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Quantum Spin Liquids Unveil the Genuine Mott State

The Widom line identifies the locus in the phase diagram where a supercritical gas crosses over from gas-like to a more liquid-like behavior. A similar transition exists in correlated electron liquids, where the interplay of Coulomb repulsion, bandwidth and temperature triggers between the Mott insulating state and an incoherent conduction regime. Here we explore the electrodynamic response of three organic quantum spin liquids with different degrees of effective correlation, where the absence of magnetic order enables unique insight into the nature of the genuine Mott state down to the most relevant low-temperature region. Combining optical spectroscopy with pressure-dependent dc transport and theoretical calculations, we succeeded to construct a phase diagram valid for all Mott insulators on a quantitative scale. In the vicinity of the low-temperature phase boundary, we discover metallic fluctuations within the Mott gap, exhibiting enhanced absorption upon cooling that is not present in antiferromagnetic Mott insulators. Our findings reveal the phase coexistence region and Pomeranchuk-like anomaly of the Mott transition, previously predicted but never observed.

The Widom line identifies the locus in the phase diagram where a supercritical gas crosses over from gas-like to a more liquid-like behavior. A similar transition exists in correlated electron liquids, where the interplay of Coulomb repulsion, bandwidth and temperature triggers between the Mott insulating state and an incoherent conduction regime. Here we explore the electrodynamic response of three organic quantum spin liquids with different degrees of effective correlation, where the absence of magnetic order enables unique insight into the nature of the genuine Mott state down to the most relevant low-temperature region. Combining optical spectroscopy with pressure-dependent dc transport and theoretical calculations, we succeeded to construct a phase diagram valid for all Mott insulators on a quantitative scale. In the vicinity of the low-temperature phase boundary, we discover metallic fluctuations within the Mott gap, exhibiting enhanced absorption upon cooling that is not present in antiferromagnetic Mott insulators. Our findings reveal the phase coexistence region and Pomeranchuk-like anomaly of the Mott transition, previously predicted but never observed.
The Mott metal-insulator transition (MIT) stands out among the key unresolved phenomena in interacting electron systems [1]. Sufficiently strong Coulomb repulsion is able to localize the charge carriers accompanied by a splitting of the nominally half-filled band into the lower (LHB) and upper (UHB) Hubbard bands separated by the Mott gap (Fig. 1b). When temperature becomes the dominant energy scale, i.e. comparable to the bandwidth W and Coulomb interaction U, the Hubbard bands are strongly blurred by thermal broadening resulting in an incoherent conduction regime (Fig. 1d). While the Mott transition is a well-defined, first-order MIT below the critical endpoint, it shows analogy to the liquid-gas crossover in the supercritical high-temperature region [2,3], recently associated with quantum critical scaling [4]. In this part of the phase diagram, the quantum Widom line (QWL) defines the boundary of the fully gapped Mott state [5]. A similar description applies to the pseudogap in doped Mott insulators, which underpins the decisive role of Mott physics for high-temperature superconductivity in cuprates [6].
At low temperatures (k B T < 10 -2 W) magnetic instabilities typically mask the Mott MIT; the antiferromagnetic ground state dominates the low-energy excitations and, due to the Clausius-Clapeyron relation dp/dT=ΔS/ΔV, the small entropy implies a negative slope of the phase transition (Fig. 1f). To circumvent this problem, we study Mott insulators that are currently under scrutiny for their quantum spin liquid ground state (QSL) as a result of large geometrical frustration and disorder [7,8,9,10,11,12]. The absence of antiferromagnetism enables us to investigate the genuine Mott state down to T  0; due to the large spin degrees of freedom, the phase transition acquires a positive slope reminiscent of the Pomeranchuk-effect in 3 He, where the solid (insulator) becomes liquid (metal) upon cooling (Fig. 1e). While from theory it is known that the phase diagram is controlled by the ratios T/W and U/W [13,5], where temperature and Coulomb repulsion are normalized to the bandwidth, experimental studies have not advanced beyond Tp plots so far, inhibiting a quantitative comparison (i) between different compounds and (ii) with theory. For several organic charge transfer salts we succeeded to determine U and W from the optical spectra and combine our findings with pressure-dependent dc transport and dynamical mean-field theory calculations. By identifying the relevant scales that control the transition, our approach allows us to put all Mott insulators --including nonfrustrated compounds and transition metal oxides --on one universal phase diagram in a quantitative way. Moreover, our optical data provide evidence for charge fluctuations close to the phase boundary on the insulating side, enhanced upon cooling due to the Pomeranchuk-like instability. Although theoretically predicted, this phase coexistence has never been observed before.
Panel d displays the prototypical phase diagram of a Mott insulator; the gap closes at the QWL (green data points determined by DMFT calculations, see Methods), which corresponds to a certain relation between U, W and T. Since both scales (temperature T/W and electronic correlations U/W) are expressed in units of W, the underlying physics is adequately described by only two parameters: the on-site Coulomb repulsion U and the bandwidth W. As we will later see, the reversal of the QWL provides a fixed point to collapse the Widom lines of different compounds onto one universal curve. Strictly speaking, the bandwidth of the Hubbard bands W is generally smaller than the bare bandwidth W 0 of the uncorrelated system. However, the effective values U/W remain a good approximation of the intrinsic U/W 0 ratio even for strong correlations [15]. In order to determine the density of states (DOS) and optical conductivity spectra for different correlation strength U/W, we performed complementary DMFT calculations utilizing the continuous time quantum Monte Carlo (CTQMC) quantum impurity solver (Methods and Supplementary information S5, S7). The results affirm in particular that the maximum of the Mott-Hubbard band is proportional to the Coulomb repulsion U (Fig. 1 c) and the U/W ratio extracted from the band shape is closely related to the bare correlations U/W 0 .

Optical Studies
Optical spectroscopy is the method of choice to experimentally determine the band parameters, as it directly maps the band shape by observing the transitions between the Hubbard bands, as illustrated in Fig. 1c. By measuring the infrared reflectivity of EtMe, AgCN and CuCN single crystals at different temperatures, we obtain the optical conductivity  1 () plotted in Fig. 2. Since the optical properties are rather isotropic within the highly-conducting plane, we focus on the polarization along the direction of the largest conductivity and present the data for the perpendicular axis in the Supplementary information S3. The most prominent feature is the Mott-Hubbard band centered around 2000 cm -1 , also seen in related materials [16,17,18,19]. Our CTQMC calculations reproduce the overall shape, asymmetry and intensity of the bands very well, establishing the dominant role of Mott physics in these compounds. The narrow peaks below 1500 cm -1 correspond to molecular vibrations and lattice phonons [20,21], which can be clearly separated from the overall change of the Mott-Hubbard band in the contour plots.
As indicated in Fig. 2d frequencies, and thus the bandwidth determined as W/ℏ =  max - 1/2 , change upon cooling. Due to the band asymmetry, we extract the bandwidth from the low-frequency wing of the Mott-Hubbard band for it is related to the intrinsic low-energy excitations of the ground state (Fig.  2g). The temperature-dependent band parameters, U and W, and the corresponding degree of electronic correlations, U/W, are plotted in Fig. S5 of the Supplementary information. While the Mott-Hubbard bands of EtMe and AgCN are subject to narrowing upon cooling, a pronounced shift towards low frequencies and unusual broadening are observed for CuCN. In order to disentangle thermal effects and to focus on quantum Mott physics, we first analyze the data at the lowest temperatures; the temperature dependence of the low-frequency conductivity will be discussed in the context of Fig. 4.
From the spectroscopic data measured at T = 5 K, we conclude that electronic correlations are largest in EtMe despite the small U. Indeed, the small bandwidth in this material implies that (i) it is located most left in the U/W phase diagram, (ii) the experimentally accessed temperatures (5 -300 K) cover a broader vertical T/W range and (iii) the electronic compressibility is largest because pressure has a larger effect on U/W. The two κ-compounds have larger energy scales and thus extend over smaller T/W and U/W ranges. Consequently, in a unified phase diagram the three materials are arranged in the order EtMe -AgCN -CuCN on the descending horizontal U/W scale (Fig. 2h), consistent with ab initio density functional theory (DFT) and extended Hückel calculations (Supplementary information S1, Table S1) [22,23,24,25,11,8].

Formation of the Mott Gap at the Quantum Widom Line
A closer inspection of the temperature-dependent optical conductivity of EtMe ( Fig. 3a and b), reveals a well-defined Mott gap that opens around 125 K and continuously grows up to 650 cm -1 when the temperature is lowered to 5 K. This coincides well with the crossover temperature from Ref. [4] extrapolated to p = 0 establishing the QWL as the true boundary between the Mott state with a well-defined charge gap and the incoherent conduction regime in the supercritical region.
Panels c and d of Fig. 3 display the temperature-dependent dc conductivity and the corresponding transport gap Δ calculated from the logarithmic resistivity derivative d(ln)/d(1/T) according to ρ(T) ∝ exp(Δ/k B T), respectively. For all three compounds Δ(T) goes through an absolute maximum at a particular temperature, which coincides with the opening of the optical gap of EtMe (Fig. 3a) and can be associated with the QWL. There is similar agreement between Ref. [4] and our pressure-dependent dc transport results for CuCN revealing the QWL at 185, 135 and 123 K at p = 0, 2.17 and 2.38 kbar, respectively (Fig. S9). Although for AgCN there is a smaller feature at 90 K, Mott physics is reflected by the global maximum of Δ(T) implying that the QWL is crossed around 245 K. Therefore, the maximum of the transport gap allows us to identify the QWL even though no clear-cut optical gap is observed for CuCN and AgCN, probably due to the smaller U/W ratio. We also notice that the size of the transport gap maximum scales with the Coulomb repulsion U determined by optical spectroscopy.

Low-Frequency Electrodynamic Response
As the relevant energy scales in the phase diagram are of the order of 100 K or less, fingerprints of the ground state are expected in the low-frequency electrodynamic response, i.e. in the THz or far-infrared spectral ranges. A closer look on the temperature evolution of the optical conductivity well below  max (Fig. 4c) illustrates the insulating nature of the ground state of EtMe, where the sub-gap absorption diminishes upon cooling as thermal band broadening is reduced (cf. Fig. S6a). The integrated spectral weight, plotted in Fig. 4d, provides a robust measure of these low-energy excitations enabling a more quantitative analysis beyond dc transport. This quantity determines the absorption strength up to a given cutoff frequency  i , chosen to be below the band edge (U -W) to probe the low-energy excitations only, and yields valuable information on correlation effects [28]. Here, N is the charge carrier density and m* indicates the effective mass, renormalized by the band structure and electronic interactions.
While the overall spectral weight grows with decreasing electronic correlations (Fig. S7c), the temperature evolution SW( i ,T) plotted in Fig. 4 reveals an unexpected behavior. Although less pronounced than in EtMe, also AgCN behaves like a typical insulator, where the in-gap states are depleted upon cooling [29]. The opposite behavior is found in the case of CuCN, which exhibits a temperature dependence characteristic of metals (panel a, cf. Fig. S6b): the lowfrequency optical conductivity increases upon lowering the temperature [30,31]. This is unexpected considering that no Drude peak is present, the hallmark of coherent transport, and that at zero frequency all compoundsincluding CuCNare electrical insulators as determined from dc transport (Fig. 3c).
This apparent contradiction can be understood by the exceptional position of CuCN in the phase diagram: this compound is more proximate to the insulator-metal boundary than EtMe and AgCN (cf. Fig. 2h). Due to the Pomeranchuk-like anomaly, below the back-bending point of the QWL, the insulator-metal boundary gets closer to the ambient pressure position as the temperature is lowered (cf. Fig. 1e). Thus, we assign the strong non-thermal enhancement of the low-energy spectral weight below T back = 70 K to metallic fluctuations in the Mott state that appear upon entering the coexistence region close to the phase boundary, which has been predicted theoretically [13,5] but never observed. From the trend of our data we expect the anomalous slope of the Mott MIT to extend to the lowest measured temperature. Going back to Fig. 3d, the transport gap of CuCN gradually decreases and seems to vanish for T  0 consistent with a weakening of the insulating behavior. Therefore, the QWL not only marks a crossover in the conduction properties at elevated temperatures but is also manifest in the sub-gap region of the low-temperature optical absorption of CuCN.
If this was a fundamental property that occurs in general in the vicinity of the metallic state, it should also be identified in the optical data of the -(BEDT-TTF) 2 Cu[N(CN) 2 ]Br x Cl 1-x series (detailed analysis in Supplementary information S6), where increasing Br content drives the bandwidth-tuned Mott MIT metallic and even superconducting [16,32,33,34,17]. Due to the substantially weaker frustration t/t, the system orders antiferromagnetically at low temperatures, leading to a phase diagram like in Fig. 1f. Above T N , however, these compounds behave similar as the frustrated Mott insulator CuCN and exhibit a comparable non-thermal enhancement of the SW corroborating our conclusions above. When antiferromagnetism is stabilized, a well-defined gap opens and the low-energy states are depleted; these intrinsic properties clearly distinguish magnetically ordered and QSL Mott states.

Unified Phase Diagram
Our comprehensive optical and transport results render possible the construction of the unified phase diagram of Fig. 5 that, on a quantitative scale, is valid for all frustrated Mott insulators. The universal quantum Widom line includes data points of EtMe and CuCN [4] and the maximum of dln()/dp calculated for AgCN [11]. On basis of our optical data, indicated by the left T/W and bottom U/W scales (cf. Fig. 2), we scaled the individual phase diagrams and merged them together (for details see Supplementary information S2). The horizontal and vertical bars on the top and right axes denote the pressure and temperature range (5 -300 K) covered by the experimental data of each compound, respectively. In Fig. S12 we illustrate that, upon proper renormalization to the intrinsic energy scales, the inorganic Mott insulator (V 1-x Cr x ) 2 O 3 fits well into the unified phase diagram, yet covering just a small range due to its large bandwidth.
The normalized pressure and temperature scales enable us to compare conveniently the experimental findings with results obtained from dynamical mean field theory (DMFT) calculations (Fig. 1d). The overall shape of the QWL, i.e. the characteristic back-bending at low temperatures, reminding of the Pomeranchuk effect, is consistently found in both experiment and theory. Even on a quantitative level, the U/W and T/W values match remarkably well, considering the present level of theory, i.e. the single-band Hubbard model as a valid effective lowenergy description for the different compounds under study.

Outlook
By combining pressure-dependent transport data with optical conductivity, we can map the quantum Widom line up to an energy range never achievable in transition metal oxides, for they have a considerably larger bandwidth compared to the organic compounds studied here. Several thousands of Kelvins and pressures beyond the accessible experimental range are required to cover the entire phase diagram in Fig. 5 by similar experiments in vanadium oxides, for instance (cf. Fig. S12). In the overlapping region close to the reversal of the quantum Widom line, however, the behaviors agree rather well [35,36,37] indicating that the bandwidth is the proper scaling parameter and the underlying physics of genuine Mott insulators is universal. To further confirm the generality of our findings, it is desirable to extend these investigations to spin liquid compounds of different symmetry, e.g. kagome and other geometries; the intrinsic properties of the Mott MIT should prevail.
For transition metal compounds charge-carrier doping is the far more common way to go from the Mott insulating to the metallic regime, most prominent in the high-temperature superconducting cuprates. Despite enormous efforts devoted to this issue over the last 30 years, no general agreement was reached on a generic phase diagram that is in-depth understood comparable to the bandwidth-controlled Mott transition. Very recently, theoretical studies identified the pseudogap in hole-doped cuprates as a Widom line and ascribe the transport, dynamic and thermodynamic crossovers to a common origin linked to the intrinsic Mott state [6,38]. A similar rescaling approach might be appropriate to reveal the inherent relationship between Coulomb repulsion, doping and superconductivity in transition metal oxides. Moreover, evaluation of the scaling properties on the metallic side will provide important information on the emergence of strongly-interacting coherent quasiparticles from the interplay of U, W and T.
We could trace the back-bending of the Mott insulator-metal boundary down to lowest temperatures and unequivocally identify the Pomeranchuk-like instability in the electrodynamics of -(BEDT-TTF) 2 Cu 2 (CN) 3 ; at finite frequencies, the compound becomes more metallic upon cooling although remaining an insulator in the dc-limit. This way we finally solved the longstanding dispute of excess conductivity and counterintuitive temperature dependence in the optical conductivity of this spin-liquid compound [30,39,31]. The anomalous slope of the Mott insulator-metal boundary seems to persist down to lowest temperatures. Such emerging metallic character may reflect the phase coexistence and spatially inhomogeneous structures caused by (weak) disorder in the vicinity of the phase transition. This interesting issue could be studied by systematically adding extrinsic disorder by heavy-ion irradiation [40,41,42]. The Anderson-Mott transition is subject of interest for decades [43], but here we provide a new possibility by exploring the dynamical properties of the fully frustrated case down to low temperatures and scaling the results in a quantitative manner.

Materials
Plate-like organic single crystals were grown during several months by air oxidation in acetone (EtMe) [23,44] and electrochemical oxidation (AgCN and CuCN) [45,25] in RIKEN, Nagoya, Stuttgart and Argonne. The samples reach typical dimensions of 3 and -(BEDT-TTF) 2 Cu 2 (CN) 3 , respectively. After selection and cleaning, the crystals are measured without further treatment; in particular the optical reflectivity is probed on as-grown crystal surfaces.

Optical Spectroscopy
Broadband optical spectra were recorded employing various Fourier-transform infrared spectrometers equipped with the corresponding sources, beam splitters, polarizers, windows, and detectors. The reflectivity was obtained by normalizing the signal off the sample to a gold mirror taking into account the tabulated frequency-dependent reflectivity of gold. To reach accurate absolute values and to account for diffraction effects in the far-infrared (/2c < 700 cm -1 ), we first measured the temperature-dependent sample reflection normalized to a big mirror, followed by evaporation of a thin gold layer (approx. 300 nm thickness) on the sample and, subsequently, a second measurement of the covered sample at the same temperatures. Eventually, the reflection was calculated by dividing the sample spectra without and with gold, both of which were normalized to the big mirror to account for temporal fluctuations of the spectrometer. The low-frequency limit typically was 30 to 100 cm -1 depending on the crystal size. The frequency resolution was set to 1 cm -1 . Several optical cryostats enabled us to cover the range 5 K < T < 300 K. Prior to cooling, the sample was aligned parallel to a gold mirror and the crystal axes were determined with an automated polarizer. Then the cryostat was evacuated to minimize thermal contact to the surroundings and avoid the formation of ice on the sample. During cooling, the typical pressure in the sample chamber was 10 -6 -10 -5 mbar. The cooling rate was 1 K/min.

Data Processing
In order to calculate the frequency-dependent conductivity by the Kramers-Kronig relations, the optical reflectivity (see Figs. S2 and S3) obtained by the various methods and spectrometers was merged. In rare cases it was required to slightly shift the raw data (several percent mismatch can occur due to alignment in different setups). Then, the low-and high-frequency reflectivity was extrapolated to 0.001 and 10 6 cm -1 , respectively. While generalized Hagen-Rubens behavior (R = a -b  n where 0 < n < 0.5) was extrapolated at low frequencies, a  -4 decay was applied towards high frequencies. For -EtMe 3 Sb[Pd(dmit) 2 ] 2 and -(BEDT-TTF) 2 Cu 2 (CN) 3 , the lowfrequency optical conductivity resulting from the Kramers-Kronig calculation was replaced by the THz data. Most importantly, the optical conductivity discussed here is not affected by the extrapolation.

Electrical Transport
Temperature-dependent measurements of the dc resistivity were performed within the plane using the standard four-probe method with a typical current of 10 µA to avoid heating of the sample. The contacts were made by pasting 25 µm gold wires with a small amount of carbon paint directly on the crystal. The specimens were mounted on a sapphire plate for good thermal contact. Temperature-dependent measurements were conducted in a 4 He exchange gas cryostat down to 2 K. The samples were slowly cooled down with approximately 0.1 K/min to avoid cracks, to ensure thermal equilibrium and to minimize effects associated with disorder.
Pressure-dependent experiments on -(BEDT-TTF) 2 Cu 2 (CN) 3 were performed in a CuBe oil pressure cell. The pressure was corrected for the temperature-dependent freezing of the medium. In Fig. 5 and S1 we incorporated the data of the pressure and temperature dependent resistivity measurements of Furukawa et al. and Shimizu et al. [4,11] on -EtMe 3 Sb [Pd(dmit) 2 ] 2 , -(BEDT-TTF) 2 Ag 2 (CN) 3 , and -(BEDT-TTF) 2 Cu 2 (CN) 3 . As sketched in Fig. 5 right insets, the inflection points of the resistivity versus pressure are taken as a crossing of the quantum Widom line at certain temperatures. It agrees well with the determination from our temperaturedependent transport studies presented in Fig. 3.

Dynamical Mean Field Theory Calculations
Dynamical Mean Field Theory (DMFT) [47] calculations were performed using a single-band Hubbard model at half-filling, and assuming a simple semi-circular density of states. We used the state-of-the-art Continuous Time Quantum Monte Carlo (CTQMC) quantum impurity solver, as first implemented by Haule [48]. The Monte Carlo step was set to 1×10 9 for precise sampling.
For evaluating the quantum Widom line (QWL) the DMFT calculation was performed on the triangular lattice to compare with the organic compounds in this report. To obtain a reliable QWL, the iterated perturbative theory (IPT) was applied as our impurity solver, which has been shown to have accurate results compared to numerically exact CTQMC but suffer from less numeric error, especially in triangular lattice systems. We employed the λ analysis to locate the QWL [13,5], where λ corresponds to the curvature of the free energy functional and shows a minimum at the QWL. More specifically, λ can be obtained from the iterative solution of DMFT equations as described in Ref. [13,5], (4) where δG (n) =G (n) -G DMFT and G DMFT is the converged DMFT Green's function. Therefore, λ corresponds to the convergence rate towards the DMFT solution. Figure 1 Electronic excitations of the Mott insulator on a half-filled triangular lattice. a, The compounds under study have a layered structure with dimers (blue circles) of organic molecules (gray bars) arranged on a triangular lattice, where t d > t (t) are the intra-and interdimer transfer integrals. b, Density of states (DOS) of the Mott insulator. The separation of the lower and upper Hubbard bands (green) is a measure of the Coulomb repulsion U (Supplementary information S1). Although generally smaller, the width W of the Hubbard bands remains a good measure of the bare bandwidth W 0 = 9t in the absence of correlations [15]. c, The band parameters are experimentally accessible through the optical conductivity, where the peak position  max corresponds to U and the half-width equals W. In the metallic case (U = 0) this transforms to a zero-frequency conductivity; for small, but finite scattering the width equals the scattering rate γ of the Drude response while for bad metals the upper bound W 0 is approached. d, Schematic phase diagram of the Mott insulator with the sketched DOS for different regions. While in the high-temperature regime charge transport is dominated by incoherent thermal excitations and the Hubbard bands are strongly blurred, a well-defined charge gap opens in the Mott insulating state and a Drude peak appears on the metallic side at low temperatures. The boundary between the conducting regime and the fully gapped Mott state is the so-called quantum Widom line (dashed pink). The back-bending at low temperatures is nicely reproduced by DMFT calculations (green points). e, Zoom to the low-temperature region around the insulator-metal boundary. For t = t geometrical frustration suppresses magnetic order down to lowest temperatures; the large entropy of fluctuating spins causes a positive slope of the Mott MIT reminiscent of the Pomeranchuk effect in 3 He. Up to the critical endpoint the phase transition is of first order while at higher temperatures the quantum Widom line defines a crossover. f, When geometrical frustration is reduced, antiferromagnetic order develops at low temperatures. The ordered state has less entropy than the metallic phase implying dT N /dp < 0.    The temperature T and Coulomb repulsion U are normalized to the bandwidth W extracted from optical spectroscopy; note that the direction of the bottom axis is reversed in order to mimic pressure. Since in these quantum spin liquids magnetic order is suppressed, the large residual entropy causes a pronounced back-bending of the quantum Widom line at low temperatures leading to metallic fluctuations (semi-transparent green area) in the Mott state close to the boundary. As the effective correlations decrease further, a metallic phase forms (green area) with Fermi liquid properties [34]. The universal phase diagram guided by the quantum Widom line is constructed on basis of our optical (

S1 Band Parameters: Definitions
Strictly speaking, the Coulomb interaction U referred to in the text and in Fig. 1    , this causes the extracted U/W to be generally larger than the calculated U/W 0 [S3,S9,S10,S11,S12,S13,S14].

S2 Construction of the Phase Diagram from Pressure-Dependent dc-Transport and Optics
Making use of our conclusions from the optical spectroscopy results presented in Fig. 2 we relate the transport results of the three frustrated Mott insulators to each other and combine them to a universal quantum Widom line (QWL). In the T-p phase diagrams shown in Furukawa et al.
[S15] the back-bending point of the QWL is located at a lower temperature for EtMe (T back < 35 K) as compared to CuCN (T back = 70 K) (Fig. S1). This agrees with the smaller bandwidth of EtMe providing evidence for the T/W scaling of the phase diagram. Considering that EtMe is located much deeper in the Mott insulating regime (cf. Fig. 2h) Fig. 1d and f from Ref. [S15], are in agreement. For AgCN, the QWL was extracted from the pressure-dependent resistivity reported in Ref. [S13] and the ambient pressure result from Fig. 3d, and   Due to the different experimental approach of Shimizu et al. [S13], who measured temperature sweeps at constant pressure rather than pressure sweeps at constant temperature, the QWL determination was not as accurate for AgCN, as indicated by the error bars in Fig. 5. Note that they concluded an absolute pressure shift of 0.6 GPa between AgCN and CuCN while our procedure results in a shift of approximately 0.9 GPa, as shown in equation (S1). Their assumption, however, is based on pressure-dependent transport studies of CuCN employing oil as pressure medium [S17]. It is known that such setups suffer from a pressure loss at low    In Fig. S4 we plot the temperature-dependent optical conductivity ) ( 1   for both in-plane polarizations of the three compounds under inspection, EtMe, AgCN and CuCN. In their overall appearance, the spectra are very similar for both polarizations with the charge-transfer band around 2000 cm -1 . In the two -salts, however, the peak splits into a low-and high-energy part for the polarization E || b. The latter corresponds to intra-dimer excitations while the former one is related to the charge transport between the dimers; due to strong Coulomb repulsion the conduction band is split into the lower and upper Hubbard bands with transitions between them right around 2000 cm -1 [S1,S5,S18]. Due to the overlapping features, the determination of  max of the inter-dimer part was less accurate as for the c-direction where the intra-dimer charge transfer is less pronounced. In the case of the EtMe compound the intra-dimer excitation band appears at much higher frequencies (around 4000 cm -1 for E || a) because of its stronger dimerization. We also extracted the optical parameters for E || b which is plotted in Fig. S5  At the lowest temperature, the absolute U/W value is slightly larger along E || b implying stronger correlations and, thus, a position more left in the phase diagram. However, this anisotropy is substantially smaller than the insecurity in combining the Widom lines from transport measurements (see Section S2). The (U/W) exp values in Table 1 are taken from E || c due to the generally smaller error bar.

S4 Low-Energy Electrodynamic Response
Metals and insulators are commonly defined according to the temperature-dependence of their resistivity. In a metal, the main changes of resistivity are related to the scattering rate γ, i.e.
, which increases with temperature and thus enhances (reduces) the resistivity (dc conductivity). For an insulator, the dominant contribution to charge transport is thermal activation; unlike in a metal the conductivity increases with temperature. As sketched in Fig. S6, a similar behavior is expected for the low-frequency optical conductivity ) , ( 1 T   at energies well below the gap and scattering rate for insulators and metals, respectively. a, For an insulator, the band is located at finite frequency and the conductivity within the gap is very small. Upon increasing the temperature the spectral weight is shifted from the maximum to the (high-and) low-energy region causing an enhancement of the low-frequency conductivity. b, Since for a metal the Drude response is centered at 0   , the low-energy contribution is suppressed with temperature. In the limit of very small frequencies the optical conductivity approaches the dc conductivity (insets), i.e. ; this is also illustrated in Fig. 4d. For an insulator the conductivity is thermally activated and thus increases with temperature; the opposite behavior is found for a metal, where scattering impedes charge transport.
As intensely discussed in the main part, for EtMe and AgCN insulating behavior is consistently found in the dc and low-frequency optical conductivity. For CuCN, however, the sub-gap spectral weight exhibits an anomalous increase upon cooling reminiscent of metallic behavior despite insulating transport properties. We interpreted this anomaly as metallic fluctuations appearing in the vicinity of the insulator-metal phase boundary.
Although there is some anisotropy in the dc response and optical conductivity, the overall temperature dependence of charge transport is similar for both in-plane crystal axes. As shown in Fig. S7a, in CuCN and AgCN the low-frequency optical conductivity is higher along the cdirection compared to E || b, similar to the dc results [S19,S20]. This anisotropy is also expressed in the maximum position of the band; similar to the transport gap, U is larger for E || b (Fig.   S5a). This observation is explained by the fact that the bandwidth is largest along the c-direction.
There is also small anisotropy for EtMe. The overall temperature evolution of the low-frequency response (Fig. S7b,c), however, is robust and does not depend on the polarization which assures our assumption that we probe the intrinsic behavior of the two-dimensional electron gas subject to strong correlations. Having two independent measurements strengthens our conclusion. Although the optical conductivity is anisotropic, there is no significant difference in the temperature dependence of the spectral weight, as indicated by the arrows. While it decreases for the former two compounds, it is enhanced for the latter one upon cooling, which is unexpected for an insulator (cf. Fig. S6). The arrows denote the frequency  i at which the temperature dependence of the spectral weight was compared in Figs. 4 and S11a.

S5 Theoretical Calculations
In Fig. S8 we present the DOS (panel a) and optical conductivity (panel b) calculated by the continuous time quantum Monte Carlo (CTQMC) method. As expected, and in agreement with the experimental spectra presented in Fig. 2   The assumed T/W = 0.0075 corresponds to the low-temperature limit of our experiments. Inset: The linear relation between the maxima of optical conductivity and the Coulomb repulsion U reassures our analysis based on Fig. 2g. Note, that due to the implicit assumptions of the theoretical model, the band edge is generally much steeper than in the materials under study.
Here we utilized a Bethe lattice, while for the iterated perturbation theory applied in Fig. 1d a triangular lattice was assumed, leading to different U/W values.    S8b. Note the reversed axis of ordinates. The discrepancy between experiment and theory is most likely due to several assumptions of the model. Still, the trends agree very well: i.e. the experimental data can be rescaled to match exactly with theory (magenta symbols). Hence, we conclude that the underlying Mott physics is reproduced reasonably well by the CTQMC calculations.

S7 Experiment vs. Theory
Finally, we compare the band shape of the experimentally determined optical conductivity with the spectra calculated by the continuous time quantum Monte Carlo (CTQMC) method.  Table S1. The -Br x Cl 1-x data were extracted similarly as for the quantum spin liquids. At first glance, there is a quantitative discrepancy between experimental and theoretical data since, apparently, the theoretical bands are narrower and the calculations imply the onset of Mott insulating behavior at a larger U C /W 0 . However, there are many assumptions of the theoretical model that may cause deviations of the absolute values, such as the Bethe lattice instead of a triangular lattice, the dimer approximation, electron-lattice coupling, screening etc. Therefore, it is reasonable to shift the experimental curve to a different onset U C /W 0 and rescale the absolute values. As the result of such a transformation, the experimental data match perfectly with the theoretical ones and their curvature compares very well. Hence, our theoretical model that only includes the charge dynamics of the Mott insulator qualitatively reproduces the electrodynamic response of our optical spectroscopy results.

S8 Comparison with Transition-Metal Oxides
In Fig. S12a, we illustrate the experimentally covered ranges in the unified phase diagram shown in Fig. 5, including our own experiments on the three frustrated Mott insulators (optics and pressure-dependent dc transport) as well as the data reported in Refs. [S15,S13]. For comparison, we add the data of (V 1-x Cr x ) 2 O 3 which exhibits a transition from a paramagnetic metal to a Mott insulator upon chromium substitution. In Fig. S12b the pressure and temperature dependence of the phase transition is plotted as obtained from dc transport measurements at variable pressure [S23]. From optical experiments [S24,S25,S26] it can be estimated that the bandwidth is approximately higher by a factor of 25 compared to EtMe; the Coulomb repulsion U is around 2.6 eV. Performing the proper scaling, we can match the phase boundary of vanadium sesquioxide with the universal QWL, covering a significantly smaller region in the phase diagram. Note that T = 300 K on the EtMe temperature scale correspond to several 10 3 or even 10 4 K for vanadium oxides, cuprates or similar materials, which is well above their melting points. Thus, the small bandwidth of the organic compounds allows us to reach a temperature regime far beyond the grasp of transition metal oxides, giving unique insight into incoherent conduction processes outside the realm of quasiparticles. vertically and horizontally. b, Experimentally accessible phase boundary of (V 1-x Cr x ) 2 O 3 as a function of pressure and temperature (reproduced from Ref. [S23]). By including these data in panel a by the green frame, it becomes obvious that a much smaller region is covered despite the larger temperature and pressure scales. This is a direct consequence of the larger energy scales, U and W, in transition-metal oxides.