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

Insights into d-wave superconductivity from quantum cluster approaches. c = -1 Perfect diamagnetism (Shielding of magnetic field) (Meissner effect). André-Marie Tremblay. CuO 2 planes. YBa 2 Cu 3 O 7- d. Experimental phase diagram. Hole doping. Electron doping. Stripes.

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

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

  2. CuO2 planes 2 YBa2Cu3O7-d

  3. Experimental phase diagram Hole doping Electron doping Stripes Optimal doping Optimal doping 1.3 1.2 1.1 1.0 0.9 0.8 0.7 3 n, electron density Damascelli, Shen, Hussain, RMP 75, 473 (2003)

  4. The Hubbard model Simplest microscopic model for Cu O2 planes. t’ t’’ m U LSCO t No mean-field factorization for d-wave superconductivity 4

  5. An effective model A. Macridin et al., cond-mat/0411092 5 Damascelli, Shen, Hussain, RMP 75, 473 (2003)

  6. A(kF,w) T A(kF,w) w U w U LHB UHB Mott transition (DMFT, exact d = ) U ~ 1.5W (W= 8t) t Effective model, Heisenberg: J = 4t2 /U Weak vs strong coupling, n=1 6

  7. U of order W at least, consider e and h doped Mott Insulator 1.3 1.2 1.1 1.0 0.9 0.8 0.7 7 n, electron density Damascelli, Shen, Hussain, RMP 75, 473 (2003)

  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! 8

  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 9

  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) 10

  11. Numerical methods that show Tc at strong coupling DCA Variational Th. Maier, M. Jarrell, Th. Pruschke, and J. Keller Phys. Rev. Lett. 85, 1524 (2000) T.A. Maier et al. PRL (2005) Paramekanti, M. Randeria, and N. Trivedi Phys. Rev. Lett. 87, 217002 (2001) 11

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

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

  14. Outside collaborators Gabi Kotliar Marcello Civelli Massimo Capone 14

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

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

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

  18. A dynamical “stationary” principleSFT : Self-energy Functional Theory With F[S] Legendre transform of Luttinger-Ward funct. is stationary with respect to S and equal to grand potential there. For given interaction, F[S] is a universal functional of S, no explicit dependence on H0(t). Hence, use solvable cluster H0(t’) to find F[S]. Vary with respect to parameters of the cluster (including Weiss fields) Variation of the self-energy, through parameters in H0(t’) 18 M. Potthoff, Eur. Phys. J. B 32, 429 (2003).

  19. Variational cluster perturbation theory and DMFT as special cases of SFT M. Potthoff et al. PRL 91, 206402 (2003). C-DMFT DCA, Jarrell et al. V- Savrasov, Kotliar, PRB (2001) Georges Kotliar, PRB (1992). M. Jarrell, PRL (1992). A. Georges, et al. RMP (1996). 19

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

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

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

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

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

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

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

  27. Effect of proximity to Mott (CDMFT ) t’= t’’=0 D-wave OP Kancharla, Civelli, Capone, Kyung, Sénéchal, Kotliar, A-M.S.T.cond-mat/0508205 Sarma Kancharla 27

  28. Gap vs order parameter t’= t’’=0 D = y Z Kancharla, Civelli, Capone, Kyung, Sénéchal, Kotliar, A-M.S.T. cond-mat/0508205 28

  29. Competition AFM-dSC – using SFT - - - + + + + + + - - - David Sénéchal See also, Capone and Kotliar, cond-mat/0603227, Macridin et al. DCAcond-mat/0411092 29

  30. Preliminary t’ = -0.3 t, t’’ = 0.2 t U = 8t 1.3 1.2 1.1 1.0 0.9 0.8 0.7 30 n, electron density Damascelli, Shen, Hussain, RMP 75, 473 (2003)

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

  32. Outline • Pseudogap 32

  33. Pseudogap (CDMFT) Armitage et al. PRL 2003 Ronning et al. PRB 2003 15% 10% 4% t’ = -0.3 t, t’’ = 0 t U = 8t Kyung, Kancharla, Sénéchal, A.-M.S. T, Civelli, Kotliar PRB (2006) Bumsoo Kyung 5% 5% 33 See also Sénéchal, AMT, PRL 92, 126401 (2004).

  34. Other properties of the pseudogap Effect of U Bumsoo Kyung Kyung, Kancharla, Sénéchal, A.-M.S. T, Civelli, Kotliar PRB in press Pseudogap size function of doping 34 See also Sénéchal, AMT, PRL 92, 126401 (2004).

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

  36. Outline • A broader perspective on d-wave superconductivity 36

  37. One-band Hubbard model for organics H. Kino + H. Fukuyama, J. Phys. Soc. Jpn 65 2158 (1996), R.H. McKenzie, Comments Condens Mat Phys. 18, 309 (1998) Y. Shimizu, et al. Phys. Rev. Lett. 91, 107001(2003) t’/t ~ 0.6 - 1.1 37

  38. Layered organics (k-BEDT-X family) ( t’ / t ) n = 1 38

  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) 39

  40. Perspective U/t d t’/t 40

  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) 41

  42. Mott transition (C-DMFT) Parcollet, Biroli, Kotliar, PRL 92 (2004) Kyung, A.-M.S.T. (2006) See also, Sénéchal, Sahebsara, cond-mat/0604057 42

  43. Mott transition (C-DMFT) Kyung, A.-M.S.T. (2006) See also, Sénéchal, Sahebsara, cond-mat/0604057 43

  44. Normal phase theoretical results for BEDT-X PI M 44 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) 45

  46. Theoretical phase diagram BEDT Sénéchal X= Cu2(CN)3 (t’~ t) Y. Kurisaki, et al. Phys. Rev. Lett. 95, 177001(2005) Y. Shimizu, et al. Phys. Rev. Lett. 91, (2003) Kyung OP Kyung, A.-M.S.T. cond-mat/0604377 Sénéchal, Sahebsara, cond-mat/0604057 46 Sahebsara

  47. AFM and dSC order parameters for various t’/t AF multiplied by 0.1 • Discontinuous jump • Correlation between maximum order parameter and Tc 47 Kyung, A.-M.S.T. cond-mat/0604377

  48. d-wave Kyung, A.-M.S.T. cond-mat/0604377 Sénéchal, Sahebsara, cond-mat/0604057 48

  49. Prediction of a new type of pressure behavior Sénéchal, Sahebsara, cond-mat/0604057 Kyung, A.-M.S.T. cond-mat/0604377 AF SL • All transitions first order, except one with dashed line dSC • Triple point, not SO(5) 49 t’/t=0.8t

  50. Références on layered organics H. Morita et al., J. Phys. Soc. Jpn. 71, 2109 (2002). J. Liu et al., Phys. Rev. Lett. 94, 127003 (2005). S.S. Lee et al., Phys. Rev. Lett. 95, 036403 (2005). B. Powell et al., Phys. Rev. Lett. 94, 047004 (2005). J.Y. Gan et al., Phys. Rev. Lett. 94, 067005 (2005). T. Watanabe et al., cond-mat/0602098. 50

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