1  = -1 Perfect diamagnetism (Shielding of magnetic field) (Meissner effect) Insights into d-wave superconductivity from quantum cluster approaches André-Marie Tremblay

2 CuO 2 planes YBa 2 Cu 3 O 7- 

3 Experimental phase diagram n, electron density Hole doping Electron doping Damascelli, Shen, Hussain, RMP 75, 473 (2003) Optimal doping Stripes

4 U t  Simplest microscopic model for Cu O 2 planes. t’ t’’ LSCO The Hubbard model No mean-field factorization for d-wave superconductivity

5 An effective model Damascelli, Shen, Hussain, RMP 75, 473 (2003) A. Macridin et al., cond-mat/

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

7 U of order W at least, consider e and h doped n, electron density Damascelli, Shen, Hussain, RMP 75, 473 (2003) Mott Insulator

8 Theoretical difficulties Low dimension –(quantum and thermal fluctuations) Large residual interactions (develop methods) –(Potential ~ Kinetic) –Expansion parameter? –Particle-wave? By now we should be as quantitative as possible!

9 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

10 Mounting evidence for d-wave in Hubbard Weak coupling (U << W) –AF spin fluctuations mediated pairing with d-wave symmetry (Bourbonnais (86), Scalapino (86), Varma (86), Bickers et al., PRL 1989; Monthoux et al., PRL 1991; Scalapino, JLTP 1999, Kyung et al. (2003)) –RG → Groundstate d-wave superconducting (Halboth, PRB 2000; Zanchi, PRB 2000, Berker 2005) Strong coupling (U >> W) –Early mean-field (Kotliar, Liu 1988, Inui et al. 1988) –Finite size simulations of t-J model Groundstate superconducting (Sorella et al., PRL 2002; Poilblanc, Scalapino, PRB 2002)

11 Th. Maier, M. Jarrell, Th. Pruschke, and J. Keller Phys. Rev. Lett. 85, 1524 (2000) T.A. Maier et al. PRL (2005) DCA Paramekanti, M. Randeria, and N. Trivedi Phys. Rev. Lett. 87, (2001) Variational Numerical methods that show Tc at strong coupling

12 Recent DCA results Finite-size studies U=4t –Maier et al., PRL 2005 Structure of pairing Kernel –Maier et al., PRL 2006

13 Bumsoo Kyung Sarma Kancharla Marc-André Marois Pierre-Luc Lavertu David Sénéchal

14 Outside collaborators Gabi Kotliar Marcello Civelli Massimo Capone

15 Outline Methodology T = 0 phase diagram –Cellular Dynamical Mean-Field Theory –Anomalous superconductivity : Non-BCS Pseudogap A broader perspective on d-wave superconductivity

16 Dynamical “variational” principle Luttinger and Ward 1960, Baym and Kadanoff (1961) …. H.F. if approximate  by first order FLEX higher order Universality Then  is grand potential Related to dynamics (cf. Ritz)

17 Another way to look at this (Potthoff) M. Potthoff, Eur. Phys. J. B 32, 429 (2003). Still stationary (chain rule)

18 A dynamical “stationary” principle SFT : Self-energy Functional Theory M. Potthoff, Eur. Phys. J. B 32, 429 (2003). With F  Legendre transform of Luttinger-Ward funct. For given interaction, F  is a universal functional of  no explicit dependence on H 0 (t). Hence, use solvable cluster H 0 (t’) to find F  Vary with respect to parameters of the cluster (including Weiss fields) Variation of the self-energy, through parameters in H 0 (t’) is stationary with respect to  and equal to grand potential there 

19 Variational cluster perturbation theory and DMFT as special cases of SFT C-DMFT V- DCA, Jarrell et al. Georges Kotliar, PRB (1992). M. Jarrell, PRL (1992). A. Georges, et al. RMP (1996). M. Potthoff et al. PRL 91, (2003). Savrasov, Kotliar, PRB (2001)

20 1D Hubbard model: Worst case scenario Excellent agreement with exact results in both metallic and insulating limits Capone, Civelli, SSK, Kotliar, Castellani PRB (2004) Bolech, SSK, Kotliar PRB (2003) Tests : CDMFT

21 Tests: Sin-charge separation d = 1 U/t = 4, = 0.2, n = 0.89 Kyung, Kotliar, Tremblay, PRB 2006

22 Test: CDMFT Recover d = infinity Mott transition Parcollet, Biroli, Kotliar, PRL (2004) /(4t)

23 Comparison, TPSC-CDMFT, n=1, U=4t TPSC CDMFT

24 Outline Methodology T = 0 phase diagram –Cellular Dynamical Mean-Field Theory –Anomalous superconductivity : Non-BCS Pseudogap A broader perspective on d-wave superconductivity

25 Outline T = 0 phase diagram –Cellular Dynamical Mean-Field Theory –Anomalous superconductivity : Non-BCS

26 CDMFT + ED Caffarel and Krauth, PRL (1994) Sarma Kancharla No Weiss field on the cluster!

27 Effect of proximity to Mott (CDMFT ) Kancharla, Civelli, Capone, Kyung, Sénéchal, Kotliar, A-M.S.T. cond-mat/ Sarma Kancharla D-wave OP t’= t’’=0

28 Gap vs order parameter Kancharla, Civelli, Capone, Kyung, Sénéchal, Kotliar, A-M.S.T. cond-mat/  t’= t’’=0

29 Competition AFM-dSC – using SFT See also, Capone and Kotliar, cond-mat/ , Macridin et al. DCA cond-mat/ David Sénéchal

30 Preliminary n, electron density Damascelli, Shen, Hussain, RMP 75, 473 (2003) t’ = -0.3 t, t’’ = 0.2 t U = 8t

31 Outline Methodology T = 0 phase diagram –Cellular Dynamical Mean-Field Theory –Anomalous superconductivity : Non-BCS Pseudogap A broader perspective on d-wave superconductivity

32 Outline Pseudogap

33 Pseudogap (CDMFT) Bumsoo Kyung Kyung, Kancharla, Sénéchal, A.-M.S. T, Civelli, Kotliar PRB (2006) 5% See also Sénéchal, AMT, PRL 92, (2004). t’ = -0.3 t, t’’ = 0 t U = 8t 15% 10% 4% Armitage et al. PRL 2003 Ronning et al. PRB 2003

34 Other properties of the pseudogap Effect of U Pseudogap size function of doping See also Sénéchal, AMT, PRL 92, (2004). Kyung, Kancharla, Sénéchal, A.-M.S. T, Civelli, Kotliar PRB in press Bumsoo Kyung

35 Outline Methodology T = 0 phase diagram –Cellular Dynamical Mean-Field Theory –Anomalous superconductivity : Non-BCS Pseudogap A broader perspective on d-wave superconductivity

36 Outline A broader perspective on d-wave superconductivity

37 One-band Hubbard model for organics Y. Shimizu, et al. Phys. Rev. Lett. 91, (2003 ) H. Kino + H. Fukuyama, J. Phys. Soc. Jpn (1996), R.H. McKenzie, Comments Condens Mat Phys. 18, 309 (1998) t’/t ~

38 Layered organics (  BEDT-X family) ( t’ / t ) n = 1

39 Experimental phase diagram for Cl F. Kagawa, K. Miyagawa, + K. Kanoda PRB 69 (2004) +Nature 436 (2005) Diagramme de phase (X=Cu[N(CN) 2 ]Cl) S. Lefebvre et al. PRL 85, 5420 (2000), P. Limelette, et al. PRL 91 (2003)

40 Perspective U/t  t’/t

41 Experimental phase diagram for Cl F. Kagawa, K. Miyagawa, + K. Kanoda PRB 69 (2004) +Nature 436 (2005) Diagramme de phase (X=Cu[N(CN) 2 ]Cl) S. Lefebvre et al. PRL 85, 5420 (2000), P. Limelette, et al. PRL 91 (2003)

42 Mott transition (C-DMFT) Parcollet, Biroli, Kotliar, PRL 92 (2004) Kyung, A.-M.S.T. (2006) See also, Sénéchal, Sahebsara, cond-mat/

43 Mott transition (C-DMFT) Kyung, A.-M.S.T. (2006) See also, Sénéchal, Sahebsara, cond-mat/

44 Normal phase theoretical results for BEDT-X PI M Kyung, A.-M.S.T. (2006)

45 Experimental phase diagram for Cl F. Kagawa, K. Miyagawa, + K. Kanoda PRB 69 (2004) +Nature 436 (2005) Diagramme de phase (X=Cu[N(CN) 2 ]Cl) S. Lefebvre et al. PRL 85, 5420 (2000), P. Limelette, et al. PRL 91 (2003)

46 Theoretical phase diagram BEDT Y. Kurisaki, et al. Phys. Rev. Lett. 95, (2005) Y. Shimizu, et al. Phys. Rev. Lett. 91, (2003) X= Cu 2 (CN) 3 (t’~ t) Kyung, A.-M.S.T. cond-mat/ OP Sénéchal Sénéchal, Sahebsara, cond-mat/ Sahebsara Kyung

47 AFM and dSC order parameters for various t’/t Discontinuous jump Correlation between maximum order parameter and Tc AF multiplied by 0.1 Kyung, A.-M.S.T. cond-mat/

48 d-wave Kyung, A.-M.S.T. cond-mat/ Sénéchal, Sahebsara, cond-mat/

49 Prediction of a new type of pressure behavior Sénéchal, Sahebsara, cond-mat/ t’/t=0.8t AF SL dSC All transitions first order, except one with dashed line Triple point, not SO(5) Kyung, A.-M.S.T. cond-mat/

50 Références on layered organics H. Morita et al., J. Phys. Soc. Jpn. 71, 2109 (2002). J. Liu et al., Phys. Rev. Lett. 94, (2005). S.S. Lee et al., Phys. Rev. Lett. 95, (2005). B. Powell et al., Phys. Rev. Lett. 94, (2005). J.Y. Gan et al., Phys. Rev. Lett. 94, (2005). T. Watanabe et al., cond-mat/

51 Summary - Conclusion Ground state of CuO 2 planes (h-, e-doped) –V-CPT, (C-DMFT) give overall ground state phase diagram with U at intermediate coupling. –Effect of t’. Non-BCS feature –Order parameter decreases towards n = 1 but gap increases. –Max dSC scales like J. –Emerges from a pseudogaped normal state (Z) (scales like t). Sénéchal, Lavertu, Marois, A.-M.S.T., PRL, 2005 Kancharla, Civelli, Capone, Kyung, Sénéchal, Kotliar, A-M.S.T. cond-mat/

52 Conclusion Normal state (pseudogap in ARPES) –Strong and weak coupling mechanism for pseudogap. –CPT, TPSC, slave bosons suggests U ~ 6t near optimal doping for e-doped with slight variations of U with doping. U=5.75 U=6.25

53 Conclusion The Physics of High-temperature superconductors d-wave) is in the Hubbard model (with a very high probability). We are beginning to know how to squeeze it out of the model! Insight from other compounds Numerical solutions … DCA (Jarrell, Maier) Variational QMC (Paramekanti, Randeria, Trivedi). Role of mean-field theories (if possible) : Physics Lot more work to do.

54 Conclusion, open problems Methodology: –Response functions –T c (DCA + TPSC) –Variational principle –First principles –…–… Questions: –Why not 3d? –Best « mean-field » approach. –Manifestations of mechanism –Frustration vs nesting

55 Bumsoo Kyung Sarma Kancharla Marc-André Marois Pierre-Luc Lavertu David Sénéchal

56 Outside collaborators Gabi Kotliar Marcello Civelli Massimo Capone

57 Mammouth, série

58 André-Marie Tremblay Sponsors:

59 Recent review articles A.-M.S. Tremblay, B. Kyung et D. Sénéchal, Low Temperature Physics (Fizika Nizkikh Temperatur), 32,561 (2006). T. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler, Rev. Mod. Phys. 77, 1027 (2005) G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C.A. Marianetti, cond-mat/ v1 3 Nov 2005

60 C’est fini… enfin C’est fini… Merci

61 MDC in CDMFT Kancharla, Civelli, Capone, Kyung, Sénéchal, Kotliar, A-M.S.T. cond-mat/ Ronning et al. PRB % 10% 4% Armitage et al. PRL % 7% t’ = -0.3 t, t’’ = 0 t U = 8t

62 AF and dSC order parameters, U = 8t, for various sizes dSC Sénéchal, Lavertu, Marois, A.-M.S.T., PRL, 2005 AF t’ = -0.3 t t’’ = 0.2t U = 8t Aichhorn, Arrigoni, Potthoff, Hanke, cond-mat/

63 Hole-doped (17%) t’ = -0.3t t”= 0.2t  = 0.12t  = 0.4t Sénéchal, AMT, PRL 92, (2004).

64 Hole-doped 17%, U=8t

65 Electron-doped (17%) t’ = -0.3t t”= 0.2t  = 0.12t  = 0.4t Sénéchal, AMT, PRL in press

66 Electron-doped, 17%, U=8t