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

1CK

1J

J JJJ

2-c

hannelK

ondo

effect:quantum

frustra

tio

n

1CK

1CK

1J

2J

J JJJJ

2-c

hannelK

ondo

effect:quantum

frustra

tio

n

1CK

1CK

1J

2J

JJ

J Jtt

J

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

flow

ofcouplin

gconstants

02

46

810

ln(D/T

k )0

0.2

0.4

0.6

0.8 1

N(0) J1

N(0) Jz

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:

flow

dia

gra

m

Pro

jectio

nonto

J−

∆pla

ne:

2CK

J>0

∆<0J*

0potentialJ=

0∆>0

∆

J

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

∆

Phase

dia

gra

min

magnetic

field

00.25

0.50.75

11.25

∆0 /(10 −3·D

0 )

0 1 2 3 4(N(0)·J)2/10

-4

B0 =

0B

0 =0.00005·D

0B

0 =0.0005·D

0

TK =

1.3·10-4D

0

TK =

1.5·10-4D

0

TK =

9.7·10-6D

0

Ballm

ann,K

roha,Ann.Phys.

(Berlin

),su

bm

itted

(2011)