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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Insights into d-wave superconductivity from quantum cluster approaches

c = -1

Perfect diamagnetism (Shielding of magnetic field)

(Meissner effect)

André-Marie Tremblay

1


Cuo 2 planes
CuO approaches2 planes

2

YBa2Cu3O7-d


Experimental phase diagram
Experimental phase diagram approaches

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 approaches

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
An effective model approaches

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

5

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


A( approacheskF,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
U approaches 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
Theoretical difficulties approaches

  • 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
Theory without small parameter: approachesHow 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
Mounting evidence for d-wave in Hubbard approaches

  • 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
Numerical methods that show Tc at strong coupling approaches

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
Recent DCA results approaches

  • Finite-size studies U=4t

    • Maier et al., PRL 2005

  • Structure of pairing Kernel

    • Maier et al., PRL 2006

12


David Sénéchal approaches

Sarma Kancharla

Bumsoo Kyung

Pierre-Luc Lavertu

Marc-André Marois

13


Outside collaborators approaches

Gabi Kotliar

Marcello Civelli

Massimo Capone

14


Outline
Outline approaches

  • 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
Dynamical “variational” principle approaches

+ ….

+

+

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
Another way to look at this (Potthoff) approaches

Still stationary (chain rule)

17

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


A dynamical stationary principle sft self energy functional theory
A dynamical “stationary” principle approachesSFT : 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
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
Tests : CDMFT cases of SFT

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
Tests: Sin-charge separation cases of SFTd = 1

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

Kyung, Kotliar, Tremblay, PRB 2006

21


Test cdmft recover d infinity mott transition
Test: CDMFT cases of SFTRecover d = infinity Mott transition

/(4t)

Parcollet, Biroli, Kotliar, PRL (2004)

22


Comparison tpsc cdmft n 1 u 4t
Comparison, TPSC-CDMFT, cases of SFTn=1, U=4t

TPSC

CDMFT

23


Outline1
Outline cases of SFT

  • Methodology

  • T = 0 phase diagram

    • Cellular Dynamical Mean-Field Theory

    • Anomalous superconductivity : Non-BCS

  • Pseudogap

  • A broader perspective on d-wave superconductivity

24


Outline2
Outline cases of SFT

  • T = 0 phase diagram

    • Cellular Dynamical Mean-Field Theory

    • Anomalous superconductivity : Non-BCS

25


Cdmft ed
CDMFT + ED cases of SFT

-

-

+

+

+

+

-

-

Sarma Kancharla

No Weiss field on the cluster!

Caffarel and Krauth, PRL (1994)

26


Effect of proximity to mott cdmft
Effect of proximity to Mott (CDMFT ) cases of SFT

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
Gap vs order parameter cases of SFT

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
Competition AFM-dSC – using SFT cases of SFT

-

-

-

+

+

+

+

+

+

-

-

-

David Sénéchal

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

Macridin et al. DCAcond-mat/0411092

29


Preliminary
Preliminary cases of SFT

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)


Outline3
Outline cases of SFT

  • Methodology

  • T = 0 phase diagram

    • Cellular Dynamical Mean-Field Theory

    • Anomalous superconductivity : Non-BCS

  • Pseudogap

  • A broader perspective on d-wave superconductivity

31


Outline4
Outline cases of SFT

  • Pseudogap

32


Pseudogap cdmft
Pseudogap (CDMFT) cases of SFT

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
Other properties of the pseudogap cases of SFT

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).


Outline5
Outline cases of SFT

  • Methodology

  • T = 0 phase diagram

    • Cellular Dynamical Mean-Field Theory

    • Anomalous superconductivity : Non-BCS

  • Pseudogap

  • A broader perspective on d-wave superconductivity

35


Outline6
Outline cases of SFT

  • A broader perspective on d-wave superconductivity

36


One band hubbard model for organics
One-band Hubbard model for organics cases of SFT

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
Layered organics ( cases of SFTk-BEDT-X family)

( t’ / t )

n = 1

38


Experimental phase diagram for cl
Experimental phase diagram for Cl cases of SFT

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
Perspective cases of SFT

U/t

d

t’/t

40


Experimental phase diagram for cl1
Experimental phase diagram for Cl cases of SFT

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
Mott transition (C-DMFT) cases of SFT

Parcollet, Biroli, Kotliar, PRL 92 (2004)

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

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

42


Mott transition c dmft1
Mott transition (C-DMFT) cases of SFT

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

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

43


Normal phase theoretical results for bedt x
Normal phase theoretical results for BEDT-X cases of SFT

PI

M

44

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


Experimental phase diagram for cl2
Experimental phase diagram for Cl cases of SFT

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
Theoretical phase diagram BEDT cases of SFT

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
AFM and dSC order parameters for various t’/t cases of SFT

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
d-wave cases of SFT

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

Sénéchal, Sahebsara, cond-mat/0604057

48


Prediction of a new type of pressure behavior
Prediction of a new type of pressure behavior cases of SFT

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


R f rences on layered organics
Références on layered organics cases of SFT

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
Summary - Conclusion cases of SFT

  • 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
Conclusion cases of SFT

  • 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


Conclusion1
Conclusion cases of SFT

  • 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
Conclusion, open problems cases of SFT

  • 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 cases of SFT

Sarma Kancharla

Bumsoo Kyung

Pierre-Luc Lavertu

Marc-André Marois

55


Outside collaborators cases of SFT

Gabi Kotliar

Marcello Civelli

Massimo Capone

56


Mammouth s rie
Mammouth, série cases of SFT

57


André-Marie Tremblay cases of SFT

Sponsors:

58


Recent review articles
Recent review articles cases of SFT

  • 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… cases of SFT

Merci

C’est fini…

enfin

60


Mdc in cdmft
MDC in CDMFT cases of SFT

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
AF and dSC order parameters, cases of SFTU = 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
Hole-doped (17%) cases of SFT

  • t’ = -0.3t

  • t”= 0.2t

  • = 0.12t

  • = 0.4t

63

Sénéchal, AMT, PRL 92, 126401 (2004).


Hole doped 17 u 8t
Hole-doped 17%, U=8t cases of SFT

64


Electron doped 17
Electron-doped (17%) cases of SFT

  • t’ = -0.3t

  • t”= 0.2t

  • = 0.12t

  • = 0.4t

Sénéchal, AMT,

PRL in press

65


Electron doped 17 u 8t
Electron-doped, 17%, cases of SFTU=8t

66


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