Unconventional critical behaviour in a quasi-two-dimensional organic conductor

F. Kagawa; K. Kanoda; K. Miyagawa

REVIEW

Unconventional critical behaviour in a quasi-two-dimensional organic conductor

F. Kagawa, K. Miyagawa, K. Kanoda

Strongly Correlated Electrons · Submitted 3 Mar 2006 · 26 references

Pith's verdict: First clean report of (δ,β,γ)≈(2,1,1) at the 2D Mott endpoint, but the uncalibrated conductance–order-parameter mapping leaves the new-universality-class claim conditional.

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arXiv IDcond-mat/0603064 (version 1)

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CitationF. Kagawa, K. Kanoda, K. Miyagawa. "Unconventional critical behaviour in a quasi-two-dimensional organic conductor." arXiv:cond-mat/0603064 (2006). https://pith.science/paper/Q5CODUZQ

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Editor's note

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The headline is simple: this is the first clean experimental report of critical exponents (δ, β, γ) ≈ (2, 1, 1) at a quasi-2D Mott endpoint, and if those exponents are genuine, they fall outside every standard universality class. But the central claim is weaker than the abstract suggests, because the mapping from conductance to order parameter is never calibrated. The observed exponents may be the true ones; they may equally be effective exponents dressed by a nonlinear conductance response.

What the paper does well is real. The measurements are careful: three single crystals, isothermal pressure sweeps, wide dynamic range in the power laws, sample-independent Tc and Pc, and a scaling collapse that holds over a broad range. The exponents were reproduced on two other crystals. That is solid experimental work, and the paper honestly flags the non-mixing approximation as tentative in the Methods.

The soft spot is the one the stress test identifies. The order parameter is taken to be G−Gc on the authority of DMFT and the prior 3D work, but no calibration of that relation is offered. A smooth nonlinear response O−Gc = Aφ^a preserves the scaling form and renormalizes the exponents, so neither the consistency check δ = 1 + γ/β nor the collapse in Fig. 4 can distinguish a new universality class from a conventional one viewed through a nonlinear conductance. The paper's own Supplementary proof addresses linear field mixing, not the observable relation. Combined with the absence of error bars on the exponents and the exclusion of the near-critical region, the evidence is suggestive rather than conclusive.

There is a sharp, falsifiable consequence: Widom scaling gives α = −1, so the specific heat should vanish linearly at the endpoint. That is not measured and is far from any known Mott critical behaviour. It is the cleanest way to settle the question. If the specific heat instead diverges, the reported exponents are effective, not true critical exponents.

Who is this for? Anyone working on correlated electron criticality, organic superconductors, or Mott transitions. It deserves a serious referee: the experiment is important and the claim is bold enough to test. I would not desk-reject it. But I would send it back with the request to either calibrate the conductance–order-parameter mapping or reframe the conclusion as effective critical behaviour. The thermodynamic prediction should be stated and tested.

Referee report

Significance

If the reported exponents truly characterize the critical behaviour of the two-dimensional Mott transition, the result would be important: it would challenge the conventional universality-class framework for correlated electron systems and could relate to the exotic physics of cuprates and organic superconductors. The paper contains a careful data analysis: the power-law fits appear internally consistent, the exponents are reproduced on two additional crystals, and the scaling collapse in Fig. 4 is shown over a wide range of variables. The paper also makes a sharp, falsifiable thermodynamic prediction (α = −1) that could be tested. However, the central claim depends on two assumptions that are not independently calibrated: the conductance–order-parameter mapping and the non-mixing approximation. The scaling collapse is constructed with exponents derived from the same data, so it is a consistency check rather than an independent validation. The uncalibrated mapping is a particular concern, because a smooth nonlinear response can renormalize the apparent exponents while preserving both the scaling relation and the quality of the collapse.

Major objections (3)

[Methods (Order parameter and scaling variables)] The identification of the order parameter φ with the conductance difference G−Gc is assumed without calibration. The text cites DMFT for the scalar nature of the order parameter, but does not establish that G−Gc is proportional to φ (or even a known function of it). A smooth nonlinear relation, e.g., O−Gc = A φ^a, preserves the scaling form of equations (1) and (2) and renormalizes the exponents to β′ = aβ, δ′ = δ/a, γ′ = γ − (a−1)β. The scaling relation δ′ = 1 + γ′/β′ remains satisfied, and the universal collapse in Fig. 4 cannot distinguish this renormalized description from the claimed novel exponents. For example, ordinary 2D Ising exponents viewed through O ∝ φ^7.5 would appear as (δ, β, γ) ≈ (2, 0.94, 0.94), close to the reported values. The paper therefore does not establish that the observed exponents are the true critical exponents of the Mott order parameter.
[Methods and text near Eq. (1) and Fig. 4] The non-mixing approximation (h ≈ P−Pc, ε ≈ T−Tc with negligible admixture) is introduced as tentative, then subsequently asserted to be valid because the scaling collapse works. This is circular: the exponents used to construct the collapse are obtained under the same approximation, and the collapse is performed on the same data set used to fit the exponents. The supplementary proof mentioned in the Methods is not available in the manuscript under review, and even a proof of weak linear-mixing effects would not address the independent issue of the conductance–order-parameter calibration. The scaling collapse is therefore a consistency check, not an independent confirmation of the non-mixing approximation.
[Widom scaling / predicted specific-heat exponent (implicit in Eqs (1)–(2))] The reported exponents imply, through the Widom relation α = 2 − β(1+δ), a specific-heat exponent α = −1, i.e., the specific heat should vanish linearly as C ∝ |T−Tc| at the critical endpoint. This is a sharp and unusual prediction that is not tested or even mentioned in the paper. If the specific heat instead diverges or remains finite, the reported exponents cannot be the thermodynamic critical exponents of the Mott transition. The authors should either provide a measurement or a clear theoretical argument for why the specific heat is expected to vanish linearly, or state that the exponents are effective rather than true critical exponents.

Minor objections (4)

[General / text] The phrase 'scaling valuables' in the Methods should be 'scaling variables'.
[Equations (1) and (2)] The notation for the scaling functions f± is not defined in the main text; the definitions in the Methods are helpful but would be clearer if placed near the equations.
[Figure 3] The comparison of exponents across universality classes is informative, but the figure would benefit from error bars on the present values, since the text quotes approximate values with no uncertainties.
[Reference 25] Reference 25 is cited as a preprint; if a published version exists, it should be cited instead.

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The paper

The paper claims

The pressure-driven Mott transition in the quasi-two-dimensional organic conductor κ-(BEDT-TTF)2Cu[N(CN)2]Cl exhibits critical exponents (δ, β, γ) ≈ (2, 1, 1), a combination outside every known universality class.

The reading

This paper reports conductance measurements under pressure sweeps across the pressure-induced Mott transition in the layered organic conductor κ-(BEDT-TTF)2Cu[N(CN)2]Cl, centred on the finite-temperature critical endpoint at $T_c\approx39.7$ K and $P_c\approx24.8$ MPa. It claims that the order parameter, identified with the metallic conductance difference $G-G_c$, follows power laws with exponents $(\delta,\beta,\gamma)\approx(2,1,1)$ that satisfy the scaling relation $\delta=1+\gamma/\beta$ and collapse all data onto two universal scaling curves. Because established universality classes have $\delta\ge3$ and $\beta\le0.5$, the combination $\delta<3$, $\beta>0.5$ places the two-dimensional Mott transition outside all known classes. If correct, this identifies a new kind of many-body criticality at a metal–insulator transition, with direct bearing on the unconventional superconductivity and pseudogap behaviour found near the Mott transition in layered organics and copper oxides.

What carries the argument

The machinery is a scaling analysis of the conductance around the critical endpoint, with the order parameter taken as $\varphi\approx G-G_c$, the conjugate scaling field as $h\approx P-P_c$, and the distance to the endpoint as $\varepsilon\approx T-T_c$ (the non-mixing approximation). The argument is carried by three power laws—$\varphi\propto h^{1/\delta}$ at $T=T_c$, $\varphi\propto|\varepsilon|^\beta$ along the first-order line, and $\partial\varphi/\partial h\propto|\varepsilon|^{-\gamma}$ on the crossover line—whose consistency is checked through the scaling relation $\delta=1+\gamma/\beta$ and through collapse of all data onto the universal form $\varphi/h^{1/\delta}=f_\pm(h/|\varepsilon|^{\delta\beta})$. The use of conductance as a scalar order parameter is justified by dynamical mean-field theory and by the earlier success of the same probe at the three-dimensional Mott transition.

Extended reading notes

Core claim

The paper's central claim is that the quasi-two-dimensional Mott transition has a critical endpoint whose exponents are $(\delta,\beta,\gamma)\approx(2,1,1)$, inconsistent with the mean-field, Ising, XY and Heisenberg universality classes. Concretely, the conductance at $T=T_c$ grows as $(P-P_c)^{1/\delta}$ with $\delta\approx2$; the metallic conductance along the first-order line vanishes as $(T_c-T)^\beta$ with $\beta\approx1$; and the pressure derivative of conductance at the crossover line diverges as $(T-T_c)^{-\gamma}$ with $\gamma\approx1$. The three exponents obey $\delta=1+\gamma/\beta$, and the whole metallic-region data set collapses onto the two branches of the universal equation of state $\varphi(h,\varepsilon)/h^{1/\delta}=f_\pm(h/|\varepsilon|^{\delta\beta})$. The authors conclude that this collective behaviour has not been seen before and that two-dimensionality is the essential ingredient, since the three-dimensional Mott transition belongs to a conventional class while the two-dimensional system deviates from mean-field behaviour in the opposite direction to ordinary fluctuations.

Load-bearing premise

The load-bearing premise is that the conductance difference $G-G_c$ faithfully measures the scalar Mott order parameter and that pressure and temperature act as unmixed scaling fields; if either fails, the quoted exponents $(2,1,1)$ are effective values rather than the true asymptotic critical exponents.

Editorial extensions

If this is right

The two-dimensional Mott transition falls outside every known universality class, so a new theoretical framework is needed to describe its critical point.
Any account of emergent behaviour near the 2D Mott transition—unconventional superconductivity, pseudogap behaviour, marginal metallicity—must be consistent with critical exponents $(\delta,\beta,\gamma)\approx(2,1,1)$.
Because $\delta<3$ and $\beta>0.5$, ordering near the 2D endpoint is more gradual than mean-field ordering, the opposite of conventional fluctuation effects; this is a quantitative fingerprint to search for in other quasi-2D Mott systems.
Conductance is confirmed as an essential probe of Mott criticality in the organics, extending the method previously applied to the three-dimensional oxide $(\mathrm{V}_{1-x}\mathrm{Cr}_x)_2\mathrm{O}_3$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct test the paper does not run: the same pressure-sweep protocol applied to the sister salts κ-(BEDT-TTF)2Cu[N(CN)2]Br and κ-(BEDT-TTF)2Cu(NCS)2 should yield the same (2,1,1) exponents if the behaviour is the generic 2D Mott universality class, and conventional exponents if it is material-specific.
The exponents are only as strong as the non-mixing approximation; a linear-mixing re-analysis of the same data would show whether $(\delta,\beta,\gamma)\approx(2,1,1)$ are asymptotic critical exponents or effective values generated by mixed scaling fields.
If the same anomalous criticality appears in cuprates, whose Mott endpoint is approached by doping rather than pressure, then the (2,1,1) exponents may be a common feature underlying the strange-metal and pseudogap physics of layered correlated metals.

Original abstract

Changing the interactions between particles in an ensemble-by varying the temperature or pressure, for example-can lead to phase transitions whose critical behaviour depends on the collective nature of the many-body system. Despite the diversity of ingredients, which include atoms, molecules, electrons and their spins, the collective behaviour can be grouped into several families (called 'universality classes') represented by canonical spin models1. One kind of transition, the Mott transition2, occurs when the repulsive Coulomb interaction between electrons is increased, causing wave-like electrons to behave as particles. In two dimensions, the attractive behaviour responsible for the superconductivity in high-transition temperature copper oxide3,4 and organic5-7 compounds appears near the Mott transition, but the universality class to which two-dimensional, repulsive electronic systems belongs remains unknown. Here we present an observation of the critical phenomena at the pressure-induced Mott transition in a quasi-two-dimensional organic conductor using conductance measurements as a probe. We find that the Mott transition in two dimensions is not consistent with known universality classes, as the observed collective behaviour has previously not been seen. This peculiarity must be involved in any emergent behaviour near the Mott transition in two dimensions.

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What the abstract itself states, as probabilities: claims first 0.27 · states limitation 0.18 · human participants 0.02 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.79 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Changing the interactions between particles in an ensemble-by varying the temperature or pressure, for example-can lead to phase transitions whose critical behaviour depends on the collective nature of the many-body system. Despite the diversity of ingredients, which include atoms, molecules, electrons and their spins, the collective behaviour can be grouped into several families (called 'universality classes') represented by canonical spin models1. One kind of transition, the Mott transition2, occurs when the repulsive Coulomb interaction between electrons is increased, causing wave-like electrons to behave as particles. In two dimensions, the attractive behaviour responsible for the superconductivity in high-transition temperature copper oxide3,4 and organic5-7 compounds appears near the Mott transition, but the universality class to which two-dimensional, repulsive electronic systems belongs remains unknown. Here we present an observation of the critical phenomena at the pressure-induced Mott transition in a quasi-two-dimensional organic conductor using conductance measurements as a probe. We find that the Mott transition in two dimensions is not consistent with known universality classes, as the observed collective behaviour has previously not been seen. This peculiarity must be involved in any emergent behaviour near the Mott transition in two dimensions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Changing the interactions between particles in an ensemble-by varying the temperature or pressure, for example-can lead to phase transitions whose critical behaviour depends on the collective nature of the many-body system.	background 1.00	0.03	0.02	asserted 0.52
s1	Despite the diversity of ingredients, which include atoms, molecules, electrons and their spins, the collective behaviour can be grouped into several families (called 'universality classes') represented by canonical spin models1.	background 1.00	0.04	0.02	asserted 1.00
s2	One kind of transition, the Mott transition2, occurs when the repulsive Coulomb interaction between electrons is increased, causing wave-like electrons to behave as particles.	background 1.00	0.03	0.02	asserted 1.00
s3	In two dimensions, the attractive behaviour responsible for the superconductivity in high-transition temperature copper oxide3,4 and organic5-7 compounds appears near the Mott transition, but the universality class to which two-dimensional, repulsive electronic systems belongs remains unknown.	gap 0.98	0.04	0.05	asserted 0.97
s4	Here we present an observation of the critical phenomena at the pressure-induced Mott transition in a quasi-two-dimensional organic conductor using conductance measurements as a probe.	contribution 0.95	0.97	0.07	asserted 1.00
s5	We find that the Mott transition in two dimensions is not consistent with known universality classes, as the observed collective behaviour has previously not been seen.	result 0.99	0.96	0.92	asserted 1.00
s6	This peculiarity must be involved in any emergent behaviour near the Mott transition in two dimensions.	interpretation 1.00	0.22	0.19	asserted 0.71

What kind of paper the classifier reports this is: empirical study 1.00 · evidence from observation or experiment 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
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T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (26)

[1]

∝ h1/δ, φ(0,ε) ∝ |ε|β and ∂φ(h,ε)/∂h|h=0 ∝ |ε|−γ. Note that these critical exponents are not independent of each other, but the scaling relation δ = 1 + (γ/β) should be fulfilled among them. To analyse the Mott critical phenomena, one should relate (φ, h, ε) of the Mott transition to the observable quantities. According to the theoretical framework of str...

View on Pith 2004
[2]

Kadanoff, L. P. et al. Static phenomena near critical points: Theory and experiment. Rev. Mod. Phys. 39, 395–431 (1967)

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Mott, N. F. Metal-Insulator Transitions (Taylor & Francis, London, 1990)

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[4]

Imada, M., Fujimori, A. & Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998)

View on Pith 1998
[5]

Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003). 6

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[6]

Mayaffre, H., Wzietek, P., Jérome, D., Lenoir, C. & Batail, P. Superconducting state of κ-(ET) CU[N(CN) ]Br studied by C NMR: Evidence for vortex-core-induced nuclear relaxation and unconventional pairing. 2 2 13 Phys. Rev. Lett. 75, 4122–4125 (1995)

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De Soto, S. M. et al. 13C NMR studies of the normal and superconducting states of the organic superconductor κ-(ET) Cu[N(CN) ]Br.2 2 Phys. Rev. B 52, 10364–10368 (1995)

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[8]

Kanoda, K., Miyagawa, K., Kawamoto, A. & Nakazawa, Y. NMR relaxation rate in the superconducting state of the organic conductor κ-(BEDT-TTF) Cu[N(CN) ]Br.2 2 Phys. Rev. B 54, 76–79 (1996)

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[9]

Limelette, P. et al. Universality and critical behavior at the Mott transition. Science 302, 89–92 (2003)

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Kino, H. & Fukuyama, H. Phase diagram of two-dimensional organic conductors: (BEDT-TTF) X.2 J. Phys. Soc. Jpn 65, 2158–2169 (1996)

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Mckenzie, R. H. Similarities between organic and cuprate superconductors. Science 278, 820–821 (1997)

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Kanoda, K. Electron correlation, metal-insulator transition and superconductivity in quasi-2D organic systems, (ET) X.2 Physica C 282, 299–302 (1997)

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[13]

Miyagawa, K., Kawamoto, A. & Kanoda, K. Proximity of pseudogapped superconductor and commensurate antiferromagnet in a quasi-two-dimensional organic system. Phys. Rev. Lett. 89, 017003 (2002)

View on Pith 2002
[14]

Lefebvre, S. et al. Mott transition, antiferromagnetism, and unconventional superconductivity in layered organic superconductors. Phys. Rev. Lett. 85, 5420–5423 (2000)

View on Pith 2000
[15]

Fournier, D., Poirier, M., Castonguay, M. & Truong, K. Mott transition, compressibility divergence, and the P-T phase diagram of layered organic superconductors: An ultrasonic investigation. Phys. Rev. Lett. 90, 127002 (2003)

View on Pith 2003
[16]

Limelette, P. et al. Mott transition and transport crossovers in the organic compound κ-(BEDT-TTF) Cu[N(CN) ]Cl.2 2 Phys. Rev. Lett. 91, 016401 (2003)

View on Pith 2003
[17]

Kagawa, F., Itou, T., Miyagawa, K. & Kanoda, K. Transport criticality of the first-order Mott transition in the quasi-two-dimensional organic conductor κ-(BEDT-TTF) Cu[N(CN) ]Cl.2 2 Phys. Rev. B 69, 064511 (2004)

View on Pith 2004
[18]

Williams, J. M. et al. From semiconductor-semiconductor transition (42 K) to the highest-T organic superconductor, κ-(ET) Cu[N(CN) ]Cl (T = 12.5 K).C 2 2 C Inorg. Chem. 29, 3272–3274 (1990). 7

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[19]

Georges, A. Strongly correlated electron materials: Dynamical mean-field theory and electronic structure. AIP Conf. Proc. 715, 3–74 (2004)

View on Pith 2004
[20]

Castellani, C., DiCastro, C., Feinberg, D. & Ranninger, J. New model Hamiltonian for the metal-insulator transition. Phys. Rev. Lett. 43, 1957–1960 (1979)

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[21]

Kotliar, G., Lange, E. & Rozenberg, M. J. Landau theory of the finite temperature Mott transition. Phys. Rev. Lett. 84, 5180–5183 (2000)

View on Pith 2000
[22]

Imada, M. Quantum Mott transition and multi-furcating criticality. J. Phys. Soc. Jpn 73, 1851–1863 (2004)

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[23]

Imada, M. Quantum Mott transition and superconductivity. J. Phys. Soc. Jpn 74, 859–862 (2005)

View on Pith 2005
[24]

Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13–125 (1996)

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Kotliar, G. & Vollhardt, D. Strongly correlated materials: Insights from dynamical mean-field theory. Phys. Today 57(3), 53–59 (2004)

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[26]

Onoda, S. Ginzburg-Landau theory and classical critical phenomena of Mott transition. Preprint at <http://xxx.lanl.gov/abs/cond-mat/0408207> (2004). Supplementary Information is linked to the online version of the paper at www.nature.com/nature. Acknowledgements We thank M. Imada, T. Itou, S. Miyashita, N. Nagaosa, S. Onoda, Y. Shimizu and N. Todoroki for...

View on Pith Pith review arXiv 2004

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

∝ h1/δ, φ(0,ε) ∝ |ε|β and ∂φ(h,ε)/∂h|h=0 ∝ |ε|−γ. Note that these critical exponents are not independent of each other, but the scaling relation δ = 1 + (γ/β) should be fulfilled among them. To analyse the Mott critical phenomena, one should relate (φ, h, ε) of the Mott transition to the observable quantities. According to the theoretical framework of str...

View on Pith 2004

[2] [2]

Kadanoff, L. P. et al. Static phenomena near critical points: Theory and experiment. Rev. Mod. Phys. 39, 395–431 (1967)

View on Pith 1967

[3] [3]

Mott, N. F. Metal-Insulator Transitions (Taylor & Francis, London, 1990)

View on Pith 1990

[4] [4]

Imada, M., Fujimori, A. & Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998)

View on Pith 1998

[5] [5]

Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003). 6

View on Pith 2003

[6] [6]

Mayaffre, H., Wzietek, P., Jérome, D., Lenoir, C. & Batail, P. Superconducting state of κ-(ET) CU[N(CN) ]Br studied by C NMR: Evidence for vortex-core-induced nuclear relaxation and unconventional pairing. 2 2 13 Phys. Rev. Lett. 75, 4122–4125 (1995)

View on Pith 1995

[7] [7]

De Soto, S. M. et al. 13C NMR studies of the normal and superconducting states of the organic superconductor κ-(ET) Cu[N(CN) ]Br.2 2 Phys. Rev. B 52, 10364–10368 (1995)

View on Pith 1995

[8] [8]

Kanoda, K., Miyagawa, K., Kawamoto, A. & Nakazawa, Y. NMR relaxation rate in the superconducting state of the organic conductor κ-(BEDT-TTF) Cu[N(CN) ]Br.2 2 Phys. Rev. B 54, 76–79 (1996)

View on Pith 1996

[9] [9]

Limelette, P. et al. Universality and critical behavior at the Mott transition. Science 302, 89–92 (2003)

View on Pith 2003

[10] [10]

Kino, H. & Fukuyama, H. Phase diagram of two-dimensional organic conductors: (BEDT-TTF) X.2 J. Phys. Soc. Jpn 65, 2158–2169 (1996)

View on Pith 1996

[11] [11]

Mckenzie, R. H. Similarities between organic and cuprate superconductors. Science 278, 820–821 (1997)

View on Pith 1997

[12] [12]

Kanoda, K. Electron correlation, metal-insulator transition and superconductivity in quasi-2D organic systems, (ET) X.2 Physica C 282, 299–302 (1997)

View on Pith 1997

[13] [13]

Miyagawa, K., Kawamoto, A. & Kanoda, K. Proximity of pseudogapped superconductor and commensurate antiferromagnet in a quasi-two-dimensional organic system. Phys. Rev. Lett. 89, 017003 (2002)

View on Pith 2002

[14] [14]

Lefebvre, S. et al. Mott transition, antiferromagnetism, and unconventional superconductivity in layered organic superconductors. Phys. Rev. Lett. 85, 5420–5423 (2000)

View on Pith 2000

[15] [15]

Fournier, D., Poirier, M., Castonguay, M. & Truong, K. Mott transition, compressibility divergence, and the P-T phase diagram of layered organic superconductors: An ultrasonic investigation. Phys. Rev. Lett. 90, 127002 (2003)

View on Pith 2003

[16] [16]

Limelette, P. et al. Mott transition and transport crossovers in the organic compound κ-(BEDT-TTF) Cu[N(CN) ]Cl.2 2 Phys. Rev. Lett. 91, 016401 (2003)

View on Pith 2003

[17] [17]

Kagawa, F., Itou, T., Miyagawa, K. & Kanoda, K. Transport criticality of the first-order Mott transition in the quasi-two-dimensional organic conductor κ-(BEDT-TTF) Cu[N(CN) ]Cl.2 2 Phys. Rev. B 69, 064511 (2004)

View on Pith 2004

[18] [18]

Williams, J. M. et al. From semiconductor-semiconductor transition (42 K) to the highest-T organic superconductor, κ-(ET) Cu[N(CN) ]Cl (T = 12.5 K).C 2 2 C Inorg. Chem. 29, 3272–3274 (1990). 7

View on Pith 1990

[19] [19]

Georges, A. Strongly correlated electron materials: Dynamical mean-field theory and electronic structure. AIP Conf. Proc. 715, 3–74 (2004)

View on Pith 2004

[20] [20]

Castellani, C., DiCastro, C., Feinberg, D. & Ranninger, J. New model Hamiltonian for the metal-insulator transition. Phys. Rev. Lett. 43, 1957–1960 (1979)

View on Pith 1979

[21] [21]

Kotliar, G., Lange, E. & Rozenberg, M. J. Landau theory of the finite temperature Mott transition. Phys. Rev. Lett. 84, 5180–5183 (2000)

View on Pith 2000

[22] [22]

Imada, M. Quantum Mott transition and multi-furcating criticality. J. Phys. Soc. Jpn 73, 1851–1863 (2004)

View on Pith 2004

[23] [23]

Imada, M. Quantum Mott transition and superconductivity. J. Phys. Soc. Jpn 74, 859–862 (2005)

View on Pith 2005

[24] [24]

Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13–125 (1996)

View on Pith 1996

[25] [25]

Kotliar, G. & Vollhardt, D. Strongly correlated materials: Insights from dynamical mean-field theory. Phys. Today 57(3), 53–59 (2004)

View on Pith 2004

[26] [26]

Onoda, S. Ginzburg-Landau theory and classical critical phenomena of Mott transition. Preprint at <http://xxx.lanl.gov/abs/cond-mat/0408207> (2004). Supplementary Information is linked to the online version of the paper at www.nature.com/nature. Acknowledgements We thank M. Imada, T. Itou, S. Miyashita, N. Nagaosa, S. Onoda, Y. Shimizu and N. Todoroki for...

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