Spin liquid and ferroelectricity close to a quantum critical point in PbCuTe2O6

Geometrical frustration among interacting spins combined with strong quantum fluctuations destabilize long-range magnetic order in favor 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 favorable 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 PbCuTe2O6, 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 behavior is revealed for the single crystals. These crystals show ferroelectric order at TFE ≈ 1 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 behavior.


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 chargetransfer 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 Na 4 Ir 3 O 8 , a 3D effective S = 1/2 spin liquid candidate system with a hyperkagome lattice 13 , evidence was reported for a nearby quantum critical point 17 -a T = 0 instability to some nearby (possibly magnetically) ordered state. More recently, PbCuTe 2 O 6 has attracted considerable interest as another 3D quantum spin liquid candidate system featuring a highly-connected 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 PbCuTe 2 O 6 , 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 longrange 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 α/C e , with α the thermal expansion coefficient and C e the electronic specific heat, probing the thermal Grüneisen parameter Γ 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 behavior. In fact, clear evidence for a nearby field-sensitive quantum critical point is observed in the electronic magnetic Grüneisen parameter Γ B,e , yielding a divergence for T → 0.

RESULTS
Material, crystal structure, and magnetic interactions PbCuTe 2 O 6 crystallizes in a non-centrosymmetric cubic structure with space group P4 1 32 (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-neighbor interaction J 1 = 1.13 meV, which are coupled via the second-nearest neighbor interaction J 2 = 1.07 meV into a hyperkagome lattice. The weaker third-and fourth-nearest neighbor interactions J 3 = 0.59 meV and J 4 = 0.12 meV couple the spins into chains. Another special feature of PbCuTe 2 O 6 , 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 SrCuTe 2 O 6 22 which is isostructural to PbCuTe 2 O 6 but features a magnetic network with predominantly 1D character.
Measurements were performed on single crystals (sc) from four different batches grown by utilizing two different techniques (see "Methods" and ref. 23 ), and pressed-powder (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, C e , measured on single crystal sc #5(b). The data reveal a pronounced λ-shape phase transition anomaly around 1 K, signaling a secondorder phase transition, on top of a broad maximum. The figure also includes data for a pressed-powder sample pd #Ι(b) showing a distinctly different behavior. Here we find the broad maximum in C e (T) centered 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 C e (T) ∝ T n dependence with n ≈ 2 at low temperatures T ≤ 0.4 K.

Thermodynamic properties
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. Fig. 2. In the inset of Fig. 2, we show data of the relative length changes, ΔL i (T)/L i, for sc #1(a) measured along two different directions. Based on structure determination at room temperature, we refer to these directions as i = [100] 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 ΔL i (T)/L i 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 ΔL i (T)/L i along both directions, consistent with a secondorder phase transition. We stress that the evolution of a nonisotropic lattice strain from a cubic high-temperature state implies the formation of structural domains. As a result, the ΔL i (T)/L i 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 α i (T) = 1/L i ·d(ΔL i (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 α [110] data represent a preferential domain orientation Fig. 1 Temperature dependence of the electronic specific heat of PbCuTe 2 O 6 . Electronic specific heat of single crystal sc #5(b) (black squares) and a pressed-powder sample pd #Ι(b) (orange spheres). Thin solid lines serve as guides to the eyes. Very similar results to those on sc #5(b) were obtained on a second single crystal #1(b) studied in this experiment, which was grown using a different technique, see "Methods" and Supplementary Fig. 2. For the nuclear contributions subtracted, see "Methods" and Supplementary Fig. 2 and Note 2. Error bars represent the standard deviation due to statistical treatment of raw data. The inset shows the entropy in units of Rln2 for both samples using the same color code. The arrow marks the position of the phase transition anomaly in C e (T). The entropy release associated with the phase transition over a narrow temperature window of about 0.2 K amounts to a few percent of Rln2. coming close to a mono-domain structure, see Fig. 6 in the Supplementary. Therefore, we confine the discussion of the temperature dependence to these α [110] data. Upon cooling, α [110] shows an extraordinarily strong increase followed by a sharp negative phase transition anomaly slightly below 1 K, and a second maximum around 0.5 K. However, a comparison with the results of C e (T) for the single crystal sc #5(b) in Fig. 1, showing a maximum around 1 K, suggests another interpretation: there is only a single huge maximum in α [110] centred slightly below 1 K to which a negative λ-shape phase transition anomaly around 1 K is superimposed. The somewhat larger width of the phase transition anomaly in α [110] (T) as compared to C e (T), may indicate some remaining domain misalignment in the thermal expansion experiment, cf. the narrowing of the phase transition anomaly in α [110] on increasing the uniaxial pressure in Fig. 6 of the Supplementary. Figure 2 also shows thermal expansion data taken on the pressed-powder sample pd #Ι(a). Whereas the ΔL(T)/L data for the powder match with the data for the single crystal for T > 1.6 K, (see inset of Fig. 2) they deviate at lower temperatures: instead of the strong contraction observed for the single crystal on approaching the phase transition, the powder sample shows a rather smooth behavior with ΔL(T)/L gradually levelling off upon cooling. The corresponding α(T) data for the powder sample shown in the main panel of Fig. 2 reveal a broad maximum around 1.4 K followed by a smooth variation α(T) ∝ T n with n ≈ 2 for T ≤ 0.4 K.

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 ε' for sc #5(b). The data reveal an enhanced background dielectric constant of ε' b ≈ 18 in the temperature range investigated and a well-pronounced peak centred at 0.97 K, signaling a ferroelectric transition at T FE = 0.97 K. This assignment is further corroborated by a Curie-Weiss-like behavior on approaching the maximum in ε' 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 behavior is typical for an order-disorder-type ferroelectric transition 25,26 where electric dipoles that are disordered at high temperature order with a net overall polarization below T FE . In contrast, for the pressed-powder sample pd #Ι(b) we find only a tiny anomaly in ε' around 1 K, see Fig. 3. This observation together with the smooth variation of ΔL(T)/L for T ≤ 1.6 K for the powder sample, reflecting the absence of an anomalous lattice distortion as revealed for the single crystals, indicate that there is no bulk ferroelectric transition in the pressed-powder sample.
To further corroborate the ferroelectric nature of the transition detected at about 1 K, we have investigated the polarization P of single-crystalline PbCuTe 2 O 6 (sc #1(a), E||[110]). Figure 4 shows P(T) as measured upon heating after poling the sample with different electrical fields during a preceding cooling run as noted in the figure. Without poling field, no significant polarization was detected, consistent with a multi-domain polar state with even distribution of polarization directions. Prepoling with positive fields between 0.1 and 2.3 kV/cm reveals a successively increasing low-temperature polarization, vanishing above T FE ≈ 1 K. The observed gradual increase of P(T) below T FE is in accordance with a second-order nature of the polar phase transition consistent with results of the specific heat and thermal expansion ( Figs. 1 and 2). Between 1.2 and 2.3 kV/cm, P exhibits only weak variation with field, pointing to a mono-domain state with saturation polarization of the . Similar measurements on the single crystals #1(a) and #4 reveal essentially the same results albeit with a somewhat reduced peak height, see Supplementary Fig. 9 and Note 6. The inset gives details of the data for sc #5(b) in a narrow temperature range around 1 K. Inset a, showing 1/(ε' − ε' b ) vs. T, reflects a Curie-Weiss-like behavior on approaching the maximum in ε' from both sides. The slopes of the linear regimes differ by a factor of about −1.9. This factor is very close to −2, expected for a mean-field secondorder ferroelectric transition 43 , but strongly departs from −8 seen at first-order ferroelectric transitions 25,44 . The lower inset b shows ε' vs. T for varying frequencies as indicated in the figure. The small shift of the background dielectric constant ε' b , visible for the data at highest frequency, indicates some contribution from extrinsic factors. Temperature dependence of the electric polarization of single crystalline PbCuTe 2 O 6 (sc #1(a)) as deduced from the pyrocurrent monitored upon heating. The measurements were performed for different poling fields (−2.3 to +2.3 kV/cm), applied to the sample along the [110] direction during the preceding cooling run. For original pyrocurrent data taken at different heating rates and poling fields, see Supplementary Fig. 10a and Fig. 10b. The results in the inset demonstrate the switching of the polarization achieved by first cooling the sample within negative field (1), applying a positive field for 1 min at 0.9 K (2), and finally reheating it without field (3). order of 5 μC/m 2 . As revealed by the lowest curve in Fig. 4, a negative field of −2.3 kV/cm results in a negative polarization, reflecting a polarization into the opposite direction, as expected for ferroelectrics. The absolute value of P, reached for −2.3 kV/cm is somewhat reduced compared to +2.3 kV/cm which may arise from strains within the crystal, slightly favoring one polarization direction.
Another defining property of ferroelectrics is the switchability of the polarization by an electrical field at temperatures below the transition. For the present material, this is demonstrated in the inset of Fig. 4. In this experiment, the sample was first polarized by cooling it to 0.9 K under negative electrical field of −2.3 kV/cm and then applying a positive field of 4.7 kV/cm. The subsequent heating run without field revealed positive polarization, unequivocally proving the in-situ switching of the polarization within the ferroelectric state.
It should be noted that the detected saturation polarization P s of PbCuTe 2 O 6 of about 5 μC/m 2 (=0.5 nC/cm 2 ) is rather small compared, e.g., to the well-established lone-pair ferroelectric BiFeO 3 27 or to classical displacive ferroelectrics like BaTiO 3 28 , revealing P s values of several tens of μC/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 ε'(T) and P s are, e.g., also found in some improper ferroelectrics like TbMnO 3 29 . However, it is conceivable that P s 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.
PbCuTe 2 O 6 has a cubic chiral but non-polar crystal structure at high temperatures (space group P4 1 32). 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 P4 1 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 solid-state 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. [30][31][32] 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 PbCuTe 2 O 6 was systematically investigated in ref. 23 . In their studies, the low-temperature 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 μm, and completely vanishes for diameters below 10 μm.
The drastic reduction of the anomaly in ε' around 1 K for the powder sample, together with the smooth behavior in the specific heat and thermal expansion (Figs. 1 and 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 PbCuTe 2 O 6 , 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 13,17,18,33,34 , shows a considerable shift from T max ≈ 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) derived from ac-susceptibility measurements on single crystal #5 (b) (see inset of Fig. 4 and Supplementary Fig. 7 as well as Note 4), still lacking any indication for long-range order for T ≥ 0.1 K, which reveal a mild reduction of M(T) with decreasing temperatures below about 0.8 K. In contrast, for powder material a slight increase was observed for M(T) on cooling below 1 K 19 . We attribute the change in the magnetic couplings to the lattice distortions-a displacement of the Pb 2+ ion out of its highsymmetry position-accompanying the ferroelectric transition, see Supplementary Fig. 1d. This displacement allows for a (sp) hybridization with oxygen p orbitals and the formation of a lone pair with asymmetric electron distribution. As these oxygen p orbitals are also involved in the dominant magnetic exchange paths ( Supplementary Fig. 1c), we expect changes in the coupling constants J 1 and J 2 of the distorted structure.
Another even more pronounced difference in the materials' low-temperature magnetic/electronic state becomes apparent by studying the ratio α [110] /C e for sc #5, a quantity which is proportional to the uniaxial thermal Grüneisen parameter Γ p[110] (see ref. 35 ) and compare this to α/C e for the pressed-powder sample pd #Ι, cf. inset of Fig. 5. Note, since for T ≤ 1 K the lattice contribution to α is very small, its influence can be neglected in discussing the critical behavior 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 low-energy excitation spectrum in the powder material is governed by a single energy scale, the pressure dependence of which is probed by Γ p . We link this energy scale to the maximum in α and C e at T max 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 behavior. 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 behavior at lowest temperatures where the influence of the phase transition anomalies in α [110] and C e around T FE ≈ 1 K is expected to be small, suggesting it is related to the magnetic sector. In fact, the data reveal a growing ratio α [110] /C e 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 behavior, especially whether or not this ratio diverges for T → 0. In general, C. Thurn et al. a diverging Grüneisen ratio Γ p = 1/T·(∂T/∂p) S , measuring temperature contours at constant entropy S, is considered a clear signature of a nearby pressure (p)-sensitive quantum critical point 35 . This defining property results from the accumulation of entropy close to the quantum critical point.
As an alternative approach for probing potential quantum critical behavior associated with the system's magnetic degrees of freedom, we performed measurements of the magnetic Grüneisen parameter Γ B = 1/T·(∂T/∂B) S 35 from which the electronic magnetic Grüneisen parameter Γ B,e has been extracted by taking into account the nuclear contributions, see "Methods" and Supplementary Note 2 as well as Supplementary Figs. 3 and 4. In fact, as shown in the main panel of Fig. 5, the data reveal a diverging Γ B,e on cooling over about one decade in temperature down to 0.06 K with an asymptotic behavior following a Γ B,e ∝ T −m dependence with m ≈ 3.9 for T ≤ 0.25 K. Note that an independent determination of Γ B,e is provided by using the identity Γ B,e = −(dM/dT)/C e 35 . In fact, the calculation of Γ B,e based on the M(T) data (inset of Fig. 5) and C e (T) data (Fig. 1), yields an almost identical behavior for Γ B,e (T) within the experimental uncertainty, see Fig. 4 in the Supplementary, that is, a divergence for T → 0. From these observations, we conclude that single crystalline PbCuTe 2 O 6 for T → 0 approaches a quantum critical point which is sensitive to magnetic fields. This is consistent with results taken on mildly increasing the magnetic field to B = 0.2 T (Fig. 5), and the variation of Γ B,e with field measured at a constant temperature of T = 0.15 K (see Supplementary Fig. 5), yielding a suppression of Γ B,e following a Γ B,e = −A/(B -B c ) dependence with A = 0.058 ± 0.005 and B c ≈ 0. According to ref. 35 this behavior is expected for a quantum paramagnetic state, that is, a non-critical spin-correlated state. This raises the question about the origin of the quantum critical behavior that gives rise to the divergence in Γ B,e (T) at small fields. In light of the strong response revealed for the uniaxial thermal Grüneisen parameter along the [110] direction, we associate the quantum critical behavior to an instability of the quantum spin liquid against uniaxial pressure p [110] and the associated lattice distortions. It remains to be shown by future experiments on mono-domain single crystals whether or not there are divergences in the uniaxial thermal Grüneisen parameters also along the orthogonal [1][2][3][4][5][6][7][8][9][10] and [001] directions. Remarkably, an anisotropic thermal expansion response with quantum critical behavior only along one direction was observed for the heavy fermion metal CeRhSn 36 featuring a frustrated 2D kagome lattice.
In conclusion, comprehensive investigations on PbCuTe 2 O 6 including thermodynamic, magnetic and dielectric probes, reveal markedly different behavior for single crystalline material as compared to pressed-powder samples: whereas the lowtemperature properties of the powder material are consistent with the recently proposed gapless quantum spin liquid state, an even more exotic behavior is observed for the single crystals. Here, we find a ferroelectric transition at T FE ≈ 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 behavior. 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 behavior 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 T max for various frustrated antiferromagnets 13 , we are inclined to assign the reduction of T max 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 behavior, 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 Pr 2 Ir 2 O 7 37 or the valence fluctuator β-YbAlB 4 38 . In contrast to these systems, PbCuTe 2 O 6 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 PbCuTe 2 O 6 offers the possibility for exploring quantum criticality resulting from a strongly frustrated hyperkagome spin system interacting with ferroelectricity-a scenario holding promise for fascinating physics. Especially, through comparison with the non-critical behavior of the undistorted low-T state realized in the pressed-powder samples of PbCuTe 2 O 6 , one may expect to find interesting magneto-electric coupling effects mediated by the lattice deformations.

Samples investigated
Single crystals of PbCuTe 2 O 6 were grown by utilizing two different techniques: a traveling 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 roomtemperature structure of the resulting single crystals can be best refined in a cubic structure with space group P4 1 32 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 Note 3 of the Supplementary of ref. 20 . The powder was pressed into a pellet of 1 cm diameter by applying hydrostatic force up to 12 tons (∼1.5 GPa). The starting material of the pressed-powder samples studied here, i.e., the ceramic powder of PbCuTe 2 O 6 , is taken from the same batch as the "polycrystalline samples" characterized and investigated via neutron scattering and synchrotron diffraction in ref. 20 . According to these characterization measurements the powder is phase pure with a roomtemperature structure of P4 1 32 space group and a lattice constant a = 12.4968(4) Å, see Supplementary of ref. 20 , consistent with the above results for the single crystals. In addition, the influence of the preparation conditions of polycrystalline materials on the thermal properties was studied in ref. 23 , and no effects of off-stoichiometry on the cation sites or oxygen deficiency could be observed.

Thermal expansion
Thermal expansion experiments were performed by using a homemade capacitive dilatometer following the design discussed in ref. 39 . In this technique, relative length changes ΔL(T)/L are measured with ΔL(T) = L(T) − L(T 0 ) and T 0 the starting temperature of the experiment. Measurements were performed at temperatures 0.05 K ≤ T ≤ 2 K, using a bottomloading 3 He-4 He dilution refrigerator and for 1.5 K ≤ T ≤ 200 K, using a 4 Hebath cryostat. In the dilatometer cell used, the spring leafs suspending the movable part exert a force F to the sample along the measuring direction with F ≈ 0.03 N-3 N, the actual strength of which depends on the chosen starting capacitance. This results in uniaxial pressure along the measuring direction p ua = F/A with A the cross-sectional area of the sample, typically ranging from p ua ≈ 0.01 MPa to 5 MPa.

Specific heat
The specific heat measurements were carried out by employing both a homemade relaxation calorimeter as well as a homemade continuousheating-type calorimeter both of which were attached to a bottom-loading 3 He-4 He dilution refrigerator. In the measurements of the specific heat at low temperatures T < 1.6 K, basically two different contributions have to be considered. The electronic contribution of interest here, C e , arises from the magnetic moments associated with the spins of the 3d electrons of the Cu 2+ ions. The second one, C n , is due to the magnetic moments of the 207 Pb, 125 Te, 63 Cu, and 65 Cu nuclei present in PbCuTe 2 O 6 . The phonon contribution C ph , however, is negligibly small and does not exceed 1 % of C e for T < 1.6 K, see ref. 18 . Below T ≈ 0.1 K the nuclear contribution dominates (see Note 2 and Fig. 2 of the Supplementary) and has to be determined quite accurately in order to extract C e reliably from the measured total specific heat C. The contributions of the various nuclei, C n,i , are Schottky anomalies resulting from the splitting of energy levels in a magnetic field. In the case of the 207 Pb and 125 Te nuclei, this field is identical to the external magnetic field, as there are no electronic magnetic moments in the immediate vicinity. Thus C n,i can be calculated easily by using the corresponding parameters from literature (refs. 40,41 ), that is, the natural abundances of the various isotopes, their nuclear spins and associated nuclear magnetic moments (see also Supplementary Table 1). For the copper nuclei, however, the local field is strongly influenced by the magnetic field caused by the magnetic moment of the 3d electrons of the Cu 2+ ion. Therefore, this local field has to be determined by a 1/T 2 fit to the low-temperature specific heat data in B = 0, yielding a local electronic field of B e ≈ 10.2 T, see also Supplementary Note 2 for details.

Magnetic Grüneisen parameter
The magnetic Grüneisen parameter, Γ B = T −1 ·(∂T/∂B) S , was measured by using a relaxation calorimeter attached to a bottom-loading 3 He-4 He dilution refrigerator. For details of the applied technique, in which Γ B is approximated by Γ B ≈T s −1 · (ΔT s /ΔB) S≈const. , see ref. 42 . Here T s is the sample temperature, and ΔT s the temperature change induced in response to changes in the applied magnetic field ΔB.
The electronic magnetic Grüneisen parameter Γ B,e can be extracted from the measured magnetic Grüneisen parameter Γ B of the total system by taking the specific heat contributions C i and magnetic Grüneisen parameters Γ B,i of all relevant subsystems into account: In the case of PbCuTe 2 O 6 , Γ B,e is then given by (see also Supplementary Note 2): ;e þ C nPb Á Γ B;nPb þ C nTe Á Γ B;nTe þ C nCu Á Γ B;nCu À Á : (1) The nuclear magnetic moments of Te and Pb 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 Γ B;nPb ¼ Γ B;nTe ¼ 1=B: For Γ B,nCu the situation is more complex due to the strong influence of the magnetic field caused by the magnetic moment of the 3d electrons on the local magnetic field acting on the nuclei In the absence of magnetic ordering, and for not too large magnetic fields, the following expression can be derived for the determination of Γ B,e (see Supplementary Note 2 for details):

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 capacitorcontact geometry as described for the dielectric measurements. To determine the ferroelectric polarization, standard pyrocurrent experiments were performed by prepoling the sample upon cooling down to 0.9 K with different applied electrical fields and subsequent monitoring of the pyrocurrent under heating beyond the transition temperature. The pyrocurrent was recorded using a Keysight B2987A electrometer and applying a heating rate of 60 mK/min. For examples of pyrocurrent measurements see Note 8 and Fig. 10 of the Supplementary. The polarization was deduced by integration of the pyrocurrent over time.
To check for the switchability of the polarization, additional experiments were performed by (i) measuring the pyrocurrent upon cooling the sample with applied electrical field, (ii) subjecting the sample to a field of opposite direction for 1 min after reaching a base temperature of 0.9 K, and (iii) monitoring the pyrocurrent upon subsequent heating in zero field. 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).