5 uation where the spectral function of the embedded dot has a significant variation on the energy scale of Γ a   which affects the resonance line-shape for large E . It should also be noted that parameters a and b have “unphysical” values a > 1 and b < 0 for small T . If the curve fitting is performed in a nar- rower energy interval [ − Γ a : +Γ a ]   the asymptotic small- E form of the resonance can be well captured  however the ex- tracted Fano parameters are factitious and therefore of little use. This demonstrates that in the presence of strong com- petition between the Fano interference and Kondo effect  the curve fitting using the Fano line-shape is not recommended since the results depend very strongly on the energy window where fitting is performed and it is thus not advisable to make any inference based on them. tance curves for U d = 0 and U d /D = 1 at t/D = 0 . 0001   where Γ a /T 0 K = 1 . 7 × 10 − 3 since the Kondo temperature (Wilson’s definition) is T 0 K = 9 . 7 × 10 − $^{5}$D . In this regime the Kondo effect on the embedded dot in no way affects the Fano interference process in the relevant range of energies and temperatures (i.e. several times Γ a ). Only for very large ϵ a and very high temperature (of the order of T 0 $_{K}$) will the differ- ences become apparent  however this is outside the parameter regime of interest here. When Γ a is equal to a considerable fraction of T 0 $_{K}$  we start to see small quantitative departure from the Fano line shape already for ϵ a and T of the order of Γ a   see Fig. 4  middle panel. This is clearly a consequence of the competition be- tween correlation and interference effects  combined with fur- ther thermal effects. As expected  the discrepancy grows with increasing ϵ a and T . Finally  for Γ a ≫ T 0 $_{K}$  the embedded dot is strongly per- turbed by the coupling to the side-dot and the differences become drastic: at finite T   the line-shape differs qualita- tively from the Fano form and  in particular  we observe the e m ergence of non-uniform E -dependence with broad humps ar o und E ∼ Γ a . -4 -2 0 2 4 E= $_{a}$/ a 0 0.2 0.4 0.6 0.8 1 G(E T) 0 0.5 1.0 1.5 0 0.5 1 T/ a 0 0.25 0.5 0.75 1 $_{/}$Γ a -0.2 0 0.2 0.4 0.6 0.8 1 1.2 a  b $_{a}$=t$^{2}$/ d T/ a a b / a t/D=0.006 $_{d}$/D=0.06 $_{a}$/D=1.67 10 -5 a) b) U$_{d}$/D=1 $_{a}$/T$_{K}$$^{0}$=6.1 0 0.2 0.4 0.6 0.8 1 G(E T) 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 G(E T) U=0 U/D=1 -4 -2 0 2 4 E= $_{a}$/ a 0 0.2 0.4 0.6 0.8 G(E T) T/ a t/D=0.0001 t/D=0.001 t/D=0.01 $_{a}$/T$_{K}$$^{0}$=0.0017 $_{a}$/T$_{K}$$^{0}$=0.17 $_{a}$/T$_{K}$$^{0}$=17 Figure 5: (Color online) Interacting model with U d = 0 and U a = 0 . Symmetric case  ϵ d = 0 . a) Conductance curves for a range of temperatures. b) Fano parameters as a function of the temperature. Curve fitting is performed in the energy window which corresp onds to the horizontal axis in the upper subfigure. Since there are two different energy scales in the problem ( T 0 K and Γ a )  the temperature-dependence of the conductance is expected to be non-monotonic  as shown in Fig. 6 where we plot the conductance at the bottom of the Fano anti-resonance (i.e. for ϵ a = 0 ). Such dependence is a consequence of the competition between the Kondo and Fano effects. The Kondo effect tends to increase the conductance through the formation of many-particle resonance at the Fermi level which opens a new conduction channel through the system. On the other hand  the Fano effect suppresses the conductance through quantum interference. For T 0 K ≫ Γ a   the conductance first increases at the higher temperature scale of T 0 K (the tempera- ture dependence being given by the universal Kondo conduc- tance curve$^{71}$)  then it decreases at the lower temperature scale set by Γ a : see results for t/D = 0 . 0001 and t/D = 0 . 001 in Fig. 6. Note that the unitary limit of full conductance quantum It is instructive to perform curve fitting on the results wher e the Fano resonance is already strongly perturbed  as shown in Fig. 5. The fits (shown using dashed lines)  performed in the energy interval [ − 4Γ a : +4Γ a ]   are clearly inade- quate. Extracted Fano parameters indicate that the resonance width is significantly reduced below Γ a and its width even decreases with increasing temperature  which reflects the s it- Figure 4: (Color online) Resonance curves for non-interacting (full lines) and interacting models (dashed lines) for a range of scaled tem- peratures T/ Γ a and for different Γ a /T 0 K ratios.