1. Theory without small parameter: How should we proceed? Identify important physical principles and laws to constrain non-perturbative approximation schemes –From weak coupling (kinetic) –From strong coupling (potential) Benchmark against “exact” (numerical) results. Check that weak and strong coupling approaches agree at intermediate coupling. Compare with experiment

2. Two-Particle Self-Consistent Approach (U < W) - How it works Vilk, AMT J. Phys. I France, 7, 1309 (1997); Allen et al.in Theoretical methods for strongly correlated electrons also cond-mat/0110130 (Mahan, third edition) General philosophy –Drop diagrams –Impose constraints and sum rules (No adjustable parameter) Conservation laws Pauli principle ( = ) Local moment and local density sum-rules Get for free: Mermin-Wagner theorem Kanamori-Brückner screening Consistency between one- and two-particle  G = U

3. Benchmarks

4. U > 0  = 5 8 X 8 QMC + cal.: Vilk et al. P.R. B 49, 13267 (1994) Notes: -F.L. parameters -Self also Fermi-liquid (0.0) (  (  Proof that it works (comparisons with QMC)

5. Calc.: Vilk et al. P.R. B 49, 13267 (1994) QMC: S. R. White, et al. Phys. Rev. 40, 506 (1989). A.-M. Daré, Y.M. Vilk and A.-M.S.T Phys. Rev. B 53, 14236 (1996) n=1

6. Double occupancy : Kyung et al. PRL 90, 099702 (2003), S. Moukouri and M. Jarrell, PRL 87, 167010 (2001)

7. Pseudogap vs DCA S. Moukouri and M. Jarrell, Phys. Rev. Lett. 87, 167010 (2001) B. Kyung, J. S. Landry, D. Poulin, and A.-M. S. Tremblay*, Phys. Rev. Lett. 90, 099702 (2003)

8. U = + 4  Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000). Proofs... TPSC

9. U = + 4 Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000).

10. Benchmark: d-wave superconductivity

11. QMC: symbols. Solid lines analytical Kyung, Landry, A.-M.S.T., Phys. Rev. B (2003)

12. QMC: symbols. Solid lines analytical Kyung, Landry, A.-M.S.T., Phys. Rev. B (2003)

13. QMC: symbols. Solid lines analytical. Kyung, Landry, A.-M.S.T. PRB (2003)

14. - - - - 1 2 3 4 - - +

15. Kyung, Landry, A.-M.S.T. PRB (2003) DCA Maier et al. cond-mat/0504529

16. Armitage et al. Phys. Rev. Lett. 87, 147003 (2001) M.B. Maple MRS Bulletin, June 1990 Electron doped:

17. Pseudogap near optimal doping in e- doped cuprates

18. TPSC U t  n  T d = 2 Hubbard model, simplest model of interacting electrons. Here U > 0 Weak coupling: U < 8t n Filling T Temperature t’= -0.175, t’’= +0.05, T=1/40 U t  n  T d = 2 Hubbard model, simplest model of interacting electrons. Here U > 0 Weak coupling: U < 8t n Filling T Temperature t’= -0.175, t’’= +0.05, T=1/40 U t  n  T d = 2 Hubbard model, simplest model of interacting electrons. Here U > 0 Weak coupling: U < 8t n Filling T Temperature t’= -0.175, t’’= +0.05, T=1/40 U t  n  T d = 2 Hubbard model, simplest model of interacting electrons. Here U > 0 Weak coupling: U < 8t n Filling T Temperature t’= -0.175, t’’= +0.05, T=1/40 U t  n  T d = 2 Hubbard model, simplest model of interacting electrons. Here U > 0 Weak coupling: U < 8t n Filling T Temperature t’= -0.175, t’’= +0.05, T=1/40 U t  n  T d = 2 Hubbard model, simplest model of interacting electrons. Here U > 0 Weak coupling: U < 8t n Filling T Temperature t’= -0.175, t’’= +0.05, T=1/40 U t  n  T d = 2 Hubbard model, simplest model of interacting electrons. Here U > 0 Weak coupling: U < 8t n Filling T Temperature t’= -0.175, t’’= +0.05, T=1/40 U t t’ t’’ Weak coupling U<8t t’=-0.175t, t’’=0.05t t=350 meV, T=200 K n=1+x – electron filling fixed

19. 15% doping: EDCs along the Fermi surface TPSC Exp Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004 U min < U< U max U max also from CPT Armitag e et al. PRL 87, 147003; 88, 257001

20. 15% doped case: EDCs in two directions Exp TPSC Exp Hankevych, Kyung, A.-M.S.T., PRL (2004).

21. TPSC Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004 EDCs along the Fermi surface TPSC Exp

22. Fermi surface plots be not too large increase for smaller doping Hubbard repulsion U has to… U=5.75 U=6.25 B.Kyung et al.,PRB 68, 174502 (2003) Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004 15% 10%

23. Recent confirmation with KR slave-bosons Yuan, Yuan, Ting, cond-mat/0503056

24. Hot spots from AFM quasi-static scattering

25. AFM correlation length (neutron) Hankevych, Kyung, A.-M.S.T., PRL, (2004). Expt: P. K. Mang et al., PRL (2004), Matsuda (1992).

26. Pseudogap temperature and QCP  Δ PG ≈10k B T* comparable with optical measurements Hankevych, Kyung, A.-M.S.T., PRL 2004 : Expt: Y. Onose et al., PRL (2001). Prediction Matsui et al. PRL (2005) Verified theo.T* at x=0.13 with ARPES

27. Pseudogap temperature and QCP  Δ PG ≈10k B T* comparable with optical measurements Prediction QCP may be masked by 3D transitions Hankevych, Kyung, A.-M.S.T., PRL 2004 : Expt: Y. Onose et al., PRL (2001). Prediction  ξ≈ξ th at PG temperature T*, and ξ>ξ th for T<T* supports further AFM fluctuations origin of PG Prediction Matsui et al. PRL (2005) Verified theo.T* at x=0.13 with ARPES

28. Observation Matsui et al. PRL 94, 047005 (2005) Reduced, x=0.13 AFM 110 K, SC 20 K

29. Weak and strong-coupling mechanism for pseudogap, see poster by Hankevych, Kyung, Daré, Sénéchal, Tremblay

30. Steve Allen François Lemay David Poulin Hugo Touchette Yury Vilk Liang Chen Samuel Moukouri J.-S. Landry M. Boissonnault

31. Alexis Gagné-Lebrun Bumsoo Kyung A-M.T. Sébastien Roy Alexandre Blais D. Sénéchal C. Bourbonnais R. Côté K. LeHur Vasyl Hankevych Sarma Kancharla Maxim Mar’enko

32. C’est fini… enfin