An illustration of the Friedel sum rule for non-interacting electrons

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Table of contents

Introduction
1 Theory of the quantumRC circuit: non-interacting case
1.1 Phenomenology
1.2 Scattering theory of the quantum RC circuit
1.2.1 The quantum capacitance
1.2.2 The charge relaxation resistance
1.2.3 The example of a quantumRC circuit with a 2DEG
1.3 Hamiltonian description of the quantum RC circuit
1.3.1 Linear response theory: basic notions for the quantum RC circuit .
1.3.2 Description of the quantum RC circuit with a resonant level model
1.4 Conclusions
2 A theory for the interacting quantumRC circuit
2.1 The Fermi liquid in the quasi static approximation
2.1.1 An illustration of the Friedel sum rule for non-interacting electrons
2.2 The Schrieffer-Wolff transformation
2.2.1 Coulomb blockade model
2.2.2 Anderson model
2.3 The quasi static approximation
2.4 Generalized formof the Korringa-Shiba relation
2.4.1 A continuum in the dot
2.5 The loss of universality
2.5.1 Dependence of Rq on the magnetic field: giant and universal peaks.
2.5.1.1 The giant peak of the charge relaxation resistance
2.5.1.2 A universal peak in the mixed-valence region
2.6 The SU(4) Anderson model
2.6.1 Determination of the SU(4) Kondo temperature
3 Effective theory of the Coulomb blockade model
3.1 Slave states and Abrikosov’s projection technique
3.2 Integration of the high energy charge sectors
3.3 The renormalization group
3.3.1 Integration of high energy degrees of freedom
3.3.2 Rescaling
3.3.3 Relevant, irrelevant andmarginal operators
3.4 Calculation of the vertex
3.4.1 Slave-boson self-energy
3.4.2 Lead/dot electrons self-energy
3.4.3 One-loop diagrams
3.4.4 The large-N limit and second order diagrams
3.4.4.1 Diagrams from combinations of the six-leg vertex
3.4.4.2 Diagrams from the ten-leg vertex
3.4.5 Total charge conservation and the Friedel sumrule
3.5 Conclusions
4 The Anderson model and the Kondo regime
4.1 The Bethe ansatz solution of the Andersonmodel
4.1.0.1 Phase diagram
4.1.1 The Bethe ansatz equations
4.1.2 Preliminary considerations on C0 and Rq
4.1.2.1 The differential capacitance C0 is proportional to the charge density of states
4.2 Kondo physics in the Anderson model
4.2.1 Path integral approach with slave states and link to the Schrieffer-Wolff transformation
4.2.2 The failure of a perturbative approach
4.2.3 A Fermi liquid theory for the Kondo Hamiltonian
4.2.4 Cragg & Lloyd’s argument for the potential scattering correction .
4.3 Calculation of the vertex in the Andersonmodel
4.3.1 Kondo temperature and agreement with the Friedel sumrule
4.4 A (giant) peak for the charge relaxation resistance
4.4.1 The giant charge relaxation resistance in the Kondo regime
4.4.2 Corrections to the Kondo scaling limit: a numerical approach
4.4.2.1 Persistence of the peak in the function ©
4.4.2.2 The ¡ U corrections to the envelope function F
4.4.2.3 Identity between χm and ∂〈 ˆ n〉 ∂H
4.4.3 Universal scaling behaviors in the valence-fluctuation regime
4.5 Conclusions
5 The SU(4) Anderson model
5.1 A new giant peak for the charge relaxation resistance
5.2 Path integral formulation of the SU(4) Andersonmodel
5.2.1 The calculation of the SU(4) renormalized vertices
5.2.1.1 Sector of charge q = 1
5.2.1.2 Sector of charge q = 2
5.2.1.3 Sector of charge q = 3
5.3 Generalization to SU(N)
5.4 Conclusions
Conclusions and perspectives
Appendix
A Results of linear response theory
A.1 Parity of the dynamical charge susceptibility
A.2 Energy dissipation in the linear response regime
B Multi resonant level model
C Scattering theory and phase-shift
D T -matrix in the potential scattering Hamiltonian
E Fundamental representation of the SU(N) group
F Contributions to V R in the Coulomb blockade model
G Bethe ansatz equations for the Anderson model
H Calculations for the SU(4) renormalized vertex
H.1 Sector with q = 1
H.1.1 Mean-field analysis
H.1.2 Calculation of the renormalized vertex
H.2 Sector with q = 2