1

Electronic correlations

André-Marie Tremblay

Sponsors:

CENTRE DE RECHERCHE SUR LES PROPRIÉTÉS ÉLECTRONIQUES

DE MATÉRIAUX AVANCÉS

Steve Allen (PhD)

Bumsoo Kyung (postdoc)

Yury Vilk (now at Chicago)

Acknowledgements:

Samuel Moukouri (postdoc)

François Lemay (PhD)

David Poulin

Hugo Touchette

Liang Chen (now at U. Ottawa)

4

1. Motivation

H = kinetic + Coulomb

- Many new ideas and concepts needed for progress(Born-Oppenheimer, H-F, Bands...)

- Successfull program - Semiconductors, metals- Magnets

- Is there anything left to do?- Unexplained materials: High Tc, Organics...- Strong correlations:

strong interactions, low dimension

Theory of solids

5

Outline 1. Motivation

2. The standard approaches : - Quasiparticles, Fermi surface, Fermi liquid- Fermi liquids and phase transitions- Heisenberg and related

3. Experimental evidence for failure :- d=1 spin-charge separation- d=2 disappearing act of Fermi surface

4. Theoretical reasons for failure :- Effects of strong coupling - Effects of low d in weak to intermediate coupling

6

5. The Hubbard model

6. A non-perturbative approach (U > 0 and U < 0 )- Proof that it works- How it works

7. Results:- Mechanism for pseudogap- Spectral weight rearrangement

8. Conclusion.

7

2. The standard approaches :

A. Quasiparticles, Fermi surface and Fermi liquids- LDA (Nobel prize 1998)

L.F. Mattheiss, Phys. Rev. Lett. 58, 1028 (1987).

La2CuO4

8

2. The standard approaches :

- Matrix elements of H in LDA basis- In practice, from general considerations:

- Short range- Single Slater determinant not eigenstate

- Phase space + Pauli restricts possible scatterings:- Quasiparticles m*, effective fields,

- «See» the quasiparticles with ARPES: ePhoton

= E + ω + µµµµ - W k2m

2

k

ph

9

R. Claessen, R.O.Anderson, J.W. Allen,C.G. Olson, C.Janowitz, W.P. Ellis,S. Harm, M. Kalning,R. Manzke,and M. Skibowski,Phys. Rev. Lett. 69,808 (1992).

10

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

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2. The standard approaches :

B. Thermodynamics and phase transitions

- Thermodynamics of Fermi liquids -particle-hole excitations

- Phase transitionsχ ∼ N(0) / ( 1 + Foa)

- Superconducting transition

C. Heisenberg model and related models

- Band for s-p- Localized (often) for d-f

Only spin degrees of freedomUse symmetry to write H

123. Experimental evidence for failure : - d=1 spin-charge separation

C. Bourbonnaiset al.(cond-mat/9903101).

13

3. Experimental evidence for failure :

- d=2 partial vanishing act of the Fermi surface.

D.S. Marshall, D.S. Dessau, A.G.Loeser, C.-H. Park,A.Y. Matsuura, J.N. Eckstein, I.Bozovic, P. Fournier,A. Kapitulnik, W.E. Spicer, andZ.X. Shen, Phys. Rev.Lett. 76, 4841 (1996).

14

Optimally doped BISCCONormal Superconductor

A. V. Fedorov, T. Valla, P. D. Johnson et al. P.R.L. 82, 2179 (1999)

15

d = 2, Mermin-Wagner

(rµ)2! q2µqµ¡q

Dµ2E/Zd2q

kT

q2!1

4. Theoretical reasons for failure :

A. Strong interactions (localized states)

B. Thermal and quantum fluctuations in d = 2

d = 1 : R.G., Bosonization, Conformal Field Theory...d = 2 : Slave bosons, R.G., strong coupling p.t., TPSC

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H = ¡ X¾

ti;j³cyi¾cj¾ + c

yj¾ci¾

´+ U

Xi

ni"ni#

5. Hubbard model (Kanamori, Gutzwiller, 1963) :

tU

- Screened interaction U- U, T, n- a = 1, t = 1, h = 1

- 2001 vs 1963: Numerical solutions to check analytical approaches

t

t

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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

ω

A(k,ω)

+ 4 t

U = 0 t = 0

U

-U/2

+U/2

0

18

19http://www.physique.usherb.ca/~tremblay/articles/PiC.pdfFor first part of this talk :

S. Lefebvre, P. Wzietek, S. Brown, C. Bourbonnais, D. Jérome, C. Mézière, M. Fourmigué, and P. Batail

20

6. A non-perturbative approach for both U > 0 and U < 0

q0.0

0.4

(0,0) (π,π)(π,0) (0,0) q0

20.0

0.4

0

2

n S c

h(q)

n S s

p(q)

U = 8

U = 8

U = 4

U = 4n = 0.26

n = 0.94n = 0.60

n = 0.45n = 0.20

n = 0.80n = 0.33n = 0.19

n = 1.0n = 0.45n = 0.20

(a)

(b) (c)

(d)

U > 0

QMC + cal.: Vilk et al. P.R. B 49, 13267 (1994)

Notes: -F.L. parameters-Self also Fermi-liquid

- Proofs that it works

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0.50 0.75 1.00T

0.0

0.5

1.0

1.5

2.0U=4.0n=0.87

Monte Carlo

8x8 (BWS)

8*8 (BW)

This work

FLEX no AL

FLEX

π,π

χ(

,0)

sp

U = + 4

QMC: Bulut, Scalapino, White, P.R. B 50, 9623 (1994).Calc.: Vilk, et al. J. Phys. I France, 7, 1309 (1997).

Proofs...

22

-5.00 0.00 5.00 -5.00 0.00 5.00ω/t

(0,0)

(0, )π

( , )π2π2

( , )π π4 4( , )π π4 2

(0,0) (0, )π

( ,0)π

Monte Carlo Many-Body

-5.00 0.00 5.00

Flex

U = + 4

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

Proofs...

23

-1 0 10.0

0.5

1.0

A(ω

)

-1 0 1 -1 0 1

Monte Carlo Many-Body Σ(s) FLEX

-0.5

0.0

G(τ

)

0 β

a) b) c)L=4

L=6

L=8

L=10

β/2

U = + 4

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

Proofs...

24

I

II ωωωω

kF

A(ωωωω)

E

k

A(ωωωω)

ωωωω

kF

III

99-aps/Explication...

E

k

A(ωωωω)

ωωωω

kF

TX

T = 0nc

T

kF

kF

2∆2∆

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U = - 4

Calc. : Kyung et al. cond-mat/0010001QMC : Moreo, Scalapino, White, P.R. B. 45, 7544 (1992)

Moving to the attractive case....

Proofs...

26

U = - 4

Calc. : Kyung et al. cond-mat/0010001QMC : Trivedi and Randeria, P.R. L. 75, 312 (1995)

2nd order perturbation theory

S.C. T-matrix

Proofs...

27

-4.0 0.0 4.0ω

0.0

0.2

0.4T = 0.25, n = 0.8, U = -4

QMCMany-Body

-4.0 0.0 4.0ω

0.0

0.4

0.8

1.2N( )ω A(k, )ω

(a) (b)

(c) (d)

-4.0 0.0 4.00.0

0.1

0.2

0.3T = 0.2, n=0.8, U = -4

Many-BodyQMC

-4.0 0.0 4.00.0

0.2

0.4

0.6

τ-0.6

-0.4

-0.2

0.0

T = 0.25, n = 0.8, U = -4QMC

Many Body

τ-0.6

-0.4

-0.2

0.0G(k, )τ G(k, )τ

(a) (b)

(c) (d)

-0.6

-0.4

-0.2

0.0

T = 0.2, n=0.8, U = -4

Many-Body

QMC

-0.6

-0.4

-0.2

0.0

Σk

U = - 4

Kyung et al. cond-mat/0010001

Proofs...

28

§(1)¾

³1; 1

´G(1)¾

³1; 2

´= AG

(1)¡¾

³1; 1+

´G(1)¾ (1; 2)

where A depends on external ¯eld and is chosen such that the exact result

§¾³1; 1

´G¾

³1; 1+

´= U

Dn"n#

Eis satis¯ed. One ¯nds

A = U

Dn"n#

EDn"E Dn#E

Functional derivative ofDn"n#

E=³Dn"E Dn#E´drops out of spin vertex

Usp = A = U

Dn"n#

EDn"E Dn#E

First step: Two-Particle Self-ConsistentHow it works...

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To close the system of equations, while satisfying conservation laws and

the Pauli principle¿³n" ¡ n#

´2À=

Dn"E+Dn#E¡ 2

Dn"n#

ET

N

Xeq

Â0(q)

1¡ 12UspÂ0(q)= n¡ 2hn"n#i (1)

Recall

Usp = U

Dn"n#

EDn"E Dn#E (2)

To have charge °uctuations that satisfy Pauli principle as well,

T

N

Xq

Â0(q)

1 + 12UchÂ0(q)= n+ 2hn"n#i ¡ n2 (3)

(Bonus: Mermin-Wagner theorem)

How it works...

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§¾³1; 1

´G¾

³1; 2

´= ¡U

DÃy¡¾

³1+´Ã¡¾ (1)þ (1)Ãy¾ (2)

EÁ

§¾³1; 1

´G¾

³1; 2

´= ¡U

"±G¾ (1; 2)

±Á¡¾ (1+; 1)¡G¡¾

³1; 1+

´G¾ (1; 2)

#

Last term is Hartree Fock (lim! ! 1). Multiply by G¡1, replace lowerenergy part results of TPSC

§(2)¾ (1; 2) = UG

(1)¡¾

³1; 1+

´± (1¡ 2)¡ UG(1)

"±§(1)

±G(1)±G(1)

±Á

#Transverse+longitudinal for crossing-symmetry

§(2)¾ (k) = Un¡¾ +

U

8

T

N

Xq

h3UspÂ

(1)(q) + UchÂ(1)(q)

iG(1)¾ (k+ q): (4)

Second step: improved self-energyHow it works...

31

= +

1

32 2

1

33

12

2

3

4

5

=Σ1 2 +

-

2

45

2

1

1

21

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Results of the analogous procedure for U < 0

Upp = Uh(1¡ n")n#ih1¡ n"ihn#i

: (5)

Â(1)p (q) =

Â(1)0 (q)

1 + UppÂ(1)0 (q)

(6)

T

N

XqÂ(1)p (q) exp(¡iqn0¡) = h¢y¢i = hn"n#i : (7)

§(1) ' U2¡ Upp (1¡ n)

2(8)

§(2)¾ (k) = Un¡¾ ¡ U T

N

XqUppÂ

(1)p (q)G

(1)¡¾(q ¡ k) (9)

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Satis¯es Pauli principle and generalization of f¡sum ruleZd!

¼ImÂ(1) (q; !) =

Dh¢q (0) ;¢

yq (0)

iE= 1¡ n ; 8q (10)

Zd!

¼! ImÂ(1) (q; !) =

24 1N

Xk

³"k + "¡k+q

´ ³1¡ 2

Dnk"

E´35 (11)¡2

µ¹(1) ¡ U

2

¶(1¡ n) ; 8q (12)

Internal accuracy check (For both U > 0 and U < 0).

1

2Trh§(2)G(1)

i= lim

¿!0¡T

N

Xk

§(2)¾ (k)G

(1)¾ (k) e

¡ikn¿ = UDn"n#

ECheck : Tr

h§(2)G(1)

i~ Tr

h§(2)G(2)

i

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U = - 47. Results:

- Mechanism for pseudogap

- Enter the renormalized-classical regime. N.B. d = 2

- (analogous to U > 0 ) : Vilk et al. Europhys. Lett. 33, 159 (1996) Pines, Schmalian (98)

32

35

U = - 47. Results:

- Mechanism for pseudogap

- Pairing correlation length larger than single-particle thermal de Broglie wavelength (vF / T)

ξ ∼ 1.3 ξth

36

0.00 0.50 1.00 1.50 2.00ω

T=1/3

T=1/4

T=1/5

Even part of the pair susceptibility at q = 0, for different temperatures

U = - 4

Allen, et al. P.R. L 83, 4128 (1999)

Mechanism for pseudogap formation in the attractive model:

d = 2 is crucial

37

U = - 47. Results:

- Spectral weight rearrangement

- Pseudogap appears first in total density of states- Fills in instead of opening up- Rearrangement over huge frequency scale compared with either T or ∆ T. (∆ T ~ 0.03, T ~ 0.2, ∆ω ∼ 1 )

38

U = - 47. Results:

- Crossover diagram

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8. Conclusion :

U > 0

- Evidence against renormalized classical regime for spin fluctuations in pseudogap regime.

Philippe Bourges cond-mat/0009373

T

δ

pseudogap

AF

SC QCP ?

40

- Quantum critical point, d = 2: - Instability at incommensurate q- Largest doping : 0.315

Freericks, Jarrell cond-mat/9405025

d = infinity

U < W

U > W

U > 0 Tδ

U SDW?

0.3

U

U

Vilk et al. P.R. B 49, 13267 (1994)

- Decreases with increasing U

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- Slightly Overdoped High-Tc Superconductor TlSr2CaCu2O6.8Guo-qing Zheng et al., P. R. L. 85, 405 (2000) - Pseudogap in Knight shift and NMR relaxation strongly H dependent, contrary to underdoped (up to 23 T).

- Underdoped in a range ∆Τ ∼ 15 Κ near Tc see evidence for renormalized classical regime (KT behavior).

Corson et al.Nature, 398, 221 (1999).

- Higher symmetry group creates large range of T where there is a pseudogap.

n = 1 n = 0

SO(2)

KT

Pseudogap

T

SO(3)Allen et al. P.R.L. 83, 4128 (1999)

U < 0 Pairing-fluctuation induced pseudogap

42

- How can we understand electronic systems that show both localized and extended character?- Why do both organic and high-temperature superconductors show broken-symmetry states where mean-field-like quasipar- ticles seem to reappear? - Why is the condensate fraction in this case smaller than what would be expected from the shape of the would-be Fermi surface in the normal state? - Are there new elementary excitations that could summarize and explain in a simple way the anomalous properties of these systems? - Do quantum critical points play an important role in the Physics of these systems?- Are there new types of broken symmetries? - How do we build a theoretical approach that can include both strong-coupling and d = 2 fluctuation effects? - What is the origin of d-wave superconductivity in the high- temperature superconductors?