Quantum-disordered state of magnetic and electric dipoles in an organic Mott system

Strongly enhanced quantum fluctuations often lead to a rich variety of quantum-disordered 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 zero-point motion of hydrogen atoms, that is, proton fluctuations in a hydrogen-bonded organic Mott insulator provides a different class of quantum spin liquids (QSLs). We find that divergent dielectric behavior associated with the approach to 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 QSL—a quantum-disordered state of magnetic and electric dipoles—through the coupling between the electron and proton degrees of freedom.


Supplementary Note 1: Sample, d.c. bias, and frequency dependence of dielectric permitivity
To check the sample dependence of the observed quantum paraelectric (QPE) behaviour in H-Cat, we measured the dielectric permittivity for several samples (the data of crystal #1 is shown in the main text). As shown in Supplementary Fig. 1a, there is no significant sample dependence in ϵ r ; the observed small difference is due to the inaccuracy in sample dimensions and is within the experimental error. We also examined the d.c. bias dependence of the dielectric permittivity in H-Cat. In addition to the a.c. electric field, a d.c. electric field of 1 kV cm −1 for both crystal #2 and crystal #3 was applied. As shown in Supplementary   Fig. 1a, the application of d.c. bias electric field has no detectable influence on the QPE behaviour in H-Cat. Moreover, we measured the frequency dependence of the dielectric permittivity in H-Cat (100 Hz to 1 MHz). As shown in Supplementary Fig. 1b, the dielectric permittivity is frequency-independent below ∼ 10 K, which supports that the observed QPE behaviour is intrinsic. In contrast, a strong frequency dependence is observed above ∼ 10 K, which originates from a large loss tangent due to the relatively high electrical conductivity in H-Cat. Such an extrinsic effect on the dielectric behaviour is often observed in organic conductors with large loss tangents. Because the loss tangent tan δ is given by 1/(ωCR p ), where ω is the angular frequency of the a.c. field, C is the lossless capacitance, and R p is the parasitic resistance, the higher the measuring frequency is, the smaller the dielectric loss becomes. Therefore, we used the dielectric permittivity data measured at 1 MHz to obtain the physical parameters such as T 0 and T 1 in the Barrett formula.

Supplementary Note 2: Resonance scattering arising from hydrogen-bond dynamics
To investigate how the phonon thermal conductivity is suppressed in H-Cat, we applied the Callaway model that describes a phonon thermal conductivity κ ph [1]: where v ph is the average sound velocity of the phonons, Θ D is the Debye temperature, τ is the total phonon relaxation rate and ω is the frequency of phonons; moreover, ℏ and k B are the reduced Planck and Bolzmann constants, respectively, and x = ℏω k B T . By fitting the Debye model to the specific-heat data [2], the values of v ph and Θ D were estimated as 1.7 km s −1 and 220 K, respectively. τ (ω, T ) was approximated as [3,4]: where the first three terms represent the phonon scatterings at the sample boundaries, dislocations and point defects, respectively, and the last two terms stand for the scatterings caused by phonon-phonon (Umklapp) processes and a resonance mode, respectively. Here  other quantum spin systems [3]. This result indicates that the resonance mode strongly suppresses the phonon thermal conductivity of H-Cat. As H-Cat possesses both proton and spin degrees of freedom, this extra resonance scattering comes from either proton-phonon scattering or spin-phonon scattering. The resonance scattering energy almost coincides with the proton fluctuations (T 1 = 7.7 K), which suggests that the resonance scattering arises from the optical mode associated with the hydrogen bonds. Here, we can safely exclude spin-phonon scattering as the origin of the resonance scattering because of the following reasons. The spin-phonon scattering should be strongly quenched by Zeeman splitting under a magnetic field, which changes the field dependence of the thermal conductivity κ(H). However, κ(H) does not strongly depend on the magnetic field up to 10 T (see Fig. 4a). Moreover, the g-value only slightly deviates from 2 and thereby supports that the spin-orbit coupling causing the spin-phonon scattering is very weak [5]. Thus, we conclude that the strong suppression of κ ph for H-Cat results from resonance scattering, which itself arises from the hydrogen-bond dynamics.

Supplementary Note 3: Specular scattering at sample surfaces
Here, we explain the observed T 2 -dependence of κ ph in H-Cat and D-Cat below ∼ 0.25 K. According to the kinetic equation, κ ph depends on the heat capacity of the phonons C ph , the sound velocity v ph and the mean free path of phonon l ph : In general, l ph and v ph become essentially constant with temperature and C ph varies as C ph ∝ T 3 at temperatures significantly lower than Θ D , leading to the relation κ ph ∝ T 3 . However, we found that κ ph ∝ T 2 holds true at sufficiently low temperatures in both H-Cat and D- behaviours are often observed in glassy materials [6] or in high-quality crystals with specular surfaces [7][8][9]; in these cases, l ph continues to increase with decreasing temperature. In glassy materials, impurity scattering reduces l ph to below the width or thickness of the sample, whereas in high-quality crystals, l ph can exceed the sample size. Here, we estimated l ph of D-Cat, for which κ = κ ph . As shown in Supplementary Fig. 3, l ph of D-Cat continues to increase with decreasing temperature and extends well beyond the sample boundaries assuming that the value of v ph (∼ 1.7 km s −1 ) remains independent of temperature. This result supports the occurrence of specular surface scattering in the present system, which is the cause of the T 2 -dependence of κ ph .

Supplementary Note 4: Thermal properties of itinerant spin excitations
We investigated the low-lying spin excitation spectrum that characterizes the quantum spin liquid (QSL) states. The spin contribution κ sp in H-Cat is extracted by subtracting the phonon contribution κ ph from the total thermal conductivity κ = κ sp + κ ph . As shown by resolved in the zero-temperature limit (Fig. 3b). We note that the residual term is also finite The residual κ/T shows that the spin excitation from the ground state is gapless.
We next estimated the mean free path of the spin excitations l sp [10]. Analogous to the case of phonons, we assumed the following relation: where C sp is the specific heat and v sp is the velocity of the spin excitations. Here, we adopted the standard formula of 2D fermions: where N F is the density of states at the Fermi surface and d = 1.47Å is the interlayer spacing. We also assumed that the linear term η 0 arises from fermionic excitations. In this case, the Fermi wave number is written as k F = 1/a, where a = 8.36Å is the nearestneighbour spin distance. From the above equations, we obtained η 0 = π 9 k 2 B ℏ lsp ad . Because the estimated residual η 0 is at least 0.06 W K −2 m −1 (see the inset of Supplementary Fig. 4), we found that l sp exceeds 120 nm at zero temperature; that is, the spin excitations are mobile to a distance 100 times the inter-spin distance without being scattered. Such highly mobile spin excitations apparently result from the extremely long spin correlation length.

Supplementary Note 5: Origin of itinerant low-energy excitations in thermal conductivity of H-Cat
The observed negligibly small field dependence of the thermal conductivity in H-Cat may raise a question as to the magnetic origin of the itinerant low-lying excitations observed in the thermal conductivity of H-Cat. However, we can exclude the possibility that the itinerant low-energy excitations, which are described by the T -linear term of the thermal conductivity as discussed in the main text, is due to either phonons or electric dipoles for the following reasons: (1) As shown in the dotted lines in Fig. 3b, the temperature dependence of κ/T of both H-Cat and D-Cat exhibits a T -linear dependence at low temperatures. Considering that D-Cat has only phonon contribution to the thermal conductivity, this result indicates that the phonon thermal conductivity in this system is described by a T 2 -dependence (owing to the influence of high-quality crystals with specular surfaces, for details, see Supplementary Note 3) rather than the conventional T 3 -dependence. Because κ ph /T ∝ T goes to zero in the zero-temperature limit, the phonon thermal conductivity cannot contribute to the itinerant low-energy excitation observed in H-Cat. (2) Because electric dipoles behave as classical Ising-like spins, a dielectric system with 1D uniaxial (Ising-type) antiferroelectric dipoles, as in the case of H-Cat, is expected to have low-energy excitations with a gap, which is inconsistent with the gapless excitations in H-Cat.

Supplementary Note 6: Origin of peak structure observed in thermal conductivity of H-CAT
The peak structure observed in κ H /T around 0.3 K originates from both κ H sp and κ H ph . We first focus on κ sp described by Supplementary Eq. (4). Recent torque measurements [5] suggest that below 2 K, the spin correlation develops in the QSL state; this implies that l sp increases with decreasing temperature below 2 K; however, l sp is eventually saturated and reaches ∼120 nm at low temperatures. In addition, C sp /T has been reported to be almost constant with temperature below 2 K [2], showing C sp ∝ T , and v sp is assumed to be essentially temperature-independent at low temperatures. Therefore, κ H sp /T below 2 K increases with decreasing temperature and should be saturated at low temperatures without showing any peak structure, which requires κ H ph in addition to κ H sp to explain the observed peak structure in κ H /T . As described in Supplementary Note 3, κ H ph /T decreases monotonically with decreasing temperature at least at low temperatures because of κ H ph /T ∝ T α−1 with α = 2-3. Consequently, we conclude that a combination of κ H sp and κ H ph results in the peak structure observed in κ H /T .

Supplementary Note 7: Estimation of distance from Mott transition
The distance from a Mott transition can be estimated from the ratio of the on-site Coulomb repulsion U to the transfer integral t, which is given by U/t ∼ t/J, where J is the spin interaction. The value of t is calculated to be 30-40 meV for H-Cat [11], ∼ 50 meV for κ-(BEDT-TTF) 2 Cu 2 (CN) 3 [12,13] and 25-40 meV for EtMe 3 Sb[Pd(dmit) 2 ] 2 [14]. Note that t is comparable among the three materials. In contrast, J/k B of H-Cat is estimated as ∼ 80 K [5] and as ∼ 1/3 in the other two compounds (J/k B ∼ 250 K for both κ-(BEDT-TTF) 2 Cu 2 (CN) 3 [15] and EtMe 3 Sb[Pd(dmit) 2 ] 2 [16]). Consequently, H-Cat has a relatively large U/t; this indicates a large distance from the Mott transition.

Supplementary Note 8: Theoretical model of electron-proton coupled system
We introduce a minimal model that describes a coupling between the π-electron and proton degrees of freedom in H-Cat and D-Cat. The model Hamiltonian is given by where the first term H e describes the π-electron system and the second term H p represents the proton system coupled with the π-electrons. The first term is given by the extended

Hubbard model (EHM)
: where c iµσ is the annihilation operator of a hole with spin σ = (↑, ↓) at the µ (= α, β) molecule of the i-th dimer, n iµ ≡ ∑ σ n iµσ ≡ ∑ σ c † iµσ c iµσ is the number operator and t d (V d ) is the intra-dimer transfer integral (Coulomb interaction) between the α and β molecules inside a dimer. Moreover, t µµ ′ ij (V µµ ′ ij ) is the inter-dimer transfer integral (Coulomb interaction) between the µ molecule of the i-th dimer and the µ ′ molecule of the j-th dimer and U 0 is the intra-molecular Coulomb interaction (see Supplementary Fig. 5a). The second term H p in Supplementary Eq. (6) is described by where t p is the proton tunneling amplitude (proton-fluctuation strength) [17], g (> 0) is the coupling constant between the hole and the hydrogen atom and P i is a pseudo-spin operator with an amplitude of 1/2 describing the proton degree of freedom at the i-th hydrogen bond.
Here, we define the eigenstates of P z as |+⟩ and |−⟩ , respectively. In these states, the hydrogen atom is localized at the right and left sides of the double potential minima, respectively (see Supplementary Fig. 5b).  Fig. 5b). This pseudo-spin representation has been applied to hydrogen-bonded systems such as the ferroelectric KH 2 PO 4 (KDP) [18,19], where displacive deformation of PO 4 tetrahedrons is coupled with O-H-O hydrogen dynamics.
We now consider a 1D chain of π-dimers and hydrogen bonds in the strong electron correlation regime, where the on-site Coulomb interaction is larger than the inter-dimer Coulomb interaction. In this situation, for relatively small t p , hydrogen-bond order occurs concomitantly with charge order inside of the π-dimer, leading to two types of ordering patterns. In one case, the hydrogen atoms are located at the same oxygen side for two O-H-O hydrogen bonds and each hole is distributed to the same molecular side of each dimer (see the left panel of Supplementary Fig. 5c). In the other case, each hydrogen atom is located at the opposite oxygen side, leading to a state where one dimer is occupied by two holes and the other is empty (see the right panel of Supplementary Fig. 5c); this state corresponds to the non-magnetic D-Cat.
In contrast, for relatively large t p , any hydrogen order does not occur owing to the strong proton tunneling effect, which corresponds to the state for H-Cat. To investigate this state more closely, we consider the simplest case, in which an isolated dimer occupied by one hole is coupled to a proton. In this case, the ground-state wave function is given by an entangled state (see Supplementary Fig. 5d) Φ 1,2 = |a⟩+|b⟩ = (|α⟩−ζ |β⟩) |+⟩+(|β⟩−ζ |α⟩) |−⟩. Here, |α⟩ and |β⟩ represent the states where the π electron is localized at the α and β molecules inside the dimer, respectively. In addition, |ζ| (< 1) expresses the degree of hybridization of |α⟩ and |β⟩ induced by t d . We stress that when t p ̸ = 0, Φ cannot be described by a Thus, proton tunneling t p is an important factor for determining the ground states in the present system.
In the present theoretical model, we have focused not on the direct proton-proton interaction arising from electrostatic and elastic couplings between two neighbouring protons, but on the proton-proton interaction through the electron-proton coupling. One of the main reasons is that the experimental results suggest that the hydrogen bonds in the present system strongly couple with the charge and spin degrees of freedom of π-electrons. In addition, there is no direct chemical bond connecting the two neighbouring protons. This structural feature causes a screening effect of the electrostatic interaction by the conducting π-electrons, as well as a weak elastic interaction between two neighbouring protons. By taking into account these aspects, we adopted the proton-electron interaction as a main interaction responsible for the observed strong correlation between the electron and proton degrees of freedom. To clarify the origin of the QSL state and the effects of the underlying proton-electron entanglement, further theoretical studies are required.

Supplementary Note 9: Possible explanation for field-independent itinerant spin excitations
To explain negligibly small field dependence of κ H sp , we consider a gapless spinon Fermi surface over the whole k-space. According to the recent torque measurements [5], spin excitations act as Pauli-paramagnetic-like low-energy excitations up to 17 T, which is a hallmark of itinerant fermions (for example, fermionic spinons) at low temperatures. The Pauli susceptibility is given by χ = (1/4)g 2 µ 2 B N F , where N F is the density of states at the Fermi energy ε F , µ B is the Bohr magneton and g is the g-factor of Cat-EDT-TTF dimer with spin-1/2. Here, we adopt the standard formula in a 2D system N F = n/ε F [20], where n is the number of fermions (spinons) per volume. Using g ∼ 2 and χ ∼ 1.2 × 10 −3 emu mol −1 obtained from the torque measurements for H-Cat [5], the Fermi temperature T F = ε F /k B is roughly estimated to be ∼ 350 K. In such a Pauli-paramagnetic state with a large T F compared to the energy scale of magnetic field, v sp becomes almost field-independent. Meanwhile, in the presence of magnetic field, the spin-up (spin-down) band shifts upward (downward) and the number of spin excitations with up-spin (downspin) increases (decreases). However, the total number of spin excitations in the applied magnetic field becomes constant in a 2D system owing to the cancellation of the spin-up and spin-down parts, resulting in that C sp is field-independent because of Supplementary Eq. (5), as confirmed by recent specific heat measurements [2]. As a result, if we simply assume that l sp is field-independent, κ sp ∝ C sp v sp l sp is essential constant with magnetic field.