the problem of true macroscopic charge quantization in ...landau100.itp.ac.ru › talks ›...

TheThe ProblemProblem ofof TrueTrue MacroscopicMacroscopic ChargeChargeQuantizationQuantization inin CoulombCoulomb BlockadeBlockade

cond-mat/0702400

Igor Burmistrov+ and Adrianus Pruisken*

+L.D. Landau Institute for Theoretical Physics*Institute for Theoretical Physics, University of Amsterdam

L.D.Landau Memorial Conference "Advances in Theoretical Physics"

June 22-26, 2008 Chernogolovka, Russia

Discussions: A.G. AbanovYu.G. Makhlin

Introduction –1

Coulomb blockade …

Introduction

Single electron transistor (SET) Equivalent circuit for SET

Single electron box (SEB)

Single charge tunneling, ed. by Grabert and Devoret, 1992

Introduction –2


Introduction

The SET parameters

Beloborodov, Efetov, Altland, Hekking, 2001Efetov, Tschersich, 2003

Number of transport channels

Tunneling matrix elements

Tunneling conductances

Charging energy

Level spacing on the island

Temperature

External charge

[in units ]

Introduction –3


Introduction

How does discreteness of the charge manifest itself at T=0 ?

Ground state energy

where h = Ec(1-2q)

where is the operator of the excess particle number on the island

-2 -1.5 -1 -0.5 0.5 1 1.5 2q

1

2EnêEc

h

At

Isolated island (gl,r=0) Periodic in q

Introduction –4


Introduction

How does discreteness of the charge manifest itself at T=0 ?

Non-isolated island

At T=0

where g = gl+gr

QQ is not quantized at is not quantized at T=0T=0

for any for any gg!!

What quantity is quantized What quantity is quantized

at at T=0T=0??

Matveev, 1991Schoeller, Schön, 1994

0.5 1q

0.5

1

Q

gπ0

g=0

ResultsR

esults -1


Novel quantity

where is current operator for the SET, V=Vr-Vl is the voltage drop

N.B. commutator

ResultsR

esults -2


Temperature dependence (explicit calculations)

Weak coupling

0.5 1q

0.5

1

q'

ResultsR

esults -3


Temperature dependence (explicit calculations)

where

0.5 1q

0.5

1

q'

T>0T=0

qq00 is quantized at is quantized at T=0T=0

Strong coupling and

ConjectureC

onjecture -1


qq00 is quantized at is quantized at T=0 T=0 independent of g independent of g

i) Decoupling Hc by the Hubbard-Stratonovich field

ii) Integration over fermions to the lowest order in HT

AES modelA

ES

model -

1


Microscopic Hamiltonian

where

AES modelA

ES

model -

2


Effective action in imaginary time

Ambegaokar, Eckern, Schön, 1982

integer

and

N.B. AES model is known also as circular brane model

Lukyanov, Zamolodchikov, 2003Lukyanov, Werner, 2006

AES modelA

ES

model -

4


Nontrivial topology and instantons

Convenient variables

Topological charge (winding number)

Instanton W>0 |za|<1

Anti-instanton W<0 |za|>1

Instanton solution (2|W| zero modes )

Classical action

Korshunov, 1987Bulgadaev, 1987Nazarov, 1999

Response to change in the boundary conditionR

esponse -1


Expansion in topological sectors

Let us formally consider Z[x] with continuous x and define


esponse -2


Analytic continuation

From W to W-n

where

Analytic continuation

where

Hence

N.B. it is solution of classical equation of motion


esponse -3


Response parameters

Finally,

where

Physical observablesP

hysical observables -1


Relation of g’ and q’ with physical quantities

Average charge Q, quantum current noise and q’

SET conductance G and g’

It can be shown in Keldysh technique

Weak coupling -

1


Weak coupling -

Perturbation theory in 1/g (W=0)

Quadratic part of the action

Correlation function

Guinea, Schön, 1986

where and

Weak coupling -

2


Weak coupling -

Instanton contribution (W=± 1)

Classical action Gaussian fluctuations around instanton


Analyticcontinuation

Panyukov, Zaikin, 1991Wang, Grabert, 1996

Weak coupling -

3


Weak coupling -

Instanton contribution (W=± 1)

At high temperatures

Notice

Altland, Glazman, Kamenev,Meyer, 2006

Oscillations in q’ is much more pronounced than in Q

Strong coupling -

1


Strong coupling –

Effective action for |q-1/2|¿ 1 and T¿ Ec

Isolated island (g=0)

where h=Ec(1-2q)

Two-level Hamiltonian

Average charge

Operators

Strong coupling -

2


Strong coupling –

Non-isolated island

Grand partition function and correlations

Larkin, Melnikov, 1971Sachdev, Ye, 1993Zhu,Si; Zarand,Demler, 2002

where – pseudofermions, chemical potential

Effective action for and (XY Bose-Kondo)

Equations for and

Strong coupling -

3


Strong coupling –

Leading logarithmic approximation –

Pseudofermion Green function renormalization

where

where and

Strong coupling -

4


Strong coupling –

Leading logarithmic approximation –

Average charge

T=0 - Matveev, 1991T>0 - Schoeller, Schön, 1994


Strong coupling -

6


Strong coupling –

(Schoeller, Schön, 1994)

For

Physical observables q’ and g’

Strong coupling -

6


Strong coupling –

Inelastic co-tunneling at

(Averin, Nazarov, 1990)

Contributions to g’ and q’

Conclusions -

1


Conclusions

Novel quantity q0 is introduced into the Coulomb blockade problem

At it is shown that q0 is quantized at T=0 (with exponential accuracy) in the leading logarithmic approximation contrary to the average charge Q on the SET island

Inelastic co-tunneling does not destroy the quantization of q’

Conclusions -

1


Experimental realization

Main difficulty is to measure non-symmetrizedcurrent noise in the SET

(Lesovik, Loosen, 1997)

(Onac, Balestro, Trauzettel, Lodewijk, Kouwenhoven, 2006)

LC curcuit inductively coupled to the SET

AES modelA

ES

model -

3


Circular brane model

where X and Y are boson fields on the surface of a half cylinder

Lukyanov, Zamolodchikov, 2003…Lukyanov, Werner, 2006

with q=0 and

Integrating out the ‘bulk’




Tunneling conductance G of the SET

Standard trick

Average current

Matsubara correlation function

Retarded correlation function




Tunneling conductance G of the SET

Average current

By definition

Ben-Jacob, Mottola, Schön, 1983

Strong coupling -

2


Strong coupling –

Pseudofermions

Spin language

Fermion language

Npf=1

Strong coupling -

5


Strong coupling –


No vertex renormalization !

Strong coupling -

6


Strong coupling –

Physical observables q’ and g’

Correlation function for

(Schoeller, Schön, 1994)

with

qq00 is quantized at is quantized at T=0T=0 independent of independent of gg !!

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