*3.1. The Local Fermi Liquid*

The local Fermi liquid approach is justified by the physical picture already presented in Section 2, namely that an incoming reservoir electron with an energy much smaller than the charging energy of the quantum dot is effectively scattered in a purely elastic way [77]. At temperatures well below the charging energy *E*c, energy conservation prevents any permanent change in the charge of the quantum dot and each electron entering the dot must be compensated by an electron leaving it within the (short) time *h*¯ /*E*c fixed by the uncertainty principle. The electron escape can occur via elastic or inelastic processes, sketched in Figure 4, depending on whether the electron energy is preserved or not. Inelastic processes cause decoherence and call for a many-body approach to be properly evaluated.

At low energy *ε* of the incoming electron, the inelastic processes are typically suppressed by the ratio (*ε*/*E*FL)<sup>2</sup> [43,44]. *E*FL is a Fermi liquid energy scale, typically of the order of the charging energy *E*c. Nevertheless, in the presence of spin-fluctuations, the emergence of strong Kondo correlations, to be discussed in Section 3.3.2, can sensibly reduce *E*FL down to the Kondo energy scale *T*K, (see Equation (26)). Therefore, in the *ε* → 0 limit, inelastic processes are ignored and the scattering is purely elastic. It is described within a single-particle formalism where the scattering by the quantum dot imprints a phase shift *δW* to the outgoing electronic wave functions. This phase shift alone incorporates all interaction and correlation effects.

The simplest model entailing these features is a free Fermi gas in which a delta barrier located at *x* = 0 (the entrance of the dot) scatters elastically quasi-particles. In the language of second quantization, the delta barrier is described by the electron operator <sup>Ψ</sup>†res(*x* = <sup>0</sup>)<sup>Ψ</sup>res(*<sup>x</sup>* = <sup>0</sup>), and its strength *W* has to depend on the parameters of the parent model (such as the orbital energy *εd*, the charging energy *E*c, etc.). Switching to momentum space, such model is a free Fermi gas with a potential scattering term:

$$\mathcal{H}\_{\rm LFL} = \sum\_{\mathbf{k}\sigma} \varepsilon\_{\mathbf{k}} c\_{\mathbf{k}\sigma}^{\dagger} c\_{\mathbf{k}\sigma} + \mathcal{W}(\varepsilon\_{\mathbf{d}}, E\_{\mathbf{c}\prime}, \dots) \sum\_{\mathbf{k}\mathbf{k}\prime\sigma} c\_{\mathbf{k}\sigma}^{\dagger} c\_{\mathbf{k}\prime\sigma} + \mathcal{O}\left(\frac{\varepsilon}{E\_{\rm FL}}\right)^2 \tag{8}$$

where the scattering potential leads, as shown in Appendix A, to the quasi-particle phase shift:

$$\delta\_{\mathcal{W}} = -\arctan\left(\pi\nu\_0\mathcal{W}\right) \,,\tag{9}$$

in which *ν*0 is the density of states of the lead electrons at the Fermi energy, see also Equation (A21) in Appendix A for its rigorous definition. In this *Local Fermi Liquid* Hamiltonian, *σ* labels either a spin polarization or a channel. The number of channels in the lead can be controlled by the opening of a quantum point contact [84]. The potential strength *W* can be cumbersome to compute, but it is nevertheless related to the occupancy of the quantum dot via the Friedel sum rule, as explained in Section 3.2.

The simplicity of the local Fermi liquid Hamiltonian (8) makes it powerful to evaluate low energy properties. Being non-interacting, it also includes the restoration of phase coherence in the scattering of electrons seen in Section 2. An important assumption that we made is that the system exhibits a Fermi liquid ground state, or Fermi liquid fixed point in the language of the renormalization group. Non-Fermi liquid fixed points exist and cannot be described by such Hamiltonian [85], but they are generally fine-tuned and unstable with respect to perturbations. Furthermore, Equation (8) is not applicable to genuine out-of-equilibrium regimes, when the perturbations are too strong or vary too fast with respect to the Fermi liquid energy scale *E*FL (typically of the order of the charging energy *E*c).

### *3.2. The Role of the Friedel Sum Rule in Local Fermi Liquid Theories*

In the process of electron backscattering by the quantum dot, or by the interacting central region, the phase shift relates the incoming and outgoing electronic wavefunctions <sup>Ψ</sup>L(<sup>0</sup>−) = *<sup>e</sup>*2*iδε*ΨR(<sup>0</sup>−), see Appendix A for an explicit illustration on the resonant level non-interacting model. The Friedel sum rule [86] establishes the relation between the average charge occupation of the dot *N* and this phase-shift *δ*. Its form in the case of *M* conducting channels reads:

$$
\langle N \rangle = \frac{1}{\pi} \sum\_{\sigma=1}^{M} \delta\_{\sigma} \,. \tag{10}
$$

The Friedel sum rule has been proven rigorously for interacting models [87,88]. It is valid as long the ground state has a Fermi liquid character. Physically, it can be understood in the following way: The derivative of the phase shift *δε* with respect to energy defines (up to *h*¯) the Wigner-Smith scattering time [89], see Equation (A38), that is the time delay experienced by a scattered electron. In the presence of a continuous flow of electrons, a time delay implies that some fraction of the electronic charge has been (pumped) deposited in the quantum dot [51,90]. Therefore the phase shift amounts to a left-over charge and it does not matter that electrons are interacting on the quantum dot as long as they are not in the leads, which is the essence of the local Fermi liquid approach.

The Friedel sum rule (10) combined with Equation (9) relates the dot occupancy to the potential scattering strength. For the single-channel case ( *M* = 1), one finds:

$$
\langle N \rangle = -\frac{1}{\pi} \arctan \left( \pi \nu\_0 \mathcal{W} \right) \,. \tag{11}
$$

This is an important result because the dot occupation *N* is a thermodynamic quantity, which can be also accessed in interacting models, allowing us to address quantitatively the close-to-equilibrium dynamics of driven settings, as we will discuss in Section 4.5.

We emphasize that the local Fermi liquid approach of Equation (8) can be extended to perturbatively include inelastic scattering and higher-order energy corrections and relate these terms to thermodynamic observables. This program has been realized in detail for the Anderson and Kondo models [57–64,91].

### *3.3. Derivation of the LFL Theory in the Coulomb Blockade and Anderson Model*

We show now how the effective theory (8) can be explicitly derived from realistic models describing Coulomb blockaded quantum dot devices [66]. We focus on the Coulomb Blockade Model (CBM) [2,3]:

$$\mathcal{H}\_{\rm CBM} = \sum\_{k} \varepsilon\_{k} c\_{k}^{\dagger} c\_{k} + t \sum\_{k,l} \left[ c\_{k}^{\dagger} d\_{l} + d\_{l}^{\dagger} c\_{k} \right] + \sum\_{l} (\varepsilon\_{d} + \varepsilon\_{l}) d\_{l}^{\dagger} d\_{l} + E\_{\rm c} \left( N - \mathcal{N}\_{\rm g} \right)^{2} \tag{12}$$

and the Anderson Impurity Model (AIM), which, in its standard form, reads [92,93]:

$$\mathcal{H}\_{\text{AIM}} = \sum\_{\mathbf{k}, \sigma} \varepsilon\_{\mathbf{k}, \sigma} c\_{\mathbf{k}, \sigma}^{\dagger} c\_{\mathbf{k}, \sigma} + t \sum\_{\mathbf{k}, \sigma} \left[ c\_{\mathbf{k}, \sigma}^{\dagger} d\_{\sigma} + d\_{\sigma}^{\dagger} c\_{\mathbf{k}, \sigma} \right] + \varepsilon\_{\mathbf{d}} \sum\_{\sigma} d\_{\sigma}^{\dagger} d\_{\sigma} + l \ln \gamma\_{\downarrow} n\_{\downarrow} \,. \tag{13}$$

Adding <sup>−</sup>*eV*g*<sup>N</sup>* + *E*cN 2 g to the AIM and for *U* = *E*c, the charging energy (1) becomes apparent in Equation (13), as in Equations (4)–(12). The CBM coincides with the Hamiltonian (4) and describes the mesoscopic capacitor in the Quantum Hall regime: A reservoir of spinless fermions *ck* of momentum *k* is tunnel coupled to an island with discrete spectrum *εl*. The AIM includes the spin degree of freedom and considers a single interacting level in the quantum dot. This model encompasses Kondo correlated regimes [49,94] and describes well the experiments [95,96].

To derive the LFL Hamiltonian (8), we rely on the Schrieffer–Wolff (SW) transformation [44,97], first devised to map the AIM [94] onto the Coqblin–Schrieffer model [98], and that we extend here to the CBM. Far from the charge degeneracy points, in the *t E*c limit, the ground-state charge configuration *n* = *N* is fixed by the gate potential *V*g and fluctuations to *n* ± 1 require energies of order *E*c. For temperatures much lower than *E*c, the charge degree of freedom of the quantum dot is frozen, acting but virtually on the low energy behavior of the system. The SW transformation is a controlled procedure to diagonalize perturbatively in *t* the Hamiltonian. The Hamiltonian is separated in two parts H = H0 + Hres−dot, in which H0 is diagonal in the charge sectors labeled by the eigenvalues *n* of the dot occupation *N*, which are mixed by the tunneling Hamiltonian Hres−dot, involving the tunneling amplitude *t*. The perturbative diagonalization consists of finding the Hermitian operator *S* (of order *t*) generating the unitary *U* = *eiS* rotating the Hamiltonian in the diagonal form H = *U*†H *U*. To leading order in *S* we find:

$$
\mathcal{H}' = \mathcal{H}\_0 + \mathcal{H}\_{\text{res}-\text{det}} + \mathrm{i} \left[ \mathcal{S}, \mathcal{H}\_0 \right] + O \left( t^2 \right). \tag{14}
$$

This Hamiltonian is block diagonal if the condition:

$$i\mathcal{H}\_{\text{res}-\text{det}} = [\mathcal{S}, \mathcal{H}\_0] \tag{15}$$

is fulfilled and Equation (14) becomes:

$$\mathcal{H}' = \mathcal{H}\_0 + \frac{i}{2} \left[ \mathcal{S}, \mathcal{H}\_{\text{res}-\text{det}} \right],\tag{16}$$

which is then projected on separated charge sectors.
