## Abstract

The nature of quantum spin liquids, a novel state of matter where strong quantum fluctuations destroy the long-range magnetic order even at zero temperature, is a long-standing issue in physics. We measured the low-temperature thermal conductivity of the recently discovered quantum spin liquid candidate, the organic insulator EtMe_{3}Sb[Pd(dmit)_{2}]_{2}. A sizable linear temperature dependence term is clearly resolved in the zero-temperature limit, indicating the presence of gapless excitations with an extremely long mean free path, analogous to excitations near the Fermi surface in pure metals. Its magnetic field dependence suggests a concomitant appearance of spin-gap–like excitations at low temperatures. These findings expose a highly unusual dichotomy that characterizes the low-energy physics of this quantum system.

Spin systems confined to low dimensions exhibit a rich variety of quantum phenomena. Particularly intriguing are quantum spin liquids (QSLs), antiferromagnets with quantum fluctuation–driven disordered ground states, which have been attracting tremendous attention for decades (*1*). The notion of QSLs is now firmly established in one-dimensional (1D) spin systems. In dimensions greater than one, it is widely believed that QSL ground states emerge when interactions among the magnetic degrees of freedom are incompatible with the underlying crystal geometry, leading to a strong enhancement of quantum fluctuations. In 2D, typical examples of systems where such geometrical frustrations are present are the triangular and kagomé lattices. Largely triggered by the proposal of the resonating-valence-bond theory on a 2D triangular lattice and its possible application to high-transition temperature cuprates (*2*), realizing QSLs in 2D systems has been a long-sought goal. However, QSL states are hard to achieve experimentally because the presence of small but finite 3D magnetic interactions usually results in some ordered (or frozen) state. Two recently discovered organic insulators, κ-[bis(ethylenedithio)-tetrathiafulvalene]_{2}Cu_{2}(CN)_{3} [κ-(BEDT-TTF)_{2}Cu_{2}(CN)_{3}] (*3*) and EtMe_{3}Sb[Pd(dmit)_{2}]_{2} (*4*, *5*), both featuring 2D spin-1/2 Heisenberg triangular lattices, are believed to be promising candidate materials that are likely to host QSLs. In both compounds, nuclear magnetic resonance (NMR) measurements have shown no long-range magnetic order down to a temperature corresponding to *J*/12,000, where *J* (~250 K for both compounds) is the nearest-neighbor spin interaction energy (exchange coupling) (*3*, *5*). In a triangular lattice antiferromagnet, the frustration brought on by the nearest-neighbor Heisenberg interaction is known to be insufficient to destroy the long-range ordered ground state (*6*). This has led to the proposals of numerous scenarios which might stabilize a QSL state: spinon Fermi surface (*7*, *8*), algebraic spin liquid (*9*), spin Bose metal (*10*), ring-exchange model (*11*), *Z*_{2} spin liquid state (*12*), chiral spin liquid (*13*), Hubbard model with a moderate onsite repulsion (*14*, *15*), and one-dimensionalization (*16*, *17*). Nevertheless, the origin of the QSL in the organic compounds remains an open question.

To understand the nature of QSLs, knowledge of the detailed structure of the low-lying elementary excitations in the zero-temperature limit, particularly the presence or absence of an excitation gap, is of primary importance (*18*). Such information bears immediate implications on the spin correlations of the ground state, as well as the correlation length scale of the QSL. For example, in 1D spin-1/2 Heisenberg chains, the elementary excitations are gapless spinons (chargeless spin-1/2 quasiparticles) characterized by a linear energy dispersion and a power-law decay of the spin correlation (*19*), whereas in the integer spin case such excitations are gapped (*20*). In the organic compound κ-(BEDT-TTF)_{2}Cu_{2}(CN)_{3}, where the first putative QSL state was reported (*3*), the presence of the spin excitation gap is controversial (*18*, *21*). In this compound, the stretched, nonexponential decay of the NMR relaxation indicates inhomogeneous distributions of spin excitations (*22*), which may obscure the intrinsic properties of the QSL. A phase transition possibly associated with the charge degree of freedom at ~6 K further complicates the situation (*23*). Meanwhile, in EtMe_{3}Sb[Pd(dmit)_{2}]_{2} (dmit-131) such a transition is likely to be absent, and a much more homogeneous QSL state is attained at low temperatures (*4*, *5*). As a further merit, dmit-131 (Fig. 1B) has a cousin material Et_{2}Me_{2}Sb[Pd(dmit)_{2}]_{2} (dmit-221) with a similar crystal structure (Fig. 1C), which exhibits a nonmagnetic charge-ordered state with a large excitation gap below 70 K (*24*). A comparison between these two related materials will therefore offer us the opportunity to single out genuine features of the QSL state believed to be realized in dmit-131.

Measuring thermal transport is highly advantageous for probing the low-lying elementary excitations in QSLs, because it is free from the nuclear Schottky contribution that plagues the heat capacity measurements at low temperatures (*21*). Moreover, it is sensitive exclusively to itinerant spin excitations that carry entropy, which provides important information on the nature of the spin correlation and spin-mediated heat transport. Indeed, highly unusual transport properties including the ballistic energy propagation have been reported in a 1D spin-1/2 Heisenberg system (*25*).

The temperature dependence of the thermal conductivity κ* _{xx}* divided by

*T*of a dmit-131 single crystal displays a steep increase followed by a rapid decrease after showing a pronounced maximum at

*T*~ 1 K (Fig. 2A). The heat is carried primarily by phonons (κ

_{g}

_{xx}^{ph}) and spin-mediated contributions (κ

_{xx}^{spin}). The phonon contribution can be estimated from the data of the nonmagnetic state in a dmit-221 crystal with similar dimensions, which should have a negligibly small κ

_{xx}^{spin}. In dmit-221, κ

_{xx}^{ph}

*/T*exhibits a broad peak at around 1 K, which appears when the phonon conduction grows rapidly and is limited by the sample boundaries. On the other hand, κ

*of dmit-131, which well exceeds κ*

_{xx}/T

_{xx}^{ph}

*/T*of dmit-221, indicates a substantial contribution of spin-mediated heat conduction below 10 K. This observation is reinforced by the large magnetic field dependence of κ

*of dmit-131, as discussed below (Fig. 3A). Figure 2B shows a peak in the κ*

_{xx}*versus*

_{xx}*T*plot for dmit-131, which is absent in dmit-221. We therefore conclude that κ

_{xx}^{spin}and κ

_{xx}^{spin}

*/T*in dmit-131 have a peak structure at

*T*~ 1 K, which characterizes the excitation spectrum.

_{g}The low-energy excitation spectrum can be inferred from the thermal conductivity in the low-temperature regime. In dmit-131, κ* _{xx}/T* at low temperatures is well fitted by κ

*κ*

_{xx}/T =_{00}

*/T + bT*

^{2}(Fig. 2C), where

*b*is a constant. The presence of a residual value in κ

*at*

_{xx}/T*T*→0 K, κ

_{00}

*/T,*is clearly resolved. The distinct presence of a nonzero κ

_{00}

*/T*term is also confirmed by plotting κ

*versus*

_{xx}/T*T*(Fig. 2D). In sharp contrast, in dmit-221, a corresponding residual κ

_{00}

*/T*is absent and only a phonon contribution is observed (

*26*). The residual thermal conductivity in the zero-temperature limit immediately implies that the excitation from the ground state is gapless, and the associated correlation function has a long-range algebraic (power-law) dependence. We note that the temperature dependence of κ

*in dmit-131 is markedly different from that in κ-(BEDT-TTF)*

_{xx}/T_{2}Cu

_{2}(CN)

_{3}, in which the exponential behavior of κ

*associated with the formation of excitation gap is observed (*

_{xx}/T*18*).

Key information on the nature of elementary excitations is further provided by the field dependence of κ* _{xx}*. Because it is expected that κ

_{xx}^{ph}is hardly influenced by the magnetic field, particularly at very low temperatures, the field dependence is governed by κ

_{xx}^{spin}(

*H*) (

*26*). The obtained

*H*-dependence, κ

*(*

_{xx}*H*), at low temperatures is quite unusual (Fig. 3A). At the lowest temperature, κ

*(*

_{xx}*H*) at low fields is insensitive to

*H*but displays a steep increase above a characteristic magnetic field

*H*~ 2 T. At higher temperatures close to

_{g}*T*, this behavior is less pronounced, and at 1 K κ

_{g}*(*

_{xx}*H*) increases with

*H*nearly linearly. The observed field dependence implies that some spin-gap–like excitations are also present at low temperatures, along with the gapless excitations inferred from the residual κ

_{00}

*/T*. The energy scale of the gap is characterized by μ

*, which is comparable to*

_{B}H_{g}*k*. Thus, it is natural to associate the observed zero-field peak in κ

_{B}T_{g}*(*

_{xx}*T*)

*/T*at

*T*with the excitation gap formation.

_{g}Next we examined a dynamical aspect of the spin-mediated heat transport. An important question is whether the observed energy transfer via elementary excitations is diffusive or ballistic. In the 1D spin-1/2 Heisenberg system, the ballistic energy propagation occurs as a result of the conservation of energy current (*25*). Assuming the kinetic approximation, the thermal conductivity is written as κ_{xx}^{spin} *= C _{s}v_{s}ℓ_{s} /3*, where

*C*is the specific heat,

_{s}*v*is the velocity, and

_{s}*ℓ*is the mean free path of the quasiparticles responsible for the elementary excitations. We tried to estimate

_{s}*ℓ*simply by assuming that the linear term in the thermal conductivity arises from the fermionic excitations, in analogy with excitations near the Fermi surface in metals. The residual term is written as κ

_{s}_{00}

*/T ~*(

*k*

_{B}^{2}

*/daħ*)

*ℓ*, where

_{s}*d*(~3 nm) and

*a*(~1 nm) are interlayer and nearest-neighbor spin distance. We assumed the linear energy dispersion ε(

*k*)

*= ħv*, a 2D density of states and a Fermi energy comparable to

_{s}k*J*(

*26*). From the observed κ

_{00}

*/T*, we find that

*ℓ*reaches as long as ~1 μm, indicating that the excitations are mobile to a distance 1000 times as long as the interspin distance without being scattered. Remarkably, the observed

_{s}*ℓ*is even longer than the mean impurity distance estimated from the susceptibility measurements (

_{s}*5*). Although our simple estimation should be scrutinized, the nearly ballistic propagation seems to be realized in this QSL state, which bears a striking resemblance to the 1D Heisenberg system (

*25*). Such a coherent motion of the excitation is only possible with an extremely long correlation length, consistent with the power-law correlation function with gapless excitations.

Our results indicate that there are two kinds of excitations in the low-temperature regime of dmit-131: One is a low-lying gapless excitation and another is a spin-gap–like excitation that couples to magnetic fields. The existence of the gapless mode imposes constraints on theories that can account for the QSL in this material. For instance, it contradicts the identification of the ground state of this system with a fully gapped short-range resonating-valence-bond state (*27*). It is intriguing that, although there has been no experimental observation, the coexistence of the gapless and gapped excitations has been theoretically predicted in highly frustrated kagomé lattice (*28*). In this case, the magnetic excitations are separated from the ground state by a spin-gap, which is filled with nonmagnetic excitations. It is a nontrivial problem, however, if such a state can be realized in a triangular lattice where less quantum degenerate states are expected.

Recently, the ground state of a QSL in the 2D triangular lattice has been discussed in terms of the spinon Fermi surface (*7*, *8*). The Fermi surface is suggested to have pairing instability at low temperatures, and a possible nodal gap structure appears that is analogous to *d*-wave superconductivity in high-transition temperature cuprates (*29*). In this scenario, the spin-gap–like behavior is attributed to the pairing gap formation, and the finite residual *T*-linear term stems from the zero-energy density of states similar to the disorder-induced normal fluid in *d*-wave superconductors. The low energy density of states is expected to be insensitive to Zeeman magnetic field as in a Fermi liquid. The nodal gap is expected to produce thermally excited quasiparticles, which may account for the observed *T*^{2} dependence of κ*/T* with a much steeper slope than κ_{xx}^{ph}*/T* (Fig. 2C). This model also predicts a thermal-Hall effect of spinons that experience the Lorentz force, akin to the conduction electrons in metals (*30*). In an attempt to observe this effect, we measured the thermal-Hall conductivity κ* _{xy}*, and the tangent of the thermal-Hall angle, tanθ(

*Η*) = κ

*(κ*

_{xy}/*– κ*

_{xx}

_{xx}^{ph}) (Fig. 3B). The observed thermal-Hall angle is orders of magnitude smaller than the prediction (

*26*), which calls for further studies. It is also an open question why continuous excitations are observed in the present compound whereas they are absent in κ-(BEDT-TTF)

_{2}Cu

_{2}(CN)

_{3}(

*18*). Such excitations may be strongly suppressed by inhomogeneity.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/328/5983/1246/DC1

Materials and Methods

Figs. S1 and S2

Table S1

References

## References and Notes

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
Materials and methods are available as supporting material on
*Science*Online. - ↵
- ↵
- ↵
- ↵
- We thank M. P. A. Fisher, S. Fujimoto, T. Itou, H. Katsura, N. Kawakami, H. Kawamura, P. A. Lee, G. Misguich, Y. Nakazawa, and A. Tanaka for valuable discussion. This work was supported by Grants-in-Aid for Scientific Research (KAKENHI) from the Japan Society for the Promotion of Science and the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), a Grant-in-Aid for the Global Centers of Excellence Program “The Next Generation of Physics, Spun from Universality and Emergence,” and a Grant-in-Aid for Scientific Research on Innovative Areas “Heavy Electrons” (No. 20102006) from MEXT, Japan.