Quantum Dots Talking through a Quantum Wire

Vladimir YudsonInstitute for Spectroscopy, RAS

Acknowledgments to: Igor Lerner and Igor Yurkevich

Support: RFBR 09-01235

September 23, 2010

“The Science of Nanostructures: New Frontiers in the Physics of Quantum Dots”

Chernogolovka, Russia

System:

Quantum Dots Coupled with a Wire

States of QDs become correlated

Limiting case: a Single Channel Wire + Single Level QDs

(spinless/spin polarized electrons)

1 2 N

x

jn

can be straightforwardly calculated for

non-interacting electrons

Quantities of interest:

averaged populations:

correlation functions:cljjl tntnttS )'()()'(

(depend on QD positions)

(equilibrium, non-equilibrium, quenches, etc.)

If not the interaction…

A Single Channel Wire with Interacting electrons (no interaction between QD’s and wire electrons)

A brief reminder: 1. LL + a single static impurity

)()(0 xUxU eff Kane & Fisher (92)

2. Luttinger Liquid + a single level QD

Furusaki & Matveev (02)

3. Chiral Luttinger Liquid + a single level QD

Lerner, V.Y., and Yurkevich (08)

6

Common wisdom:

e-e interaction:

Chiral Toy Model:

1 2 N

x

(only right movers)

]0)'()'()['(]0)()([2

10 xxxxVxxdx RRRR

The only effect – renormalization of Fermi-velocity: vvF

To begin with…

Non-interacting electrons(for comparison, notations, etc.)

Generating functional integral (for Matsubara technique):

],;,[];,[],[],[],[][

aaSJaaSS hybreaaDDJZ

)()]([)(];,[ jjjj

j

aJadJaaS

),(])[,(],[ xivxdxdS xF

)](),(),()([],;,[ jjjj

j

jhybr axxadtaaS

1

0

gpvi

ipgFn

n

1

),(0

8

QD action (after integration over wire electrons):

Extremely anisotropic disorder!

less anisotropic

Here:

1)0( jlG

)'(])',;,())(()'([)(];,[ 0

,

llljjjjjlj

lj

atxxgtJadJaaS

333231

2221

11

1)0(

)()(

0)(

00

)0(

iiiiii

iiii

ii

iG

nnn

nn

n

njl

|)(||)(|)( ljnljn xx

jl

xx

F

lj

njl eev

tti

F

j

jv

t

2

2

Some consequences:

jnEquilibrium populations do not depend on

positions of QDs (in contrast to the non-chiral model)

Triangular structure of the correlation function:

ljatnnTSnilj

R

jl

,0)]'()([)()0(

Example (two QDs), low-temperature limit:

22

0

22

0

2

0

2

/)(2

2121

)(ln

2

1

)()( 12

i

iie

iS FvxxiR

Accounting for interaction

(Grishin, Yurkevich, & Lerner (04))

)(1

02

1

02

1 idxdVdxd

eDeVdxd

1. H-S decoupling:

2. Gauge transformation: ii ee ;

Fermionic action:

),()],()[,(];,[ xxiivxdxdS xF

3. Accounting for the Jacobian: 1

0

11

0 VVV

][ xFiv

Bosonic field correlation functions (for the toy

model):

QDs functional (after integration over wire electrons):

];;,[][ ],[][][ JaaSS eaaDeDJZ

)'(])',;,(

))(()'([)(];,[

)',(

0

),(

,

ll

xi

lj

xi

j

jjjlj

lj

atexxget

JadJaaS

lj

))((),(),( 0

qvivqi

Viqiq

Fnn

nn

gvv

Vvv F

F

F

01

Constv

vF )()( 0

vvljS

xi

lj

xi

lj F

lj xxgexxgexxg

)]',;,([)',;,()',;,( 0

][

)',(

0

),(

)()'(

)()'(ln)]',(),([

2

1 2

lj

ljF

ljxxiv

xxivxx

In low-temperature limit:

I. Wire Green’s function for the chiral model:

)]()'([2

1)',;,(0

ljF

ljxxiv

xxg

As a result, DOS of the “chiral toy model”:

The “toy model” is not the Luttinger Liquid

(“Common wisdom”)

II. One QD coupled with a toy chiral mode:

][

)'()',0;,0()(' )',0(0

),0(2

S

aegetadd lii

e

In the n-th order of a formal expansion

v

vn F

nn ln)]'(...)'()(...)([2

1 2

11

Exactly as many as required to renormalize the product

of n bare Green’s functions!

Resume:

A system of a single QD coupled to an toy chiral mode

is trivial (the only effect: )vvF

III. Several QD’s coupled with a chiral mode:

][

)'()',;,()(')',(

0

),(2

S

aexxgetadd llxi

ljjxi

j

e

Already in the 2nd order of a formal expansion

)()''(

)()''(lnln

)()(

)()(lnln

)()'(

)()'(ln

)()'(

)()'(ln

)]',()',(),()([2

1

2121

2121

2121

2121

1222

1222

2111

2111

2

21122211

xxiv

xxiv

v

v

xxiv

xxiv

v

v

xxiv

xxiv

xxiv

xxiv

xxxx

FFFF

FF

Too many terms!

System of several QD coupled to an interacting toy

chiral mode is apparently “non-trivial”.

In fact, this is the property of the chiral model itself:

averaging of a product of several Green’s functions over

bosons generates more additional factors than the

number of Green’s functions.

In case of the chiral toy model, cancellation of extra

terms might be provided by an “almost complete”

vanishing of point interaction effects for a single mode.

For the chiral Luttinger model (two types of movers) no

cancellation is expected.

System of several QDs coupled with a chiral mode

needs a proper field-theoretical treatment.

One of ways – a gauge transformation in QD action.

Gauge transformation in QD’s action:

)'(])',;,(

))(()'([)(];,[

)',(

0

),(

,

ll

xi

lj

xi

j

jjjlj

lj

atexxget

JadJaaS

lj

j

xi

j

xi

jj aeaeaa jj ),(),(;

)()],([)(];,[ 1

0 jjj

j

axGadJaaS

Transformed action:

Correlations of bosonic field

)]([][||2),(),(/|)(|/|)(|

ljn

vxxvxx

nnlnj xxeeiqAiqA Fljnljn

:),()( jj xA

(for the toy chiral model with only right movers)

Correlations of bosonic field

)()],([)(];,[ 1

0 jjj

j

axGadJaaS

:),()( jj xA

}2

)(2)]([

)(2)]([{

||),(),(

/|)(|

22

/|)(|

/|)(|0

vxx

F

Fvxx

F

Fljn

vxx

F

Fljn

nnlnj

ljnljn

ljn

evv

vve

vv

vvxx

evv

vvxx

v

ViqAiqA

Correlators of bosonic field for chiral

Luttinger model (both right and left interacting movers):

Lowest order (in ).0V

vr

F

Fvr

F

Fn

nnn

nn evv

vve

vv

vv

v

ViqAiqA

/

22

/012

2

)(2][),(),(

2 1)()(),( 2121 nnn iGi

0)]'()([)(

,0)]'()([)(

)0(2112

)0(1221

n

n

i

R

i

R

nnTS

nnTS

Summary

1. System of several QDs coupled with a single chiral

branch of an interacting electron wire allows a simpler

(as compared to coupling with the both branches)

analysis with the use of a gauge field in the QD’s action.

2. This allows one to develop a perturbation theory in the

gauge field to calculate various correlation functions of

QDs.

3. The triangular structure of Green’s functions and

correlation function of population in different QDs is

preserved for the chiral LL in the first order in

interaction (at least).

20

Thank you!