André-Marie Tremblay

Sponsors:

CENTRE DE RECHERCHE SUR LES PROPRIÉTÉS ÉLECTRONIQUES

DE MATÉRIAUX AVANCÉS

I. All is not well with the theory of solids

II. A microscopic model

III. Inching our way up the weak-coupling regime.

IV. What was the competition up to?

V. Conclusion

Mathematics, Physics, and Computers : strongly correlated electrons intwo dimensions as a case study.

H = Kinetic + Coulomb

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

- Successful program - Semiconductors, metals and superconductors- Magnets

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

strong interactions, low dimensionstrong fluctuations

Theory of solids

I. All is not well with the theory of solids

4

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

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

La2CuO4

The standard approaches :

ePhoton

= E + ω + µµµµ - W k2m

2

k

ph

Angle-Resolved Photoemission (ARPES)

Quasi 2-d material

6

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

n = 1, Metal according to bandAFM insulator in reality

8

Optimally doped BISCCONormal Superconductor

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

- d=2 partial vanishing actof the Fermi surface awayfrom n = 1.

κ−(BEDT)2X

II. A microscopic model

Ut

•Compute [ ][ ]

[ ][ ]Tr Oe

Tr e

H k T

H k T

B

B

−

−

/

/

•Size of Hilbert space : 4 N (N = 16)

µ

Simplest microscopic model for Cu Oplanes.

11

H = ¡ X¾

ti;j³cyi¾cj¾ + c

yj¾ci¾

´+ U

Xi

ni"ni#

Hubbard model (Kanamori, Gutzwiller, 1963) :

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

- 2001 vs 1963: Numerical solutions to check analytical approaches

t

t

Ut

12

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

13

T

U

A(kF,ω)

A(kF,ω)A(kF,ω)

A(kF,ω)

A(kF,ω)

∆∆

U U

U

D. Sénéchal

ω

ωω

ω

ω

Weak vs strong coupling

•QMC•André Reid, Christian Boily, Hugues Nélisse

L. Chen, C. Bourbonnais, T. Li, and A.–M. S. TremblayPhys. Rev. Lett. 66, 369-372 (1991)

Liang Chen

Claude Bourbonnais

III. Inching our way up the weak coupling regime

•Problems:

•Cannot compute charge structure factor with satisfactory accuracy

•Predicts a finite T antiferromagnetic phase transition in d = 2

•Contradicts Mermin-Wagner theorem

( )∇ → → → ∞∫θ θ θ θ α2 2 2 2 2q k T d q k Tqq q BB

•Yury Vilk, 1993

•Forget diagrams

•Keep RPA form since satisfies conservation laws

•Determine renormalized interaction from sum-rule (Singwi)

•(Double-occupancy determined self-consistently)

•Get the charge fluctuations from Pauli principle

•Mermin-Wagner theorem automatically satisfied

Yury Vilk

17

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 alsoFermi-liquid

- Proofs that it works

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...

0.0 0.2 0.4 0.6 0.8 1.0T

0

4

8

12

16S s

p(π

,π)

Monte Carlo

4x4

6x6

8x8

10x10

12x12

ξ = 5

U=4n=1

TX Tmf

Calc.: Vilk et al. P.R. B 49, 13267 (1994) QMC: S. R. White, et al. Phys. Rev. 40, 506 (1989).

)( ∞=NO A.-M. Daré, Y.M. Vilk and A.-M.S.T Phys. Rev. B 53, 14236 (1996)

( )ξ ~ exp ( ) /C T T

20

= +

1

32 2

1

33

12

2

3

4

5

=Σ1 2 +

-

2

45

2

1

1

21

What about 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

Quantitative agreement with QMC

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

!ωsf Bk Tξ ξ ξ

ξ ξ ξ π> ≡th th F Bv k T( / )!

∆ ∆ ∆ε ε≈ ∇ ⋅ ≈ =k F Bk v k k T!

Σ" , ln /R

F thUk 0 02a f a f d i∝ − ξ ξ ξ

TcY.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);

M. Vekic and S. R. White, Phys. Rev. B 47, 1160–1163 (1993)

J. J. Deisz, D. W. Hess, and J. W. SerenePhys. Rev. Lett. 76, 1312-1315 (1996)

FLEXQMC

IV. What was the competition up to?

State of the art analytical tools

Φ [G] = +

Σ [G] = δΦ [G] / δG = +

12

+ ...

Γ [G] = δΣΣΣΣ [G] / δG = +

+ ...

- Thermodynamically consistent:dF/dµ = Tr[G]

- Satisfies Luttinger theorem (Volume of Fermi surface at T = 0 preserved)

- Satisfies Ward identities (conservation laws):G2(1,1;2,3) appropriately related to G(1,2)

Gordon Baym, Phys. Rev. 127, 1391 (1962).N.E. Bickers and D.J. Scalapino, Annals of Physics, 193, 206 (1989)

Advantages

FLEX

- Integration over coupling constant of potential energydoes not give back the starting Free energy.

- The Pauli principle in its simplest form is not satisfied(It is used in defining the Hubbard model in the first place)

- There is an infinite number of conserving approximations(How do we pick up the diagrams?)

- Inconsistency:Strongly frequency-dependent self-energy, constant vertex

No Migdal theorem, sovertex corrections should be included

Disdvantages

27

-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).

D. PoulinH. Touchette

Back to us

-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 = + 4

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

S. Moukouri F. Lemay S. Allen Y. Vilk B. Kyung

29

I

II ωωωω

kF

A(ωωωω)

E

k

A(ωωωω)

ωωωω

kF

III

99-aps/Explication...

E

k

A(ωωωω)

ωωωω

kF

TX

c

kF

kF

2∆2∆

T = 0n

T

30

§(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...

S. Allen

Y. Vilk

31

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...

32

§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

�3UspÂ

(1)sp (q) + UchÂ

(1)ch (q)

¸G(1)¾ (k+ q): (4)

Second step: improved self-energy

[ ]

How it works...

-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

Proof that generalization for U < 0 works

S. Allen

B. Kyung

34

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

35

U > 0- Renormalized classical regime

for spin fluctuations in pseudogap regime?

Philippe Bourges cond-mat/0009373

T

δ

pseudogap

AF

SC QCP ?

36

- 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

V. Conclusion•What is happening now?

•Methods for thermodynamics and for crossed channels.

•Computers : •Small sizes, not all relevant parameter regimes are accessible

•Analytical studies :•No small parameter, no perfect approximation

•Limiting cases, physical intuition•Not always reliable

Steve Allen

François Lemay

David Poulin Hugo Touchette

Yury VilkLiang Chen

Samuel MoukouriBumsoo Kyung

David Sénéchal A.-M.T.

Michel Barrette

Alain Veilleux

Mehdi Bozzo-Rey

Elix

Jean-Séastien Landry

Bumsoo Kyung

A-M.T.

Sébastien Roy Alexandre Blais

D. Sénéchal

Claude Bourbonnais

R. Côté

41

- How can we understand electronic systems that showboth 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 fromthe shape of the would-be Fermi surface in the normal state?

- Are there new elementary excitations that couldsummarize and explain in a simple way the anomalousproperties 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 bothstrong-coupling and d = 2 fluctuation effects?

- What is the origin of d-wave superconductivity in the high-temperature superconductors?