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

2. Some basic Solid State Physics : non-interacting electrons  k  a  a  r kxkx kyky

3. e Photon = E +  - W k 2m 2 k ph Angle Resolved Photoemission Spectroscopy (ARPES) Electronic states in d=2

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

5. Electron-doped, non-interacting MDC

6. Interacting case: The Fermi liquid A(k,  f  Damascelli, Shen, Hussain, RMP 75, 473 (2003)

7. A Fermi liquid in d = 2 Perfetti, Grioni et al. Phys. Rev. B 64, 115102 (2001) U / W = 0.8 T-TiTe 2

8. Destroying the Fermi liquid at half-filling: Lattice + interactions A-Long-range order Q  k  a  a Introduce “frustration”  Will “resist” LRO until critical U

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

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

11. 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

12. CuO 2 planes YBa 2 Cu 3 O 7- 

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

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

15. Fermi surface, hole-doped case 10%

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

17. T U U A(k F,  )   Weak vs strong coupling, n=1 Mott transition U ~ 1.5W (W= 8t) LHB UHB t Effective model, Heisenberg: J = 4t 2 /U

18. 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.

19. 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