7 ca se, the differences are even qualitative: while G ( T ) exhibits two ranges of logarithmic behavior (the first Kondo screenin g st a ge from 10 − 4 < T < 10 − $^{2}$, and the second Kondo screen- in g stage from 10 − 6 < T < 10 − $^{4}$), the zero-temperature sp ectral function is more complex and even attains a minimum at ω ≈ 2 × 10 − 6 before increasing back to its asymptotic value at the Fermi level. This complex behavior is not reflected in th e temperature dependence of the transport properties. Ad- di tional calculations (results not shown) indicate that G ( T ) an d A d ( ω, T = 0) are very similar only in vicinity of the pa rticle-hole symmetric point, and even there only when the two Kondo temperature scales are well separated. This should se rve as a caveat: since G ( T ) is a convolution of the finite- te mperature spectral function, see Eq. (6), it is not expected th at, in general, all the details of the energy-dependence of th e zero-temperature spectral function will manifest as similar va riations in the temperature-dependence of the conductance at the related temperatures. 0.2 0.4 0.6 0.8 1 G(E,T=0) 0 10 -5 10 -4 10 -3 10 -2 10 -1 1 -4 -2 0 2 4 E= $_{a}$/ a 0 0.5 1 1.5 2 n a -2 0 2 4 E= $_{a}$/ a t/D=0.001 U$_{d}$= $_{d}$=0 $_{a}$=1.67 10 -5 U$_{d}$/D=1 $_{d}$=0 U$_{a}$/D VI. RELATION TO THE EXPERIMENTAL RESULTS Figure 8: (Color online) Conductance curves and level-occupancies at T = 0 for a range of electron-electron interaction strengths U a . Left panels: non-interacting embedded impurity. Right panels: in- teracting embedded impurity. The arrows show the direction of in- creasing U a . Since the experiment is performed at finite temperatures, the results for the zero-temperature conductance can be very misleading in problems with very low energy scales$^{50}$. This commonly occurs in the side-coupling geometry and in clusters of quantum dot with weak inter-dot tunnel coupling$^{27,28,48,50}$. In Fig. 10a) we thus plot the conductance for a range of finite temperatures, which reveals completely different transport properties when compared with the zero- temperature limit. U a occurs for rather small values of U a , therefore it is not ap- propriate to model the side-coupled dot as a non-interacting impurity, since it is unlikely that the electron-electron repul- sion in two similar quantum dots would differ by orders of the magnitude. We now face a dilemma: assuming a non- interacting side-coupled dot, we obtain conductance curves which appear to agree with the experimental results (albeit we know that they correspond to a change of occupancy by 2, in contradiction with the experimental situation), while assum- ing an interacting side-coupled dot, we obtain very different results which are in strong disagreement with those actually obse rved. When both quantum dots are interacting and the parameters are set in such a way that local moments form on both dots, there is a possibility that the Kondo screening will proceed in two steps$^{26,28,29,48,50,52,73}$: at higher Kondo temperature T 0 $_{K}$, the moment on the first dot will be screened, while another stage of Kondo screening of the moment on the side-coupled second dot occurs at another much lower energy scale T 2 K $^{28,73}$: where c 1 and c 2 are some constants of the order 1. Such two- stageKondoscreeningoccurs if the effective Kondo exchange coupling J eff of the spin on the second dot with the quasipar- ticles resulting from the first stage is lower than T 0 $_{K}$. In this case, the second Kondo scale may be pushed to exponentially low temperature T 2 $_{K}$. The exchange coupling T 2 K = c 1 T 0 K exp ( − c 2 T 0 $_{K}$/J eff ) , (15) J eff = 2 t 2 ϵ a − ϵ d + U a + 2 t 2 ϵ d − ϵ a + U d (16) 10 -8 10 -6 10 -4 10 -2 10 0 T/D,  /D 0 0.2 0.4 0.6 0.8 1 G(T), $^{πΓ}$$^{d}$ A d ( ,T=0) G(T) A( ,T=0) $_{d}$=0, E= $_{a}$/ $_{a}$=-2 U$_{a}$=U$_{d}$=1 t/D=0.001 depends on the value of ϵ a and decreases as ϵ a is reduced, attaining its minimum for ϵ a = δ d − U a / 2 . In turn, this im- plies that T 2 K becomes very low for small ϵ a , thus at the ex- perimentally relevant temperatures the second stage of Kondo screening does not occur, the conductance suppression does not arise and the conductance curve takes a form similar to that of the Fano resonance. In experiments, the temperature is roughly one order of the magnitude lower than the Kondo In Fig. 9 we compare the temperature dependence G ( T ) with the energy-dependence of the T = 0 spectral function A d ( ω, T = 0) . As before (cf. Fig. 7 and the related dis- cussion), we observe that in spite of a certain general simi- larity between the curves, the differences are notable. In this Figure 9: (Color online) Comparison of the temperature-dependence of the conductance G ( T ) with the energy-dependence of the zero- temperature spectral function on the embedded dot A d ( ω,T = 0) .