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Dynamic Variational Principle and the Phase Diagram of High-Temperature Superconductors

This study explores the dynamic variational principle and phase diagram of high-temperature superconductors using the perfect diamagnetism and shielding of magnetic field (Meissner effect) in CuO2 planes. It examines the experimental phase diagram and the theoretical difficulties in low dimensions with quantum and thermal fluctuations.

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Dynamic Variational Principle and the Phase Diagram of High-Temperature Superconductors

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

  2. CuO2 planes YBa2Cu3O7-d

  3. Experimental phase diagram - + + - 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 LCO t No mean-field factorization for d-wave superconductivity

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

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

  7. Methodology

  8. Heuristics C-DMFT V- M. Potthoff et al. PRL 91, 206402 (2003).

  9. 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)

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

  11. SFT : 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’) M. Potthoff, Eur. Phys. J. B 32, 429 (2003).

  12. The T = 0 ordered phases

  13. Different clusters L = 8 L = 6 L = 10 V-CPT David Sénéchal The mean-fields decrease with system size

  14. Order parameters for competing d-SC and AF

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

  16. AF order parameter, size dependence e h U = 8t t’ = -0.3t t’’ = 0.2t e-doped h-doped

  17. AF order parameter, size dependence

  18. AF order parameter, size dependence

  19. dSC order parameter, size dependence Lichtenstein et al. 2000

  20. dSC order parameter, size dependence

  21. dSC order parameter, size dependence

  22. AF and dSC order parameters, U = 8t, for various sizes dSC AF Sénéchal, Lavertu, Marois, A.-M.S.T., PRL, 2005

  23. AF and dSC order parameters, L=8, for various U dSC AF

  24. Other numerical results • Th. Maier et al. PRL 85, 1524 (2000) • A. Paramekanti et al. PRL 87 (2001). • S. Sorella et al. PRL 88, 2002 • D. Poilblanc et al. cond-mat/0202180 (2002) • Th. Maier et al.cond-mat/0504529 • Review: Th. Maier, RMP (2005) Here AFM and d-SC for both hole and electron doping

  25. The ARPES spectrum

  26. Fermi surface plots, U = 8t, L = 8 15% 10% 4% MDC at the Fermi energy Wave-particle

  27. Single-particle (ARPES) spectrum, e-dopedn = 1.1 For coexisting AF and d-SC

  28. Comparison with mean-fieldn = 1.1 AF

  29. Single-particle (ARPES) spectrum, e-dopedn = 1.1

  30. Observation of in-gap states H.Kusunose et al., PRL 91, 186407 (2003) Armitage et al. PRL 87, 147003; 88, 257001

  31. Single-particle (ARPES) spectrum, e-dopedn = 1.1 Atomic limit AF Mott gap

  32. Dispersion, U = 8t, L = 8, n =1.1 d-wave gap

  33. e-doped h-doped

  34. Conclusion • With V-CPT • t-t’-t’’-U Hubbard model has AF and d-SC where expected from experiment on both h-, e-doped cuprates. (fixed t-t’-t’’) • ARPES in agreement with experiment (40 meV) • Homogeneous coexistence of AF + dSC is stable on e-doped side. • D. Sénéchal, Lavertu, Marois AMST PRL 2005

  35. Conclusion • The Physics of High-temperature superconductors is in the Hubbard model (with a very high probability). • We are beginning to know how to squeeze it out of the model!

  36. Mammouth, série

  37. C’est fini… enfin

  38. André-Marie Tremblay Sponsors:

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

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

  41. 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

  42. Starting from weak coupling, U << 8t

  43. Theory difficult even at weak to intermediate coupling! 4 2 1 1 1 2 2 - + = 2 3 3 3 3 5 S = • RPA (OK with conservation laws) • Mermin-Wagner • Pauli • Moryia (Conjugate variables HS f4 = <f2> f2 ) • Adjustable parameters: c and Ueff • Pauli • FLEX • No pseudogap • Pauli • Renormalization Group • 2 loops X X X Vide X Rohe and Metzner (2004) Katanin and Kampf (2004)

  44. Two-Particle Self-Consistent Approach (U < 8t) - How it works • General philosophy • Drop diagrams • Impose constraints and sum rules • Conservation laws • Pauli principle ( <ns2> = <ns > ) • Local moment and local density sum-rules • Get for free: • Mermin-Wagner theorem • Kanamori-Brückner screening • Consistency between one- and two-particle SG = U<ns n-s> 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)

  45. TPSC approach: two steps (a) spin vertex: 2. Factorization I: Two-particle self consistency 1. Functional derivative formalism (conservation laws) (b) analog of the Bethe-Salpeter equation: (c) self-energy:

  46. TPSC… Kanamori-Brückner screening 3. The F.D. theorem and Pauli principle II: Improved self-energy Insert the first step results into exact equation:

  47. Benchmark for TPSC :Quantum Monte Carlo • Advantages of QMC • Sizes much larger than exact diagonalizations • As accurate as needed • Disadvantages of QMC • Cannot go to very low temperature in certain doping ranges, yet low enough in certain cases to discard existing theories.

  48. Proofs... U = + 4 b = 5 TPSC Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000).

  49. Moving on to experiment : pseudogap

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