1. Full Current Statistics in Multiterminal Mesoscopic Conductors
Dmitry A. Bagrets
Universität Karlsruhe
Collaboration: Yuli V. Nazarov

2. Outline What is Full Current Statistics? (N-terminal case)
Circuit theory of FCS (non-interacting regime)
Master equation approach to FCS
(strong Coulomb blockade limit)
Weak charge quantization and FCS
(weak Coulomb interaction)
Conclusions

3. Full Current Statistics
3-terminal case
Current conservation
number of electrons transferred

4. FCS in N-terminal case Generating function Currents Noise correlations
counting fields
Action
Probability
Currents
Noise correlations
3-terminal circuit
Higher order correlations

5. Keldysh Method Effective Hamiltonian Generating function
Keldysh path
Generating function
( measuring device [spin] )
Conventional Green function technique

6. Circuit Theory Semiclassical approach Boundary conditions
Non-interacting systems
Conductance
Semiclassical approach
Boundary conditions
Gauge transform

7. Minimal Action Principal
Total action
Action
Normalization condition
Saddle point
( “1st Kirchhoff’s rule“ )
Matrix current
( “2nd Kirchhoff’s rule“ )

8. Chaotic Quantum Dot Connectors Big current fluctuations! 1-tunnel
2-diffusive
3-ballistic

9. Coulomb Blockade Dot
Coulomb blockade system

10. Markov process Master Equation “Orthodox” theory Limits of validity
Rates
“Orthodox” theory
( charge quantized ! )
Limits of validity
( No co-tunneling ! )

11. FCS + Master Equation Effective Master equation Generating function
Time arrow
Effective Master equation
Generating function

12. Coulomb Blockade Dot Big current fluctuations! Coulomb blockade
5 - no interaction

13. Weak Charge Quantization
Panyukov, Zaikin, ’91
Flensberg ’93, Matveev ’95
Nazarov ‘99
Free energy
- “Effective” charging energy
-Tunnel junctions
- Diffusive contacts

14. Coulomb Island - Conductance - Charging energy - Thouless energy
- inverse RC-time
Phase
Relations between energy scales

15. “Effective” Keldysh Action
Interaction
Electrons
Electrostatic energy

16. ! Quantum corrections Weak localization correction:
( Random-matrix theory )
Interaction correction
Inelastic ( )
Elastic ( )
Divergent ! ?
Dissipation!

17. Renormalization Group
“Poor man’s” scaling
( 1-Loop order )
RG Equations
“Running” cut-of
RG diverges:
Metallic saddle point is unstable ! At
Charging energy:
Equivalence to instanton calculation !
Large phase fluctuations
Charge quantization !
[Q,f]=ie

18. Onset of Coulomb Blockade
Critical conductance

19. Full Current Statistics
Connectors
1-tunnel
2-diffusive
3-ballistic

20. Conclusions The theory of FCS in N-terminal circuits:
- Non-interacting regime
- Coulomb blockade regime
- Weakly interacting regime
Evaluation of probability of big current fluctuations
Suppression of current fluctuation at
Renormalization of transmission eigenvalues
Crossover to Coulomb blockade
regime at