01-Pres specialisee boston - usherbrooke.ca...Pairing vs antiferromagnetism in the cuprates 1....
Transcript of 01-Pres specialisee boston - usherbrooke.ca...Pairing vs antiferromagnetism in the cuprates 1....
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