And

ré-M

arie

Tre

mbl

ay

Spon

sors

:CEN

TRE

DE

REC

HER

CH

E SU

R L

ES P

RO

PRIÉ

TÉS

ÉLEC

TRO

NIQ

UES

DE

MA

TÉR

IAU

X A

VA

NC

ÉS

Pairi

ng v

sant

iferr

omag

netis

min

the

cupr

ates

1. M

otiv

atio

n, m

odel

2. T

hest

anda

rd a

ppro

ach

: -w

hat i

t is

-a q

ualit

ativ

ely

inco

rrec

t res

ult

-lim

itatio

ns o

f the

app

roac

h3.

A n

on-p

ertu

rbat

ive

appr

oach

(U >

0an

dU

< 0

)-P

roof

that

it w

orks

-How

it w

orks

4. R

esul

ts:

-Mec

hani

smfo

r pse

udog

ap-S

pect

ral w

eigh

t rea

rran

gem

ent

-Pai

ring

in th

e re

puls

ive

mod

el

5. P

ersp

ectiv

es.

3

-Her

e, -W

< U

< W

with

(W =

8t)

-Rel

evan

t for

hig

h T c

whe

reU

> W

?-A

que

stio

n of

thre

shol

d(a

nd c

ontin

uity

)T

δU

SDW

-Loc

atio

n of

QC

P-N

o fe

rrom

agne

tism

-Im

porta

nce

of q

uant

itativ

e pr

edic

tions

-Eff

ectiv

e U

< 0

mod

el m

ay b

eno

t too

stro

ngly

inte

ract

ing

-Sup

pose

we

find

new

qua

sipa

rticl

esin

stro

ng

coup

ling.

How

do

we

stud

y re

sidu

alin

tera

ctio

ns?

-Do

we

give

up c

alcu

latin

gLa

ndau

par

amet

ers?

-How

to p

redi

ct w

hen

the

theo

ry is

bad

?-S

tand

ard

met

hod

give

s qua

litat

ivel

yin

corr

ect r

esul

ts

1. M

otiv

atio

n, m

odel

4

Φ [G

] =

+ 1 2

+ ..

.

Γ [G

] = δ

Σ[G

] /δG

= +

Σ [G

] = δ

Φ[G

] /δG

= +

+ ..

.

2. T

hest

anda

rd a

ppro

ach

: -w

hat i

t is(

FLEX

, sel

f-co

nsis

tent

T-m

atrix

...)

5

2. T

hest

anda

rd a

ppro

ach

: -w

hat i

t is(

FLEX

, sel

f-co

nsis

tent

T-m

atrix

...)

-The

rmod

ynam

ical

lyco

nsis

tent

:dF

/dµ

= T

r[G

]

-Sat

isfie

s Lut

tinge

r the

orem

(V

olum

e of

Fer

mi s

urfa

ce a

t T

= 0

pre

serv

ed)

-Sat

isfie

sWar

d id

entit

ies(

cons

erva

tion

law

s):

G2(

1,1;

2,3)

appr

opria

tely

rela

ted

to G

(1,2

)

6

-5.0

00.

005.

00-5

.00

0.00

5.00

ω/t

(0,0

)

(0,

)π

( ,

)π 2

π 2

( ,

)π

π4

4

( ,

)π

π4

2

(0,0

)(0

, ) π

( ,0

)π

Mon

te C

arlo

Man

y-Bo

dy

-5.0

00.

005.

00

Flex

2. T

hest

anda

rd a

ppro

ach

: -a

qua

litat

ivel

yin

corr

ect r

esul

t

Cal

c. +

QM

C: M

ouko

urie

t al.

P.R

. B 6

1, 7

887

(200

0).

U =

+ 4

T =

0.2

n=1

8 X

8

7

I

99-a

ps/E

xplic

atio

n...

E

k

A(ω

)

ωkF

T X

c

kF

E

k

A(ω

)

ωkF

III

kF

2∆2∆

T =

0n

T

IIωk

FA

(ω)

8

2. T

hest

anda

rd a

ppro

ach

: -l

imita

tions

of t

he a

ppro

ach

-Int

egra

tion

over

cou

plin

gco

nsta

nt o

f pot

entia

l ene

rgy

does

not g

ive

back

the

star

ting

Free

ene

rgy.

-The

Paul

i prin

cipl

ein

its s

impl

est f

orm

isno

t sat

isfie

d(I

t is u

sed

in d

efin

ing

the

Hub

bard

mod

el in

the

first

plac

e)

-The

re is

an in

finite

num

bero

f con

serv

ing

appr

oxim

atio

ns(H

ow d

o w

e pi

ckup

the

diag

ram

s?)

-Inc

onsi

sten

cy:

Stro

ngly

freq

uenc

y-de

pend

ents

elf-

ener

gy, c

onst

ant v

erte

x

No

Mig

dal t

heor

em, s

ove

rtex

corr

ectio

ns sh

ould

be

incl

uded

9

Σ(kF,ikn)≈U 4

T N

X qUspχsp(q, 0)

1

ikn−�² k+q−Σ(kF+q,ikn)

Σ(ikn)=

∆2

ikn−Σ(ikn)

ReΣR(ω)=ω 2−

ω

2|ω|θ(|ω|−

2∆)³ ω2

−4∆2´ 1/ 2

ImΣR(ω)=−1 2θ(2∆−|ω|)³ 4∆

2−ω2´ 1/ 2

Sing

ular

Non

Fer

mi-l

iqui

dbu

t not

sing

ular

atω

= 0

Vilk

, et a

l. J.

Phys

. I F

ranc

e, 7

, 130

9 (1

997)

.

10

2. T

hest

anda

rd a

ppro

ach

: -p

robl

em...

-We

do n

ot k

now

how

to p

rope

rly so

lve

even

the

«sta

ndar

d m

odel

» fo

r hea

vyfe

rmio

ns (C

olem

an).

-Lee

-Ric

e-A

nder

son

sim

pler

app

roac

h se

ems

qual

itativ

ely

bette

r, w

hy?

-GW

app

roac

hto

impr

ove

band

stru

ctur

e ca

lcul

atio

ns?

11q

0.0

0.4 (0

,0)

(π,π

)(π

,0)

(0,0

)q

020.

0

0.4

02

n Sch(q)

n Ssp(q)

U =

8

U =

8

U =

4

U =

4n

= 0.

26

n =

0.94

n =

0.60

n =

0.45

n =

0.20

n =

0.80

n =

0.33

n =

0.19

n =

1.0

n =

0.45

n =

0.20

(a)

(b)

(c)

(d)

U >

0

QM

C +

cal

.: V

ilket

al.

P.R

. B49

, 132

67 (1

994)

Not

es:

-F.L

.pa

ram

eter

s-S

elf a

lso

Ferm

i-liq

uid

3. A

n ap

proa

chfo

r bo

thU

> 0

and

U <

0-P

roof

s tha

t it w

orks

0.50

0.75

1.00

T

0.0

0.5

1.0

1.5

2.0

U=4

.0n=

0.87

Mon

te C

arlo

8x8

(BW

S)

8*8

(BW

)

This

wor

k

FLEX

no

AL

FLEX

π,π χ( ,0) sp

U =

+ 4

QM

C: B

ulut

, Sca

lapi

no, W

hite

, P.R

. B 5

0, 9

623

(199

4).

Cal

c.: V

ilk, e

t al.

J. Ph

ys. I

Fra

nce,

7, 1

309

(199

7).

Proo

fs...

0.0

0.2

0.4

0.6

0.8

1.0

T

0481216Ssp(π,π)

Mon

te C

arlo

4x4

6x6

8x8

10x1

0

12x1

2

ξ =

5

U=4

n=1

TX

Tm

f

Cal

c.: V

ilket

al.

P.R

. B49

, 132

67 (1

994)

Q

MC

: S. R

. Whi

te, e

t al.

Phys

. Rev

. 40 ,

506

(198

9).

)(

∞=

NO

A.-M

. Dar

é, Y

.M. V

ilkan

dA

.-M.S

.T P

hys.

Rev

. B 5

3 , 1

4236

(199

6)()

ξ~

exp

()/

CT

T

14

-5.0

00.

005.

00-5

.00

0.00

5.00

ω/t

(0,0

)

(0,

)π

( ,

)π 2

π 2

( ,

)π

π4

4

( ,

)π

π4

2

(0,0

)(0

, ) π

( ,0

)π

Mon

te C

arlo

Man

y-Bo

dy

-5.0

00.

005.

00

Flex

U =

+ 4

Cal

c. +

QM

C: M

ouko

urie

t al.

P.R

. B 6

1, 7

887

(200

0).

Proo

fs...

15

-10

10.

0

0.5

1.0

A(ω)

-10

1-1

01

Mon

te C

arlo

M

any-

Bod

y Σ

(s)

FLEX

-0.50.0

G(τ)

0β

a)b)

c)L=

4

L=6

L=8

L=10

β /2

U =

+ 4

Cal

c. +

QM

C: M

ouko

urie

t al.

P.R

. B 6

1, 7

887

(200

0).

Proo

fs...

C.H

ushc

rof,

M. J

arre

llet

al.

P.R

.L 8

6, 1

39 (2

001)

.

Oth

erap

proa

chD

CA

Proo

fs...

17

U =

-4

Cal

c. :

Kyu

ng, A

llen,

A.M

.S.T

. P.R

.B 6

4, 0

7511

6/1-

15 (2

001)

Q

MC

: Mor

eo, S

cala

pino

, Whi

te, P

.R. B

. 45,

754

4 (1

992)

Mov

ing

to th

eat

tract

ive

case

....

Proo

fs...

18

U =

-4

Cal

c. :

Kyu

ng, A

llen,

A.M

.S.T

. P.R

.B 6

4, 0

7511

6/1-

15 (2

001)

QM

C: T

rived

i and

Ran

deria

, P.R

. L. 7

5, 3

12 (1

995)

2ndor

derp

ertu

rbat

ion

theo

ry

S.C

. T-m

atrix

Proo

fs...

19

-4.0

0.0

4.0

ω

0.0

0.2

0.4

T =

0.25

, n =

0.8

, U =

-4Q

MC

Man

y-B

ody

-4.0

0.0

4.0

ω

0.0

0.4

0.8

1.2

N(

)ω

A(k,

) ω

(a)

(b)

(c)

(d)

-4.0

0.0

4.0

0.0

0.1

0.2

0.3

T =

0.2,

n=0

.8, U

= -4

Man

y-B

ody

QM

C

-4.0

0.0

4.0

0.0

0.2

0.4

0.6

τ-0

.6

-0.4

-0.20.0

T =

0.25

, n =

0.8

, U =

-4Q

MC

Man

y B

ody

τ-0

.6

-0.4

-0.20.0

G(k

, )τ

G(k

, )

τ

(a)

(b)

(c)

(d)

-0.6

-0.4

-0.20.0

T =

0.2,

n=0

.8, U

= -4

Man

y-B

ody

QM

C

-0.6

-0.4

-0.20.0

Σ k

U =

-4

Kyu

ng, A

llen,

A.M

.S.T

. P.R

.B 6

4, 0

7511

6/1-

15 (2

001)

Proo

fs...

20

3. A

n ap

proa

chfo

r bo

thU

> 0

and

U <

0-H

ow it

wor

ks

Gen

eral

resu

lts(D

yson

equa

tion)

:

1,1

G

1,2

U

1

1

1

2

.

1,1

G

1,1

Un

1

n 1

Har

tree-

Fock

appr

oxim

atio

nH

1,1

GH

1,2

U

GH1

,1G

H1

,2.

Step

one

ofou

rapp

roxi

mat

ion

1

1,1

G1

1,2

AG

1

1,1

G1

1,2

A

Un

n

nn

.

Find

nn

fr

om(a

)flu

ctua

tion-

diss

ipat

ion

theo

rem

(b)P

auli

prin

cipl

e

Usp

1

G1

1

G1

Un

n

nn

.

n

n 2

n

n

2 n

n

T N q

0

q1

1 2U

sp

0q

n

2n

n

T N q

0

q1

1 2U

ch

0q

n

2n

n

n2 .

23

=+

1 32

21 3

312

2 3

4 5

-

=Σ

12

+

2

45

2

1 1

21

A b

ette

r ap

prox

imat

ion

for

sing

le-p

artic

le p

rope

rtie

s(R

ucke

nste

in)

Y.M

. Vilk

and

A.-M

.S. T

rem

blay

, J. P

hys.

Che

m. S

olid

s56,

176

9 (1

995)

.Y

.M. V

ilk a

ndA

.-M.S

. Tre

mbl

ay, E

urop

hys.

Lett.

33,

159

(199

6);

N.B

.: N

o M

igda

l the

orem

Impr

oved

self-

ener

gy(s

tep

(2))

2

k sy

m

U

n

U 8T N q

3Usp

sp1

q

Uch

ch1

q G

1

k

q .

Con

sist

ency

chec

k1 2

Tr

2 G

1

lim

0T N k

2

k G

1

k e

ikn

Un

n

.

25

Res

ults

of th

e an

alog

ous p

roce

dure

for U

< 0

Upp=Uh(1

−n↑)n↓i

h1−n↑ihn↓i.

(5)

χ(1)

p(q)=

χ(1)

0(q)

1+Uppχ(1)

0(q)

(6)

T N

X qχ(1)

p(q)exp(−iqn0− )=h∆

� ∆i=

hn↑n↓i.

(7)

Σ(1)'U 2−Upp(1−n)

2(8)

Σ(2)

σ(k)=Un−σ−UT N

X qUppχ(1)

p(q)G(1)

− σ(q−k)

(9)

26

SatisÞesPauliprincipleandgeneralizationoff−sumrule

Z dω πImχ(1)( q,ω)=Dh ∆

q(0), ∆

� q(0)iE

=1−n

;∀ q

(10)

Z dω πωImχ(1)( q,ω)=

1 N

X k

³ ε k+ε −k+q

´³ 1−2D n k

↑E´ (11)

−2µ µ

(1)−U 2

¶ (1−n)

;∀ q

(12)

Internalaccuracycheck(ForbothU>0andU<0).

1 2Trh Σ(2

) G(1)i =

lim

τ→0−T N

X k

Σ(2)

σ(k)G(1)

σ(k)e−iknτ=UD n ↑

n↓E

Check

:Trh Σ(2

) G(1)i

�Trh Σ(2

) G(2)i

27

U =

-4

-Ent

er th

e re

norm

aliz

ed-c

lass

ical

regi

me.

N

.B. d

= 2

-(an

alog

oust

o U

> 0

) : V

ilket

al.

Euro

phys

. Let

t. 33

, 159

(199

6)Pi

nes,

Schm

alia

n(9

8)

29

4. R

esul

ts:

-Mec

hani

smfo

r pse

udog

ap

Kyu

ng, A

llen,

A.M

.S.T

. P.R

.B 6

4, 0

7511

6/1-

15 (2

001)

28

U =

-4

4. R

esul

ts:

-Mec

hani

smfo

r pse

udog

ap

-Pai

ring

corr

elat

ion

leng

th la

rger

than

si

ngle

-par

ticle

ther

mal

de

Bro

glie

wav

elen

gth

(vF/ T

)

ξ ∼

1.3

ξ th

Kyu

ng, A

llen,

A.M

.S.T

. P.R

.B 6

4, 0

7511

6/1-

15 (2

001)

29

0.00

0.50

1.00

1.50

2.00

ω

T=1/3

T=1/4

T=1/5

Even

par

t of t

he p

air s

usce

ptib

ility

at q

= 0

, for

diff

eren

t tem

pera

ture

s

U =

-4

Alle

n, e

t al.

P.R

. L 8

3, 4

128

(199

9)

Mec

hani

smfo

r pse

udog

ap fo

rmat

ion

in th

eat

tract

ive

mod

el:

d =

2is

cruc

ial

Mec

hani

smfo

rpse

udog

apre

gim

e

k,

ikn

Un

U 4T N q

0 1

ikn k

q

Larg

eco

rrel

atio

nle

ngth

limit

clk

,ikn

UT

2 02

dd q 2

d1

q2

21

ikn k

q

v k.

Ana

lytic

cont

inua

tion

clRk

F,ik

n

UT

2 02

dd q 2

d1

q2

21

i

q

v F.

inR

C

Upp

ppq

,

Imag

inar

ypa

rt:co

mpa

reFe

rmil

iqui

d,lim

T0

Rk

F,0

0

Rk

F,0

T v Fd

d1 q

1

q 2

2

T v f3

d

d2

th

Why

lead

sto

pseu

doga

p

A k,

2

R

k

R2

R2

32

U =

-4

4. R

esul

ts:

-Spe

ctra

l wei

ght r

earr

ange

men

t

-Pse

udog

ap a

ppea

rs fi

rsti

n to

tal d

ensi

tyof

stat

es-F

illsi

n in

stea

dof

ope

ning

up-R

earr

ange

men

t ove

r hug

e fr

eque

ncy

scal

e co

mpa

red

with

ei

ther

Tor

∆T.

(∆T

~ 0

.03,

T ~

0.2

, ∆ω

∼ 1

)

Kyu

ng, A

llen,

A.M

.S.T

. P.R

.B 6

4, 0

7511

6/1-

15 (2

001)

33

U =

-4

4. R

esul

ts:

-Cro

ssov

er d

iagr

am

Kyu

ng, A

llen,

A.M

.S.T

. P.R

.B 6

4, 0

7511

6/1-

15 (2

001)

4. R

esul

ts:

-Pai

ring

in th

e re

puls

ive

mod

el

His

tori

cal r

emin

der:

Spin

fluc

tuat

ion

indu

ced

d-w

ave

supe

rcon

duct

ivity

D. J

. Sca

lapi

no, E

. Loh

, Jr.,

and

J. E.

Hirs

chP.

R. B

34,

819

0-81

92 (1

986)

.B

éal�

Mon

od, B

ourb

onna

is, E

mer

yP.

R. B

. 34,

771

6 (1

986)

.M

iyak

e, S

chm

itt�R

ink,

and

Var

ma

P.R

. B 3

4, 6

554-

6556

(198

6)K

ohn,

Lut

tinge

r, P.

R.L

. 15,

524

(196

5).

Bic

kers

, Sca

lapi

no, W

hite

P.

R.L

62,

961

(198

9).

U/t

= 4

1.0

0.8

0.6

0.4

0.2

0.0

43

21

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

U =

0 U

= 4

U =

6 U

= 8

U =

10d x2 -y

2-wav

e su

scep

tibili

ty fo

r 6x

6 la

ttic

e

χdx2-y

2

δ

β

Land

ry, A

.-M.S

.T. u

npub

lishe

d

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

L =

6 U

= 0

,

β =

4

U =

4

χd

Dop

ing

β =

4 β

= 3

β =

2 β

= 1

QM

C: s

ymbo

ls.

Solid

line

sana

lytic

al.

Kyu

ng, L

andr

y, A

.-M.S

.T.

=-

-+

--

--

G

GG

1

G

1

G01

d

G

~-

--

--

-12

3 4

d

G +

-- 12

3 4

--

--

--

-- 12

3 4

s+

=

0.0

0.2

0.4

0.6

0.8

1.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

U =

4,

β =

4

χd

Dop

ing

x

Ful

l sus

cept

ibilit

y, L

=32

Sus

cept

ibilit

y w

ithou

t ver

tex,

L=3

2 F

ull s

usce

ptib

ility,

L=6

Ana

logo

us p

roce

dure

for a

ttrac

tive

case

can

be

used

to c

ompu

tepa

rticl

e-ho

le p

rope

rties

(Spi

n an

d ch

arge

susc

eptib

ilitie

s)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

QM

C

Cro

ssin

g

Self-

cons

iste

nt T

-mat

rix

U=-

4, n

=0.2

, N

=12X

12 (F

ig.1

in th

e pa

per)

χsp(T)

Tem

pera

ture

5. P

ersp

ectiv

es

Phen

omen

olog

ical

mod

el

Ht

x i,

j,

c i,

c j,

h.c

J i,

j

S i

S j

1 4n i

n j i,

c i,

c i,

.

Com

puta

tion

oflo

calc

orre

latio

nfu

nctio

nfr

omth

em

ean-

field

solu

tion

HM

F k ,

k

c k ,

c k ,

4Jm

m

Jdd

d

Jy y

F0

Com

pute

dyna

mic

alsu

scep

tibili

tiesw

ithre

norm

aliz

edve

rtice

sobt

aine

dfr

omT N q

ppqe

i m

0|

d0

|2 .

Com

pute

sing

le-pa

rticl

epr

oper

tiesf

rom

ppk

1 4V

effV

ppT N q

d

k

dq

k 2

ppqG

0 q

k .

T=0

orde

rpar

amet

ers

From

par

amag

netic

self-

ener

gy:

Ope

ning

of p

seud

ogap

(RC

regi

me)

From

dyna

mic

al su

scep

tibili

ty,

ξ sc

= 4

B. K

yung

, P.R

.B. 6

4, 1

0451

2 (2

001)

.

Pseu

doga

p an

d to

tal g

ap

B. K

yung

, P.R

.B. 6

4, 1

0451

2 (2

001)

.K

yung

and

A.-M

.S.T

. unp

ublis

hed

t/J =

3 ,

J =

125

meV

Prop

ertie

s of t

he c

onde

nsat

e

Som

e sp

ecul

atio

ns

T

δ

pseu

doga

p

AF

fluct

uatio

ns

quas

i d =

2

S.C

.

T

δ

AF

d =

3

S.C

. ?

Wea

k -c

oupl

ing

T

δ

pseu

doga

p

AF

Stro

ng-c

oupl

ing

quas

i d =

2

T

δ

AF

Stro

ng-c

oupl

ing

d =

3

S.C

. ?

Stro

ng-c

oupl

ing

Dav

id S

énéc

hal

A.-M

.T.

Mic

hel B

arre

tte

Ala

in V

eille

ux

Meh

di B

ozzo

-Rey

Elix

Stev

e A

llen

Fran

çois

Lem

ay

Yur

y V

ilk

Dav

id P

oulin

Hug

o To

uche

tte

Lian

g C

hen

Sam

uel M

ouko

uri

Bum

soo

Kyu

ng

Jean

-Séa

stie

n La

ndry

Bum

soo

Kyu

ng

A-M

.T.

Séba

stie

n R

oyA

lexa

ndre

Bla

is

D. S

énéc

hal

Cla

ude

Bou

rbon

nais

R. C

ôté