The The Problem Problem of of True True Macroscopic Macroscopic Charge Charge Quantization Quantization in in Coulomb Coulomb Blockade Blockade 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. Abanov Yu.G. Makhlin
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Let us formally consider Z[x] with continuous x and define
Response to change in the boundary conditionR
esponse -2
Coulomb blockade …
Analytic continuation
From W to W-n
where
Analytic continuation
where
Hence
N.B. it is solution of classical equation of motion
Response to change in the boundary conditionR
esponse -3
Coulomb blockade …
Response parameters
Finally,
where
Physical observablesP
hysical observables -1
Coulomb blockade …
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
Coulomb blockade …
Weak coupling -
Perturbation theory in 1/g (W=0)
Quadratic part of the action
Correlation function
Guinea, Schön, 1986
where and
Weak coupling -
2
Coulomb blockade …
Weak coupling -
Instanton contribution (W=± 1)
Classical action Gaussian fluctuations around instanton
Correlation function
Analyticcontinuation
Panyukov, Zaikin, 1991Wang, Grabert, 1996
Weak coupling -
3
Coulomb blockade …
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
Coulomb blockade …
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
Coulomb blockade …
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
Coulomb blockade …
Strong coupling –
Leading logarithmic approximation –
Pseudofermion Green function renormalization
where
where and
Strong coupling -
4
Coulomb blockade …
Strong coupling –
Leading logarithmic approximation –
Average charge
T=0 - Matveev, 1991T>0 - Schoeller, Schön, 1994
Correlation function
Strong coupling -
6
Coulomb blockade …
Strong coupling –
(Schoeller, Schön, 1994)
For
Physical observables q’ and g’
Strong coupling -
6
Coulomb blockade …
Strong coupling –
Inelastic co-tunneling at
(Averin, Nazarov, 1990)
Contributions to g’ and q’
Conclusions -
1
Coulomb blockade …
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
Coulomb blockade …
Experimental realization
Main difficulty is to measure non-symmetrizedcurrent noise in the SET