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## Hamiltonian description of the quantum RC circuit

The aim of this Thesis is to address the effects of interactions on the quantum capacitance and the charge relaxation resistance. To do this, we cannot look anymore at the quantum dot as a sort of “black-box”, butmodels must be considered that give an exact description of electron-electron interactions within it. In this section, the results of Section 1.2 are derived from an Hamiltonian approach. We neglect for the moment interactions, which allows for straightforward calculations. We establish the physical origin of the quantum capacitance and the charge relaxation universality from this alternative point of view.

Considering spinless electrons, the Coulomb blockade model Eq. (iv) accounts explicitly for the driving of charges on the dot and interactions. This can be understood by inspecting the part of the Hamiltonian describing the capacitor HCapa = Ec ( ˆ n −N0)2 . (1.28) If the square is expanded and constant contributions are neglected, it results in a renormalization of orbital energies εl in Eq. (iv) and in an interaction term, namely HCapa = −eVg (t ) ˆ n +Ec ˆ n2 .

### Linear response theory: basic notions for the quantum RC circuit

Linear response theory [89, 90] addresses the calculation of observables in time-dependent problems which can be written in the form H = H0−λf (t )ˆA . (1.31) ˆA is a generic operator, f (t ) any function of time of order 1 and λ a perturbation parameter. To first order in λ, the time-dependent corrections to any observable 〈ˆO 〉 can be written as a functional of operators averaged at equilibrium 〈ˆO 〉(t ) = 〈ˆO 〉0+λ Z ∞ −∞ dt ′χ(t −t ′) f (t ′) . (1.32).

The notation 〈·〉0 means that the averages are carried out by taking traces involving the timeindependent part H0 of the Hamiltonian Eq. (1.31). For instance 〈ˆO 〉0 = Z−1Tr £ˆO e−βH0 ¤ . Z =0.

#### Description of the quantumRC circuit with a resonant level model

In this section, we use a resonant level model to describe the quantumdot and the lead HRes = X k εkc† kck +t X k ³ c† kd +d†ck ´ +εdd†d . (1.41) We restrict for simplicity to the single-channel and the single-level case. The generalization of the following calculations to the many-channel case is straightforward. In Appendix B, we carry out explicitly the calculation of the differential capacitance in the multi-level case. The model Eq. (1.41) describes the situation pictured in Fig. 1.4. It constitutes a simplification of the Coulomb blockade model Eq. (iv), where the interacting term of Eq. (1.29) is neglected.

This is equivalent to consider Cg =∞, recovering the mean-field analysis of Section 1.2. We take advantage of this section to introduce the notations thatwe shall use in the path integral formalism in the following chapters. The partition function associated to the Hamiltonian Eq. (1.41) can be written in the form[91] Z = Z D h c,c†,d,d† i e−S £ c,c†,d,d†¤ .

**The Fermi liquid in the quasi static approximation**

As already discussed in Section 1.3.1, linear response theory states the possibility to study a time-dependent problem by looking at dynamical correlators considered at equilibrium. So, neglecting for the moment the time-dependence through εd (t ) of our problem, we consider the situation at equilibrium. It is a well established fact that both the Coulomb blockade model [60, 92] and the Anderson model [93] have a Fermi liquid behavior at low energy. We refer to textbooks [94, 95] for a complete review about Fermi liquid theory. What is important for the following discussion is that a fermionic system behaving like a Fermi liquid deploys the same qualitative physics as that of a free Fermi gas. Its constituents are then called quasi-particles, which are still fermions whose mass and spectrum are renormalized by the presence of interactions and which still carry the same quanta of charge and spin.

The effective Hamiltonian describing the behavior of these quasi-particles has then to be a non-interacting Hamiltonian HQP = X kσ εka† kσak′σ .

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

The Friedel sum rule [77] relates the charge occupation of the dot 〈 ˆ n〉 to the phase-shift δ(ε) that the presence of the quantum dot causes on the wave function of the lead electrons at the Fermi surface. It reads 〈 ˆ n〉 = δ(0) π (2.11) and the phase-shift is considered at the Fermi energy EF = 0. In this section, we provide an illustration of the Friedel sum-rule for the Anderson model Eq. (v) in the absence of interactions, U = 0. The extension to the interacting case is more difficult and it is given in Ref. [96, 97]. As the σ =↑, ↓ spin sectors are decoupled, we can neglect the electron spin and calculate the occupation of the quantum dot described by a resonant level as in Eq. (1.41). To do this, the total electron occupation 〈 ˆN 〉 of the electron gas has to be calculated. It is given by 〈 ˆN 〉 = X α Z ∞ −∞ dωAα(ω) f (ω) , (2.12) the sum on the label α running over all the eigenstates of the Hamiltonian. Aα(ω) is the spectral function of the state α and it is defined as Aα(ω) = − 1 π ImGαα(ω+i0+) , (2.13).

Gαα being the retarded Green’s function associated to the state α. In this section, we adopt the notation Gdk(t − t ′) = −iθ(t − t ′) 〈d(t )c† k(t ′)〉. In the absence of the quantum dot, described by the operators d in the AndersonHamiltonian Eq. (v),Gkk(ω+i0+) = (ω+i0+−εk)−1 and Eq. (2.12) reduces to a sum over the Fermi function P k f (εk) giving the total number of electrons 〈 ˆN 〉 composing the system. When the quantum dot is taken into account, a further resonant level is introduced. Working at fixed chemical potential, the total number of electrons Eq. (2.12) will be modified, and the difference with the previous one will give the amount of electrons displaced by the presence of the quantumdot. We already calculated in Section 1.3.2 the retarded Green’s function of the quantum dot electrons Eq. (1.54). The retarded Green’s function of the lead electrons can be also obtained writing down the equation of motions (ω−εd )Gdd (ω) = 1+t X k Gkd (ω) .

**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