4 -4 -2 0 2 4 E= $_{a}$/ a 0 0.2 0.4 0.6 0.8 1 G(E,T) 0 0.25 0.5 0.75 1.0 1.25 1.5 1.75 0 0.5 1 T/ a 1 1.5 2 2.5 $_{/}$Γ a 0 0.2 0.4 0.6 0.8 1 a, b, q,  r $_{/}$Γ a $_{a}$=(4/5)t$^{2}$/ d T/ a a b / a t/D=0.001 $_{d}$/D=0.06 $_{a}$/D=1.33 10 -5 a) b) q $_{d}$/ $_{d}$=0.5 $_{r}$/ a B. Asymmetric case, ϵ d = 0 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 : Γ a ≡ γ a (0) = t 2 Γ d 1 ( ϵ d / Γ d ) 2 + 1 . (13) ϵ r = ϵ d , Γ = Γ a , q = ϵ d / Γ d , a = 1 1 + ( ϵ d / Γ d ) $_{2}$, b = 0 . (14) Figure 3: (Color online) Non-interacting model with U a = U d = 0 , asymmetric case, ϵ d = 0 . a) Conductance curves for a range of tem- peratures. Full curves: NRG results. Dashed lines: Fano resonance line fits. b) Fano parameters as a function of the temperature . Pa- rameters are extracted by curve fitting in the energy window w hich corresponds to the horizontal axis in the upper subfigure. 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). (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- 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. 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 IV. INTERACTING CASE WITH U d = 0 AND U a = 0