Main

Mutually interacting electrons with sufficiently strong Coulomb repulsion U fall into the Mott insulating state when the carrier density corresponds to an electron per site (a half-filled band)3. As the bandwidth W is increased by pressure or chemical substitution, the electrons gain kinetic energy and become itinerant at a critical value of W/U. The Mott transition, a marked phase transition between a metal and an insulator, is a collective manifestation of imbalance in the particle–wave duality of electrons. As one of the main issues in the quantum physics of condensed matter, the quantum-critical nature of the Mott transition awaits clarification. In contrast to intensive theoretical studies4,5,6, however, this issue has not yet been addressed experimentally because most Mott transitions in real systems have critical points at finite temperatures7,8,9,10,11; thus, they are not genuine quantum phase transitions.

In general, quantum criticality is observed at the temperature T sufficiently lower than the competing energy scales underlying the phase transition1,2, which are the bandwidth W and on-site Coulomb energy U in the case of the Mott transition. Thus, even if the system’s critical point, Tc, is finite, unlike the genuine quantum phase transition, in the case that Tc is orders of magnitude lower than W and U, there is a vast temperature region of Tc < T ≪ U, W, where the system can experience quantum criticality (Fig. 1a). Indeed, using dynamical mean field theory (DMFT), which can properly describe the Mott transition12, the authors of refs 4, 13 have suggested the scaling of transport for quantum criticality in an intermediate temperature range well above Tc.

Figure 1: The structure and phase diagram of organic Mott systems.
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a, Schematic phase diagram of the Mott transition. b,c, Structure of a conducting layer of κ-Cu2(CN)3 andκ-Cl (b) and EtMe3Sb-dmit (c). The conducting layers formed with (ET)2 and [Pd(dmit)2]2 dimer are stacked with non-magnetic insulating layers, alternately. One (ET)2 ([Pd(dmit)2]2) dimer has one hole (electron) and corresponds to a lattice site of an anisotropic triangular lattice. Strong on-site Coulomb repulsion and a half-filled band make the system a Mott insulator. d–f, Pressure–temperature phase diagram of κ-Cu2(CN)3 (d), κ-Cl (e) and EtMe3Sb-dmit (f). The red line represents the first-order Mott transition line terminating at a critical end point. The open circles indicate the metal–insulator crossover pressure Pc(T) defined in the text. The colour represents the magnitude of | log10, where is the normalized resistivity defined in the text. The lowest temperature experimentally accessible for EtMe3Sb-dmit is limited to 35 K, below which the He medium is solidified at pressures of interest for EtMe3Sb-dmit.

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To explore the possible Mott quantum criticality from the experimental side, we performed pressure studies of the electron transport for three different quasi-two-dimensional organic Mott insulators with anisotropic triangular lattices, κ-(ET)2Cu2(CN)3, κ-(ET)2Cu[N(CN)2]Cl and EtMe3Sb[Pd(dmit)2]2 (hereafter abbreviated to κ-Cu2(CN)3, κ-Cl and EtMe3Sb-dmit, respectively), where ET and dmit represent bis(ethylenedithio)tetrathiafulvalene and 1,3-dithiole-2-thione-4,5-dithiolate, respectively (Fig. 1b, c). In the Mott insulating phases, κ-Cu2(CN)3 and EtMe3Sb-dmit host quantum spin liquids (QSLs), whereas κ-Cl is an antiferromagnet14,15,16,17,18 (AFM). In the metallic phases, κ-Cu2(CN)3 and κ-Cl are superconducting (SC) at low temperatures, whereas EtMe3Sb-dmit remains a paramagnetic metal9,10,11,19,20,21 (PM). Then, the three systems have different types of Mott transition in their ground states, for example, QSL–SC, AFM–SC and QSL–PM transitions (Fig. 1d–f). Clear first-order Mott transitions are observed in κ-Cu2(CN)3 and κ-Cl up to Tc values of 20 K (ref. 22) and 38 K (ref. 11), respectively, whereas there is no clear first-order nature in the Mott transition in EtMe3Sb-dmit21; its critical temperature, if any, is well below 30 K. The critical temperatures of the three compounds are two or three orders of magnitude lower than the values of W, U, which are several thousand Kelvin or more19 (Fig. 1a); the orders-of-magnitude difference between Tc and U, W preserves the possibility of quantum criticality in the intermediate temperature region (Tc < T ≪ U, W). We measured resistivity curves ρ(P, T) under continuously controlled He-gas pressure P at various fixed temperatures to cover the metal–insulator crossover region and tested the quantum-critical scaling of the ρ(P, T) data. Figure 1d–f present coloured contour plots of the normalized ρ(P, T) (explained later in detail), which is shown to follow the quantum-critical scaling almost perfectly in the fan-shaped region where the colour changes, as described in the following section.

First, we define metal–insulator crossover pressures at a given temperature, Pc(T), as inflection points in the experimental log ρ(P, T) versus P curve (Supplementary Information). The Pc(T) determined at different temperatures forms a bow-shaped crossover line, which corresponds to the Widom line of the Mott transition4,13. The Widom line divides the insulating (δP ≡ P − Pc(T) < 0) and metallic sides (δP > 0), as observed in Fig. 1d–f. Figure 2 shows the normalized resistivity , T) ≡ ρ(δP, T)/ρc(T) of κ-Cu2(CN)3 as a function of δP, where ρc(T) ≡ ρ(δP = 0, T) is the crossover resistivity along the Widom line. It is observed that , T) crosses continuously from the insulating state (δP < 0) to the metallic state (δP > 0). As a result of the normalization, all curves cross at a single point for which δP = 0 and , and the slope at the inflection point is steeper at lower temperatures. It is noted that the volume change of the sample, which can be large particularly near the Mott transition, has no practical influence on the ρ(P, T) values, as explained in the Supplementary Information.

Figure 2: The normalized resistivity (δP, T) of κ-Cu2(CN)3 measured from the metal–insulator crossover line against δP = P − Pc(T).
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For a quantum phase transition1,2, as a system approaches a quantum-critical point while remaining at zero temperature, not only the spatial correlation length ξ but also the correlation time τ diverges as ξ ∝ | g − gc | −ν and τ ∝ ξz ∝ | g − gc | −zν, where g is the parameter controlling the quantum fluctuations, gc is the critical value of g, ν is the critical exponent of the correlation length, and z is the dynamical exponent. At finite temperatures, another timescale, LT ∝ T−1 appears because of a thermal effect; LT corresponds to the system size in the temporal direction, where LT → ∞ when T → 0. At finite temperatures, finite-size scaling is available, such that the ratio of LT to τ determines the development of critical fluctuations. Consequently, the singular parts of physical quantities are governed by one scaling parameter, τ/LT ∝ T/ | g − gc | zν. In the present study, considering the bow-shaped Widom line, it is natural to adopt δP = P − Pc(T) as the parameter controlling the quantum fluctuations instead of P − Pc(T → 0). Accordingly, if can be expressed as , T) = F [T/ | cδP | zν], where c is an arbitrary constant, and F(y) is a scaling function of the scaling variable y = T/ | cδP | zν, such that F(y) → 1 when y → ∞, we can conclude that the system is in the quantum-critical regime.

Turning to the experimental data for κ-Cu2(CN)3, all data for , T) in a wide temperature range (35 K ≤ T ≤ 90 K) collapse onto bifurcating scaling curves against T/T0 ≡ T/ | cδP | zν, with a critical exponent of zν = 0.62 ± 0.02, as shown in Fig. 3a. The scaling is fulfilled over several orders of magnitude. For κ-Cl and EtMe3Sb-dmit, similar scaling is observed for 75 K ≤ T ≤ 115 K with zν = 0.49 ± 0.01 (Fig. 3b) and for 35 K ≤ T ≤ 90 K with zν = 0.68 ± 0.04 (Fig. 3c), respectively (Supplementary Information). The scaling curves for the three systems nearly coincide with each other (Supplementary Information). It is remarkable that these three systems with different ground states show common scaling behaviour; this implies that the present Mott quantum criticality in the intermediate temperatures comes from the high-energy itinerant-localized competition, no matter which ground state the system falls into at low temperatures. These scaling curves clearly reveal the perfect ‘mirror symmetry’ of , T) between the insulating () and metallic () branches at T/T0 ≳ 1. The mirror symmetry indicates that the scaling function F(y) exhibits duality at y ≳ 1 between the metallic (δP > 0) and insulating (δP < 0) sides as Fmetal(y) = 1/Finsulator(y). It is noticeable that this duality of scaling is also observed in a metal–insulator transition in the two-dimensional electron gas system of a metal–oxide–semiconductor field-effect transistor23 (MOSFET), where the electron correlation is argued to play a key role in the metal–insulator transition, possibly a Wigner crystallization24.

Figure 3: Scaling analysis.
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a–c, The scaling plot of the normalized resistivity (δP, T) versus T/T0 = T/ | cδP | zν with the present values, zν = 0.62 and c = 20.9 for κ-Cu2(CN)3 (a), zν = 0.49 and c = 289 for κ-Cl (b), and zν = 0.68 and c = 13.1 for EtMe3Sb-dmit (c). The upper and lower curves are the insulating and metallic branches, respectively. The dashed line in a indicates that . The reason why the κ-Cl data are not extended in the insulating branch is that the Mott critical pressure of κ-Cl is so low that the insulating region would be extended at negative pressures experimentally inaccessible.

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The form of scaling function is explained by a typical quantum-critical regime. At T/T0 ≫ 1, the system does not ‘know’ whether it resides in the metallic or insulating side because of the large quantum-critical fluctuations. As T/T0 decreases, the system gradually ‘notices’ the regime to which it belongs. The , T) of the κ-Cu2(CN)3 versus T/T0 plot in Fig. 3a is shown using logarithmic scales in Fig. 4, which illustrates that for further decreases in T/T0, the metallic branch deviates from the quantum-critical behaviour following towards the Fermi-liquid T2 behaviour (Fig. 3a), followed by saturation to the residual resistivity, and the insulating branch also deviates from quantum-critical behaviour towards near-activation behaviour fitted by the form , T) = exp[(T/1.4T0)−1.1] with a gap of the order of T0. Similar behaviour is also observed for the other two compounds (see Supplementary Fig. 5). These behaviours at low T/T0 indicate the entrance of the high-temperature quantum-critical state into the low-temperature regimes of the Fermi liquid or the gapped Mott insulator. The T0 values for the three compounds, which characterize quantum-critical regions, are shown in Supplementary Fig. 4.

Figure 4: Crossover behaviour of the scaling curve.
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Plot of log10 | log10 versus T/T0 for κ-Cu2(CN)3. The upper and lower panels are the insulating and metallic branches, respectively. The dashed and dotted lines correspond to quantum-critical behaviour (zν = 0.62, + : insulating branch, − : metallic branch) and nearly activation-type behaviour , respectively. The arrows indicate the characteristic value of T/T0, represented as (T/T0)∗, for the entrance to the low-temperature regimes of the gapped Mott insulator or the Fermi liquid, determined by eye.

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The normalized resistivity , T) is represented by a range of colour in Fig. 1d–f, to visualize the quantum-critical region (the red, yellow and green regions), where the mirror symmetry in Fig. 3a–c is near-perfect. A depiction of in the δP–T plane instead of the P–T plane highlights the mirror symmetry of the quantum-critical region, as shown in Supplementary Fig. 6.

The scaling features of the present data surprisingly coincide with those of recent DMFT calculations4,13. The most remarkable point is that both the experimental and theoretical scaling curves have the same functional form, with the mirror symmetry. Moreover, the critical exponent zν = 0.57 in DMFT (ref. 4) is close to the present values zν = 0.62 for κ-Cu2(CN)3, 0.49 for κ-Cl, and 0.68 for EtMe3Sb-dmit. It is also noticeable that these experimental values are comparable to zν = 0.67, a value expected for the genuine quantum Mott transition25. Although there seem to be appreciable differences between the zν values (0.62 ± 0.02, 0.49 ± 0.02 and 0.68 ± 0.04), we note that they are in a window of 0.6 ± 0.1 and that the differences between the present zν values are much smaller than the possible range of zν values for quantum metal–insulator transitions; for example, a value of zν = 2 (z = 4, ν = 1/2) is proposed theoretically for the marginal quantum Mott transition in ref. 6 and a value of zν = 1.6 (z = 1, ν = 1.6) is deduced experimentally in a Si-MOSFET in ref. 26. Thus, the present results suggest a possible universality class with a zν value around 0.6. However, we do not exclude a possibility that the small distribution in zν values is a signature of an intriguing case with non-universal critical exponents such as the Berezinskii–Kosterlitz–Thouless transition27, where the universality can be characterized by lines of fixed points in renormalization group theory, and the variation of zν comes from the microscopic parameters of material such as lattice geometry. In the DMFT work, the temperature range in which the scaling is valid is approximately 2Tc < T < 0.12t, where t is the nearest-neighbour transfer integral in the Hubbard model. Both the lower and upper limits nearly coincide with the present results. Furthermore, the bow-shaped crossover line and the fan-shaped quantum-critical region are common to both the present and the DMFT works. As DMFT can treat only fluctuations independent of wavenumber k, the present agreement may indicate that the fluctuations of Mott quantum criticality, where electrons are neither particle-like nor wave-like objects, can be of the k-independent local nature.

In general, quantum-critical behaviour springs from a quantum phase transition associated with symmetry breaking. However, the Mott quantum criticality in question is associated with the charge delocalization transition without symmetry breaking and has been actually demonstrated here to occur irrespectively of the presence/absence of symmetry breaking in spin degrees freedom. Thus, the present quantum criticality is beyond the conventional symmetry-breaking framework. Unconventional quantum criticality has also been discussed for heavy-electron systems in light of the Kondo breakdown, where the reconstruction of Fermi surfaces or an orbital-selective Mott transition due to itinerant-localized competition of f electrons supposedly causes unconventional quantum criticality28,29. Moreover, theoretical investigations of the holographic duality of current interest have proposed that strange metals in heavy electrons and cuprates are in yet-unspecified quantum-critical regimes30,31. It is likely that correlated quantum systems of organics, heavy electrons and cuprates carry a new class of quantum criticality that originates from the itinerant-localized competition rather than symmetry breaking.

Methods

Single crystals of κ-(ET)2Cu2(CN)3 and κ-(ET)2Cu[N(CN)2]Cl were grown by conventional electrochemical oxidation. Single crystals of EtMe3Sb[Pd(dmit)2]2 were obtained by air oxidation of (EtMe3Sb)2[Pd(dmit)2] in acetone containing acetic acid32. The typical sample sizes are ∼0.3 mm × 0.3 mm × 0.006 mm for κ-(ET)2Cu2(CN)3, ∼1.0 mm × 0.5 mm × 0.2 mm for κ-(ET)2Cu[N(CN)2]Cl, and ∼0.8 mm × 0.9 mm × 0.05 mm for EtMe3Sb[Pd(dmit)2]2. The in-plane electrical resistivity was measured using the standard d.c. four-probe method under an isothermal pressure sweep (descending processes), using helium gas as the pressure medium. To examine the sample dependence, measurements were performed for two samples of the same compound simultaneously in the same run. The reproducibility of the scaling behaviour was basically ensured for each compound. Gold wires of 25 μm in diameter were glued on the crystal faces with carbon paste as electrodes. Throughout the experiments, we confirmed that the resistivity was independent of the applied current.