Abstract
Strongly enhanced quantum fluctuations often lead to a rich variety of quantumdisordered states. Developing approaches to enhance quantum fluctuations may open paths to realize even more fascinating quantum states. Here, we demonstrate that a coupling of localized spins with the zeropoint motion of hydrogen atoms, that is, proton fluctuations in a hydrogenbonded organic Mott insulator provides a different class of quantum spin liquids (QSLs). We find that divergent dielectric behavior associated with the approach to hydrogenbond order is suppressed by the quantum proton fluctuations, resulting in a quantum paraelectric (QPE) state. Furthermore, our thermaltransport measurements reveal that a QSL state with gapless spin excitations rapidly emerges upon entering the QPE state. These findings indicate that the quantum proton fluctuations give rise to a QSL—a quantumdisordered state of magnetic and electric dipoles—through the coupling between the electron and proton degrees of freedom.
Introduction
The nature of QSLs has been well established in onedimensional (1D) spin systems. However, it still remains unclear how QSLs emerge in dimensions greater than one. The celebrated resonatingvalencebond theory on a 2D triangular lattice^{1,2} puts forward the possibility that geometrical frustration plays an important role in stabilizing QSLs. In fact, a few candidate materials hosting QSLs have now been reported in materials with 2D triangular lattices^{3,4,5,6,7,8}. Nevertheless, according to subsequent theoretical studies^{9,10}, the effect of geometrical frustration in the triangular lattice is insufficient to stabilize QSLs, leading to a number of proposed mechanisms that may stabilize the QSL states found in the candidate materials^{11}. One of the most promising approaches is to utilize a coupling of spins with charges and orbitals; the former has been discussed near a Mottinsulatortometal transition where the charge degrees of freedom begin to delocalize^{12,13,14,15,16,17,18} and the latter has been considered in the framework of a spin–orbital coupling^{19}. Such strategies, however, have been limited to the utilization of internal degrees of freedom of electrons.
The hydrogenbonded organic Mott insulator κH_{3}(CatEDTTTF)_{2} (hereafter abbreviated as HCat)^{5,20,21,22} may serve as a candidate for a different class of QSLs, where H_{2}CatEDTTTF is catecholfused ethylenedithiotetrathiafulvalene (see Fig. 1a–c). HCat forms a 2D spin1/2 Heisenberg triangular lattice of CatEDTTTF dimers^{20} (Fig. 1e). Despite the antiferromagnetic interaction energy J/k _{B} of ~80 K, no magnetic order has been observed down to 50 mK; this indicates the realization of a QSL state^{5}. A distinct feature of HCat is that the 2D πelectron layers are connected by hydrogen bonds^{20,21} (Fig. 1a, c), which is in marked contrast to other 2D organic QSL materials such as κ(BEDTTTF)_{2}Cu_{2}(CN)_{3} (ref. ^{3}) and EtMe_{3}Sb[Pd(dmit)_{2}]_{2} (ref. ^{4}), where the 2D spin systems are separated by nonmagnetic insulating layers. This structural feature of HCat is highlighted by deuteration of the hydrogen bonds^{21}; specifically, in the deuterated analog of HCat, κD_{3}(CatEDTTTF)_{2} (denoted as DCat), deuterium localization occurs at T _{c} = 185 K, accompanied by charge disproportionation within the CatEDTTTF layers, resulting in a nonmagnetic ground state (Fig. 1d, f). This demonstrates that the hydrogen bonds in this system strongly couple with the charge and spin degrees of freedom of the πelectrons in the CatEDTTTF dimers.
In contrast to DCat, the hydrogen atoms in HCat do not localize down to low temperatures^{21}. This is inconsistent with the fact^{23} that the potential energy curve of the hydrogen bonds calculated for an isolated supramolecule has a double minimum potential with a large energy barrier of ~800 K (see Fig. 1c, g), which should localize the hydrogen atoms in HCat at low temperatures. Recent theoretical calculations^{22,23} have pointed out that the potential energy curve has a singlewell structure and its bottom becomes very shallow and anharmonic (see Fig. 1g) owing to a manybody effect arising from the network of hydrogen bonds and πelectrons. In this case, the zeropoint motion of the hydrogen atoms (termed “proton fluctuations”) can be strongly enhanced by the anharmonic potential curve. In contrast to DCat, the enhanced proton fluctuations may delocalize the hydrogen atoms down to absolute zero, providing an opportunity for realizing a QSL state through strong coupling between the hydrogen bonds and the πelectrons. However, it has not been established whether such strong quantum proton fluctuations are indeed present in HCat, and if so, how the quantum fluctuations affect the QSL state.
Here we show, by using a combination of dielectric permittivity and thermal conductivity measurements, that the quantum proton fluctuations presented in HCat provide a quantumdisordered state of magnetic and electric dipoles through the coupling between πelectrons and hydrogen atoms. These methods are particularly suitable because the dielectric permittivity is sensitive to local electricdipole moments arising from the hydrogenbond dynamics^{24}, whereas the thermal conductivity is a powerful probe to detect itinerant lowlying energy excitations associated with the nature of QSL states^{25,26}.
Results
Dielectric permittivity measurements
Figure 2 shows the temperature dependence of the dielectric constant \(\epsilon _{\mathrm{r}}\)(T) for HCat and DCat. In HCat, \(\epsilon _{\mathrm{r}}\)(T) steeply increases with decreasing temperature and then saturates below ~2 K. In sharp contrast, \(\epsilon _{\mathrm{r}}\)(T) of DCat is temperatureindependent owing to deuterium localization (Fig. 2a). The temperature dependence of \(\epsilon _{\mathrm{r}}\) for HCat is a typical dielectric behavior observed in quantum paraelectric (QPE) materials such as SrTiO_{3} (ref. ^{27}), in which longrange electric order is suppressed by strong quantum fluctuations. In the QPE state, \(\epsilon _{\mathrm{r}}\)(T) is described by the socalled Barrett formula^{28}:
Here, A is a constant offset, C = nμ ^{2}/k _{B} is the Curie constant (where n is the density of dipoles, μ is the local dipole moment, and k _{B} is the Boltzmann constant), T _{0} is the Curie–Weiss (CW) temperature in the classical limit (that is, a temperature at which (anti)ferroelectric order occurs in the absence of strong quantum fluctuations) and T _{1} is the characteristic crossover temperature from the classical CW regime to the QPE regime. As shown in the solid line in Fig. 2b, \(\epsilon _{\mathrm{r}}\)(T) of HCat is well fitted by the Barrett formula with T _{0} = −6.4 K and T _{1} = 7.7 K; this confirms that strong quantum fluctuations suppress longrange electric order. The relative strength of quantum fluctuations among different QPE materials can be evaluated by the ratio of T _{1} to T _{0}. The value of T _{1}/T _{0} in HCat is 1.2, which is smaller than that of the typical QPE material SrTiO_{3} (T _{1}/T _{0} = 2.3, see ref. ^{27}). This is consistent with the experimental fact that the QPE behavior of SrTiO_{3} is more significant than that of HCat. The obtained negative value of T _{0} immediately indicates the presence of an antiferroelectric (AFE) interaction in HCat, which is consistent with the AFE configuration resulting from deuterium localization in DCat (see the inset of Fig. 2b). Therefore, the observed quantum paraelectricity in HCat clearly shows that strong quantum fluctuations that suppress the hydrogenbond order as observed in DCat arise from the potential energy curve of HCat, consequently leading to the persistence of enhanced proton fluctuations down to low temperatures. The presence of strong quantum fluctuations is consistent with the recent theoretical calculations that highlight the importance of strong manybody effects imposed by the proton–πelectron network on the potential energy curve of HCat^{22,23}.
Thermal conductivity measurements
Using thermal conductivity measurements, we next examine how the proton dynamics in the QPE state affects the nature of the QSL state in HCat. Figure 3 shows the temperature dependence of the thermal conductivity of HCat (κ ^{H}) and DCat (κ ^{D}). The heat in HCat is carried by the phonons \(( {\kappa _{{\mathrm{ph}}}^{\mathrm{H}}} )\) and the spin excitations \(( {\kappa _{{\mathrm{sp}}}^{\mathrm{H}}} )\), whereas in nonmagnetic DCat, it is transported only by phonons \(( {\kappa _{{\mathrm{ph}}}^{\mathrm{D}}} )\). Assuming that HCat and DCat share almost identical phonon thermal conductivity \(( {\kappa _{{\mathrm{ph}}}^{\mathrm{H}}\sim \kappa _{{\mathrm{ph}}}^{\mathrm{D}}} )\), the relation \(\kappa ^{\mathrm{H}} = \kappa _{{\mathrm{ph}}}^{\mathrm{H}} + \kappa _{{\mathrm{sp}}}^{\mathrm{H}} \ge \kappa _{{\mathrm{ph}}}^{\mathrm{D}} = \kappa ^{\mathrm{D}}\) holds. Unexpectedly, however, we find that κ ^{H} < κ ^{D} above 2 K (see Fig. 3a), indicating that \(\kappa _{{\mathrm{ph}}}^{\mathrm{H}}\) is much more suppressed than \(\kappa _{{\mathrm{ph}}}^{\mathrm{D}}\).
To investigate the origin of this suppression, we employ the Callaway model^{29}, which describes the heat transport of acoustic phonons. Above 2 K, κ ^{H} is reproduced by the model including a single resonance scattering mode with a resonance energy of ħω _{0}/k _{B} ~ 5–10 K in addition to standard scattering processes (Supplementary Fig. 2; Supplementary Table 1; Supplementary Note 2). This energy scale is close to the proton fluctuations (T _{1} = 7.7 K), indicating that resonance scattering arises between the acoustic phonons and the optical mode from the hydrogen bonds. Here, it should be noted that we can rule out the possibility of a spin–phonon scattering for the suppression of \(\kappa _{{\mathrm{ph}}}^{\mathrm{H}}\), because the spin–orbit coupling of HCat is very weak, as confirmed by the small field dependence of κ ^{H} (Fig. 4a). Thus, it appears that the thermal fluctuations of the hydrogen bonds strongly suppress \(\kappa _{{\mathrm{ph}}}^{\mathrm{H}}\) above 2 K.
Below 2 K, κ ^{H} rapidly increases and eventually exceeds κ ^{D} (Fig. 3a). This rapid increase of κ ^{H} may come from increases in \(\kappa _{{\mathrm{sp}}}^{\mathrm{H}}\) as well as \(\kappa _{{\mathrm{ph}}}^{\mathrm{H}}\). We now investigate the behavior of κ ^{H} at lower temperatures where \(\kappa _{{\mathrm{sp}}}^{\mathrm{H}}\) becomes dominant over \(\kappa _{{\mathrm{ph}}}^{\mathrm{H}}\); this provides essential information on the lowlying excitation spectrum characterizing the QSL state^{25,26}. As shown in the dotted lines in Fig. 3b, both \(\kappa _{{\mathrm{ph}}}^{\mathrm{H}}\) and \(\kappa _{{\mathrm{ph}}}^{\mathrm{D}}\) exhibit a T ^{2}dependence rather than the conventional T ^{3}dependence. This originates from the influence of highquality crystals with specular surfaces (Supplementary Fig. 3; Supplementary Note 3). The zerotemperature extrapolation of κ ^{H}/T shows a finite residual (Fig. 3b; Supplementary Fig. 4), thereby demonstrating a gapless spin excitation with high mobility (the mean free path of the gapless spin excitations l _{sp} is estimated to be ~120 nm; See Supplementary Note 4). This result is consistent with recent magnetic torque measurements^{5} of HCat. Here, we stress that we can exclude the possibility that the itinerant lowenergy excitations are due to either phonons or electric dipoles (see Supplementary Note 5).
Discussion
A key question raised here is how the gapless QSL state is stabilized in HCat. In organic QSL candidates with a triangular lattice charge fluctuations near a Mott transition^{12,13,14,15,16,17,18} have been pointed out to play an important role for stabilizing the QSLs. However, HCat is located deeper inside the Mottinsulating phase compared to the other organic QSL candidates, κ(BEDTTTF)_{2}Cu_{2}(CN)_{3} (ref. ^{3}) and EtMe_{3}Sb[Pd(dmit)_{2}]_{2} (refs. ^{4,30}). The distance from a Mott transition is inferred from the ratio of the onsite Coulomb repulsion U to the transfer integral t, which is given by \(U{\mathrm{/}}t\sim t{\mathrm{/}}J\). Whereas the transfer integrals are comparable among the three materials, J for HCat is ~1/3 compared to that for the other two (see Supplementary Note 7); this means that U/t is significantly large in HCat. Indeed, HCat sustains an insulating behavior even at 1.6 GPa (ref. ^{20}), whereas the other two compounds become metallic at 0.4–0.6 GPa (ref. ^{31}); this result also supports that HCat is far from the Mott transition. Therefore, the QSL in HCat should be stabilized using a different mechanism.
Figure 4b shows the temperature dependence of the dielectric constant \(\epsilon _{\mathrm{r}}\), the thermal conductivity divided by temperature κ/T and the magnetic susceptibility χ (ref. ^{5}) for HCat. Below 2 K, the thermal conductivity increases upon entering the QPE state, where \(\epsilon _{\mathrm{r}}\) saturates. The characteristic temperature coincides with the temperature at which the susceptibility becomes constant; this occurs when the spin correlation develops in the QSL state^{5}. The coincidence of the QPE and QSL states is surprising and strongly suggests that the development of the quantum proton fluctuations triggers the emergence of the QSL. We now theoretically analyze the effects of proton dynamics on the QSL state. In HCat, the charge degrees of freedom of the hydrogen bonds and the πelectrons are strongly coupled because of the charge neutrality within the H_{3}(CatEDTTTF)_{2} supramolecule (see Fig. 1c, d). A minimal and realistic model that describes the coupling between πelectrons and hydrogen bonds is the extended Hubbard model coupled with the proton degree of freedom; this model captures the essence of this system, namely that the proton and charge degrees of freedom are strongly entangled in HCat (Supplementary Fig. 5; Supplementary Note 8). According to the present model, charge fluctuations inside a dimer and/or between two neighboring dimers are affected by proton fluctuations for the following reasons: the magnetic exchange processes are governed by the virtual electron hopping (mainly the second term in Supplementary Eq. (7)), which is of the order of 100 meV in the present system^{20,21,22}. In contrast, the time scale of the proton fluctuations T _{1} = 7.7 K (~1 meV) is two orders of magnitude slower than that of the electron hopping. Such lowenergy proton fluctuations modulate the amplitude of the electron transfers and the energy levels of the molecular orbitals. These effects may induce a dynamical modulation of J as well as a reduction of the onsite Coulomb repulsion U due to the bipolaron effect^{32}, both of which appear to destabilize the magnetic longrange order, that is, induce a QSL state.
Finally, we discuss the magnetic field dependence of the thermal conductivity in the QSL state of HCat. As shown in Fig. 4c, the field dependence of the thermal conductivity at low temperatures in HCat is negligibly small (a slight increase of 1–2% against the magnetic field of 10 T). In contrast, in κ(BEDTTTF)_{2}Cu_{2}(CN)_{3} (ref. ^{25}) and EtMe_{3}Sb[Pd(dmit)_{2}]_{2} (ref.^{26}), the magnetic field dependence of the thermal conductivity shows a gaplike behavior, which has been discussed in terms of an inhomogeneous QSL^{3,25,33,34,35} and a dichotomy of gapless and gapped excitations^{26,30}, respectively. Therefore, the observed negligibly small field dependence in HCat indicates the absence of gapped excitations with magnetic field, which may suggest a more globally homogeneous QSL with gapless excitations. One of the possible explanations is a gapless spinon Fermi surface over the whole kspace^{36} (for details, see Supplementary Note 9). Recent torque measurements^{5} of HCat have shown that spin excitations behave as Pauliparamagneticlike lowenergy excitations where the Fermi temperature T _{F} is estimated to be 350 K. In such a case, in the regime where \(T_{\mathrm{F}} \gg g\mu _{\mathrm{B}}H\) (here, μ _{B} is the Bohr magneton and g is the gfactor of CatEDTTTF dimer with spin1/2), the total number of spin excitations in the applied magnetic field becomes constant in a 2D system, and the velocity of the spin excitations v _{sp} is assumed to be almost fieldindependent. As a result, κ _{sp} becomes essentially fieldindependent.
The homogeneous gapless QSL insensitive to magnetic fields may originate from the structure of the present system; the 2D πelectron layers of HCat are connected by hydrogen bonds, whereas in the other organic QSL candidates they are separated by anion layers that may induce randomness in the πelectron system^{37}. This structural difference enables a different mechanism to stabilize the QSL state in HCat compared to previous organic QSL candidates. The QSL realized in HCat can be induced by quantum proton fluctuations rather than charge fluctuations near a Mott transition. Thus, our findings suggest that a quantumdisordered state of magnetic and electric dipoles emerges in HCat from cooperation between the electron and proton degrees of freedom. Utilizing such a strong coupling between multiple degrees of freedom will advance our explorations of quantum phenomena such as orbital–spin liquids^{19,38} and electric–dipole liquids^{17,39}.
Methods
Sample preparation
Single crystals of κH_{3}(CatEDTTTF)_{2} and κD_{3}(CatEDTTTF)_{2} were prepared by the electrochemical oxidation method, as described in refs^{20,21}. A typical sample size for both compounds is ~0.03 × 0.12 × 1.0 mm^{3}. In HCat, the anisotropy parameter t′/t at 50 K is estimated to be 1.48 by the extended Hückel method^{5} or 1.25 by the first principles DFT calculations^{22} (see Fig. 1e). In contrast, the value of t′/t for DCat is estimated to be 1.36 at 270 K by the extended Hückel method^{21} (see Fig. 1f), which is close to t′/t = 1.37 at 298 K for HCat obtained by the same method^{40}.
Dielectric measurements
The dielectric permittivity measurements were carried out down to 0.4 K in a ^{3}He cryostat using an LCR meter (Agilent 4980A) operated at 100 Hz–1 MHz along the a* direction, which is perpendicular to the bc plane. The experiment was limited to the a*axis direction by the platelike shape of the sample. The applied a.c. voltage was 2 V. We confirmed the voltageindependent response of \(\epsilon _{\mathrm{r}}\) up to 2 V below 20 K. The dielectric permittivity was measured by sweeping both temperature and frequency. The electrical contacts were made using carbon paste. The open/short correction was performed before connecting the sample to the measurement system.
Thermal conductivity measurements
The thermaltransport measurements were performed by a standard steadystate heatflow technique in the temperature range from 0.1 to 10 K using a dilution refrigerator. The heat current was applied along the c axis. The magnetic field was applied perpendicularly to the bc plane up to 10 T. Two RuO_{2} thermometers precisely calibrated in the magnetic field and one heater were attached on the sample through gold wires.
Data availability
The data that support the findings of this study are available on request from the corresponding authors (M.S. or K.H.).
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Acknowledgements
We thank J. Müller, M. Oshikawa, H. Seo, T. Shibauchi, M. Tachikawa, Y. Tada, T. Tsumuraya, H. Watanabe, and K. Yamamoto for fruitful discussions. We also thank K. Torizuka and Y. Uwatoko for providing technical assistance. This work was supported by GrantsinAid for Scientific Research (Grants Nos. 24340074, 26287070, 26610096, 15H00984, 15H00988, 15H02100, 15K13511, 15K17691, 16H00954, 16H04010, 16K05744, 16K17731, 17H05138, 17H05143, and 17K18746) from MEXT and JSPS, by a GrantinAid for Scientific Research on Innovative Areas “πFiguration” (No. 26102001), by the Canon Foundation and by Toray Science Foundation.
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M.S., K.H., and M.Y. conceived the project. M.S., Y.S., K.S., S.Y., Y.I. and M.Y. performed the thermal conductivity measurements. K.H., R.K., K.I., S.Iguchi, and T.S. performed the dielectric permittivity measurements. A.U. and H.M. carried out sample preparation. M.N. and S.Ishihara gave the theoretical model. M.S., K.H., and M.Y. analyzed the data and wrote the manuscript with inputs from A.U., M.N. and S.Ishihara. All authors discussed the experimental results.
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Shimozawa, M., Hashimoto, K., Ueda, A. et al. Quantumdisordered state of magnetic and electric dipoles in an organic Mott system. Nat Commun 8, 1821 (2017). https://doi.org/10.1038/s4146701701849x
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