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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. 

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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 

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. 

## VI. RELATION TO THE EXPERIMENTAL RESULTS 

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. 

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 

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 

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