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

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

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

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

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

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

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

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

Coulomb Blockade Dot Coulomb blockade system

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

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

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

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

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

“Effective” Keldysh Action Interaction Electrons Electrostatic energy

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

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

Onset of Coulomb Blockade Critical conductance

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

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