Quantum-disordered state of magnetic and electric dipoles in a hydrogen-bonded Mott system

TL;DR

This study reveals a new quantum spin liquid state in a hydrogen-bonded Mott insulator, where proton fluctuations induce a quantum-disordered state of magnetic and electric dipoles, linking electron and proton degrees of freedom.

Contribution

It demonstrates that coupling between localized spins and proton zero-point motion creates a novel quantum spin liquid in a hydrogen-bonded Mott system.

Findings

Quantum proton fluctuations suppress hydrogen-bond order.

Emergence of a gapless quantum spin liquid state.

Observation of quantum paraelectric behavior.

Abstract

Strongly enhanced quantum fluctuations often lead to a rich variety of quantum-disordered states. A representative case is liquid helium, in which zero-point vibrations of the helium atoms prevent its solidification at low temperatures. A similar behaviour is found for the internal degrees of freedom in electrons. Among the most prominent is a quantum spin liquid (QSL), in which localized spins are highly correlated but fluctuate even at absolute zero. Recently, a coupling of spins with other degrees of freedom has been proposed as an innovative approach to generate even more fascinating QSLs such as orbital--spin liquids. However, such ideas are limited to the internal degrees of freedom in electrons. Here, we demonstrate that a coupling of localized spins with the zero-point motion of hydrogen atoms (proton fluctuations) in a hydrogen-bonded organic Mott insulator provides a new class of QSLs. We find that a divergent dielectric behaviour towards a hydrogen-bond order is suppressed by the quantum proton fluctuations, resulting in a quantum paraelectric (QPE) state. Furthermore, our thermal-transport 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 novel QSL --- a quantum-disordered state of magnetic and electric dipoles --- through the coupling between the electron and proton degrees of freedom.

Equations

$${ϵ}_r(T)=A+\frac{C}{\frac{T_1}{2}\coth (\frac{T_1}{2T})-T_0},$$

Full text

Quantum-disordered state of magnetic and electric dipoles

in a hydrogen-bonded Mott system

M. Shimozawa1,∗,†

  

K. Hashimoto2,∗,†

  

A. Ueda1

  

Y. Suzuki1

  

K. Sugii1

  

S. Yamada1

  

Y. Imai1

  

R. Kobayashi2

  

K. Itoh2

  

S. Iguchi2

  

M. Naka3

  

S. Ishihara3

  

H. Mori1

  

T. Sasaki2

  

M. Yamashita1

1The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan

2Institute for Materials Research, Tohoku University, Aoba-ku, Sendai 980-8577, Japan

3Department of Physics, Tohoku University, Sendai 980-8578, Japan

∗These authors contributed equally to this work.

*†*To whom correspondence should be addressed.

E-mail: shimo@issp.u-tokyo.ac.jp (M.S.); hashimoto@imr.tohoku.ac.jp (K.H.)

**Strongly enhanced quantum fluctuations often lead to a rich variety of quantum-disordered states. A representative case is liquid helium, in which zero-point vibrations of the helium atoms prevent its solidification at low temperatures. A similar behaviour is found for the internal degrees of freedom in electrons. Among the most prominent is a quantum spin liquid (QSL)Balents10 , in which localized spins are highly correlated but fluctuate even at absolute zero. Recently, a coupling of spins with other degrees of freedom has been proposed as an innovative approach to generate even more fascinating QSLs such as orbital–spin liquidsFeiner97 . However, such ideas are limited to the internal degrees of freedom in electrons. Here, we demonstrate that a coupling of localized spins with the zero-point motion of hydrogen atoms (proton fluctuations) in a hydrogen-bonded organic Mott insulatorIsono13 ; Isono14 ; Ueda14 ; Tsumuraya15 provides a new class of QSLs. We find that a divergent dielectric behaviour towards a hydrogen-bond order is suppressed by the quantum proton fluctuations, resulting in a quantum paraelectric (QPE) stateMuller79 ; Shen15 ; Barrett52 . Furthermore, our thermal-transport 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 novel QSL — a quantum-disordered state of magnetic and electric dipoles — through the coupling between the electron and proton degrees of freedom. **

The nature of QSLs has been well established in one-dimensional (1D) spin systems. However, it still remains elusive how QSLs emerge in dimensions greater than one. The celebrated resonating-valence-bond theory on a 2D triangular lattice Anderson73 puts forward the possibility that geometrical frustration plays an important role for stabilizing QSLs. In fact, a few candidate materials hosting QSLs have now been reported in materials with 2D triangular lattices Isono14 ; Shimizu03 ; Itou08 ; Shimizu16 ; Shen16 ; Paddison16 . Nevertheless, according to subsequent theoretical studies Bernu92 ; Capriotti99 , the effect of geometrical frustration in the triangular lattice is insufficient to stabilize QSLs, leading to a number of mechanisms that may stabilize the QSL states found in the candidate materials. One of the most promising approaches is to utilize a coupling of spins with charges and orbitals; the former has been discussed near a Mott-insulator-to-metal transition where the charge degrees of freedom begin to delocalize Morita02 ; Yoshioka09 ; Misguich99 ; Motrunich05 ; Mross11 ; Hotta10 ; Naka16 and the latter has been considered in the framework of a spin–orbital coupling Feiner97 . Such strategies, however, have been limited to the utilization of internal degrees of freedom of electrons. Here, we demonstrate a novel approach for stabilizing a QSL realized in a hydrogen-bonded organic Mott insulator, in which a coupling between $\pi$-electrons and hydrogen atoms provides a novel quantum-disordered state of magnetic and electric dipoles.

$\kappa$-H3(Cat-EDT-TTF)2 (hereafter abbreviated as H-Cat) is a hydrogen-bonded organic Mott insulator, where Cat-EDT-TTF is catechol-fused ethylenedithiotetrathiafulvalene (refs Isono13, ; Isono14, ; Ueda14, ; Tsumuraya15, ). H-Cat forms a 2D spin-1/2 Heisenberg triangular lattice of Cat-EDT-TTF dimers Isono13 (Figs 1 a and d). Despite the antiferromagnetic interaction energy $J/k_{\rm B}$ of $\sim 80$ K, no magnetic order has been observed down to 50 mK; this indicates the realization of a QSL state Isono14 . A distinct feature of H-Cat is that the 2D $\pi$-electron layers are connected by hydrogen bonds Isono13 ; Ueda14 (Figs 1 a and b), which is in marked contrast to other 2D organic QSL materials such as $\kappa$-(BEDT-TTF)2Cu2(CN)3 (ref. Shimizu03, ) and EtMe3Sb[Pd(dmit)2]2 (ref. Itou08, ) where the 2D spin systems are separated by non-magnetic insulating layers. This structural feature of H-Cat is highlighted by deuteration of the hydrogen bonds Ueda14 ; specifically, in the deuterated analogue of H-Cat, $\kappa$-D3(Cat-EDT-TTF)2 (denoted as D-Cat), deuterium localization occurs at $T_{c}=185$ K, accompanied by charge disproportionation within the Cat-EDT-TTF layers, resulting in a non-magnetic ground state (Figs 1c and e). This demonstrates that the hydrogen bonds in this system strongly couple with the charge and spin degrees of freedom of the $\pi$-electrons in the Cat-EDT-TTF dimers.

In contrast to D-Cat, the hydrogen atoms in H-Cat do not localize down to low temperatures Ueda14 . This is inconsistent with the fact Yamamoto16 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 $\sim 800$ K (see Figs 1b and f), which should localize the hydrogen atoms in H-Cat at low temperatures. Recent theoretical calculations Tsumuraya15 ; Yamamoto16 have pointed out that the potential energy curve has a single-well structure and its bottoms become very shallow and anharmonic (see Fig. 1f) owing to a many-body effect arising from the network of hydrogen bonds and $\pi$-electrons. In this case, the zero-point motions of the hydrogen atoms (termed “proton fluctuations”) can be strongly enhanced by the anharmonic potential curve. In contrast to D-Cat, the enhanced proton fluctuations may delocalize the hydrogen atoms down to absolute zero, providing a playground for realizing the QSL state through the strong coupling between the hydrogen bonds and the $\pi$-electrons. However, it has been elusive whether such strong quantum proton fluctuations are indeed present in H-Cat, and if so, how the quantum fluctuations affect the QSL state.

To answer these questions, we demonstrate a combination of dielectric permittivity and thermal conductivity measurements. These methods are particularly suitable because the dielectric permittivity is sensitive to local electric-dipole moments arising from the hydrogen-bond dynamics Horiuchi03 , whereas the thermal conductivity is a powerful probe to detect itinerant low-lying energy excitations associated with the nature of QSL states Yamashita09 ; Yamashita10 .

Figure 2 shows the temperature dependence of the dielectric constant $\epsilon_{r}(T)$ for H-Cat and D-Cat. In H-Cat, $\epsilon_{r}(T)$ steeply increases with decreasing temperature and then saturates below $\sim 2$ K. In sharp contrast, $\epsilon_{r}(T)$ of D-Cat is temperature-independent owing to deuterium localization (Fig. 2a). The temperature dependence of $\epsilon_{r}$ for H-Cat is a typical dielectric behaviour observed in quantum paraelectric (QPE) materials such as SrTiO3 (ref. Muller79, ), in which long-range electric order is suppressed by strong quantum fluctuations. In the QPE state, $\epsilon_{r}(T)$ is described by the so-called Barrett formula Barrett52 :

[TABLE]

where $A$ is a constant offset, $C$ the Curie constant, $T_{0}$ the Curie-Weiss temperature in the classical limit (proportional to the effective dipole–dipole coupling strength) and $T_{1}$ the characteristic crossover temperature from the classical to the quantum regime. As shown in the solid line in Fig. 2b, $\epsilon_{r}(T)$ of H-Cat 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 long-range electric order. The obtained negative value of $T_{0}$ immediately indicates the presence of an antiferroelectric (AFE) dipole–dipole interactions in H-Cat, which is consistent with the static AFE dipole moments arising from deuterium localization in D-Cat (see the inset of Fig. 2b). Therefore, the observed quantum paraelectricity in H-Cat clearly shows that strong quantum fluctuations that suppress the hydrogen-bond order as observed in D-Cat arise from the potential energy curve of H-Cat, 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 many-body effects imposed by the proton–$\pi$-electron network on the potential energy curve of H-Cat Tsumuraya15 ; Yamamoto16 .

Using thermal conductivity measurements, we next examine how the proton dynamics in the QPE state affects the nature of the QSL state in H-Cat. Figure 3 shows the temperature dependence of the thermal conductivity of H-Cat ($\kappa^{\rm H}$) and D-Cat ($\kappa^{\rm D}$). The heat in H-Cat is carried by the phonons ($\kappa_{\rm ph}^{\rm H}$) and the spin excitations ($\kappa_{\rm sp}^{\rm H}$), whereas in non-magnetic D-Cat, it is transported only by phonons ($\kappa_{\rm ph}^{\rm D}$). Assuming that H-Cat and D-Cat share almost identical phonon thermal conductivity ($\kappa_{\rm ph}^{\rm H}\sim\kappa_{\rm ph}^{\rm D}$), the relation $\kappa^{\rm H}=\kappa_{\rm ph}^{\rm H}+\kappa_{\rm sp}^{\rm H}\geq\kappa_{\rm ph}^{\rm D}=\kappa^{\rm D}$ holds. Unexpectedly, however, we find that $\kappa^{\rm H}<\kappa^{\rm D}$ above 2 K (see Fig. 3a), indicating that $\kappa_{\rm ph}^{\rm H}$ is much more suppressed than $\kappa_{\rm ph}^{\rm D}$.

To investigate the origin of this suppression, we employ the Callaway model Berman76 , which describes the heat transport of acoustic phonons. Above 2 K, $\kappa^{\rm H}$ is reproduced by the model including a single resonance scattering mode with a resonance energy of $\hbar\omega_{0}/k_{\rm B}\sim$ 5–10 K in addition to standard scattering processes (see Supplementary Information Sec. I). 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 $\kappa_{\rm ph}^{\rm H}$, because the spin–orbit coupling of H-Cat is very weak, as confirmed by the small field dependence of $\kappa^{\rm H}$ (see the inset of Fig. 4b). Thus, it appears that the thermal fluctuations of the hydrogen bonds strongly suppress $\kappa^{\rm H}$ above 2 K.

Below 2 K, $\kappa^{\rm H}$ rapidly increases and eventually exceeds $\kappa^{\rm D}$ (Fig. 3a). This rapid increase of $\kappa^{\rm H}$ may come from an increase in $\kappa_{\rm sp}^{\rm H}$. We now investigate the behaviour of $\kappa_{\rm sp}^{\rm H}$ at lower temperatures, which provides essential information on the low-lying excitation spectrum characterizing the QSL state Yamashita09 ; Yamashita10 . As shown in the dotted lines in Fig. 3b, both $\kappa_{\rm ph}^{\rm H}$ and $\kappa_{\rm ph}^{\rm D}$ exhibit a $T^{2}$-dependence rather than the conventional $T^{3}$-dependence. This is originated from the influence of high-quality crystals with specular surfaces (see Supplementary Information Sec. II). The zero-temperature extrapolation of $\kappa^{\rm H}/T$ shows a finite residual (Fig. 3b), thereby demonstrating a gapless spin excitation with high mobility (the mean free path of the gapless spin excitations $l_{\rm sp}$ is estimated to be $\sim 120$ nm; See Supplementary Information Sec. III). This result is consistent with recent magnetic torque measurements Isono14 of H-Cat.

A key question raised here is how the gapless QSL state is stabilized in H-Cat. In organic QSL candidates with a triangular lattice, charge fluctuations near a Mott transition Morita02 ; Yoshioka09 ; Misguich99 ; Motrunich05 ; Mross11 ; Hotta10 ; Naka16 have been pointed out to play an important role for stabilizing the QSLs. However, H-Cat is located deeper inside the Mott-insulating phase as compared to the other organic QSL candidates, $\kappa$-(BEDT-TTF)2Cu2(CN)3 (ref. Shimizu03, ) and EtMe3Sb[Pd(dmit)2]2 (refs Itou08, ; Itou10, ). The distance from a Mott transition is inferred from the ratio of the on-site Coulomb repulsion $U$ to the transfer integral $t$, which is given by $U/t\sim t/J$. Whereas the transfer integrals are comparable among the three materials, $J$ for H-Cat is $\sim 1/3$ compared to that for the other two (see Supplementary Information Sec. IV); this means that $U/t$ is significantly large in H-Cat. Indeed, H-Cat sustains an insulating behaviour even at 1.6 GPa (ref. Isono13, ), whereas the other two compounds become metallic at 0.4–0.6 GPa (ref. Furukawa15, ); this result also supports that H-Cat is far from the Mott transition. Therefore, the QSL in H-Cat should be stabilized using a completely different mechanism.

Figure 4a shows the temperature dependence of the dielectric constant $\epsilon_{r}$, the thermal conductivity divided by temperature $\kappa/T$ and the magnetic susceptibility $\chi$ (ref. Isono14, ) for H-Cat. Below 2 K, the thermal conductivity increases upon entering the QPE state, where $\epsilon_{r}$ saturates. This characteristic temperature coincides with the temperature at which the susceptibility becomes constant; this occurs when the spin correlation develops in the QSL state Isono14 . This 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 H-Cat, the charge degrees of freedom of the hydrogen bonds and the $\pi$-electrons are strongly coupled because of the charge neutrality within the H3(Cat-EDT-TTF)2 supramolecule (see Figs 1b and c). A minimal and realistic model that describes the coupling between $\pi$-electrons 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 H-Cat (see Supplementary Information Sec. V). According to this model, charge fluctuations between the two molecules inside a dimer are governed by proton fluctuations ($\sim$ 1 meV), which are comparable to the magnetic exchange interaction $J$ in the $\pi$-electron spin system. Such low-energy charge fluctuations lead to a dynamical modulation of $J$ as well as a reduction of the on-site Coulomb repulsion $U$ due to the bi-polaron effect Alexandrov96 , both of which may destabilize the magnetic long-range order, that is, induce a QSL state.

Finally, we discuss the magnetic field dependence of the thermal conductivity $\kappa(H)$ in the novel QSL state of H-Cat. As shown in Fig. 4b, the field dependence of $\kappa(H)$ is much weaker in H-Cat than in $\kappa$-(BEDT-TTF)2Cu2(CN)3 (ref. Yamashita09, ) and EtMe3Sb[Pd(dmit)2]2 (ref. Yamashita10, ). The magnetic field dependence of the thermal conductivity in $\kappa$-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2 has been discussed in terms of an inhomogeneous QSL Pratt11 and a dichotomy of gapless and gapped excitations Yamashita10 , respectively. Conversely, the observed small field dependence of the thermal conductivity in H-Cat suggests that a more globally homogeneous QSL with gapless excitations is realized in this system. This difference may highlight the unique feature of H-Cat, namely, that the QSL is induced by quantum proton fluctuations rather than charge fluctuations near a Mott transition, as discussed above. Thus, our findings suggest that a novel quantum entangled state, that is, a quantum-disordered state of magnetic and electric dipoles emerges in H-Cat 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 Feiner97 ; Nakatsuji12 and electric-dipole liquids Shen15 ; Hotta10 .

Methods

Sample preparation. Single crystals of $\kappa$-H3(Cat-EDT-TTF)2 and $\kappa$-D3(Cat-EDT-TTF)2 were prepared by the electrochemical oxidation method, as described in refs Isono13, and Ueda14, . A typical sample size for both compounds is $\sim 0.03$ mm $\times\,0.12$ mm $\times\,1.0$ mm.

Dielectric measurements. The dielectric permittivity measurements were carried out down to 0.4 K in a 3He cryostat using an LCR meter (Agilent 4980A) operated at 1 MHz along the $a^{*}$ direction, which is perpendicular to the $b$-$c$ plane. The experiment was limited to the $a^{*}$-axis direction by the plate-like shape of the sample.

Thermal conductivity measurements. The thermal-transport measurements were performed by a standard steady-state heat-flow 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 $b$-$c$ plane up to 10 T. We attached two precisely calibrated RuO2 thermometers in the magnetic field and one heater on the sample.

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 Grants-in-Aid for Scientific Research (Grants No. 24340074, No. 26287070, No. 26610096, No. 15H00984, No. 15H00988, No. 15H02100, No. 15K13511, No. 15K17691, No. 16H00954, No. 16H04010, No. 16K05744 and No. 16K17731) from MEXT and JSPS, by a Grant-in-Aid for Scientific Research on Innovative Areas “$\pi$-Figuration” (No. 26102001), by the Canon Foundation and by Toray Science Foundation.

Author contributions

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 theoretical advice. M.S., K.H. and M.Y. analyzed the data and wrote the manuscript with inputs from M.N. and S.Ishihara. All authors discussed the experimental results.