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For ϵ d ̸ = 0 , the d -level Lorentzian is no longer fixed at the Fermi level, which has a number of significant consequences. The effective level broadening on the side-coupled dot is re- duced and it is given by 

Furthermore, the background phase shift is no longer π/ 2 , thus the asymmetry parameter q is different from 0. From Eq. (9) it is clear that the zero-temperature conductance is al- ways zero at ϵ a = 0 , however the actual resonance energy ϵ r is actually shifted away from zero. In fact, within an error of order t $^{2}$/ Γ 2 $_{d}$, the following Fano parameters hold at T = 0 for general ϵ d : 

The results of a temperature-dependent calculation for ϵ d / Γ d = 0 . 5 are shown in Fig. 3. We observe a departure from the Fano line-shape which grows as the temperature is increased; nevertheless, the departure is small and it is still meaningful to perform curve fitting using the Fano formula. Similar to the symmetric ϵ d = 0 case, the main effect of finite temperature is to increase the width of the Fano resonance be- yond Γ a and to reduce the amplitude of the resonance. The asymmetry parameter q is, however, nearly constant: it re- duces from q = 0 . 5 at zero temperature to q = 0 . 47 at T = Γ a , while in the same temperature interval the amplitude is reduced by half. The resonance position, as given by ϵ r , is also affected rather weakly. Such behavior is characteristic of the t ≪ Γ d limit. For larger t , the extracted Fano parameters q and ϵ r are more strongly temperature dependent, however the deviation from the Fano profile remains small (results no t shown). 

We conclude, fully in line with the expectations, that in the non-interacting case the main effect of finite temperature i s to reduce the amplitude and increase the width of the resonance, while the Fano line-shape is largely preserved. As we show in the following, this is no longer the case in the presence of interactions, except in some special limits. 

## IV. INTERACTING CASE WITH U d ̸ = 0 AND U a = 0 

The occurrence of the Fano resonance in transport prob- lems is always associated with changes of the charge state of the weakly coupled discrete state. For this reason, there is a significant difference between the U a ≪ Γ a and U a ≫ Γ a situations. In the first limit, the occupancy changes by 2 

## B. Asymmetric case, ϵ d ̸ = 0 

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(by one electron for each spin orientation) as ϵ a crosses the Fermi level, thus the phase change in each spin channel is π : this corresponds to the usual Fano-resonance scenario. In the second limit, the total occupancy changes only by one due to the electron-electron repulsion preventing the second elec- tron from entering the quantum dot. This corresponds to a π/ 2 phase shift in each spin channel and, roughly speaking, only one half of the Fano-resonance-like feature is expected to be seen. In this section we discuss the first case, i.e., the U a ≪ Γ a limit, which we study by setting U a to zero. 

When the embedded dot is tuned to its particle-hole sym- metric point ( δ d = 0 ), the impurity spectral function A 0 d is pinned at the Fermi level to the value of 1 /π Γ d irrespective of the value of the electron-electron repulsion U d . This is true even for moderate departures from the particle-hole symmet- ric point, | δ d | ≲ U d / 2 , since the Kondo resonance remains near the Fermi level, unlike the Lorentzian peak in the non- interacting case. For small t , the characteristic energy scale of the side-coupled quantum dot is therefore still Γ a = t $^{2}$/ Γ d as in the symmetric non-interacting case. For larger t the broadening will, however, eventually become comparable to the width of the Kondo resonance (of order T 0 $_{K}$). The behav- ior therefore depends on the ratio between the hybridization of the side-coupled dot Γ a and the Kondo temperature of the decoupled embedded dot, T 0 $_{K}$. If Γ a ≪ T 0 $_{K}$, the physics of the embedded dot will not be affected much by the side-coupled dot (and vice versa), thus the conductance as a function of ϵ a will differ only slightly from the non-interacting case. This is illustrated in Fig. 4, upper panel, where we compare conduc- 

