units of PbCuTe2O6
PbCuTe2O6 crystallizes in a non-centrosymmetric cubic structure with space group P4132 (No. 213), see Fig. 1. According to density functional theory calculations [1] based on room-temperature structural data, the magnetic lattice can be described by isolated equilateral S = 1/2 triangles with nearest-neighbour interaction J1 = 1.13 meV, which are coupled via the second-nearest neighbour interaction J2 = 1.07 meV into a hyperkagome lattice (Fig. 1b). The weaker third-and fourth-nearest neighbour interactions J3 = 0.59 meV and J4 = 0.12 meV couple the spins into chains. Another unique feature of PbCuTe2O6, which has been largely ignored until now, relates to its dielectric degrees of freedom [2]. The material contains polar building blocks originating from the free electron pairs (lone pairs) of the Te 4+ ions in the oxotellurate tetrahedrons and the asymmetrically coordinated Pb 2+ ions. These characteristics together with the non-centrosymmetric structure imply the possibility of ferroelectric order interacting with the strongly frustrated quantum spin system.

Specific heat and magnetic Grüneisen parameter
At low temperatures (T < 1.6 K) there are basically two different contributions to the specific heat, C, of PbCuTe2O6. The first one, Ce, arises from the magnetic moments associated with the spins ⃗ of the 3d electrons of the Cu 2+ ions, and the second one, Cn, is due to the magnetic moments of the 207 Pb, 125 Te, 63 Cu and 65 Cu nuclei. The phonon contribution Cph [3], however, is negligibly small and does not exceed 1 % of Ce for T < 1.6 K. Below T  0.1 K the nuclear contribution dominates and has to be determined quite accurately in order to extract Ce reliably from the measured total specific heat C.
The contributions Cn,i of the above listed nuclei to the specific heat are Schottky anomalies with equidistant energy levels produced by the different orientations of the nuclear magnetic moments ⃗ in the local magnetic fields ⃗⃗ * : Here The magnetic Grüneisen parameter ,e of the electron spins can be extracted from the measured magnetic Grüneisen parameter B = T -1 (T/B)S of the total system by taking the specific heat contributions Ci and magnetic Grüneisen parameters B,i of all relevant subsystems into account. In general, the following relation holds: with = ∑ the total specific heat. In the case of PbCuTe2O6, ,e is then given by = 1 • ( e • ,e + nPb • ,nPb + nTe • ,nTe + nCu • ,nCu ) .
The nuclear magnetic moments of tellurium and lead behave as Langevin paramagnets (for which the entropy S = S(T/B)) with the local magnetic field ⃗⃗ * being identical to the external magnetic field ⃗⃗ , so that ,nPb = ,nTe = 1 .
For B,nCu the situation is less simple due to the strong influence of ⃗⃗ e on ⃗⃗ * . As the magnetic moments of the copper nuclei practically do not interact with one another, their / * ratio is constant in an adiabatic process: * = .
This can be simplified for the case ≪ e by a series expansion and taking into account only the terms which are linear in B/Be: As the copper spins show no indications of ordering in a small external magnetic field ⃗⃗ , they can be assumed to be oriented completely randomly relatively to ⃗⃗ and thus the same holds for ⃗⃗ e too. Therefore, (8) has to be averaged evenly over the whole solid angle: (1) The first technique applied [6] follows the definition B = T -1 (T/B)S, by measuring temperature changes T * of the sample, which is in weak thermal contact to a bath at temperature Tb, in response to changes of the magnetic field B, i.e., B  T -1 (T * /B)Sconst.. A typical measuring cycle is shown in Fig. 3. Since this technique measures the total Grüneisen parameter B, which is the sum of various contributions (see eq. (3)), the determination of the electronic part B,e requires a careful consideration of the nuclear contributions according to eq. (10).
2) As an alternative approach, we took advantage of the identity B,e = -Ce -1 (M/T) by using the electronic specific heat data, Ce, of sc #5(b) shown in Fig. 1

Magnetic susceptibility
The main panel of figure 7 exhibits the ac-susceptibility of crystal #5(b) as a function of temperature for 0.1 K  T  1.6 K in a magnetic field of 0.1 T. The data were taken by using a home-made ac-susceptometer adapted to a top-loading dilution refrigerator.
The ac-susceptometer was calibrated via magnetization measurements up to 5 T by comparing the results with data obtained by using a SQUID magnetometer (Quantum Design MPMS). For the ac measurements the external field was aligned along the orientations. The data reveal a small anomaly around 6 K of unknown origin. A more pronounced anomaly is also visible in the same temperature range for sc #2 (not shown). In this sample sc #2 there is also a small but distinct difference visible between field-cooled and zero-field cooled data. Similar anomalies in this temperature range were also observed in other frustrated triangular-lattice systems and were ascribed to a spin freezing induced by quenched disorder [9]. upon cooling from room temperature down to approximately 10 K where it adopts a broad maximum. Below 10 K cL decreases moderately strongly down to 1.3 K, the lowest temperature of our experiment. Below approximately 2 K the softening becomes stronger which we interpret -in analogy to the thermal expansion -as a precursor of the ferroelectric transition.

Dielectric constant
The dielectric constant ' was measured on 3 single crystals of PbCuTe2O6 in the temperature range 0.25 K  T  1.3 K, see Figure 9. In these measurements the

Grain-size effect of ferroelectric order
The observation made here of a ferroelectric transition in PbCuTe2O6 single crystals and the suppression of this transition in pressed-powder samples is consistent with results on grain-size effects in ferroelectric ceramics [10][11][12], yielding a critical grain size below which the transition disappears. Whereas for an isolated grain the instability of the ferroelectric phase is mainly due to the surface effect, the situation becomes more complex for ceramics where beside intrinsic grain-size effects other factors, which may change with the size of the system, can be of relevance as well, see ref.
[12] and references cited therein. heating. As expected, the pyrocurrent is enhanced for higher poling field, corresponding to stronger polarization. Figure 10b shows the pyrocurrent for two different heating rates and identical, negative prepoling fields. As expected, the current is enhanced for higher rate. For conducting materials, thermally stimulated discharge currents can also lead to pyrocurrent peaks.

46
The fact that the peak temperature in Fig. 10b is unaffected by the heating rate excludes this non-intrinsic effect, which should lead to a significant shift of the peak to higher temperatures for higher heating rates [13,14]. pair associated with Pb2 in its symmetrically coordinated environment, forming bonds of similar length to six O1 ions. Therefore, in the non-distorted high-temperature structure the dipole moments originating from the tellurates and Pb1 within each unit compensate each other. Thus a distortion along the three-fold rotation axis, i.e., a displacement of the Pb2 out of its high-symmetry position, is necessary to allow for a (sp) hybridization [15] resulting in the formation of a lone pair with asymmetric electron distribution prerequisite to the formation of ferroelectric order. This process can be described as a second-order Jahn-Teller instability [15]. As the O1 p orbitals are involved in the dominant magnetic exchange paths, we expect that the distorted structure (with the modified electron distribution) will also result in a change in the magnetic coupling constants J1 and J2.  By extrapolating the exponential decay back to times at which the field change started, the increase in temperature T * can be determined from an equal-areas construction as indicated in the inset of Fig. 3.    for T  2 K measured by using a commercial SQUID magnetometer (Quantum Design).