3.3.1. Coulomb Blockade Model

To derive the effective low-energy form of the CBM model, it is useful, following Grabert [99,100], to decouple the charge occupancy of the dot from the fermionic degree of freedom of the electrons. This is achieved by adding the operator *n*ˆ = ∑*n* |*nn*|, measuring to the dot occupation number. The fermionic operators *dl* in Equation (12) are replaced by new operators describing a non-interacting electron gas in the dot. The Hamiltonian (12) acquires then the form:

$$\mathcal{H}\_{\rm CRM} = \sum\_{k} \varepsilon\_{k} c\_{k}^{\dagger} c\_{k} + t \sum\_{n,k,l} \left[ d\_{l}^{\dagger} c\_{k} \left| n+1 \right> \left< n \right| + \text{h.c.} \right] + \sum\_{l} \varepsilon\_{l} d\_{l}^{\dagger} d\_{l} + \varepsilon\_{d} \hbar + E\_{\rm c} \left( \hbar - \mathcal{N}\_{\rm g} \right)^{2}. \tag{17}$$

The operator *S* = *s* + *s*† fulfilling the condition (15) reads:

$$s = it \sum\_{k,l,n} s\_{kln} c\_k^\dagger d\_l \left| n - 1 \right\rangle \left\langle n \right|, \tag{18} \\ s\_{kln} = \frac{1}{\varepsilon\_l - \varepsilon\_k + E\_\varepsilon(2n - 1) + \varepsilon\_d}. \tag{18}$$

This operator, when inserted into Equation (16), also generates higher order couplings between sectors of charge *n* and *n* ± 2, which we neglect in the present discussion. The Hamiltonian becomes then block diagonal in the sectors given by different values of *n*. For (Ng − *<sup>ε</sup>dC*g/*e*) ∈ [−1/2, 1/2], the lowest energy sector corresponds to *n* = 0 and the effective Hamiltonian reads <sup>H</sup>CBM = H0 + HB, with:

$$\mathcal{H}\_{\rm B} = \frac{\hbar^2}{2} \sum\_{\mathbf{k}\mathbf{k}'\mathbf{l}\mathbf{l}'} \left( s\_{\mathbf{k}l0} d\_{l'}^{\dagger} c\_{k'} c\_k^{\dagger} d\_l - s\_{\mathbf{k}l1} c\_k^{\dagger} d\_l d\_{l'}^{\dagger} c\_{k'} + \text{h.c.} \right). \tag{19}$$

This interaction can be simplified by a mean-field treatment:

$$d\_l^\dagger c\_k c\_{k'}^\dagger d\_{l'} = \left\langle d\_l^\dagger d\_{l'} \right\rangle c\_k c\_{k'}^\dagger + \left\langle c\_k c\_{k'}^\dagger \right\rangle d\_l^\dagger d\_{l'} = \delta\_{ll'} \theta(-\varepsilon\_l) c\_k c\_{k'}^\dagger + \delta\_{kk'} \theta(\varepsilon\_k) d\_{l}^\dagger d\_{l'},\tag{20}$$

allowing to carry out part of the sums in Equation (19). Notice that the orbital energy *εd* does not appear in Equation (20) as it is now only associated to the charge degree of freedom *n*, while the Fermi gases corresponding to *ck* and *dl* have the same Fermi energy *EF* = 0. One thus finds the effective low energy model, which, to leading order, reads [66]:

$$\mathcal{H}'\_{\rm CEM} = \mathcal{H}\_0 + \frac{g}{\nu\_0} \ln \left( \frac{E\_c - \varepsilon\_d}{E\_c + \varepsilon\_d} \right) \left[ \sum\_{ll'} d\_l^{\dagger} d\_{l'} - \sum\_{kk'} c\_k^{\dagger} c\_{k'} \right] \tag{21}$$

in which we have introduced the dimensionless conductance *g* = (*<sup>ν</sup>*0*<sup>t</sup>*)2, corresponding to the conductance of the Quantum Point Contact (QPC) connecting dot and lead in units of *<sup>e</sup>*2/*h*. This Hamiltonian describes two decoupled Fermi gases, but affected by potential scattering with opposite amplitudes. Equation (21) coincides with the LFL Hamiltonian (8) for the lead electrons. The phase-shift *δW* (9) allows for the calculation of the charge occupation of the dot to leading order by applying the Friedel sum rule (10):

$$
\langle N \rangle = \frac{\delta\_W}{\pi} = \lg \ln \left( \frac{E\_\varepsilon - \varepsilon\_d}{E\_\varepsilon + \varepsilon\_d} \right) . \tag{22}
$$

This result reproduces the direct calculation of the dot occupation [99–101], showing the validity of the LFL model (8), with Friedel sum rule for the CBM. The extension to *M* channels is obtained by replacing *g* → *<sup>M</sup>*(*<sup>ν</sup>*0*<sup>t</sup>*)<sup>2</sup> in Equation (22). The extended proof to next-to-leading order in *g* is given in Ref. [66].
