Abstract
In Mott insulators, the strong electron–electron Coulomb repulsion localizes electrons. In dimensions greater than one, their spins are usually ordered antiferromagnetically at low temperatures. Geometrical frustrations can destroy this longrange order, leading to exotic quantum spin liquid states. However, their magnetic ground states have been a longstanding mystery. Here we show that a quantum spin liquid state in the organic Mott insulator EtMe_{3}Sb[Pd(dmit)_{2}]_{2} (where Et is C_{2}H_{5}−, Me is CH_{3}−, and dmit is 1,3dithiole2thione4,5dithiolate) with twodimensional triangular lattice has Pauliparamagneticlike lowenergy excitations, which are a hallmark of itinerant fermions. Our torque magnetometry down to low temperatures (30 mK) up to high fields (32 T) reveals distinct residual paramagnetic susceptibility comparable to that in a halffilled twodimensional metal, demonstrating the magnetically gapless nature of the ground state. Moreover, our results are robust against deuteration, pointing toward the emergence of an extended 'quantum critical phase', in which lowenergy spin excitations behave as in paramagnetic metals with Fermi surface, despite the frozen charge degree of freedom.
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Introduction
At sufficiently low temperatures, condensed matter systems generally tend to order. Quantum spin liquids (QSLs)^{1} are a prominent exception, in which no local order parameter is formed while the entropy vanishes at zero temperature. In two or threedimensional (2D or 3D) systems, it is widely believed^{1} that QSL ground states may emerge when interactions among the magnetic degrees of freedom are incompatible with the underlying crystal geometry. Typical 2D examples of such geometrically frustrated systems can be found in triangular and kagome lattices. Largely triggered by the proposal of the resonatingvalencebond theory on spin1/2 degrees of freedom residing on a 2D triangular lattice^{2} and its possible application to hightransitiontemperature superconductivity^{3}, realizing/detecting QSLs has been a longsought goal. Until now, quite a number of QSLs with various types of ground states have been proposed^{2,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}, but the lack of real materials had prevented us from understanding the nature of QSLs.
The two recently discovered organic Mott insulators, κ(BEDTTTF)_{2}Cu_{2}(CN)_{3} (refs 20,21,22,23,24,25) and EtMe_{3}Sb[Pd(dmit)_{2}]_{2} (refs 26,27,28,29) (see Fig. 1a), are very likely to be the first promising candidates to host a QSL in real bulk materials^{30}. In these systems, cations or anions are strongly dimerized and spin1/2 units of (BEDTTTF)_{2} and [Pd(dmit)_{2}]_{2} form 2D triangular structure. In both systems, nuclear magnetic resonance (NMR)^{20,26} and muon spin rotation^{24,25} measurements exhibit no sign of longrange magnetic ordering down to a very low temperature whose energy scale corresponds to J/10,000 (J/k_{B}~250 K for both compounds), where J is the nearestneighbour exchange coupling energy. These findings aroused great interest because even in a quantum spin1/2 triangular lattice antiferromagnet, the frustration brought on by the nearestneighbour Heisenberg interaction is known to be insufficient to destroy the longrange ordered ground state^{31}. Several classes of QSL states have been put forth to explain these exotic spin states, yet their ground state remains puzzling. Between the two compounds, EtMe_{3}Sb[Pd(dmit)_{2}]_{2} appears to be more ideal to single out genuine features of the QSL, because more homogeneous QSL state can be attained at low temperatures^{28}. In contrast, κ(BEDTTTF)_{2}Cu_{2}(CN)_{3} has been reported to have a structural transition at 6 K (ref. 23) and an inhomogeneous, phaseseparated spin state^{20,24}.
What is remarkable in the QSL state of EtMe_{3}Sb[Pd(dmit)_{2}]_{2} is the presence of the gapless excitations, which are highly mobile with long meanfree path, as reported by thermal conductivity^{27} and heat capacity measurements^{29}. However, there are two fundamental questions that still remain open. The first one is the magnetic nature of the gapless excitations, which cannot be obtained from the above measurements. Whether the excitations are magnetic or nonmagnetic bears direct implications on the spin–spin correlation function. The second question concerns the phase diagram; how the QSL varies when tuned by the nonthermal parameter, such as frustration, is the key to understanding the ground state. The uniform susceptibility in the lowtemperature limit provides pivotal information on the magnetic character of the ground state. However, conventional measurements using SQUID magnetometer are susceptible to paramagnetic contributions due to impurities. Indeed, for EtMe_{3}Sb[Pd(dmit)_{2}]_{2}, the temperature region of importance is below ~5 K, but SQUID signal is dominated by the paramagnetic contributions residing on the nonmagnetic cation EtMe_{3}Sb layers^{27}.
To resolve the intrinsic magnetic susceptibility of the QSL in singlecrystalline samples, we performed the magnetic torque τ measurements by using a microcantilever method^{32,33} (Fig. 1b). The torque is given by the cross product of the applied field H and the magnetization M (, where χ_{ij} is the spin susceptibility tensor diagonalised along the magnetic principal axes, see Fig. 1c,d) as τ=μ_{0}VM×H, where μ_{0} is the permeability of vacuum and V is the sample volume. The most advantage of this method is that the large Curie contribution from impurity spins is cancelled out (see Methods). Our torque magnetometry down to low temperatures (30 mK) up to high fields (32 T) reveals distinct residual paramagnetic susceptibility comparable to that in a halffilled 2D metal, showing 'magnetic' gapless excitations in the QSL of EtMe_{3}Sb[Pd(dmit)_{2}]_{2}. We further demonstrate that changing the degree of frustration by a deuteration has a small effect on the susceptibility, indicating the emergence of an extended quantum critical phase in which the lowlying excitations are fermions with a Fermi surface.
Results
Electron paramagnetic resonance measurements
To determine the gfactor anisotropy, which is a fundamental property to discuss the magnetic anisotropy, we first measure electron paramagnetic resonance (EPR) as a function of field angle. The principal axes of the gyromagnetic ratio (gfactor) of EtMe_{3}Sb[Pd(dmit)_{2}]_{2}, p, q and r (see Fig. 1c,d), are deduced from the EPR measurements at 4.2 K. The angular dependence of the gfactor is shown in Fig. 1e,f (open circles). As the principle axes of the gtensor virtually coincide with that of the molecule, the anisotropy of the gfactor is likely given by the spinorbit coupling as found in other organic materials^{34,35} (see Methods).
Magnetic torque measurements
Figure 1e,f depicts the magnetic torque curves of EtMe_{3}Sb[Pd(dmit)_{2}]_{2} (h_{9}dmit) when the magnetic field of μ_{0}H=7 T is rotated within the ac* and bc* plane, where θ is the polar angle measured from the c* axis. We note that although the principle axes of the gtensor do not coincide with the crystallographic axes (Fig. 1c,d), the torque crosses zero exactly at the minimum and maximum positions of the gfactor for both field rotation at all temperatures, indicating that the torque signal reflects the anisotropy of the gfactor. Assuming that the gfactor is temperature independent at low temperatures, we determined and (see Methods).
Figure 2a,b depicts the temperature dependence of at μ_{0}H=7 T and the field dependence of at T=30 mK, respectively, for pristine crystal (h_{9}dmit) and its deuterated compound (d_{9}dmit), which has different degrees of geometrical frustration^{29}. The absolute value of is nearly 1.5 times larger than χ of polycrystalline samples of h_{9}dmit determined by the SQUID magnetometer (Fig. 2a).
We also measured the magnetic torque of the pristine sample (h_{9}dmit (C)) in magnetic fields up to 32 T to search a quantum oscillation. After subtracting the background magnetization signal determined by a linear approximation to the data >12 T, we plot the data as a function of 1/H as the red line in the upper inset of Fig. 2b. No discernible oscillation is observed within the experimental resolution.
Discussion
The most notable features are (i) is temperature independent <2 K and remains finite in zero temperature limit (Fig. 2a) and (ii) increases nearly linearly with the applied field up to 17 T (Fig. 2b). In the dmit materials the orbital paramagnetic (Van Vleck) susceptibility is very small (see Methods), and thus the spin (Pauli) paramagnetism should be responsible for the observed temperatureindependent and fieldlinear paramagnetic response. These results in the lowtemperature limit provide direct evidence of the presence of lowlying gapless 'magnetic' excitations. It is also clear that the gapless excitations observed in the thermal conductivity^{27} and the heat capacity^{29} measurements contain the magnetic ones. As the spin gap Δ is inversely proportional to the magnetic correlation length ξ, this spingapless QSL state has infinite ξ. Therefore, the presence of the gapless magnetic excitations is of crucially importance, as it immediately indicates that the QSL is in a critical state where the spin–spin correlation function decays with distance r in an anomalous nonexponential manner: a simple example is the powerlaw decay with η>0 for the socalled algebraic spin liquid^{5}.
There are two possible phase diagrams for the QSL as a function of frustration, as illustrated in Fig. 3a,b. With increasing frustration, a magnetically ordered phase with broken symmetry is destroyed at a critical value, beyond which a QSL phase without broken symmetry appears. The first case is that a gapless QSL with infinite ξ emerges only at a quantum critical point^{36}, beyond which a spingapped QSL with finite ξ appears (Fig. 3a). In the gapped QSL, its spin correlation function decays exponentially with distance as exp(−r/ξ) except at the quantum critical point. The second case is that there is an extended critical 'phase', where the gapless QSL with infinite correlation length is stably present (Fig. 3b). These two cases can be distinguished by comparing of h_{9}dmit and d_{9}dmit at , because deuteration of the Me groups in EtMe_{3}Sb[Pd(dmit)_{2}]_{2} changes the degrees of geometrical frustration by reducing t′/t (ref. 29). As shown in Fig. 2a,b, both h_{9} and d_{9}dmit crystals exhibit essentially the same paramagnetic behaviours with no spin gap, indicating that both materials are in a critical state down to k_{B}T~J/10,000. These results lead us to conclude the presence of an extended quantum critical phase of the spingapless QSL (Fig. 3b).
The presence of a stable QSL 'phase' with gapless magnetic excitations is hard to explain in the conventional bosonic picture. Within this picture, when the magnetic longrange order is established, the gapless Nambu–Goldstone bosons, that is, magnons, would arise as the consequence of the spontaneous symmetry breaking of the spin SU(2) symmetry. When the system evolves into the QSL state with no symmetry breaking, the common wisdom is that both of the magnetic order and Nambu–Goldstone bosons disappear at the same time. This implies that the presence of gapless QSL phase has some exotic elementary excitations. Among them, 'spinon' excitation, which is a chargeless spin1/2 quasiparticle (half of the magnons), has been discussed extensively. In theories of the QSL with the bosonic spinons, the magnetic order is described by the Bose–Einstein condensation^{37} of the bosonic spinons. When the magnetic order is destroyed or the Bose–Einstein condensation disappears, the bosonic spinons in the resulting QSL can only be gapless at the critical point beyond which they become gapped^{6}. The presence of an extended region of spingapless QSL phase at zero or very low temperatures, therefore, would suggest that the underlying QSL phase may be better described by the QSL with fermionic spinons^{7,9}.
Indeed, the presence of the gapless magnetic excitations bears some resemblance to the elementary excitations in the spin channel of metals with Fermi surface, that is, Pauli paramagnetism. Here, we examine a thermodynamic test simply by analysing the data in accordance with the assumptions that the elementary quasiparticles in the QSL phase possess Fermi surface, even though the system is an insulator. The Pauli susceptibility is given by , where is the Fermi energy, D() is the density of states and μ_{B} is the Bohr magneton. In 2D systems, D()=n/, where n is the number of fermion (spin) per volume. Using =8.0(5)×10^{−4} emu mol^{−1} (Fig. 2a), the specific heat coefficient of the fermionic excitations, which corresponds to the Sommerfeld constant in metals, yielded by the equation , is estimated to be γ≈56 mJ K^{−2} mol^{−1}. This value is of the same order of magnitude as the values reported by the heat capacity measurements^{29}. Provided that a substantial portion of the gapless excitations observed by specific heat is magnetic, the γ and values result in a Wilson ratio R_{w} close to unity (R_{w}=2.83 (1.41) for the pristine (deuterated) sample), which is a basic properties of metals. Moreover, it has been predicted^{38} that the Fermi temperature of the spinon is roughly of the order of magnitude of J/k_{B}(~250 K). The Fermi temperature, given by the relation , is estimated to be 480±30 K, which is in the same order as J/k_{B}. Thus, although the present simple estimations should be scrutinized, all thermodynamic quantities appear to be quantitatively consistent with the theory of the QSL that possesses a spinon Fermi surface. In d_{9}dmit, while γvalue is enhanced by nearly twice, is reduced by nearly 15%, compared with h_{9}dmit. This may indicate that nonmagnetic spinsinglet excitations, which enter not in but in γ, are enhanced very close to the magnetic end point. This might be a novel phenomenon unanticipated by present theories, which will deserve further studies.
There are several types of theories that predict the gapless QSL with spinon Fermi surface. Some of these theories, which include Amperian pairing^{13}, dwave pairing^{14} and gapless triplet pairing^{15}, predict an instability of the spinon Fermi surface at low temperatures, which accompany a phase transition. Here, we comment on these theories. Although the NMR measurements report an anomaly at 1 K (ref. 28) and discuss possible instabilities at low temperatures^{13,14,15}, no anomaly is observed in the magnetic susceptibility, which is consistent with the thermal conductivity^{27} and specific heat measurements^{29}. This indicates that the NMR anomaly at 1 K is not a signature of the thermodynamic transition, but a phenomenon that may be related to the slow spin dynamics of the QSL. Nevertheless, more stringent test of phase transition should be examined using experimental probes sensitive to rotational symmetry breaking^{32,33}, which deserves further studies. We also note that the recent theory of Majorana fermions with a Fermi surface^{16} predicts the Wilson ratio that is close to our observation. However, the suppression of the susceptibility in high magnetic fields predicted by this theory is not observed in this experiment at least up to 17 T.
The most direct evidence of the presence of the Fermi surface may be given by the quantum oscillation measurements^{39}. However, as shown in the inset in Fig. 2b, we find an extended paramagnetic response up to the highest field, but no discernible oscillation is observed within the experimental resolution (upper inset in Fig. 2b). We find that our estimation of the mobility of the gapless excitation strongly suggests that the impurity scattering effect is not the main origin of the absence of the quantum oscillations (see Methods). A possible explanation for this is that the gauge flux, which causes the quantum oscillation for the spinons, only weakly couples to the applied magnetic field or becomes unstable under a small magnetic field^{39}. This seems to be consistent with the fact that our torque curves can be described only by the gfactor anisotropy, without showing any contribution from the orbital (spinon) motion in the 2D plane. We also point out that the thermal Hall effect in this material^{27} is negligibly small compared with a theoretical suggestion^{38}, which may be originated from the same physics.
The presently revealed extended Pauliparamagnetic quantum phase bears striking resemblance to metals with Fermi surface rather than insulators even though the charge degrees of freedom are frozen. We also emphasise that the presence of such a novel quantum phase might provide profound implications on the physics of other class of strongly interacting manybody systems in the vicinity of the end point of longrange order as well as the exotic quantum spin systems.
Methods
Sample preparation
Single crystals of EtMe_{3}Sb[Pd(dmit)_{2}]_{2} (Et=C_{2}H_{5}, Me=CH_{3}, dmit=1,3dithiole2thione4,5dithiolate) were grown by the airoxidation method. This material has a layered structure (space group C2/c) with the lattice parameters a=14.515 Å, b=6.408 Å, and the interlayer distance d=c*/2=18.495 Å. The deuterated crystals of d_{9}EtMe_{3}Sb[Pd(dmit)_{2}]_{2} were prepared by the same method using (Et(CD_{3})_{3}Sb)_{2}[Pd(dmit)_{2}]. The deuterium atoms were introduced by CD3I (99.5%D). Typical sample size is ~1×1×0.05 mm.
Torque measurements
All magnetic torque measurements were carried out by attaching one single crystal to a cantilever with a tiny amount of grease. Piezoresistive cantilevers were used in the measurements at the Kyoto University (μ_{0}H<7 T, T>0.3 K) and at Tsukuba Magnet Laboratories, NIMS (μ_{0}H<17 T, 30 mK ≤T≤1 K). In the measurement at high magnetic field facility at Grenoble (30 mK ≤T≤1 K), magnetic fields up to 32 T were provided by a resistive magnet and a capacitancesensing metallic cantilever was used. To determine the absolute value of (Fig. 2a), sensitivities of the each cantilever were calibrated in situ by detecting sin θ oscillation owing to the sample mass at zero field. Thermal equilibrium between samples and cryostat at the lowest temperature was ensured by immersing the whole setup into the mixture of the dilution refrigerator.
Electron paramagnetic resonance measurements
All EPR measurements have been performed in RIKEN by using a conventional Xband EPR spectrometer (JEOL, JESRE3X) equipped with a continuous Heflow cryostat. All EPR spectra have a symmetric Lorentzian line shape with the line width of 3 mT. The angular dependence measurements of the gvalue was carried out using the same single crystal at 4.2 K, and the gvalue for each angle has been precisely obtained from the correction using the Mn^{2+} field marker. It should be noted that detail mechanism of the anisotropy of the gtensor has been discussed in other Pd(dmit)_{2} compounds^{34}, (TMTTF)_{2}X (ref. 35), and inorganic spin chain materials^{40} from the dependence of the line width on the sample angle, the frequency and the temperature. Such comprehensive EPR studies of EtMe_{3}Sb[Pd(dmit)_{2}]_{2} remain as a future work.
Determination of susceptibility from the torque measurements
Here, we describe how we determine the uniform susceptibility and the magnetization from our torque measurements.
By adopting the principal axes of the gfactor (p, q and r=p×q as shown in Fig. 1c,d) as orthogonal coordinate system, the spin susceptibility tensor can be diagonalized as
The torque detected by the cantilever method is a component perpendicular to the fieldrotating plane (Fig. 1b). For the c*−a rotation this is given as
and for the c*−b rotation we obtain
These equations show that the torque should have a sinusoidal shape with twofold oscillation crossing zero at the maximum and minimum positions of the gfactor. This is exactly what we observed (Fig. 1e,f). This indicates that the susceptibility can be diagonalized as in equation (1) and that the anisotropy of the gfactor is the origin of the anisotropy of the susceptibility in this spin liquid system.
Having established that the magnetic susceptibility reflects the anisotropy of the gfactor, we assume that the susceptibility is proportional to the square of the gfactor () as in conventional paramagnetic systems. Then χ_{ij} can be obtained by the reduced susceptibility , which is determined from and as
where the principal values of the gfactor and θ_{0} are provided by the EPR measurements (Fig. 1e,f) as g_{pp}=1.9963, g_{qq}=2.0325, g_{rr}=2.0775, and θ_{0}=32 degree. The susceptibility for the magnetic field perpendicular to the basal plane is, therefore, given by
The two independent measurements and give quantitatively consistent results for (red and green circles in Fig. 2a, respectively), confirming the validity of our analysis.
Origin of the paramagnetic susceptibility
The obtained (T) in the spinliquid state of EtMe_{3}Sb[Pd(dmit)_{2}]_{2} is nearly temperatureindependent at low temperatures below ~2 K. This immediately indicates that in the torque measurements, the Curie contribution from the free spins of the impurities is almost exactly cancelled out. This indicates that such free spins are not subject to the gfactor anisotropy, which may be explained if the impurity spins reside at the EtMe_{3}Sb cation site between the dmit layers. The absence of the Curie term is also confirmed from the fieldlinear magnetization without saturation in a wide field range covering large B/T ratios, which cannot be fitted by the Brillouin function.
The paramagnetic temperatureindependent susceptibility, in general, may be originated from the orbital Van Vleck susceptibility and the spin Pauli susceptibility. However, the Van Vleck term is found to be negligibly small in the closely related isostructural material Et_{2}Me_{2}Sb[Pd(dmit)_{2}]_{2} with slightly different cations Et_{2}Me_{2}Sb. In this material, a large spin gap is formed below the firstorder transition at ~70 K, which has been attributed to the charge disproportionation in [Pd(dmit)_{2}]_{2} anions. The susceptibility jumps from the value close to the present EtMe_{3}Sb[Pd(dmit)_{2}]_{2} case at high temperature to almost zero below the transition, indicating the absence of the Van Vleck paramagnetic susceptibility^{41}. This strongly suggests that the sizable paramagnetic temperatureindependent susceptibility in the spin liquid state of EtMe_{3}Sb[Pd(dmit)_{2}]_{2} can be fully attributed to the spin Pauli susceptibility. We also note that the diamagnetic Landau susceptibility, if present, should give a very minor correction to the Pauli susceptibility, because we did not observe the quantum oscillations at high magnetic fields.
Estimation of the mobility of the gapless excitation
An important requirement to observe the quantum oscillation in the torque (de Haasvan Alphen effect) in a metal is a long meanfree path (v_{F} is the Fermi velocity and τ is the scattering time) to satisfy that exceeds unity, where ω_{c} is the cyclotron frequency, m* is the effective mass and μ is the mobility. In a simple metal, the mobility can be estimated by using the residual thermal conductivity κ_{0}/T, the Sommerfeld constant γ, and the Fermi energy through
By using κ_{0}/T≈0.2 W K^{−2}m^{−1} (ref. 27), γ≈20 mJ K^{−2}mol^{−1} (ref. 29) and /k_{B}~480 K estimated by the present susceptibility measurements, we can roughly estimate μ~0.18 T^{−1} in the spinliquid state of EtMe_{3}Sb[Pd(dmit)_{2}]_{2}. This yields ω_{c}τ>5 at 32 T, which strongly suggests that the impurity scattering effect is not the main origin of the absence of the quantum oscillations.
Additional Information
How to cite this article: Watanabe, D. et al. Novel Pauliparamagnetic quantum phase in a Mott insulator. Nat. Commun. 3:1090 doi: 10.1038/ncomms2082 (2012).
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Acknowledgements
We thank fruitful discussion with L. Balents, S. Fujimoto, T. Goto, M. Imada, N. Kawakami, Y.B. Kim, P.A. Lee, R. Moessner, N. Nagaosa, T. Sasaki, T. Senthil and K. Totsuka. This research has been supported through GrantinAid for the Global COE program 'The Next Generation of Physics, Spun from Universality and Emergence' from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, KAKENHI from JSPS and grantinaid for Scientific Research on Innovative Areas (nos. 20110003 and 23110716) from the MEXT. Part of this work has been supported by EuroMagNET II under the EU contract.
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D.W., M.Y., S.T., I.S., T.T. and S.U. made torque measurements. Y.O., H.M.Y. and R.K. grew and characterized the crystals. D.W. and M.Y. analysed the data. T.S. and Y.M. conceived and designed the project. M.Y., K.B., T.S. and Y.M. wrote the manuscript with input from all the authors. D.W. and M.Y. contributed equally to this work.
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Watanabe, D., Yamashita, M., Tonegawa, S. et al. Novel Pauliparamagnetic quantum phase in a Mott insulator. Nat Commun 3, 1090 (2012). https://doi.org/10.1038/ncomms2082
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DOI: https://doi.org/10.1038/ncomms2082
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