Insights into d-wave superconductivity from quantum cluster approaches
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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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Insights into d-wave superconductivity from quantum cluster approaches

c = -1

Perfect diamagnetism (Shielding of magnetic field)

(Meissner effect)

André-Marie Tremblay

1


CuO2 planes

2

YBa2Cu3O7-d


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)


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


An effective model

A. Macridin et al., cond-mat/0411092

5

Damascelli, Shen, Hussain, RMP 75, 473 (2003)


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


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)


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


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


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


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


Recent DCA results

  • Finite-size studies U=4t

    • Maier et al., PRL 2005

  • Structure of pairing Kernel

    • Maier et al., PRL 2006

12


David Sénéchal

Sarma Kancharla

Bumsoo Kyung

Pierre-Luc Lavertu

Marc-André Marois

13


Outside collaborators

Gabi Kotliar

Marcello Civelli

Massimo Capone

14


Outline

  • Methodology

  • T = 0 phase diagram

    • Cellular Dynamical Mean-Field Theory

    • Anomalous superconductivity : Non-BCS

  • Pseudogap

  • A broader perspective on d-wave superconductivity

15


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


Another way to look at this (Potthoff)

Still stationary (chain rule)

17

M. Potthoff, Eur. Phys. J. B 32, 429 (2003).


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).


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


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


Tests: Sin-charge separation d = 1

U/t = 4, = 0.2, n = 0.89

Kyung, Kotliar, Tremblay, PRB 2006

21


Test: CDMFT Recover d = infinity Mott transition

/(4t)

Parcollet, Biroli, Kotliar, PRL (2004)

22


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

TPSC

CDMFT

23


Outline

  • Methodology

  • T = 0 phase diagram

    • Cellular Dynamical Mean-Field Theory

    • Anomalous superconductivity : Non-BCS

  • Pseudogap

  • A broader perspective on d-wave superconductivity

24


Outline

  • T = 0 phase diagram

    • Cellular Dynamical Mean-Field Theory

    • Anomalous superconductivity : Non-BCS

25


CDMFT + ED

-

-

+

+

+

+

-

-

Sarma Kancharla

No Weiss field on the cluster!

Caffarel and Krauth, PRL (1994)

26


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


Gap vs order parameter

t’= t’’=0

D = y Z

Kancharla, Civelli, Capone, Kyung, Sénéchal, Kotliar,

A-M.S.T. cond-mat/0508205

28


Competition AFM-dSC – using SFT

-

-

-

+

+

+

+

+

+

-

-

-

David Sénéchal

See also, Capone and Kotliar, cond-mat/0603227,

Macridin et al. DCAcond-mat/0411092

29


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)


Outline

  • Methodology

  • T = 0 phase diagram

    • Cellular Dynamical Mean-Field Theory

    • Anomalous superconductivity : Non-BCS

  • Pseudogap

  • A broader perspective on d-wave superconductivity

31


Outline

  • Pseudogap

32


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).


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).


Outline

  • Methodology

  • T = 0 phase diagram

    • Cellular Dynamical Mean-Field Theory

    • Anomalous superconductivity : Non-BCS

  • Pseudogap

  • A broader perspective on d-wave superconductivity

35


Outline

  • A broader perspective on d-wave superconductivity

36


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


Layered organics (k-BEDT-X family)

( t’ / t )

n = 1

38


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


Perspective

U/t

d

t’/t

40


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


Mott transition (C-DMFT)

Parcollet, Biroli, Kotliar, PRL 92 (2004)

Kyung, A.-M.S.T. (2006)

See also, Sénéchal, Sahebsara, cond-mat/0604057

42


Mott transition (C-DMFT)

Kyung, A.-M.S.T. (2006)

See also, Sénéchal, Sahebsara, cond-mat/0604057

43


Normal phase theoretical results for BEDT-X

PI

M

44

Kyung, A.-M.S.T. (2006)


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


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


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


d-wave

Kyung, A.-M.S.T. cond-mat/0604377

Sénéchal, Sahebsara, cond-mat/0604057

48


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


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


Summary - Conclusion

  • Ground state of CuO2 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/0508205

51


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

U=6.25

U=5.75

52


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.

53


Conclusion, open problems

  • Methodology:

    • Response functions

    • Tc (DCA + TPSC)

    • Variational principle

    • First principles

  • Questions:

    • Why not 3d?

    • Best « mean-field » approach.

    • Manifestations of mechanism

    • Frustration vs nesting

54


David Sénéchal

Sarma Kancharla

Bumsoo Kyung

Pierre-Luc Lavertu

Marc-André Marois

55


Outside collaborators

Gabi Kotliar

Marcello Civelli

Massimo Capone

56


Mammouth, série

57


André-Marie Tremblay

Sponsors:

58


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/0511085 v1 3 Nov 2005

59


C’est fini…

Merci

C’est fini…

enfin

60


MDC in CDMFT

Armitage et al.

PRL 2003

Ronning et al. PRB 2003

15%

10%

4%

t’ = -0.3 t, t’’ = 0 t

U = 8t

7%

7%

4%

Kancharla, Civelli, Capone, Kyung, Sénéchal, Kotliar,

A-M.S.T. cond-mat/0508205

61


AF and dSC order parameters, U = 8t, for various sizes

1.3 1.2 1.1 1.0 0.9 0.8 0.7

AF

t’ = -0.3 t

t’’ = 0.2t

U = 8t

dSC

Sénéchal, Lavertu, Marois, A.-M.S.T., PRL, 2005

62

Aichhorn, Arrigoni, Potthoff, Hanke, cond-mat/0511460


Hole-doped (17%)

  • t’ = -0.3t

  • t”= 0.2t

  • = 0.12t

  • = 0.4t

63

Sénéchal, AMT, PRL 92, 126401 (2004).


Hole-doped 17%, U=8t

64


Electron-doped (17%)

  • t’ = -0.3t

  • t”= 0.2t

  • = 0.12t

  • = 0.4t

Sénéchal, AMT,

PRL in press

65


Electron-doped, 17%, U=8t

66


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