Introduction

A quantum spin liquid (QSL) is an intriguing exception for the Landau theory of phase transitions; at sufficiently low temperatures, condensed-matter systems form an ordered state characterized by broken symmetries and corresponding order parameters. However, the order can be suppressed when there exist strong quantum-mechanical fluctuations enhanced by low dimensionality and/or geometrical frustration. The resulting exotic quantum liquids such as the QSL are not described by any broken symmetry or order parameter1. The QSL is also of great interest in connection with a mechanism of high-temperature superconductivity2 and application to quantum computation3.

A spin-1/2 triangular-lattice antiferromagnet with nearest-neighbor (NN) antiferromagnetic (AF) exchange interactions J is one of the most typical example of two-dimensional frustrated spin systems, in which Anderson first proposed a QSL ground state more than 40 years ago4. Although the ground state of the triangular-lattice Heisenberg AF system is now known to be 120° AF order5, its ordered state can be suppressed by ring-exchange interactions6, next NN interactions7, or a spatial distribution of an exchange coupling constant8, consequently leading to QSL ground states. One of the most fundamental properties of a QSL is the presence of charge neutral excitations carrying spin-1/2 quantum number, spinons. These fractional excitations are clearly distinct from spin-1 magnon excitations in magnetically ordered states. Depending on the theoretical model, the spinon excitations may be gapped or gapless, and may obey Bose or Fermi statistics1,6,9,10,11.

In 2003, the first evidence of a QSL was reported in an organic triangular-lattice antiferromagnet, κ-(BEDT-TTF)2Cu2(CN)3 (ref. 12), where BEDT-TTF stands for bis(ethylenedithio)tetrathiafulvalene. In this material, a spin-1/2 is located on a (BEDT-TTF)\(_2^ +\) dimer, which is arranged on a triangular lattice. Despite the large NN AF interactions, J/kB ~ 250 K, no magnetic long-range order happens down to T ∼ 30 mK12,13,14, which is four orders of magnitude lower than J/kB. This suggests that the QSL state is realized in κ-(BEDT-TTF)2Cu2(CN)3. The nature of the magnetic excitations of the QSL state, spinons, has been intensively studied and discussed. A finite value of the specific heat divided by temperature C/T for T → 0, and Pauli-like magnetic susceptibility are hallmarks of gapless spin excitations15,16. Conversely, Arrhenius behavior of the thermal conductivity suggests the presence of a small gap, Δ/kB ~ 0.5 K17. There have been many debates about the spinon excitations not only in κ-(BEDT-TTF)2Cu2(CN)3, but also in the other QSL candidates such as a triangular-lattice system, YbMgGaO418,19, and a kagome-lattice system, ZnCu3(OH)6Cl2 (refs. 20,21).

Here we report magnetocaloric-effect (MCE) measurements on κ-(BEDT-TTF)2Cu2(CN)3, which unveils a characteristic thermal relaxation of the QSL state. At very low temperatures in a magnetic field, the thermal relaxation time between the electron spins and lattice is dramatically increased, indicating that the spins are decoupled from the lattice bath. The spin–lattice decoupling can explain the seeming discrepancy in the nature of the spin-excitation spectrum; the spin excitations are gapless. Moreover, we show by combining the present MCE results with our recent magnetic-susceptibility study that as a zero-field quantum critical point (QCP) is approached, the number of spin states is strongly enhanced. This is compatible with a model of a strongly correlated QSL near a QCP.

Results

Magnetocaloric effect

In the present study, we have measured the MCE to resolve the discrepancy between the gapped and gapless features of the spin excitations in κ-(BEDT-TTF)2Cu2(CN)3. The MCE, ΔT, is a thermal response of a sample to magnetic-field changes, dH/dt, given by

$${\mathrm{\Delta }}T = - \tau \frac{{{\mathrm{d}}({\mathrm{\Delta }}T)}}{{{\mathrm{d}}t}} - \frac{T}{{K_{\mathrm{B}}}}\frac{{{\mathrm{d}}H}}{{{\mathrm{d}}t}}\left( {\frac{{\partial S}}{{\partial H}}} \right)_T,$$
(1)

where KB represents the thermal conductance between the sample and heat bath. The second term in Eq. (1) describes heating or cooling of the sample by the magnetic entropy change, dS/dH. When S decreases (increases) with increasing field, the sample is heated up (cooled down). Resulting ΔT is relaxed with the relaxation time, τ = C/KB, as described in the first term. Here, C represents the heat capacity of the sample. Figure 1a, b show heating and relaxation processes of the sample at the bath temperature, TB, of 0.26 K (see also Supplementary Fig. 1). As the field increases up to 0.8 T (t < 10 s) (Fig. 1a), the sample temperature increases, ΔT > 0 K. We observe positive ΔT in the whole field and temperature range up to 17 T between 0.1 and 1.6 K, indicating that the magnetic entropy is monotonously suppressed by applying a magnetic field. This is a clear contrast to a quantum disordered state with a singlet–triplet gap being closed in high fields, where the increase of the entropy results in negative ΔT22,23. Hereafter, we call the heating by the MCE the spin heating. At μ0H = 0.8 T (t = 10 s), the field is stopped [dH/dt = 0 in Eq. (1)] and then we observe the relaxation of ΔT, which is well represented as a single exponential decay, ΔT ~ exp(−t/τ), with relaxation time τ = 2.2 s (solid line). By applying the magnetic field of 3 T, τ dramatically increases to 47 s (Fig. 1b). By contrast, when the sample is heated by an electric heater (called the lattice heating below), τ remains very short in the whole field region, as shown in Fig. 1c, d: τ = 0.5 and 0.2 s at μ0H = 0.8 and 3 T, respectively. Here, we note that τ ~ 2 s for the spin heating is attributed to the time constant of a superconducting magnet used in the present measurements, as shown in Supplementary Fig. 2; actual τ is shorter than 2 s.

Fig. 1
figure 1

Thermal relaxation curve at the temperature of 0.26 K. Thermal relaxation curve for the spin (magnetocaloric) heating, when the magnetic field is swept up from a 0.6 to 0.8 T, and b 2 to 3 T. The thermal relaxation curve for the lattice (Joule) heating at c μ0H = 0.8 T and d 3 T. In each figure, the solid line represents a single exponential decay, ΔT = A exp(−t/τ), with the relaxation time, τ, and a constant, A

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The magnetic-field dependence of the relaxation time for the spin heating is summarized in Fig. 2a. At TB = 0.26 K, τ is shorter than 2 s up to μ0H ~ 0.9 T. Above this field, τ exceeds 2 s and rapidly increases more than one order of magnitude, followed by a slow increase without showing saturation behavior above ~2 T. Here, we determine the onset field of the τ increase, H*, based on the linear-scale plot of τ(H) (inset of Fig. 2a). With temperature elevation, H* monotonically shifts to a higher value. Figure 2b depicts the relaxation time as a function of temperature. At μ0H = 0.1 T, τ is shorter than 2 s down to TB ~ 0.12 K, below which a rapid increase of τ is observed. The onset temperature of the τ increase is raised by applying a magnetic field. At 1.5 T, τ is rapidly increased below 0.5 K, and after that τ shows a rather slow increase. At 10 T, we observe power-law behavior, τ ~ T−1.3, in the wide temperature range. Figure 2c shows the temperature dependence of (τT1.3)−1. As the temperature goes to zero, (τT1.3)−1 approaches a constant value (dashed lines). This value is monotonically decreased as a magnetic field is increased.

Fig. 2
figure 2

Magnetic field and temperature dependence of the relaxation time. a Magnetic-field dependence of the relaxation time, τ, for the spin heating at various temperatures. The dashed line represents constant τ of 2.1 s. At low fields, τ is shorter than 2.1 s (thick arrows). The thin arrows indicate the onset field of the τ increase, H*, which is determined based on the linear-scale plot, shown in the inset. b Temperature variation in τ for the spin heating at various magnetic fields. The black- and purple-dashed lines denote the power-law behavior with an exponent of −1.3, and constant τ of 2.1 s, respectively. At each field, τ is shorter than 2.1 s above the temperature indicated by the thick arrows. c Temperature dependence of (τT1.3)−1 at various magnetic fields. The dashed line represents a zero-temperature extrapolation of the low-temperature data, \((\tau T^{1.3})_0^{ - 1}\)

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Simulation results

An important question here is why the slow relaxation takes place only for the spin heating. To address this question, we simulate the thermal relaxation curve for the spin and lattice heatings, based on a simple spin–lattice coupling model, as shown in Fig. 3a (details of the simulations are shown in Methods). Our MCE measurements probe the temperature of the lattice system, TL. In a conventional case, the thermal coupling between the spins and lattice, KSL, is sufficiently stronger than lattice–bath coupling, KB: KSL \(\gg\) KB. In this case, the temperature of the spin system, TS, is identical to TL for both the spin and lattice heatings (PS and PL, respectively), as shown in Fig. 3b, Supplementary Fig. 3(a), and its inset. The relaxation time is given by τ = (CS + CL)/KB = 0.4 s, which is confirmed by the exponential decay fit to the relaxation curve. Here, CS and CL represent the heat capacities of the spins and lattice, respectively. When the spins are decoupled from the lattice (KSL \(\ll\) KB), TL is rapidly raised by the lattice heating, whereas TS slowly increases because of small KSL (Supplementary Fig. 3(b)). After stopping the heating, TL is quickly relaxed, given by \(\tau \sim C_{\mathrm{L}}{\mathrm{/}}K_{\mathrm{B}}\) = 0.1 s, as shown in Fig. 3c. In case of the spin heating, by contrast, supplied heat can hardly be relaxed to the lattice, and consequently, TS becomes much higher than TL, as shown in Supplementary Fig. 3(c) and its inset. After stopping the heating, TS and TL are slowly relaxed to the bath temperature, TB. The relaxation time is given by \(\tau \sim C_{\mathrm{S}}{\mathrm{/}}K_{{\mathrm{SL}}}\) = 40 s, which is much longer than that for the lattice heating (Fig. 3d and its inset). The above spin–lattice decoupling model reasonably explains the significant difference of τ between the spin and lattice heatings shown in Fig. 1b, d.

Fig. 3
figure 3

Simulation results of the thermal relaxation. a Schematic diagram of the experimental situation. The heat supplied to the spin system first flows to the lattice system through the thermal conductance, KSL, and then to the heat bath through KB. PS (PL) and CS (CL) represent the heating power of the spin (lattice) heating and the heat capacity of the spin (lattice) system, respectively. TS, TL, and TB denote the temperature of the spins, lattice, and bath, respectively. b The thermal relaxation curve when the spins are strongly coupled to the lattice [ΔTS(t) = TS(t) − TB and ΔTL(t) = TL(t) − TB]. The thermal relaxation curve when the spins are decoupled from the lattice for c the lattice heating and d the spin heating. Inset of d: the enlarged figure of the main panel. Magenta and cyan lines represent ΔTS(t) and ΔTL(t), respectively

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Discussion

The spin–lattice decoupling requires reconsideration of the data analysis in the heat-capacity measurement on κ-(BEDT-TTF)2Cu2(CN)3. The heat capacity has been measured in a limited temperature and field region, T > 0.8 K and μ0H < 8 T15. In this region, C is obtained by measuring the thermal relaxation in a long time scale, 10–100 s. This time scale is much longer than our lattice-heating result, 0.1–1 s, and even comparable to the spin-heating one, <20 s (Fig. 2b). Therefore, CS as well as CL is measured in ref. 15, and consequently the finite C/T value for T → 0 is observed. An important implication of our MCE study is that the electronic spin entropy is monotonously decreased as a magnetic field is increased (\({\mathrm{\Delta }}T\) ~ −dS/dH > 0). From the Maxwell’s relation (dS/dH) T  = (dM/dT) H , we obtain (dM/dT) H  < 0, where M represents the magnetization. This is consistent with the temperature dependence of the magnetic susceptibility below about 4 K and 3 T16. By contrast, no remarkable field dependence of the specific heat has been observed down to T ~ 0.8 K, suggesting dS/dH ~ 015. The heat-capacity measurement in a magnetic field at lower temperatures, where the field dependence should become more pronounced, is highly required in the future.

Not only the heat capacity, but also the heat-transport measurement is affected by the spin–lattice decoupling. It has been reported that in cuprate superconductors, a poor contact between electrons and a lattice prevents a heat transfer between them, which leads to the strong suppression of the electronic thermal conductivity24. Likewise, once the spins are decoupled from the lattice, the spin contribution to the thermal conductivity will be significantly reduced. This decoupling is likely the origin of the rapid decrease of κ/T at very low temperatures for κ-(BEDT-TTF)2Cu2(CN)317. On the other hand, κ is gradually increased by applying a magnetic field. A possible scenario is a model of a strongly correlated QSL (SCQSL) located near a fermion-condensation quantum-phase-transition point, where the QSL plays the role of heavy-fermion liquids placed into insulators25; as a magnetic field is increased away from the QCP, the effective mass of spinons, m*, is reduced, leading to the increase of the spin thermal conductivity. In fact, we have recently determined the H–T phase diagram based on a scaling analysis of χ, where a QCP is present near the zero field (Fig. 4a)16. In the quantum critical (QC) region (yellow area), χ diverges for T → 0, while χ shows almost T-independent (Pauli-paramagnetic-like) behavior in the QSL region (dark-red area). The Pauli-like susceptibility is commonly observed in organic triangular-lattice QSL materials26,27, and its values, χP, are explained by a model of a QSL with a spinon Fermi surface6,27. In this model, fermionic spinons play the role of metallic electrons placed into insulators; χP comes from the spinon density of states at the Fermi level, N(EF), being proportional to m*. In this context, we examine the magnetic-field dependence of χP for κ-(BEDT-TTF)2Cu2(CN)3 in Fig. 4b, where the χP values are determined based on the susceptibility data reported by us16. By the application of a magnetic field, χP is decreased following a power law, \(\chi _{\mathrm{P}} \sim H^{ - a}\), with the exponent a = 0.7–0.8, suggesting that as the system is away from the QCP, N(EF) ∝ m* is strongly suppressed. This is compatible with the SCQSL model.

Fig. 4
figure 4

Comparison of the MCE results with the susceptibility data. a Contour plot of the magnetic susceptibility multiplied by the power of temperature, χT0.83, in the T–H plane, determined by us in ref. 16. The yellow and dark-red areas represent the quantum critical (QC) and QSL states, respectively, which are separated by the crossover region (light-red area). The onset field of the τ increase, H*, shown by the circles, falls on the contour line (dashed line), χT0.83 = 2.3 mJ K0.83 T−2 mol−1. The blue solid line indicates a linear fit of H*. b The zero-temperature extrapolation of the (τT1.3)−1 values raised to the power p, \((\tau T^{1.3})_0^{ - p}\), and the Pauli-like susceptibility, χP, as a function of magnetic field. χP is determined based on the susceptibility data in ref. 16. The solid line represents a power-law fit of χP with the exponent −0.73

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In order to compare the MCE results with the susceptibility data, we plot H* on the H–T phase diagram, determined by us in ref.16 (Fig. 4a). A striking finding is that the data points of H* fall on the contour line, χT0.83 = 2.3 mJ K0.83 T−2 mol−1 (dashed line), in the crossover region (light-red area). This coincidence implies that low-energy spin excitations giving χP are also responsible for the thermal relaxation phenomena through coupling to lattice vibrations. In Fig. 4b, we further examine this relationship by comparing the field dependence of χP with the zero-temperature extrapolations of (τT1.3)−1(see Fig. 2c), raised to different powers, \((\tau T^{1.3})_0^{ - p}\). As the zero-field QCP is approached, \((\tau T^{1.3})_0^{ - p}\) with the exponent p = 0.5 is strongly increased in a manner similar to χP. This suggests that \((\tau T^{1.3})_0^{ - 0.5}\) depends on N(EF) as well; the electron spin–lattice relaxation rate could be given by \(\tau ^{ - 1} \sim [N(E_{\mathrm{F}})]^2T^{1.3}\) in the QSL state. The enhancement of N(EF) for H → 0 is intuitively consistent with the thermal-decoupling phenomena observed here; the application of a magnetic field reduces the number of spin states that interact with lattice vibrations, leading to the weak spin–lattice coupling.

In relation to the spin–lattice decoupling phenomena, several theoretical works have studied the interaction of spinons with phonons28,29. In a QSL with a spinon Fermi surface, spinons undergo pairing instability30,31, similar to the Cooper pairing in metals. The resulting gapped state reduces N(EF), leading to the weak spinon–phonon interactions28. However, this is not applicable to our case because no sign of a phase transition near H* has been found in thermodynamic quantities such as the specific heat and the magnetic torque15,16.

The main findings of the present MCE study are summarized in the following two points. First, we find the spin–lattice decoupling phenomena in the QSL state, which can explain the seeming discrepancy between the gapped17 and gapless15,16 features of spin excitations in κ-(BEDT-TTF)2Cu2(CN)3; spin excitations are gapless. Recently, the inorganic triangular-lattice antiferromagnet, YbMgGaO4, has been discussed as a candidate material for the QSL with the spinon Fermi surface. In this material, the gapless nature of spinons has been reported by the neutron-scattering and specific-heat experiments18,19, whereas the spin thermal conductivity appears to be absent19. The MCE study on YbMgGaO4 may resolve the discrepancy. Second, as the system is away from the QCP, the number of spin states is rapidly decreased, and consequently the spin–lattice interaction is weakened. This is compatible with the SCQSL model25, where the QSL has much similarity with heavy-fermion liquids.

Methods

Sample preparation and MCE measurements

Single-crystalline samples were prepared by electrochemical oxidation of BEDT-TTF molecules. In magnetocalorimetry, several single-crystalline samples of 227 μg were attached to a small thermometer (Cernox, Lake Shore) by a grease (Apiezon N Grease), and then the composition was enclosed in a home-made miniature vacuum cell, together with a reference thermometer. A temperature difference between the two thermometers, ΔT, was measured with a magnetic field swept up to 17 T at a sweep rate of 0.5 T min−1. All the measurements were made using a 20 T superconducting magnet with a dilution refrigerator at Tsukuba Magnet Laboratory, NIMS. A magnetic field is applied approximately perpendicular to a two-dimensional triangular-lattice plane.

Thermal relaxation measurements and simulations

We have applied two heating methods for the thermal relaxation measurements, spin (magnetocaloric) and lattice (Joule) heatings. A simplified diagram of the experimental configuration is shown in Fig. 3a. Here, we assume that the lattice is strongly coupled to a thermometer; the ‘lattice’ in the figure includes addenda (a thermometer and grease). This condition is well satisfied when a thermometer is tightly attached to samples by a grease, as in the present case. Our relaxation measurements probe the lattice temperature, TL.

In the spin heating, spins are directly heated up by sweeping a magnetic field when dS/dH in Eq. (1) has a negative value. The heat supplied to the spins first flows to the lattice through the thermal conductance, KSL, and then to a heat bath through KB. After stopping the heating (field sweep), the spin temperature, TS, and TL are relaxed to the bath temperature, TB. In the case of the lattice heating, the lattice is directly heated up by an electrical heater, while TS is raised by a heat transfer from the lattice. The thermometer is also used as a heater in our experimental set-up. After stopping the heating, TS and TL are relaxed to TB.

In order to simulate the thermal relaxation curves for the two heating methods, we begin with heat balance equations for the above model (Fig. 3a),

$$P_{\mathrm{S}} = C_{\mathrm{S}}\frac{{{\mathrm{d}}T_{\mathrm{S}}}}{{{\mathrm{d}}t}} + K_{{\mathrm{SL}}}(T_{\mathrm{S}} - T_{\mathrm{L}}),$$
(2)
$$P_{\mathrm{L}} = C_{\mathrm{L}}\frac{{{\mathrm{d}}T_{\mathrm{L}}}}{{{\mathrm{d}}t}} + K_{{\mathrm{SL}}}(T_{\mathrm{L}} - T_{\mathrm{S}}) + K_{\mathrm{B}}(T_{\mathrm{L}} - T_{\mathrm{B}}).$$
(3)

Here, PS (PL) and CS (CL) represent the heating power of the spin (lattice) heating, and the heat capacity of the spins (lattice), respectively. By substituting dT/dt = [T(t + Δt) − T(t)]/Δt into Eqs. (2) and (3), we obtain

$$T_{\mathrm{S}}(t + {\mathrm{\Delta }}t) = T_{\mathrm{S}}(t) - \frac{{{\mathrm{\Delta }}t}}{{C_{\mathrm{S}}}}\{ K_{{\mathrm{SL}}}[T_{\mathrm{S}}(t) - T_{\mathrm{L}}(t)] - P_{\mathrm{S}}\},$$
(4)
$$\begin{array}{*{20}{l}}{T_{\mathrm{L}}(t + {{\mathrm{\Delta}}}t)} \hfill & = \hfill & T_{\mathrm{L}}(t) - \frac{{{\mathrm{\Delta}}t}}{{C_{\mathrm{L}}}}\left\{K_{{\mathrm{SL}}}\left[T_{\mathrm{L}}(t) - T_{\mathrm{S}}(t)\right]\right.\\ \hfill & {} \hfill & \left. + K_{\mathrm{B}}[T_{\mathrm{L}}(t) - T_{\mathrm{B}}(t)] - P_{\mathrm{L}}\right\}.\end{array}$$
(5)

Based on Eqs. (4) and (5), we simulate the thermal relaxation curve at TB = 0.26 K and μ0H = 3 T. The spin and lattice contribution to the heat capacity of κ-(BEDT-TTF)2Cu2(CN)3 have been reported to be 730 and 90 pJ K−1 at T = 0.26 K, respectively15. The heat capacity of addenda (thermometer and grease) is about 60 pJ K−1, and then CS = 730 pJ K−1 and CL = 150 pJ K−1. When the sample is heated up by PL = 14 pW, TL is raised about 7 mK, as shown in Fig. 1d. These values together with a relation ΔTL = PL/KB give KB = 2 nW K−1. Here, we consider two limiting conditions, conventional and decoupling cases. In the former case, the thermal coupling between the spins and lattice is sufficiently stronger than the lattice–bath coupling, KSL \(\gg\) KB. In this case, TS is identical to TL during the heating and relaxation processes, regardless of the heating methods (Supplementary Fig. 3(a), its inset, and Fig. 3(b)). The simulations are made using KSL = 100 KB, PL = 14 pW for the lattice heating, and PS = 14 pW for the spin heating. The relaxation time is given by τ = (CS + CL)/KB = 0.4 s. In the latter case, the spins are decoupled from the lattice, KSL \(\ll\) KB. In case of the lattice heating, TL is first raised, and after that TS slowly increases owing to a small heat transfer from the lattice (Supplementary Fig. 3(b)). After stopping the heating, TL is quickly relaxed, given by \(\tau \sim C_{\mathrm{L}}{\mathrm{/}}K_{\mathrm{B}}\) = 0.1 s, as shown in Fig. 3c. The magnetic specific heat no longer contributes to the relaxation. In case of the spin heating, by contrast, the supplied heat is accumulated in the spins, because the heat can hardly flow to the lattice. Consequently, TS becomes much higher than TL, as shown in Supplementary Fig. 3(c) and its inset. After stopping the heating, TS and TL are slowly relaxed, following \(\tau \sim C_{\mathrm{S}}/K_{{\mathrm{SL}}}\) = 40 s (Fig. 3d and its inset), which is much longer than that for the lattice heating. The simulations are made using KSL = KB/100, PL = 14 pW for the lattice heating, and PS = 1.4 pW for the spin heating. The spin–lattice decoupling model (Fig. 3c, d) reasonably explains why the slow thermal relaxation is observed only for the spin heating (Fig. 1b, d).

Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.