for quantum impurities perturbative renormalization group
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Ha
ns K
roh
a
Un
ive
rsity o
f Bo
nn
INES W
inte
r Sc
ho
ol
Mo
ha
np
ur, 0
4 J
an
ua
ry, 2
012
Pe
rturb
ativ
e R
en
orm
aliza
tion
Gro
up
for Q
ua
ntu
m Im
pu
rities
Fa
r from
Eq
uilib
rium
Overv
iew
•Q
uantu
mim
puritie
s,quantu
mdots
Stro
ng
on-site
repulsio
n,charg
ing
energ
y
Localiz
ed
spin
Kondo
effect
•Renorm
aliz
atio
ngro
up
Scale
invaria
nce
Renorm
aliz
atio
nofcouplin
gconsta
nts
near
the
Ferm
ienery
(gro
und
state
)
RG
equatio
ns
•Non-e
quilib
rium
(non-z
ero
DC
bia
s)
Ele
ctro
ns
inhig
hly
excite
dsta
tes
Inela
sticre
laxatio
nra
tes
•Applic
atio
ns
2-c
hannelK
ondo
effect
RG
treatm
ent
ofth
eth
ree-le
velsy
stem
out
ofequilib
rium
Quantum
impurit
ies:experim
entalsystem
s
Localiz
ed
quantu
mdegre
eoffre
edom
couple
dto
afe
rmio
nic
contin
uum
many others
Crom
mie, B
erndt, Schneider,
Yu, N
atelson, NanoLett. 4, 79 (2004)
Park, M
cEuen, R
alph et al., Nature, 417, 722 (2002)
Goldhaber−
Gordon et al.,
Nature, 391, 156 (1998)
Weis (1998)
Van der W
iel, Tarucha,
Kouw
enhoven et al., Nature (2000)
Ehm
, Reinert, H
üfner et al.,PR
B 76, 045117 (2007)
Quantum
impurit
ies:lo
calm
om
ents
Localiz
ed
quantu
mdegre
eoffre
edom
couple
dto
afe
rmio
nic
contin
uum
T =
D e
K
πE
d2 N
0( )
+ΓR
ΓL
2B
B =
0B
> T
B
TK
A ( )
ωd
ω
A ( )
ωd
ω
Ed
E +
Ud
ΓL
ΓR
µL
µR
K
Quantum
impurit
ies:K
ondo
effect
H=
∑kσ
εk
c†kσckσ
+J
∑
kk′σ
σ′ c
†kσ
(
~S·~σ
)
ck′σ
′
JJ
JJ
Jt( ) =ω
++
t(ω)
=N
0J
[1−
2N
0J
lnωD
]
Resu
mm
atio
nof
log
term
s:
T(ω
)=
∑nt(ω
)n
=12
1
lnωTK
Kondo
tem
pera
ture
:T
K=
De−1/(2N
0J)
−→
Sin
gula
rpro
ble
m.
Scale
invaria
nce!
Renorm
aliz
atio
ngro
up
•Scale
invaria
nce:
Physic
alquantitie
sat
low
energ
ies
(ω<
TK)
do
not
explic
itlyde-
pend
on
hig
h-e
nerg
ypara
mete
rs(c
ouplin
gconsta
nts,
band
cuto
ff).
•How
must
couplin
gconsta
nt
Jbe
changed
under
are
ductio
nof
the
band
cuto
ffD
,so
that
physic
al
quantitie
s(
t(ω)
)re
main
unchanged?
•−→
Renorm
aliz
atio
ngro
up
equatio
ns
(βfu
nctio
n)
dJ
dln
D=
−2N
0J2
Renorm
aliz
atio
ngro
up
Localiz
ed
spin
ina
meta
l:
H=
∑kσ
εk
c†kσckσ
+J
∑
kk′σ
σ′ c
†kσ
(
~S·~σ
)
ck′σ
′
EF
ED
J0
J
Renorm
aliz
atio
ngro
up
Localiz
ed
spin
ina
meta
l:
H=
∑kσ
εk
c†kσckσ
+J
∑
kk′σ
σ′ c
†kσ
(
~S·~σ
)
ck′σ
′
EF
ED
J0
J
Renorm
aliz
atio
ngro
up
Localiz
ed
spin
ina
meta
l:
H=
∑kσ
εk
c†kσckσ
+J
∑
kk′σ
σ′ c
†kσ
(
~S·~σ
)
ck′σ
′
0
EF
EJ0
D=
D
JJ
J
J
Renorm
aliz
atio
ngro
up
Localiz
ed
spin
ina
meta
l:
H=
∑kσ
εk
c†kσckσ
+J
∑
kk′σ
σ′ c
†kσ
(
~S·~σ
)
ck′σ
′
0
EF
ED
J0
D
JJ
J
J
Renorm
aliz
atio
ngro
up
Localiz
ed
spin
ina
meta
l:
H=
∑kσ
εk
c†kσckσ
+J
∑
kk′σ
σ′ c
†kσ
(
~S·~σ
)
ck′σ
′
0
EF
ED
J0
D
JJJ
J
Renorm
aliz
atio
ngro
up
Localiz
ed
spin
ina
meta
l:
H=
∑kσ
εk
c†kσckσ
+J
∑
kk′σ
σ′ c
†kσ
(
~S·~σ
)
ck′σ
′
0
EF
ED
0
D
JJ
JJ
J
Renorm
aliz
atio
ngro
up
Localiz
ed
spin
ina
meta
l:
H=
∑kσ
εk
c†kσckσ
+J
∑
kk′σ
σ′ c
†kσ
(
~S·~σ
)
ck′σ
′
KT
= D
e02N
J0
1
0
EF
ED
J0
D
Kondo
singlet
JJ
J
Non-e
quilib
rium
renorm
aliz
atio
ngro
up
D−
D
D
D−
D
D
V
EE
δ
δ
Fermi sea
Fermi sea
R
L
Rosc
h,Paask
e,K
roha,W
olfl
e,
PRL
90,076804
(2003)
Paask
e,Rosc
h,K
roha,W
olfl
e,
PRB
70,155301
(2004)
Couplin
gconsta
nt
RG
:
T(D
,g,ω
)=
T(D
′,g′,ω
)
dg
lnD
=β(g
)
Nonequilib
rium
:
•Ele
ctro
ntra
nsfe
rin
[−V
/2,V
/2]:
couplin
gfu
nctio
ns
g(ω
)
dependin
gon
ele
ctro
nenerg
yω
•Cuto
ffs
Dsy
mm
etric
al
about
Ferm
iedge
ineach
rese
rvoir
•D
iscre
tedegre
es
offre
edom
on-sh
ell
•In
ela
sticsc
atte
ring
at
finite
bia
s:
RG
flow
ofeach
scatte
ring
pro
cess
cut
off
by
decay
rate
Γℓ
ofin
term
edia
telo
calsta
te
Non-e
quilib
rium
RG
:β
functio
n
dg(j
)α
βm
n(ω
)
dln
D=
2∑jℓγ
−1≤
j+
n−
ℓ≤1 (1−
δm
ℓ δnℓ )
×
[g(j
+n−
ℓ)α
γm
ℓ(Ω
nℓ )
g(j
)γβ
ℓn( ω
)Θ
γnℓ−
g(j
+m−
ℓ)α
γℓn
(Ω
mℓ )
g(j
)γβ
mℓ
( ω)
Θγm
ℓ
]
ln
mm
ln
−
Θγnℓ
=Θ
D−
√
(
Ωnℓ−
γV2
)
2
+Γ
2ℓ
Ωγnl
=ω
+(|n
|−
|ℓ|)∆
(D)A
rnold
,Langenbru
ch,K
roha,PRL
99,186601
(2007)
Non-e
quilib
rium
RG
:energ
ydependence
−200−100
0100
200
ω / T
K
0.08
0.1
0.12
0.14
0.16
gz / ⊥ (ω)g
z, (ω)
gz, (ω
)g
⊥ (ω)
|V−B
|V
V+
B
Experim
ent:
Ralp
h,Buhrm
an,PRL
72,3401
(1994).
Theory
:
Rosc
h,Paask
e,K
roha,W
olfl
e,PRL
90,076804
(2003).
-4-3
-2-1
01
23
4
V / B
0 0.05
0.1
0.15
0.2
0.25
0.3
0.35
G/G0
B=
108TK
B=
72TK
B=
36TK
-4-3
-2-1
01
23
4
V / B
0
0.1
G/G0
RG
O(J
3)
O(J
2)
2-c
hannelK
ondo
effect:quantum
frustra
tio
n
2CK
1CK
1CK
1J
2J
JJ
J Jtt
J
T(w
c)K
=D
e−
12N
0J
T(sc)K
=D
e−
γN
0J/2
J=
t 2/J
Duality
J←→
1/γJ
Nozie
res,
Bla
ndin
,J.Phys.(
P)41,193
(1080)
Kolf,
Kro
ha,PRB
75,045129
(2007)
2-c
hannelK
ondo
effect:Q
uantum
frustra
tio
n
2CK
1CK
1CK
1J
2J
JJ
J Jtt
J
T(w
c)K
=D
e−
12N
0J
T(sc)K
=D
e−
γN
0J/2
J=
t 2/J
Duality
J←→
1/γJ
Nozie
res,
Bla
ndin
,J.Phys.(
P)41,193
(1080)
Kolf,
Kro
ha,PRB
75,045129
(2007)
Experim
ental2CK
sig
nature
s
Conducta
nce
anom
alie
sin
ultra
small
meta
llicpoin
tconta
cts
Ralp
het
al.,
PRL
69,2118
(1992)
~10nm
Cu
Cu
d
Experim
ental2CK
sig
nature
s
2-c
hannelK
ondo
scalin
g
••
Ralp
het
al.,
PRL
72,1064
(1994)
Hettle
r,K
roha,Hersh
field
,PRL
73,1967
(1994)
theory
experiment
Experim
ental2CK
sig
nature
s
2-c
hannelK
ondo
scalin
g
Altsh
ule
r-Aro
nov
DO
Sanom
aly
?
•Ela
sticm
ean
free
path
from
wid
thofanom
aly
:
ℓ=
vF
τ≫
d
•D
opin
gdependence
inconsiste
nt
Ralp
het
al.,
PRL
72,1064
(1994)
Hettle
r,K
roha,Hersh
field
,PRL
73,1967
(1994)
theory
experiment
1/τ
Experim
ental2CK
sig
nature
s
Diff
ere
ntia
lconducta
nce:sp
ikes
at
ele
vate
dbia
s
Ralp
h,Buhrm
an,PRB
51,3554
(1995)
Rotatio
naldefe
ct
modelfo
rthe
2CK
effect
Musta
kas,
Fish
er,
PRB
(1995)
Wuerg
er
(1990)
02π
ϕ
/aπ
−π/a
∆m
=+
1 _
ϕ
m=
0•
Pro
ton
at
inte
rstitialsite
•M
ole
cule
with
rota
tional
degre
eoffre
edom
Arn
old
,Langenbru
ch,K
roha,PRL
99,186601
(2007)
Ψ(ϕ
)=
eim
ϕu
m(ϕ
)
Rotatio
naldefe
ct
modelfo
rthe
2CK
effect
m=
+1
∆
m=
+2
∆
m=
−1
∆
M’
Mm
’
m
m=
0
m=
−1
m=
+1
m=
−2
∆
m=
−1
∆m
=+
1∆
m=
0
m=
−1
m=
+1
..
Rotatio
naldefe
ct
modelfo
rthe
2CK
effect
M: 0
−−
> 1
−1
−−
> 0
m=
+1
∆M: 0
−−
> −
1
1 −
−>
0
m=
+2
∆
m=
−1
∆M
: 0 −
−>
1
−1
−−
> 0
M: 0
−−
> −
1
1 −
−>
0
M’
Mm
’
m
m=
0
m=
−1
m=
+1
m=
−2
∆
m=
−1
∆m
=+
1∆
M: −
1 −
−>
1
m=
0
m=
−1
m=
+1
M: 1
−−
> −
1
..
Rotatio
naldefe
ct
modelfo
rthe
2CK
effect
M: 0
−−
> 1
−1
−−
> 0
m=
+1
∆M: 0
−−
> −
1
1 −
−>
0
m=
+2
∆
m=
−1
∆M
: 0 −
−>
1
−1
−−
> 0
M: 0
−−
> −
1
1 −
−>
0
M’
Mm
’
m
m=
0
m=
−1
m=
+1
m=
−2
∆
m=
−1
∆m
=+
1∆
M: −
1 −
−>
1
m=
0
m=
−1
m=
+1
M: 1
−−
> −
1
..
SU(3
)Ham
iltonia
nw
ith“pse
udom
agnetic
field
”∆
0:
H=
∑
kσm
α
′εkcα
†kσm
cαkσm
+∆
0
∑
m=±1
f†m
fm
+∑
σα
β
[
Jz2S
z sσ
αβ
z+
J⊥
(
S1,−
1sσ
αβ
−1,1
+S−1,1
sσ
αβ
1,−
1
)
]
+∑kk′
σα
β
∑m,n
−1≤
n−
m≤1
[g(n
)m
0S
m,0
sσ
αβ
n−
m,n
+H
.c.
]
Energ
ydependence
ofcouplin
gconstants
-12-8
-40
48
12ω
/ TK
0.01
0.02
g(0)
01(ω)V
= 2 T
k
V =
5 Tk
V =
10 Tk
-1-0.5
00.5
1ν / D
0
-30
-20
-10 0
Σ(ν) / J⊥2
∆(D
=0)
ImΣ
0Im
Σ1
ReΣ
1
ReΣ
0
V =
10 Tk
RG
Flo
wofcouplin
gconstants
02
46
810
ln(D/T
k )0
0.05
0.1
0.15
0.2
0.25N
(0) G1
N(0) G
2
g10
g(0)10 (1)
02
46
810
ln(D/T
k )0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
N(0) A
1N
(0) A2
01g
01g
(0)
(−1)
Corre
latio
n-in
duced
levelcro
ssin
g
Σ =m
m’
m
G0(ν
)=
1
ν−
Σ0(ν
)
G±1(ν
)=
1
ν−
∆−
Σ±1(ν
)
Levelcro
ssing:
Σ±1(∆
)+
∆<
Σ0(0
)
-0.20
0.2
-12
-10 -8 -6
Re[Σ(ν)] , |M
|=1
Re[Σ(ν)] , M
=0
-4-2
02
4
ν / D0
-12 -8 -4 0 4 8 12
Re[Σ(ν)] / J 2, |M
|=1
Re[Σ(ν)] / J 2, M
=0
∆
pseudospin relaxationrate (M
=0)
Corre
latio
n-in
duced
levelcro
ssin
g
46
810
ln(D/T
K )-20
-10 0 10 20
∆/ΤΚ
Levelre
norm
aliz
atio
n:
•Levelcro
ssing
not
forb
idden
by
sym
metry
!
•Levelcro
ssing
occurs
generic
ally
inth
epertu
rbaiv
ere
gim
e.
SU(3)
model:
phase
dia
gra
m
00.001
0.0020.003
0.0040.005
∆0 / D
0
0
0.0001
0.0002
0.0003
0.0004(N(0) J
1)2
G/J1 =
0.6G
/J1 = 0.8
G/J1 =
1.0G
/J1 = 1.2
Jz / J1 = 0.8
SU(3)
model:
phase
dia
gra
m
00.001
0.0020.003
0.0040.005
∆0 / D
0
0
0.0001
0.0002
0.0003
0.0004(N(0) J
1)2
G/J1 =
0.6G
/J1 = 0.8
G/J1 =
1.0G
/J1 = 1.2
Jz / J1 = 1.0
SU(3)
model:
phase
dia
gra
m
00.001
0.0020.003
0.0040.005
∆0 / D
0
0
0.0001
0.0002
0.0003
0.0004(N(0) J
1)2
G/J1 =
0.6G
/J1 = 0.8
G/J1 =
1.0G
/J1 = 1.2
Jz / J1 = 1.2
Conductance
spikes
-2-1
01
2V
/ TK
-0.6
-0.4
-0.2 0
1.61.8
2V
/ TK
-0.3
-0.2
-0.1
dI/dV [e2/h]
dI/dV [e /h]414040804020
1510
50
−5
−10
V [m
V]
−15
2
Γ1 I ~
g ln |V/D
| δ
At
least
two
exponentia
llydiff
ere
nt
energ
ysc
ale
s:
TK
≃D
e−
12M
J(0)
2CK
Kondo
tem
pera
ture
T⋆K
≃D
e−
12M
g01(∆
)sp
ike
wid
th
Arn
old
,Langenbru
ch,K
roha,PRL
99,186601
(2007)
Magnetic
field
dependence
B-fi
eld
couplin
gto
lattic
eangula
rm
om
entu
mofth
edefe
ct
−gµ
BL·B
m=
0
m=
−1
m=
1B
V
dI/dV
∆
top related