= -1

Perfect diamagnetism (Shielding of magnetic field)

(Meissner effect)

Dynamic variational principle and the phase diagram of high-temperature superconductors

André-Marie Tremblay

CuO2 planes

YBa2Cu3O7-

Experimental phase diagram

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

++-

-

Ut

Simplest microscopic model for Cu O2 planes.

t’t’’

LCO

H ij t i,j c i c j c j

c i Uinini

The Hubbard model

No mean-field factorization for d-wave superconductivity

T

UU

A(kF,)A(kF,)

Weak vs strong coupling, n=1

Mott transitionU ~ 1.5W (W= 8t)

LHB UHB

tEffective model, Heisenberg: J = 4t2 /U

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!

Methodology

Heuristics

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

C-DMFTV-

Dynamical “variational” principle

tG G TrG0t 1 G 1 G Tr ln G

GG

tG

G G0t

1 G 1 0

G 1G0t

1

Luttinger and Ward 1960, Baym and Kadanoff (1961)

G + + + ….

H.F. if approximate by first order

FLEX higher order

Universality

Then is grand potentialRelated to dynamics (cf. Ritz)

Another way to look at this (Potthoff)

t G Tr G Tr ln G0t 1

GG

tG G TrG0t 1 G 1 G Tr ln G

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

t F Tr ln G0t 1

Still stationary (chain rule)

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 Legendre transform of Luttinger-Ward funct.

t F Tr ln G0 1 1 #

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

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 and equal to grand potential there

The T = 0 ordered phases

Different clusters

L = 6

L = 8

L = 10The mean-fields decrease with system size

David Sénéchal

V-CPT

Order parameters for competing d-SC and AF

HM M a 1ana na

HD a,b

abcacb H.c.

Experimental Phase diagram

Optimal dopingOptimal doping

n, electron density

Hole dopingElectron doping

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

Stripes

AF order parameter, size dependence

U = 8tt’ = -0.3tt’’ = 0.2t

e-dopedh-doped

e h

AF order parameter, size dependence

AF order parameter, size dependence

dSC order parameter, size dependence

Lichtenstein et al. 2000

dSC order parameter, size dependence

dSC order parameter, size dependence

AF and dSC order parameters, U = 8t, for various sizes

dSC

AF

Sénéchal, Lavertu, Marois, A.-M.S.T., PRL, 2005

AF and dSC order parameters, L=8, for various U

dSC

AF

Other numerical results

• Th. Maier et al. PRL 85, 1524 (2000)

• A. Paramekanti et al. PRL 87 (2001).

• S. Sorella et al. PRL 88, 2002

• D. Poilblanc et al. cond-mat/0202180 (2002)

• Th. Maier et al. cond-mat/0504529

• Review: Th. Maier, RMP (2005)

Here AFM and d-SC for both hole and electron doping

The ARPES spectrum

Fermi surface plots, U = 8t, L = 8

15%

10%

4%

MDC at the Fermi energy

Wave-particle

Single-particle (ARPES) spectrum, e-dopedn = 1.1

For coexisting AF and d-SC

Comparison with mean-field n = 1.1

AF

Single-particle (ARPES) spectrum, e-doped n = 1.1

Armitage et al. PRL 87, 147003; 88, 257001

H.Kusunose et al., PRL 91, 186407 (2003)

Observation of in-gap states

Single-particle (ARPES) spectrum, e-dopedn = 1.1

Mott gap

AF

Atomic limit

Dispersion, U = 8t, L = 8, n =1.1

d-wave gap

e-doped

h-doped

Conclusion

• With V-CPT • t-t’-t’’-U Hubbard model

has AF and d-SC where expected from experiment on both h-, e-doped cuprates. (fixed t-t’-t’’)

• ARPES in agreement with experiment (40 meV)

• Homogeneous coexistence of AF + dSC is stable on e-doped side.

• D. Sénéchal, Lavertu, Marois AMST PRL 2005

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!

Mammouth, série

C’est fini…

enfin

André-Marie Tremblay

Sponsors:

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

Steve Allen

François Lemay

David Poulin Hugo Touchette

Yury VilkLiang Chen

Samuel Moukouri

J.-S. Landry M. Boissonnault

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

Starting from weak coupling, U << 8t

Theory difficult even at weak to intermediate coupling!

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

• Moryia (Conjugate variables HS <– 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 2

13 3

12

2

3

4

5

-

Vide

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

• Conservation laws

• Pauli principle ( <n> = <n > )

• Local moment and local density sum-rules

• Get for free: • Mermin-Wagner theorem

• Kanamori-Brückner screening

• Consistency between one- and two-particle G = U<n n->

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)

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

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 2nn

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.

-5.00 0.00 5.00 -5.00 0.00 5.00/ t

(0,0)

(0, )

( , )2

2

( , ) 4 4

( , ) 4 2

(0,0) (0, )

( ,0)

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...

TPSC

Moving on to experiment : pseudogap

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

15%

10%

4%

15%10%4%Pseudogap at hot spots

15% doping: EDCs along the Fermi surfaceTPSC

Exp

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

Umin< U< Umax

Umax also from CPT

Hot spots from AFM quasi-static scattering

AFM correlation length (neutron)

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

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

Observation

Matsui et al. PRL 94, 047005 (2005)

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

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.

What about d-wave superconductivity?

1.00.8

0.60.4

0.2

0.0

43

21

0.100.150.200.250.300.350.400.450.500.55

0.60

0.65

0.70

0.75

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

dx2-y2-wave susceptibility for 6x6 lattice

d x2 -y2

0.0 0.2 0.4 0.6 0.8 1.00.1

0.2

0.3

0.4

0.5

0.6

0.7 L = 6 U = 0, = 4

U = 4

d

Doping

= 4 = 3 = 2 = 1

QMC: symbols. Solid lines analytical Kyung, Landry, A.-M.S.T., Phys. Rev. B (2003)

QMC: symbols. Solid lines analytical Kyung, Landry, A.-M.S.T., Phys. Rev. B (2003)

1.00.8

0.60.4

0.2

0.0

43

21

0.100.150.200.250.300.350.400.450.500.55

0.60

0.65

0.70

0.75

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

dx2-y2-wave susceptibility for 6x6 lattice

d x2 -y2

0.0 0.2 0.4 0.6 0.8 1.00.1

0.2

0.3

0.4

0.5

0.6

0.7 L = 6 U = 0, = 4

U = 4

d

Doping

= 4 = 3 = 2 = 1

QMC: symbols. Solid lines analytical. Kyung, Landry, A.-M.S.T. PRB (2003)

1.00.8

0.60.4

0.2

0.0

43

21

0.100.150.200.250.300.350.400.450.500.55

0.60

0.65

0.70

0.75

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

dx2-y2-wave susceptibility for 6x6 lattice

d x2 -y2

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, = 4

U = 4

d

Doping

= 4 = 3 = 2 = 1

0.00 0.05 0.10 0.15 0.20 0.250.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

L = 256

Tc

Doping

U = 4 U = 6

Kyung, Landry, A.-M.S.T. PRB (2003)

DCA Maier et al. cond-mat/0504529

What about the pseudogap in large U?

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

Fermi surface, hole-doped case 10%

Hole-doped (17%)

t’ = -0.3tt”= 0.2t

= 0.12t= 0.4t

Sénéchal, AMT, PRL 92, 126401 (2004).

Hole doped, 75%, U = 16 t

Hole-doped 17%, U=8t

Electron-doped (17%)

t’ = -0.3tt”= 0.2t

= 0.12t= 0.4t

Sénéchal, AMT, PRL in press

4x4

Electron-doped 12.5%, U=8t

15%

10%

4%

12.5%

Electron-doped, 17%, U=4t

Strong coupling pseudogap (U > 8t)

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

• Pseudogap tied to and only in regions nearly connected by (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).

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) ~ d/dn U=4

n

Sénéchal, AMT, PRL 92, 126401 (2004).

Conclusion

• Ground state of CuO2 planes (h-, e-doped)

– V-CPT, (C-DMFT) give overall ground state phase diagram with U at intermediate coupling.

– TPSC reconciles QMC with existence of d-wave ground state.

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, = 4

U = 4

d

Doping

= 4 = 3 = 2 = 1

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.

U=5.75

U=5.75

U=6.25

U=6.25