125
1 χ = -1 Perfect diamagnetism (Shielding of magnetic field) (Meissner effect) What does this have to do with quantum magnetism? André-Marie Tremblay

0606 Les Houches bref - University of British Columbia · 1 χ= -1 Perfect diamagnetism (Shielding of magnetic field) (Meissner effect) What does this have to do with quantum magnetism?

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1

χ = -1

Perfect diamagnetism(Shielding of magnetic field)

(Meissner effect)

What does this have to do with quantum magnetism?

André-Marie Tremblay

2

CuO2 planes

YBa2Cu3O7-δ

3

Recall some basic Solid State Physics

ω

k

π/a−π/a

ω

r

4

n = 1,Metal according to band theoryAntiferromagnetic insulator in reality

Failure of the « Fermi liquid approach »in High Tc

5

High-temperature superconductivity

Away from half-filling

6

7

- -

+-

+ -

+ -

Type s (n = 1)

Type p (n = 2)

Type d (n = 3)

n=1n=2n=3

Unusual experimental facts about HTSC: several possibilities for Cooper pairs

8

d-wave (n = 3)

n=1n=2n=3

- -

Superconductivity is d-wave

UBC : Bonn, Hardy, Liang

9

An « anomalous » normal state

10

ePhoton

= E + ω + μ - W k2m

2

k

ph

Angle Resolved Photoemission Spectroscopy (ARPES)

Observing electronic states in d = 2

11

Synchrotron radiation

Damascelli, Shen, Hussain, 2002.

12

Recall some basic Solid State Physics

ω

k

π/a−π/a

ω

r kx

ky

13

Non-interacting case

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

EDC

14

With interactions : the Fermi liquid

A(k,ω)f(ω)

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

15

T. Valla, A. V. Fedorov, P. D. Johnson, and S. L. HulbertP.R.L. 83, 2085 (1999).

16

Pseudogap: Fermi surface of an e-doped high TcArmitage et al. PRL 87, 147003; 88, 257001

15%

10%

4%

17

Experimental phase diagram

Optimal dopingOptimal doping

n, electron density

Hole dopingElectron doping

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

Stripes

Mott Insulator

18

Outline of all three hours

• The Hubbard model : itinerant vs localizedbehavior

• Methodology (non-perturbative)

• Results and concordance between methods.

• Comparisons with experiment

19

Ut

μ

Simplest microscopic model for Cu O2planes.

t’t’’

LSCO

H −∑ij t i,j ci cj cj

ci U∑ i ni↑ni↓

The Hubbard model

No mean-field factorization for d-wave superconductivity

20

An effective model

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

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

21

One band Hubbard model

Direct exchange is ferromagnetic !

22

One-band Hubbard model for organics

Y. Shimizu, et al. Phys. Rev. Lett. 91, 107001(2003)

meV 50≈t meV 400≈⇒ U

H. Kino + H. Fukuyama, J. Phys. Soc. Jpn 65 2158 (1996), R.H. McKenzie, Comments Condens Mat Phys. 18, 309 (1998)

t’/t ~ 0.6 - 1.1

23

Layered organics (κ−BEDT-X family)

( t’ / t )

n = 1

25

Experimental phase diagram for Cl

40

30

20

10

0

T

(K)

6004002000

P (bar)

AF

U-SCAF/SC

PI

M

F. Kagawa, K. Miyagawa, + K. KanodaPRB 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)

26

Perspective

U/t

δ

t’/t

t

tt’

t

tt’

27

« Solutions » of the Hubbard model

• Bethe ansatz in one dimension (correlationfunctions?).

• Renormalization group in one dimension or quasi-one dimension (spin-charge separation, Luttingerliquid)– Solyom, Bourbonnais

• Nagaoka theorem• In two or three dimension :

– Gutzwiller approximation– Various forms of slave bosons (+ gauge fields) mean-

field theory.• Infinite dimension (Dynamical Mean-Field

Theory)• ……

28

Ut

μ

Simplest microscopic model for Cu O2planes.

t’t’’

LSCO

H −∑ij t i,j ci cj cj

ci U∑ i ni↑ni↓

The Hubbard model

No mean-field factorization for d-wave superconductivity

29

A(k,ω)

ω- U/2 U/2

ω

k

π/a−π/a

ω

k

π/a−π/a

ω

A(k,ω)

- 4 t

A(k,ω)

+ 4 t

U = 0 t = 0

U

-U/2

+U/2

μ

ω

30

T

U U

A(kF,ω)A(kF,ω)

ωω

Weak vs strong coupling, n=1

Mott transitionU ~ 1.5W (W= 8t)

LHB UHB

tEffective model, Heisenberg: J = 4t2 /U

31

A hierarchy of effective models (see Schrödinger vs Dirac)

• t – J model : applying second orderdegenerate perturbation theory

32

n = 1 the Heisenberg model

33

n = 1 the Heisenberg model

Worthwhile decrease in size of Hilbert space !

34

Is this model enough for the AFM at n = 1?

Coldea et al. PRL (2001)

35

Including t(t/U)3 terms

Coldea et al. PRL (2001)

U / t ~ 7 - 9

36

What about t’,t’’ ?

Delannoy et al. + thesis 2006

37

Delannoy et al. + thesis 2006

38

Alternate set of parameters

Delannoy et al. + thesis 2006

39

General methods for the derivation of effective spin Hamiltonians : all canonically equivalent

• Brillouin-Wigner degenerate perturbation theory– Expansion of denominators

• Resolvent operator (for Hamiltonian)• Canonical transformation

Chernyshev et al. PRB 2004

40

Canonical transformation

H H0 VD VX

S SD SX

SD 0

eiSAe−iS A iS,A 12 iS, iS,A …

SX S1 S2 …

eiSH0 VD VX e−iS H0 VD

VX iS1,H0

iS2,H0 iS1,VD

iS1,VX 12 iS1, iS1,H0

VD T0

VX T1 T−1

H0,Tm mUTm

⟨′ |H| ⟨ ′ |e iSHe−iS |

41

Beware ! When spin is not spin…

• A paradox

42

Back to the general problem: 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!

43

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

44

Question

• Is the Hubbard model rich enough to contain the essential physics of the cupratesand the organics? (e- and h-doped and layered?)

• Yes (in part): – New theoretical approaches – Increase in computing power– Theoretical approaches (numerical, analytical)

give consistent results even if the starting points are very different.

45

Mounting evidence for d-wave in Hubbard

• Weak coupling (U << W)– AF spin fluctuations mediated pairing with d-wave

symmetry• (Bickers et al., PRL 1989; Monthoux et al., PRL 1991;

Scalapino, JLTP 1999, Kyung et al. (2003))– RG → Groundstate d-wave superconducting

• (Bourbonnais (86), 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, Scalapnio, PRB 2002)

46

Th. Maier, M. Jarrell, Th. Pruschke, and J. KellerPhys. Rev. Lett. 85, 1524 (2000)T.A. Maier et al. PRL (2005)

DCA

Paramekanti, M. Randeria, and N. TrivediPhys. Rev. Lett. 87, 217002 (2001)

Variational

Numerical methods that show Tc at strong coupling

47

Methodology

Weak-coupling approaches

48

Theory difficult even at weak to intermediate coupling!

• RPA (OK with conservation laws)– Mermin-Wagner– Pauli

• Moryia (Conjugate variables HS φ4 = <φ2> φ2 )– Adjustable parameters: c and Ueff– Pauli

• FLEX– No pseudogap– Pauli

• Renormalization Group– 2 loops

XXX

Rohe and Metzner (2004)Katanin and Kampf (2004)X

= +1

32 21

3 3

12

2

3

4

5-

Vide

49

TPSC

50

• General philosophy– Drop diagrams– Impose constraints and sum rules

• Conservation laws• Pauli principle ( <nσ

2> = <nσ > )• Local moment and local density sum-rules

• Get for free: • Mermin-Wagner theorem• Kanamori-Brückner screening• Consistency between one- and two-particle ΣG = U<nσ n-σ>

Two-Particle Self-Consistent Approach (U < 8t)- How it works

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)

51

TPSC approach: two steps

I: Two-particle self consistency

1. Functional derivative formalism (conservation laws)

(a) spin vertex:

(b) analog of the Bethe-Salpeter equation:

(c) self-energy:

2. Factorization

52

TPSC…

II: Improved self-energy

3. The F.D. theorem and Pauli principle

Kanamori-Brückner screening

Insert the first step resultsinto exact equation:

n ↑ − n↓2 ⟨n↑ ⟨n↓ − 2⟨n↑n↓

53

= +

1

32 2

1

33

12

2

3

4

5

-

=Σ1 2 +

2

45

2

1

1

21

A better approximation for single-particle properties (Ruckenstein)

Y.M. Vilk and A.-M.S. Tremblay, J. Phys. Chem. Solids 56, 1769 (1995).Y.M. Vilk and A.-M.S. Tremblay, Europhys. Lett. 33, 159 (1996);

N.B.: No Migdal theorem

54

Benchmarks for TPSC

55

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.

56

-5.00 0.00 5.00 -5.00 0.00 5.00ω/t

(0,0)

(0, )π

( , )π2

π2

( , )π π4 4

( , )π π4 2

(0,0) (0, )π

( ,0)π

Monte Carlo Many-Body

-5.00 0.00 5.00

Flex

U = + 4β = 5

Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000).

Proofs...

TPSC

57QMC: symbols. Solid lines analytical. Kyung, Landry, A.-M.S.T.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7L = 6 U = 0, β = 4

U = 4

χ d

Doping

β = 4 β = 3 β = 2 β = 1

1.00.8

0.60.4

0.20.0

43

21

0.100.150.200.250.300.350.400.450.500.550.600.650.700.75

U = 0 U = 4 U = 6 U = 8 U = 10

dx2-y2-wave susceptibility for 6x6 lattice

χ d x2 -y2

δ

β

58

Methodology

Strong-coupling approaches

59

Cluster perturbation theory

60

Vary cluster

shape and size

D. Sénéchal et al., PRL. 84, 522 (2000); PRB 66, 075129 (2002), Gross, Valenti, PRB 48, 418 (1993).

W. Metzner, PRB 43, 8549 (1991).Pairault, Sénéchal, AMST, PRL 80, 5389 (1998).

Cluster perturbation theory (CPT)

David Sénéchal

61

Different clusters

L = 6

L = 8

L = 10

David Sénéchal

62

63

Self-energy functional approach and special cases

65

Dynamical “variational” principle

tG G − TrG0t−1 − G−1G Tr ln−G

GG

tGG − G0t

−1 G−1 0

G 1G0t−1−

Luttinger and Ward 1960, Baym and Kadanoff (1961)

G + + + ….

H.F. if approximate Φby first order

FLEX higher order

Universality

Then Ω is grand potentialRelated to dynamics (cf. Ritz)

66

Another way to look at this (Potthoff)

t G − TrG − Tr ln−G0t−1

GG

tG G − TrG0t−1 − G−1G Tr ln−G

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

t F − Tr ln−G0t−1

Still stationary (chain rule)

67

SFT : Self-energy Functional Theory

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

t t′ − Tr ln−G0′ −1 − −1 Tr ln−G0

−1 − −1.

With F[Σ] Legendre transform of Luttinger-Ward funct.

t F Tr ln−G0−1 − −1

For given interaction, F[Σ] is a universal functional of Σ, no explicit dependence on H0(t). Hence, use solvable cluster H0(t’) to find F[Σ].

Vary with respect to parameters of the cluster (including Weiss fields)

Variation of the self-energy, through parameters in H0(t’)

is stationary with respect to Σ and equal to grand potential there.

68

Variational cluster perturbation theory and DMFT as special cases of SFT

C-DMFTV-DCA, Jarrellet al.

Georges Kotliar, PRB

(1992).M. Jarrell,

PRL (1992).A. Georges,

et al.RMP (1996).

M. Potthoff et al. PRL 91, 206402 (2003).

Savrasov, Kotliar,

PRB (2001)

69

Cellular dynamical mean-field theory and Dynamical cluster approximation

70

Benchmarks for quantum cluster approaches

71

0 0.5 1 1.5 2

-0.5

-0.4

-0.3

-0.2

-0.1

0

Im G

11

0

-0.5

-0.4

-0.3

-0.2

-0.1

0DMRGCDMFTPCDMFTDMFT

0 2 4 6 8-0.5

-0.4

-0.3

-0.2

-0.1

0

0-0.5

-0.4

-0.3

-0.2

-0.1

0

0 0.5 1 1.5 2ω ν

0

0.2

0.4

0.6

0.8

1

Re

G12

00

0.2

0.4

0.6

0.8

1

0 2 4 6 8ω ν

0

0.1

0.2

0.3

0.4

0.5

00

0.1

0.2

0.3

0.4

0.5

U/t=1 U/t=7

U/t=7U/t=1

-2 -1 0 1 2μ

0

0.25

0.5

0.75

1

n

CDMFTPCDMFTBADMFT

-0.5 -0.45 -0.40.55

0.56

0.57

0.58

1 1.25 1.5 1.75 20.85

0.9

0.95

1

1D Hubbard model: Worst case scenario

Excellent agreement with exact results in both metallic and insulatinglimits

Capone, Civelli, SSK, Kotliar, Castellani PRB (2004)Bolech, SSK, Kotliar PRB (2003)

0 5 10U/t

0

2

4

6

8

Gap

ExactLISACDMFT

Quality of approximation made by CDMFT

72

DCA vs CDMFT

G. Biroli and G. Kotliar, Phys. Rev. B 65, 155112 (2002)Aryanpour et al. cond-mat/0301460

Maier et al. cond-mat/0205460

Quantitative aspects of the dynamical impurity approach, Pozgajciccond-mat/0407172

73

Results and concordance between different methods

74

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

tt’TPSC CDMFT

75

CPT

76

Sénéchal, AMT, cond-mat/0308625

Hole-doped (17%)

77

Hole-doped (17%)

t’ = -0.3tt”= 0.2t

η= 0.12tη= 0.4t

Sénéchal, AMT, PRL 92, 126401 (2004).

78

Electron-doped (17%)

79

Electron-doped (17%)

t’ = -0.3tt”= 0.2t

η= 0.12tη= 0.4t

Sénéchal, AMT, PRL in press

80

TPSC : Fermi surface plots

be not too large

increase for smaller doping

Hubbard repulsion U has to…

U=5.75

U=5.75

U=6.25

U=6.25

B.Kyung et al.,PRB 68, 174502 (2003)Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004

15%

10%

81

I

II ω

kF

A(ω)

E

k

A(ω)

ω

kF

III

99-aps/Explication...

E

k

A(ω)

ω

kF

TX

c

kF

kF

2Δ2Δ

T = 0n

T

82

Qualitatively new result: effect of critical fluctuations on particles (RC regime)

hωsf Bk T<<

in 2D:

in 3D: marginal case

in 4D: quasiparticle survives up to .

( )Σ kk q

F nd

nik T d q

q q ik,

||∝

+ + +⊥−

− +∫

1 12 2 2ξ ε

Δ Δ Δε ε≈ ∇ ⋅ ≈ =k F Bk v k k Th

Tc

( )Im ,Σ RF

F

dTv

k 0 3∝ − −ξ

ξ ξ ξ π> ≡th th F Bv k T( / )h

( ) ( )Im , /ΣRF thUk 0 10

2∝ − >ξ ξ ξ

Y.M. Vilk and A.-M.S. Tremblay, J. Phys. Chem. Solids 56, 1769 (1995).Y.M. Vilk and A.-M.S. Tremblay, Europhys. Lett. 33, 159 (1996);

83

Imaginarypart: compare Fermi liquid, limT→0 R′′kF, 0 0

R′′kF, 0 T

vF dd−1q 1q2 −2

Tvf3−d

d2

th

Why leads to pseudogap

Ak, −2R′′

− k − R′ 2 R

′′2

Analytically : effect of critical fluctuations on particles (RC regime)

hωsf Bk T<<

Y.M. Vilk and A.-M.S. Tremblay, J. Phys. Chem. Solids 56, 1769 (1995).Y.M. Vilk and A.-M.S. Tremblay, Europhys. Lett. 33, 159 (1996);

( ) ( )Im , /ΣRF thUk 0 10

2∝ − >ξ ξ ξ

84

Comparisons with experiment

85

ARPES spectrum, an overview

86

Strong coupling pseudogap (U > 8t)t’= -0.3t , t’’ = 0.2t

• Different from Mott gap that is local (all k) not tied to ω=0.

• Pseudogap tied to ω=0 and only in regions nearly connected by (π,π). (e and h),

• Pseudogap is independent of cluster shape (and size) in CPT.

• Not caused by AFM LRO– No LRO, few lattice spacings.– Not very sensitive to t’– Scales like t.

Sénéchal, AMT, PRL 92, 126401 (2004).

87

Weak-coupling pseudogap

• In CPT – is mostly a depression in

weight – depends on system size

and shape.– located precisely at

intersection with AFM Brillouin zone

• Coupling weaker because better screened U(n) ~ dμ/dn U=4

μ

n

Sénéchal, AMT, PRL 92, 126401 (2004).

88

The pseudogap in electron-doped cuprates

• Near optimal doping U ~ 6t– Optical gap 1.3eV vs 2eV– Weight near the diagonal

• CPT, Slave-bosons, t-J

– dTc/dp negative.– Simple screening

89

TPSC

Ut

t’

t’’

Weak coupling U<8t

t’=-0.175t, t’’=0.05tt=350 meV, T=200 K

n=1+x – electron filling

fixed

H −∑ij t i,j ci cj cj

ci U∑ i ni↑ni↓

90

Fermi surface, electron-doped caseArmitage et al. PRL 87, 147003; 88, 257001

15%

10%

4%

15%10%4%Pseudogap at hot spots

91

15% doping: EDCs along the Fermi surfaceTPSC

Exp

Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004

Umin< U< Umax

Umax also from CPT

92

Hot spots from AFM quasi-static scattering

93

AFM correlation length (neutron)

Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004 Expt: P. K. Mang et al., cond-mat/0307093, Matsuda (1992).

94

Pseudogap temperature and QCP

ΔPG≈10kBT* comparable with optical measurements

Prediction QCPmay be masked by 3D transitions

Hankevych, Kyung, A.-M.S.T., PRL 2004 : Expt: Y. Onose et al., PRL (2001).

Prediction

ξ≈ξth at PG temperature T*, and ξ>ξth for T<T*

supports further AFM fluctuations origin of PG

Prediction

Matsui et al. PRL (2005)Verified theo.T* at x=0.13with ARPES

95

Observation

Matsui et al. PRL 94, 047005 (2005)

Reduced, x=0.13AFM 110 K, SC 20 K

96

Precursor of SDW state(dynamic symmetry breaking)

• Y.M. Vilk and A.-M.S. Tremblay, J. Phys. Chem. Solids 56, 1769-1771 (1995).

• Y. M. Vilk, Phys. Rev. B 55, 3870 (1997).• J. Schmalian, et al. Phys. Rev. B 60, 667 (1999).• B.Kyung et al.,PRB 68, 174502 (2003).• Hankevych, Kyung, A.-M.S.T., PRL, sept 2004 • R. S. Markiewicz, cond-mat/0308469.

97

The phase diagram for high-temperature superconductors

98

Competition between antiferromagnetism and superconductivity

99

Competition AFM-dSC – using SFT

See also, Capone and Kotliar, cond-mat/0603227, Macridin et al. DCA cond-mat/0411092

++

-

-

++

-

-

David Sénéchal

-

-

++

100

Preliminary

n, electron density Damascelli, Shen, Hussain, RMP 75, 473 (2003)

1.3 1.2 1.1 1.0 0.9 0.8 0.7

t’ = -0.3 t, t’’ = 0.2 tU = 8t

101

Anomalous superconductivity near the Mott transition

102

Effect of proximity to Mott

Sarma Kancharla

103

Gap vs order parameter

104

One-band Hubbard model for organics

Y. Shimizu, et al. Phys. Rev. Lett. 91, 107001(2003)

meV 50≈t meV 400≈⇒ U

H. Kino + H. Fukuyama, J. Phys. Soc. Jpn 65 2158 (1996), R.H. McKenzie, Comments Condens Mat Phys. 18, 309 (1998)

t’/t ~ 0.6 - 1.1

105

Perspective

U/t

δ

t’/t

t

tt’

t

tt’

106

Experimental phase diagram for Cl

40

30

20

10

0

T

(K)

6004002000

P (bar)

AF

U-SCAF/SC

PI

M

F. Kagawa, K. Miyagawa, + K. KanodaPRB 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)

107

Mott transition (C-DMFT)

4 6 8 10U/t

0

0.05

0.1

0.15

0.2

D

0.5 0.6 0.7 0.8 0.9 1t’/t

6

7

8

9

Uc/t

solid: t’/t=0.7dashed: t’/t=1.0

(a) (b)

Parcollet, Biroli, Kotliar, PRL 92 (2004)Kyung, A.-M.S.T. (2006)

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

108

Mott transition (C-DMFT)

(0,0)

(0,π)

(−π,π)

(0,0) -6 -4 -2 0 2 4 6ω

U/t=6.9, t’/t=0.7(a)

-6 -4 -2 0 2 4 6ω

U/t=6.95, t’/t=0.7(b)

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

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

109

Normal phase theoretical results for BEDT-X

0 0.2 0.4 0.6 0.8 1 1.2t’/t

0

2

4

6

8

10

12U

/t

AF

M

SL

d−wave SC

PI

M

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

110

Experimental phase diagram for Cl

40

30

20

10

0

T

(K)

6004002000

P (bar)

AF

U-SCAF/SC

PI

M

F. Kagawa, K. Miyagawa, + K. KanodaPRB 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)

111

Theoretical phase diagram BEDT

40

30

20

10

0

T

(K)

6004002000

P (bar)

AF

U-SCAF/SC

PI

M

Y. Kurisaki, et al. Phys. Rev. Lett. 95, 177001(2005) Y. Shimizu, et al. Phys. Rev. Lett. 91, (2003)

X= Cu2(CN)3 (t’~ t)

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

OP

Sénéchal

Sénéchal, Sahebsara, cond-mat/0604057

Sahebsara

Kyung

112

AFM and dSC order parameters for various t’/t

0.1 0.15 0.2 0.25t/U

0

0.04

0.08

0

0.04

0.08

0

0.04

0.08

0.1 0.15 0.2 0.25t/U

t’/t=0.5 t’/t=0.6

t’/t=0.7 t’/t=0.8

t’/t=0.9 t’/t=1.0

AF

dSC

AF

dSC

3.910.411.6Tc

1.060.840.68t’/t

Cu2(CN)3Cu(NCS)2Cu[N(CN)2]BrX

3.910.411.6Tc

1.060.840.68t’/t

Cu2(CN)3Cu(NCS)2Cu[N(CN)2]BrX

•Discontinuous jump

•Correlation betweenmaximum orderparameter and Tc

AF multiplied by 0.1

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

113

d-wave

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

Sénéchal, Sahebsara, cond-mat/0604057

114

Prediction of a new type of pressure behaviorSénéchal, Sahebsara, cond-mat/0604057

0 0.2 0.4 0.6 0.8 1 1.2t’/t

0

2

4

6

8

10

12

U/t

AF

M

SL

d−wave SC

t’/t=0.8t

AF

SL

dSC•All transitions first order, except one

with dashed line•Triple point, not SO(5)

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

115

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.

116

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 = 1but 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

117

Conclusion

• Normal state (pseudogap in ARPES)– Strong and weak

coupling mechanism for pseudogap.

– CPT, TPSC, slave bosons suggests U ~ 6tnear optimal doping for e-doped with slight variations of U with doping.

U=5.75

U=5.75

U=6.25

U=6.25

118

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!

• Insight from other compounds

• Numerical solutions … DCA (Jarrell, Maier) Variational QMC (Paramekanti, Randeria, Trivedi).

• Role of mean-field theories : Physics

• Lot more work to do.

119

Conclusion, open problems

• Methodology:– Response functions– Tc– Variational principle– First principles– …

• Questions:– Why not 3d?– Best « mean-field » approach.– Manifestations of mechanism– Frustration vs nesting

120

Steve Allen

François Lemay

David Poulin Hugo Touchette

Yury VilkLiang Chen

Samuel Moukouri

J.-S. Landry M. Boissonnault

121

Alexis Gagné-Lebrun

Bumsoo Kyung

A-M.T.

Sébastien Roy

Alexandre Blais

D. Sénéchal

C. Bourbonnais

R. Côté

K. LeHur

Vasyl Hankevych

Sarma Kancharla

Maxim Mar’enko

122

Bumsoo KyungSarma Kancharla

Marc-André Marois Pierre-Luc Lavertu

David Sénéchal

123

Outside collaborators

Gabi Kotliar Marcello Civelli Massimo Capone

124

Mammouth, série

125

André-Marie Tremblay

Sponsors:

126

Recent review articles

• A.-M.S. Tremblay, B. Kyung et D. Sénéchal, 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 Nov2005

127

C’est fini…

enfin