3.3.2. Anderson Impurity Model

Considering the internal spin degree of freedom in the AIM (13), it does not fundamentally affect the effective LFL behavior. Nevertheless, in the case where a single electron is trapped in the quantum dot ( − *U εd* 0), the derivation of Equation (8) is more involved, and sketched in Figure 5. The SW transformation maps the AIM onto a Kondo Hamiltonian including a potential scattering term [94]:

$$\mathcal{H}'\_{\rm AM} = \mathcal{H}\_0 + f\mathbf{S} \cdot \mathbf{s} + W \sum\_{kk'\sigma} \mathbf{c}\_{kr}^\dagger \mathbf{c}\_{k'\sigma} \,. \tag{23}$$

The spin of the electron in the quantum dot **S** is coupled anti-ferromagnetically to the local spin of the lead electrons **s** = ∑*kkττ c*† *kτ σττ* 2 *ckτ* , with *σττ* the vector composed of the Pauli matrices, and

*J* = 2Γ *πν*0 1 *εd* + *U* − 1 *εd* , *W* = − Γ 2*πν*0 1 *εd* + *U* + 1 *εd* , (24)

in which we introduced the hybridization energy Γ = *πν*0*t* 2, corresponding to the width acquired by the orbital level when coupled to the lead and which depends on the density of states of the lead electrons at the Fermi energy *ν*0, see also Equation (A21) in Appendix A for its rigorous definition. Neglecting for the moment the Kondo anti-ferromagnetic coupling controlled by *J*, the LFL Hamiltonian (8) is directly recovered. Nevertheless, the potential scattering term is absent ( *W* = 0) at the particle-hole symmetric point *εd* = − *U*/2. At this point, the charge on the dot is fixed to one by symmetry, and the absence of potential scattering allows to derive various rigorous results, for instance concerning the ground state properties relying on Bethe ansatz [102,103]. It is a well-established fact that the system described by Equation (23) behaves as a LFL at low energies [48,104,105] and that the Friedel sum rule applies [87]. As a consequence, the Kondo coupling is responsible for the phase-shift of the low energy quasi-particles. Particle-hole symmetry, spin degeneracy, and Friedel sum rule fix the Kondo phase-shift to *δK* = *π*/2. The Friedel sum rule states that:

$$
\langle N \rangle = 2 \frac{\delta\_{\mathbf{K}}}{\pi} \,' \tag{25}
$$

*N* = 1 because of particle-hole symmetry and the factor 2 signals spin degeneracy, fixing *δ*K = *π*/2. The detailed description of the Kondo effect is far beyond the scope of this review and we direct the interested reader to Ref. [49] for a comprehensive review and to Refs. [106–111] for the description of the low energy fixed point relying on boundary conformal field theory. For the scopes of this review it is enough to mention that below the Kondo temperature: [102,103,112,113]

$$T\_{\mathbf{K}} = \frac{e^{\frac{1}{4}}\gamma}{2\pi} \sqrt{\frac{2LI\Gamma}{\pi}} e^{\frac{\pi e\_d(e\_d+lI)}{2LI\Gamma}}\,\mathrm{},\tag{26}$$

the spin-exchange coupling *J* in Equation (23) flows to infinity in the renormalization group sense. The relevance of this interaction brings the itinerant electrons to screen the local spin-degree of freedom

of the quantum dot and phase-shifts the resulting quasi-particles by *δ*K, see Figure 5. The phase-shift *π*/2 acquires thus a simple interpretation in one dimension [44,106]: Writing ΨR = *<sup>e</sup>*2*iδ*ΨL for a given spin channel at the impurity site, *δ* = *δ*K = *π*/2 leads to ΨR + ΨL = 0. The fact that the wave-function is zero at the impurity site, corresponds to the situation in which an electron screens the impurity spin, leading to Pauli blockade (due to Pauli principle), thus preventing other electrons with the same spin to access the impurity site. This dynamical screening of the impurity spin forms the so called "Kondo cloud" [114–117], see Figure 5. It is responsible for increasing the local density of states and leads to the Abrikosov–Suhl resonance [118], which causes the increase, below *T*K, of the dot conductance in Coulomb blockaded regimes [93,119,120]. Remarkably, the Kondo phase-shift *δ*K = *π*/2 and the Kondo screening cloud have been also directly observed in two recent distinct experiments [121,122]. In Section 4.7, we illustrate how such phenomena also affect the dynamical properties of the mesoscopic capacitor in a non-trivial way.

**Figure 5.** Modification of the physical scenario of Figure 3 in the presence of spin-exchange interactions between the dot and lead electrons in the Anderson Impurity Model (AIM). Spin-exchange interactions trigger the formation of the Kondo singlet below the Kondo temperature *T*K, which is responsible for an additional elastic *δ*K = *π*/2 phase-shift of lead electrons in the effective Local Fermi Liquid (LFL) theory.

It remains to establish the combined role of spin-exchange and potential scattering on the low-energy quasi-particles. Remarkably, the phase-shift *δW*, caused by potential scattering, is additive to *δ*K [123–125]:

$$
\delta = \delta\_\mathsf{K} + \delta\_\mathsf{W}.\tag{27}
$$

and can thus be calculated independently. The validity of the above expression is demonstrated by comparison with the exact Bethe ansatz solution of the AIM [102,103]. Inserting the expression (9) for the phase-shift caused by the potential scattering in the Friedel sum rule one finds:

$$
\langle N \rangle = \frac{2}{\pi} \left[ \delta\_K - \arctan(\pi \nu\_0 \mathcal{W}) \right] = 1 + \frac{\Gamma}{\pi} \left( \frac{1}{\varepsilon\_d + lI} + \frac{1}{\varepsilon\_d} \right) \tag{28}
$$

This expression is consistent with the condition *N* = 1, imposed by particle-hole symmetry, but also with the static charge susceptibility *χ*c, which was derived with the Bethe ansatz [126]:

$$\chi\_{\mathbb{C}} = -\frac{\partial \left< N \right>}{\partial \varepsilon\_d} \Big|\_{\varepsilon\_d = -\frac{\mathcal{U}}{2}} = \frac{8\Gamma}{\pi \mathcal{U}^2} \left( 1 + \frac{12\Gamma}{\pi \mathcal{U}} + \dots \right). \tag{29}$$

The extension of this proof to next-to-leading order in *t*, is given in Refs. [66,68] and it shows how LFL approaches are effective in providing analytic predictions out of particle-hole symmetry as well, extending Bethe ansatz results.

This discussion concludes our demonstration of the persistence of elastic and coherent effects triggered by interactions at equilibrium. Local Fermi liquids provide a general framework to describe interacting and non-interacting systems at low energy, within an effective elastic scattering theory. Nevertheless, it is important to stress that LFL theories can fail in specific cases, such as overscreened Kondo impurities [127,128], and that their validity is limited to close-to-equilibrium/low-energy limits. It is thus expected that interactions become crucial as soon as such systems are driven out of equilibrium. We will illustrate now how the LFL theory allows to describe exotic, but still coherent in nature, dynamical effects in a paradigmatic setup such as the mesoscopic capacitor.
