Does the Hubbard model have the right stuff?

1
Insights into d-wave superconductivity from
quantum cluster approaches
André-Marie Tremblay
2
CuO2 planes
YBa2Cu3O7-d
3
Experimental phase diagram
1.3 1.2 1.1 1.0
0.9 0.8 0.7
n, electron density
Damascelli, Shen, Hussain, RMP 75, 473 (2003)
4
The Hubbard model
No mean-field factorization for d-wave
superconductivity
5
An effective model
A. Macridin et al., cond-mat/0411092
Damascelli, Shen, Hussain, RMP 75, 473 (2003)
6
Weak vs strong coupling, n1
7
U of order W at least, consider e and h doped
1.3 1.2 1.1 1.0
0.9 0.8 0.7
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!

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 ltlt 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 gtgt 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
Numerical methods that show Tc at strong coupling
12
Recent DCA results

• Finite-size studies U4t
• Maier et al., PRL 2005
• Structure of pairing Kernel
• Maier et al., PRL 2006

13
David Sénéchal
Sarma Kancharla
Bumsoo Kyung
Pierre-Luc Lavertu
Marc-André Marois
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
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)
17
Another way to look at this (Potthoff)
Still stationary (chain rule)
M. Potthoff, Eur. Phys. J. B 32, 429 (2003).
18
A dynamical stationary principleSFT
Self-energy Functional Theory
With FS
Legendre transform of Luttinger-Ward funct.
is stationary with respect to S and equal to
grand potential there.
For given interaction, FS is a universal
functional of S, no explicit dependence on H0(t).
Hence, use solvable cluster H0(t) to find FS.
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).
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).
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)
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
/(4t)
Parcollet, Biroli, Kotliar, PRL (2004)
23
Comparison, TPSC-CDMFT, n1, U4t
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
Sarma Kancharla
No Weiss field on the cluster!
Caffarel and Krauth, PRL (1994)
27
Effect of proximity to Mott (CDMFT )
t t0
D-wave OP
Kancharla, Civelli, Capone, Kyung, Sénéchal,
Kotliar, A-M.S.T. cond-mat/0508205
Sarma Kancharla
28
Gap vs order parameter
t t0
D y Z
Kancharla, Civelli, Capone, Kyung, Sénéchal,
Kotliar, A-M.S.T. cond-mat/0508205
29
Competition AFM-dSC using SFT
David Sénéchal
See also, Capone and Kotliar, cond-mat/0603227,
Macridin et al. DCA cond-mat/0411092
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
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

32
Outline

• Pseudogap

33
Pseudogap (CDMFT)
t -0.3 t, t 0 t U 8t
Kyung, Kancharla, Sénéchal, A.-M.S. T, Civelli,
Kotliar PRB (2006)
Bumsoo Kyung
5
5
See also Sénéchal, AMT, PRL 92, 126401 (2004).
34
Other properties of the pseudogap
Bumsoo Kyung
Kyung, Kancharla, Sénéchal, A.-M.S. T, Civelli,
Kotliar PRB in press
Pseudogap size function of doping
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

36
Outline

• A broader perspective on d-wave superconductivity

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
38
Layered organics (k-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 (XCuN(CN)2Cl) S. Lefebvre
et al. PRL 85, 5420 (2000), P. Limelette, et al.
PRL 91 (2003)
40
Perspective
U/t
d
t/t
41
Experimental phase diagram for Cl
F. Kagawa, K. Miyagawa, K. Kanoda PRB 69
(2004) Nature 436 (2005)
Diagramme de phase (XCuN(CN)2Cl) S. Lefebvre
et al. PRL 85, 5420 (2000), P. Limelette, et al.
PRL 91 (2003)
42
Mott transition (C-DMFT)
Kyung, A.-M.S.T. (2006)
See also, Sénéchal, Sahebsara, cond-mat/0604057
43
Mott transition (C-DMFT)
Kyung, A.-M.S.T. (2006)
See also, Sénéchal, Sahebsara, cond-mat/0604057
44
Normal phase theoretical results for BEDT-X
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 (XCuN(CN)2Cl) S. Lefebvre
et al. PRL 85, 5420 (2000), P. Limelette, et al.
PRL 91 (2003)
46
Theoretical phase diagram BEDT
Kyung
OP
Kyung, A.-M.S.T. cond-mat/0604377
Sénéchal, Sahebsara, cond-mat/0604057
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

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

t/t0.8t
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.
51
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
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.

U5.75
U6.25
U5.75
U6.25
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
• Tc (DCA TPSC)
• Variational principle
• First principles
• Questions
• Why not 3d?
• Best  mean-field  approach.
• Manifestations of mechanism
• Frustration vs nesting

55
David Sénéchal
Sarma Kancharla
Bumsoo Kyung
Pierre-Luc Lavertu
Marc-André Marois
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/0511085 v1 3 Nov 2005

60
Cest fini Merci
Cest fini enfin
61
MDC in CDMFT
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
62
AF and dSC order parameters, U 8t, for various
sizes
t -0.3 t t 0.2t U 8t
dSC
Sénéchal, Lavertu, Marois, A.-M.S.T., PRL, 2005
Aichhorn, Arrigoni, Potthoff, Hanke,
cond-mat/0511460
63
Hole-doped (17)

• t -0.3t
• t 0.2t
• 0.12t
• 0.4t

Sénéchal, AMT, PRL 92, 126401 (2004).
64
Hole-doped 17, U8t
65
Electron-doped (17)

• t -0.3t
• t 0.2t
• 0.12t
• 0.4t

Sénéchal, AMT, PRL in press
66
Electron-doped, 17, U8t