Abstract
Superconductivity in the cuprates exhibits many unusual features. We study the twodimensional Hubbard model with plaquette dynamical meanfield theory to address these unusual features and relate them to other normalstate phenomena, such as the pseudogap. Previous studies with this method found that upon doping the Mott insulator at low temperature a pseudogap phase appears. The lowtemperature transition between that phase and the correlated metal at higher doping is firstorder. A series of crossovers emerge along the Widom line extension of that firstorder transition in the supercritical region. Here we show that the highly asymmetric dome of the dynamical meanfield superconducting transition temperature , the maximum of the condensation energy as a function of doping, the correlation between maximum and normalstate scattering rate, the change from potentialenergy driven to kineticenergy driven pairing mechanisms can all be understood as remnants of the normal state firstorder transition and its associated crossovers that also act as an organizing principle for the superconducting state.
Introduction
In holedoped cuprate hightemperature superconductors, dwave superconductivity shows unusual features that cannot be explained by theoretical methods based on weak correlations^{1,2}. This has motivated the hypothesis that such unusual features emerge from doping a twodimensional Mott insulator. Advances in this regard were enabled by the development of new theoretical methods such as cluster extensions^{3,4} of dynamical meanfield theory^{5}. A collective effort over the last decade has shown that the key aspects of the phenomenology of cuprates are contained in the twodimensional Hubbard model. Within this theoretical framework, here we show that these key aspects rest with a single organizing principle, namely a normalstate firstorder transition between pseudogap and correlated metal beneath the superconducting dome, identified in ref. 6. Our analysis indicates that this emerging phase transition at finite doping shapes not only the normalstate phase diagram, but strikingly leaves its mark on the complex structure of the superconducting condensate that is born out of this unusual normal state.
Model and Method
The two dimensional Hubbard model on a square lattice reads
where and c_{iσ} operators create and destroy an electron of spin σ on site i, n_{iσ} = c_{iσ} is the number operator, μ is the chemical potential, U the onsite Coulomb repulsion and t_{ij} is the nearest neighbor hopping amplitude. Neglecting secondneighbor hopping, necessary to capture the correct Fermi surface, minimizes the MonteCarlo signproblem and does not alter our main findings (see supplementary Fig. S7). Unless specified, the lattice spacing, Planck’s constant, Boltzmann’s constant and t are unity.
We solve this model using cellular dynamical meanfield theory^{3,4} (CDMFT) on a 2 × 2 plaquette immersed in an infinite selfconsistent bath of noninteracting electrons. This plaquette is the minimal cluster that includes all twodimensional shortrange charge, spin and superconducting dynamical correlations. We do not take into account longrange chargedensity waves in light of the recent experimental results where this transition is removed by pressure^{7}. Longrange antiferromagnetism concomitant with longrange superconductivity has been treated at T = 0 in previous work^{8,9,10}. Since we are interested in large values of U, i.e. a doped Mott insulator, the most appropriate method to solve the impurity (cluster plus bath) problem is the hybridization expansion continuoustime quantum Monte Carlo method^{11}. Sign problems prevent the study of large U with alternate quantum Monte Carlo methods^{11}. We use two recent algorithmic improvements to speed up the calculations: a fast rejection algorithm with skiplist data structure^{12} and four point updates that are necessary for broken symmetry states like dwave superconductivity^{13}.
Let us first consider the superconducting phase diagram. We then discuss features of the normal state that determine its shape.
Superconducting dome
Previous studies show that both at halffilling and at finite doping the metallic state close to the Mott insulator is unstable to dwave superconductivity^{8,9,10,14,15,16,17,18,19,20,21,22,23}. In Fig. 1 we map out the superconducting state in the U − T plane for the undoped case and in the δ − T plane for different values of U. The superconducting region is defined as the region of nonzero superconducting order parameter (where the cluster momentum K is (π, 0)). The boundary, , is obtained from the mean of the two temperatures where Φ changes from finite to a small value (here Φ = 0.002). While there is no continuous symmetry breaking in two dimensions at finite temperature, physically denotes the temperature below which the superconducting pairs form within the cluster^{14}. The actual T_{c} can be reduced (because of long wavelength thermal or quantum fluctuations^{24} or of competing long range order^{1}) or increased (because of pairing through long wavelength antiferromagnetic fluctuations^{25}), but still remains a useful quantity marking the region where Mott physics and shortrange correlations produce pairing.
As a function of U, changes from finite to zero discontinuously at the firstorder Mott metalinsulator transition (red shaded region in panel a). Superconductivity appears in the metastable metallic state near the Mott insulator, never in the Mott insulator itself (panels a, b). As a function of doping, forms a dome as long as U is larger than the critical value necessary to obtain a Mott insulator at halffilling (panels c–g). In our previous studies^{13,14} we left opened two possibilities: as a function of δ, either superconductivity is separated from the Mott insulator at δ = 0 by a firstorder transition or there is an abrupt fall of (δ). By increasing the resolution in doping near δ = 0, here we find the latter, namely T_{c}^{d}(δ) plummets with decreasing δ.
The superconducting dome is highly asymmetric. (δ) is zero at δ = 0, initially rises steeply with increasing δ, reaching a peak at the optimal doping δ_{opt} and then declines more gently with further doping. The global maximum T_{c}^{max} of in the U − δ − T space occurs just above U_{MIT} and at finite doping δ_{opt}. Further increase of U leads to a decrease in , as expected if (δ) scales with the superexchange energy J = 4t^{2}/U for large enough U^{10,26}. As a function of U, the optimal doping δ_{opt} departs from δ = 0 for U > U_{MIT}, increasing with increasing U and saturating around δ ≈ 0.04 for large U (see also supplementary Fig. S2).
The range of doping where superconductivity occurs at the lowest temperature is consistent^{13} with results obtained with CDMFT at T = 0^{10}. The asymmetric superconducting dome with an abrupt fall of with decreasing δ is also consistent with dynamical cluster approximation results on larger clusters^{22}. In the latter calculations, the increased accuracy in momentum space leads to a T_{c} that vanishes before halffilling.
Superconducting order parameter
To analyse the shape of the superconducting phase we turn to the superconducting order parameter Φ, whose magnitude is colorcoded in Fig. 1 (the raw data is in Fig. S1). While occurs at finite doping, the overall maximum Φ_{max} is found in the undoped model close to the Mott insulator. But as a function of doping, for U > U_{MIT}, Φ forms a dome that reaches a peak at δ_{Φ max}. At our lowest temperature, δ_{Φmax} increases with increasing U, and saturates around δ ≈ 0.11^{10} for large values of U. Notice that δ_{Φ max} at our lowest temperature does not coincide with δ_{opt}, i.e. the doping that optimizes . Hence, (δ) does not scale with Φ(δ, T → 0). Instead, the locus of the maxima of Φ in the δ − T plane at fixed U traces a negatively sloped line within the superconducting dome (lines with blue triangles) that separates the superconducting dome in two regions. The sharp asymmetry of the superconducting dome is thus linked to this negatively sloped line, which in turn is related to the phase transition between pseudogap and correlated metal in the underlying normal state, as we discuss below.
Superconductivity and pseudogap
Understanding the normal state has long been considered a prerequisite to a real understanding of hightemperature superconductivity. This comes out clearly from our results. Previous normalstate CDMFT studies show that for U > U_{MIT} and small δ, large screened Coulomb repulsion U and the emergent superexchange J lead at low T to a state with strong singlet correlations. That phase has the characteristics of the pseudogap phase^{6}. The fall of the Knight shift as a function of temperature^{27} is usually associated with T*(δ) the onset temperature for the pseudogap. The line with orange filled circles in Fig. 2(a–c) ^{28} indicates the onset of the drop of the spin susceptibility and of the density of states as a function of T and the minimum in the T dependence of the caxis resistivity^{28} and is thus T*(δ) in our calculation. From our point of view, it is just a precursor to a more fundamental phenomenon. T*(δ) exists only if the doping is less than a critical value δ < δ_{p} which is the doping for the critical endpoint (δ_{p}, T_{p}) of a firstorder transition that appears in Fig. 2a. A number of crossover lines are associated with this firstorder transition. We will discuss them in turn. For larger values of U, Fig. 2b,c, the firstorder transition is nolonger visible at accessible temperatures, but the crossovers that are left suggest that it is still present^{29}.
The normalstate firstorder transition separating a pseudogap phase and a correlated metal persists up to the critical endpoint, beyond which only a single normalstate phase exists. Quite generally, different response functions have maxima defining crossover lines emerging from the critical endpoint^{30}. The Widom line is known as the line where these maxima join asymptotically close to the critical endpoint^{30}. Here we estimate that line, (red open triangles) T_{W} in the upper panels of Fig. 2, as the line where the isothermal electronic compressibility has a maximum^{6,29,31}. Let us briefly consider the other crossover lines. A scan in doping at fixed T shows that the local density of states at the Fermi energy, the spin susceptibility and the caxis DC conductivity go through an inflection point at T_{W}(δ)^{28}. The firstorder transition is also a source of anomalous scattering^{29,31}. The blue open diamonds indicate the maximum Γ_{max} of the normal state scattering rate Γ. Its magnitude, estimated from the zerofrequency extrapolation of the imaginary part of the (π, 0) component of the cluster selfenergy, is colorcoded in Fig. 2a–c. The region where Γ is large is dark blue. It originates at the transition, extends well above and is tilted towards the Mott insulator. This large Γ is suppressed upon entering the superconducting state^{21,32} (see supplementary Fig. S3).
Even though the firstorder transition is absent in the superconducting state, the structure it imposes on the normal state shapes the superconducting phase diagram: (a) the maximum of the superconducting order parameter Φ_{max} (line with blue filled triangles in Fig. 2a–c) parallels T_{W} and Γ_{max}, hence the highly asymmetric shape of the superconducting dome is correlated with the slope of the firstorder transition and of its supercritical crossovers in the T − δ plane; (b) Γ_{max} crosses the superconducting dome approximately at δ_{opt}, hence a region of anomalous scattering broadens as it comes out of the dome; (c) since T* can be detected for doping smaller than δ_{p} only, superconductivity and pseudogap are intertwined phenomena: superconductivity can emerge from a pseudogap phase below δ_{p}, or from a correlated metal above δ_{p}^{14}; (d) the normal state also controls the source of condensation energy, as we now discuss.
Condensation energy
The superconducting state clearly has a lower free energy than the normal state out of which it is born. In the ground state, the energy difference between both states is known as the condensation energy. The origin of the condensation energy is unambiguous only within a given model^{33,34}. In the BCS model, superconductivity occurs because of a decrease in potential energy. The kinetic energy increase due to particlehole mixing in the ground state is not large enough to overcome the potential energy drop. In the cuprates, analysis of inelastic neutron scattering^{35} has suggested that superconductivity arises because of a gain in exchange energy in the t – J model. Analysis of ARPES^{36} and optical data^{37,38,39,40} in the context of the Hubbard model has suggested that superconductivity is kineticenergy driven in the underdoped regime^{34,35,41,42,43}.
In the lower panels of Fig. 2 we plot, for the Hubbard model Eq. 1, the difference in kinetic and potential energies between the superconducting and normal states (ΔE_{kin} and ΔE_{pot}; blue and red lines respectively) as a function of doping. The results for the two different temperatures are close enough to suggest we are close to ground state values. The net condensation energy, shown by the green line, is always negative, as expected. The doping dependence of ΔE_{kin} and ΔE_{pot} on the other hand shows two striking features: it is non monotonic and can display a sign change. For U = 6.2,7, Fig. 2d,e, superconductivity is kineticenergy driven at small doping and potential energy driven, as in BCS theory, at large doping. For U = 9, Fig. 2f, superconductivity is kinetic energy driven for all dopings, although the potential energy difference ΔE_{pot} can change sign.
Previous investigations^{23,39,44} have revealed a complex behavior that remained to this day a puzzle, with ΔE_{kin} going from negative to positive depending on T and U. What has beeno missing to make sense of this complexity is the existence of the normal state firstorder transition and its associated supercritical crossovers. By considering different values of U, we provide a unified picture of a host of apparently contradictory results. For all U considered, the largest condensation energy (see green line in bottom panels of Fig. 2 and green squares in top panels of Fig. 2) is concomitant with the largest superconducting order parameter Φ(δ) (but not with the maximum ) and hence correlates with the normalstate pseudogaptocorrelated metal firstorder transition, and its associated supercritical crossovers. For all U, the sign changes are also close to the maximum condensation energy and hence also correlated with the same normalstate features. The influence of Mott and superexchange physics extends unambiguously all the way to the normalstate firstorder transition terminating at the critical endpoint, from which supercritical crossovers emerge^{31}. This reflects itself in the superconducting state in a decisive manner: the changes in sign of the different sources of condensation energy occur for dopings similar to those where the normalstate transition occurs.
Source of condensation energy
Bottom panels of Fig. 2 (see also Fig. S5) show that in the underdoped region, the kineticenergy change in the superconducting state is close to minus twice the potential energy change. This is what is expected if superexchange^{45} J drives superconductivity there^{26}. The decrease with U of the maximum , of the magnitude of the individual kinetic and potential energy contributions to condensation energy, and of the maximum value of the T = 0 order parameter^{8,9,10,18}, are also all consistent with the importance of J in the effective model that arises from the Hubbard model at large U. The BCSlike behavior in the overdoped regime for U = 6.2, 7 probably arises from leftover of the weakcoupling longwavelength antiferromagnetic spinwave pairing mechanism^{46}, although the effect of the selfconsistent rearrangement of the spinfluctuation spectrum in the superconducting state has not been studied yet.
Discussion
Our findings further broaden our understanding of the CDMFT solution of the Hubbard model in the doped Mott insulator regime by showing how and to what extent the organizing principle for both the normal state and the superconducting state is the finitedoping firstorder transition that determines the shape and the properties of both phases, even though the transition itself is invisible in the superconducting state. In the T − δ plane, the loci of the maximum order parameter, of the extremum condensation energy, of the maximum normal state scattering relative to the maximum , all correlate with crossover lines of the underlying normal state that is unstable to dwave superconductivity.
We speculate that the application of a magnetic field strong enough to suppress T_{c} and pressures large enough to remove density waves may reveal the underlying transition. We also speculate that sound anomalies associated with the large compressibility in the underlying normal state above the critical endpoint could appear, in analogy with what is observed near the halffilled Mott transition in layered organics^{47,48,49,50,51,52}. The appearance of large electronic compressibility near the normal state firstorder transition suggests that further studies of ubiquitous bonddensity waves^{7} should be undertaken with the same set of methods.
Additional Information
How to cite this article: Fratino, L. et al. An organizing principle for twodimensional strongly correlated superconductivity. Sci. Rep. 6, 22715; doi: 10.1038/srep22715 (2016).
References
 1
Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to hightemperature superconductivity in copper oxides. Nature 518, 179–186 (2015).
 2
Tremblay, A.M. S. Strongly correlated superconductivity. In Pavarini, E., Koch, E. & Schollwöck, U. (eds.) Emergent Phenomena in Correlated Matter Modeling and Simulation, vol. 3, chap. 10 (Verlag des Forschungszentrum, 2013).
 3
Kotliar, G. et al. Electronic structure calculations with dynamical meanfield theory. Rev. Mod. Phys. 78, 865 (2006).
 4
Maier, T., Jarrell, M., Pruschke, T. & Hettler, M. H. Quantum cluster theories. Rev. Mod. Phys. 77, 1027–1080 (2005).
 5
Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J. Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13 (1996).
 6
Sordi, G., Sémon, P., Haule, K. & Tremblay, A.M. S. Pseudogap temperature as a Widom line in doped Mott insulators. Sci. Rep. 2, 547 (2012).
 7
CyrChoinière, O. et al. Suppression of charge order by pressure in the cuprate superconductor YBa2Cu3O y : Restoring the full superconducting dome. ArXiv eprints 1503.02033. (2015).
 8
Sénéchal, D., Lavertu, P.L., Marois, M.A. & Tremblay, A.M. S. Competition between antiferromagnetism and superconductivity in highT c cuprates. Phys. Rev. Lett. 94, 156404 (2005).
 9
Capone, M. & Kotliar, G. Competition between d wave superconductivity and antiferromagnetism in the twodimensional hubbard model. Phys. Rev. B 74, 054513 (2006).
 10
Kancharla, S. S. et al. Anomalous superconductivity and its competition with antiferromagnetism in doped mott insulators. Phys. Rev. B 77, 184516 (2008).
 11
Gull, E. et al. Continuoustime monte carlo methods for quantum impurity models. Rev. Mod. Phys. 83, 349–404 (2011).
 12
Sémon, P., Yee, C.H., Haule, K. & Tremblay, A.M. S. Lazy skiplists: An algorithm for fast hybridizationexpansion quantum monte carlo. Phys. Rev. B 90, 075149 (2014).
 13
Sémon, P., Sordi, G. & Tremblay, A.M. S. Ergodicity of the hybridizationexpansion monte carlo algorithm for brokensymmetry states. Phys. Rev. B 89, 165113 (2014).
 14
Sordi, G., Sémon, P., Haule, K. & Tremblay, A.M. S. Strong coupling superconductivity, pseudogap, and mott transition. Phys. Rev. Lett. 108, 216401 (2012).
 15
Maier, T., Jarrell, M., Pruschke, T. & Keller, J. dwave superconductivity in the hubbard model. Phys. Rev. Lett. 85, 1524–1527 (2000).
 16
Lichtenstein, A. I. & Katsnelson, M. I. Antiferromagnetism and dwave superconductivity in cuprates: A cluster dynamical meanfield theory. Phys. Rev. B 62, R9283–R9286 (2000).
 17
Kyung, B. & Tremblay, A.M. S. Mott transition, antiferromagnetism, and dwave superconductivity in twodimensional organic conductors. Phys. Rev. Lett. 97, 046402 (2006).
 18
Aichhorn, M., Arrigoni, E., Potthoff, M. & Hanke, W. Antiferromagnetic to superconducting phase transition in the hole and electrondoped hubbard model at zero temperature. Phys. Rev. B 74, 024508 (2006).
 19
Balzer, M., Hanke, W. & Potthoff, M. Importance of local correlations for the order parameter of highT c superconductors. Phys. Rev. B 81, 144516 (2010).
 20
Maier, T. A., Jarrell, M., Schulthess, T. C., Kent, P. R. C. & White, J. B. Systematic study of dwave superconductivity in the 2d repulsive hubbard model. Phys. Rev. Lett. 95, 237001 (2005).
 21
Haule, K. & Kotliar, G. Strongly correlated superconductivity: A plaquette dynamical meanfield theory study. Phys. Rev. B 76, 104509 (2007).
 22
Gull, E., Parcollet, O. & Millis, A. J. Superconductivity and the pseudogap in the twodimensional hubbard model. Phys. Rev. Lett. 110, 216405 (2013).
 23
Gull, E. & Millis, A. J. Energetics of superconductivity in the twodimensional hubbard model. Phys. Rev. B 86, 241106 (2012).
 24
Emery, V. J. & Kivelson, S. A. Superconductivity in bad metals. Phys. Rev. Lett. 74, 3253–3256 (1995).
 25
BealMonod, M. T., Bourbonnais, C. & Emery, V. J. Possible superconductivity in nearly antiferromagnetic itinerant fermion systems. Phys. Rev. B 34, 7716–20 (1986).
 26
Kotliar, G. & Liu, J. Superconducting instabilities in the largeU limit of a generalized hubbard model. Phys. Rev. Lett. 61, 1784–7 (1988).
 27
Alloul, H., Mendels, P., Casalta, H., Marucco, J. F. & Arabski, J. Correlations between magnetic and superconducting properties of Znsubstituted YBa2Cu3O6 + x . Phys. Rev. Lett. 67, 3140–3143 (1991).
 28
Sordi, G., Sémon, P., Haule, K. & Tremblay, A.M. S. caxis resistivity, pseudogap, superconductivity, and widom line in doped mott insulators. Phys. Rev. B 87, 041101 (2013).
 29
Sordi, G., Haule, K. & Tremblay, A.M. S. Mott physics and firstorder transition between two metals in the normalstate phase diagram of the twodimensional Hubbard model. Phys. Rev. B 84, 075161 (2011).
 30
Xu, L. et al. Relation between the Widom line and the dynamic crossover in systems with a liquid liquid phase transition. Proc. Natl. Acad. Sci. USA 102, 16558–16562 (2005).
 31
Sordi, G., Haule, K. & Tremblay, A.M. S. Finite Doping Signatures of the Mott Transition in the TwoDimensional Hubbard Model. Phys. Rev. Lett. 104, 226402 (2010).
 32
Haule, K. & Kotliar, G. Avoided criticality in nearoptimally doped hightemperature superconductors. Phys. Rev. B 76, 092503 (2007).
 33
Chester, G. V. Difference between normal and superconducting states of a metal. Phys. Rev. 103, 1693–1699 (1956).
 34
Leggett, A. A. “midinfrared” scenario for cuprate superconductivity. Proceedings of the National Academy of Sciences 96, 8365–8372 (1999).
 35
Scalapino, D. J. & White, S. R. Superconducting condensation energy and an antiferromagnetic exchangebased pairing mechanism. Phys. Rev. B 58, 8222–8224 (1998).
 36
Norman, M. R., Randeria, M., Jankó, B. & Campuzano, J. C. Condensation energy and spectral functions in hightemperature superconductors. Phys. Rev. B 61, 14742–14750 (2000).
 37
Molegraaf, H. J. A., Presura, C., van der Marel, D., Kes, P. H. & Li, M. Superconductivityinduced transfer of inplane spectral weight in Bi2Sr2CaCu2O8 + δ . Science 295, 2239–2241 (2002).
 38
Deutscher, G., SantanderSyro, A. F. & Bontemps, N. Kinetic energy change with doping upon superfluid condensation in hightemperature superconductors. Phys. Rev. B 72, 092504 (2005).
 39
Carbone, F. et al. Doping dependence of the redistribution of optical spectral weight in Bi2Sr2CaCu2O8 + δ . Phys. Rev. B 74, 064510 (2006).
 40
Giannetti, C. et al. Revealing the highenergy electronic excitations underlying the onset of hightemperature superconductivity in cuprates. Nature Communications 2, 353 (2011).
 41
Anderson, P. W. The theory of Superconductivity in the High Tc cuprates (Princeton University Press, Princeton, 1997).
 42
Hirsch, J. & Marsiglio, F. Where is 99% of the condensation energy of Tl2Ba2CuO y coming from? Physica C: Superconductivity 331, 150–156 (2000).
 43
Demler, E. & Zhang, S.C. Quantitative test of a microscopic mechanism of hightemperature superconductivity. Nature 396, 733–735 (1998).
 44
Maier, T. A., Jarrell, M., Macridin, A. & Slezak, C. Kinetic energy driven pairing in cuprate superconductors. Phys. Rev. Lett. 92, 027005 (2004).
 45
Fazekas, P. Lecture Notes on Electron Correlation and Magnetism (World Scientific, Singapore, 1999).
 46
Scalapino, D. The case for d x 2  y 2 pairing in the cuprate superconductors. Physics Reports 250, 329–365 (1995).
 47
Fournier, D., Poirier, M., Castonguay, M. & Truong, K. D. Mott transition, compressibility divergence, and the P − T phase diagram of layered organic superconductors: An ultrasonic investigation. Phys. Rev. Lett. 90, 127002 (2003).
 48
Hassan, S. R., Georges, A. & Krishnamurthy, H. R. Sound velocity anomaly at the mott transition: Application to organic conductors and V2O3 . Phys. Rev. Lett. 94, 036402 (2005).
 49
Rozenberg, M. J., Chitra, R. & Kotliar, G. Finite temperature mott transition in the hubbard model in infinite dimensions. Phys. Rev. Lett. 83, 3498–3501 (1999).
 50
Furukawa, T., Miyagawa, K., Taniguchi, H., Kato, R. & Kanoda, K. Quantum criticality of Mott transition in organic materials. Nature Physics 3, 221 (2015).
 51
Terletska, J., Vucicevic, D., Tanaskovic, & Dobrosavljevic V. Phys. Rev. Lett. 107, 026401 (2011) [http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.107.026401]
 52
Hebert. C.D., Semon P., & Tremblay A.M. S. Phys. Rev. B. 92, 195112 (2015) [http://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.195112]
Acknowledgements
We acknowledge D. Sénechal, L. Taillefer, C. Bourbonnais and H. Alloul for useful discussions. This work was partially supported by the Natural Sciences and Engineering Research council (Canada), and by the Tier I Canada Research Chair Program (A.M.S.T.). Simulations were performed on computers provided by CFI, MELS, Calcul Québec and Compute Canada.
Author information
Affiliations
Contributions
L.F. obtained and analysed the data. P.S. wrote the main codes. G.S. and A.M.S.T. supervised the project and wrote the manuscript, and all authors discussed the results and commented on the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Fratino, L., Sémon, P., Sordi, G. et al. An organizing principle for twodimensional strongly correlated superconductivity. Sci Rep 6, 22715 (2016). https://doi.org/10.1038/srep22715
Received:
Accepted:
Published:
Further reading

Strainengineering Mottinsulating La2CuO4
Nature Communications (2019)

Entropic Origin of Pseudogap Physics and a MottSlater Transition in Cuprates
Scientific Reports (2017)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.