c = -1 Perfect diamagnetism (Shielding of magnetic field) (Meissner effect)

Dynamic variational principle and the phase diagram of high-temperature superconductors c = -1 Perfect diamagnetism (Shielding of magnetic field) (Meissner effect) André-Marie Tremblay

ky w w kx k r -p/a p/a Some basic Solid State Physics : non-interacting electrons

e Photon 2 k = E +w + m- W ph 2m k Electronic states in d=2 Angle Resolved Photoemission Spectroscopy (ARPES)

The non-interacting case EDC Damascelli, Shen, Hussain, RMP 75, 473 (2003)

Electron-doped, non-interacting MDC

Interacting case: The Fermi liquid A(k,w)f(w) Damascelli, Shen, Hussain, RMP 75, 473 (2003)

A Fermi liquid ind = 2 T-TiTe2 U / W = 0.8 Perfetti, Grioni et al. Phys. Rev. B64, 115102(2001)

w Q D k -p/a p/a Destroying the Fermi liquid at half-filling:Lattice + interactions A-Long-range order Introduce “frustration” Will “resist” LRO until critical U

w w w U W W r r r U w w w W W U U DMFT- Georges, Kotliar, Rosenberg, 1986. r r r Destroying the Fermi liquid at half-filling:Lattice + interactions B-Strong on-site repulsion (Mott transition)

Question: What happens away from n = 1? A- Long-Range Order (U large enough) Hole pockets: Still FL B- Mott transition : DMFT If gapped, gapped everywhere

Two ways to destroy a Fermi liquid: hole and electron-doped cuprates. • I. Introduction • Fermi liquid • II. Experimental results from cuprates + model • III. Strong and weak coupling pseudogap (CPT) • IV. Weak coupling pseudogap (QMC,TPSC) • V. d-wave superconductivity • VI. Conclusion

CuO2 planes YBa2Cu3O7-d

Hole doping Electron doping Optimal doping Optimal doping Phase diagram n, electron density Damascelli, Shen, Hussain, RMP 75, 473 (2003)

15% 10% 10% 15% 4% Pseudogap at hot spots 4% Fermi surface, electron-doped case Armitage et al. PRL 87, 147003; 88, 257001

Fermi surface, hole-doped case 10%

Simplest microscopic model for Cu O planes. t’ t’’ m U LSCO t • Size of Hilbert space : • With N=16, It takes 4 GigaBits just to store the states (N = 16) The « Hubbard model »

A(kF,w) A(kF,w) LHB UHB t Effective model, Heisenberg: J = 4t2 /U Weak vs strong coupling, n=1 T w U w U Mott transition U ~ 1.5W (W= 8t)

Question: quantitative and qualitative • How do we go from a Mott insulator to a conductor as a function of doping? • Hot spots and pseudogaps in the Hubbard model (like experiment) ? • Close to understood in e-doped case.

Two ways to destroy a Fermi liquid: hole and electron-doped cuprates. • I. Introduction • Fermi liquid • II. Experimental results from the cuprates and model • III. Strong and weak coupling pseudogap (CPT) • IV. Weak coupling pseudogap (QMC,TPSC) • V. d-wave superconductivity • VI. Conclusion

c = -1 Perfect diamagnetism (Shielding of magnetic field) (Meissner effect)

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