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theoretical works consider the zero-bias conductance in the zero-temperature limit, while the experiment is decidedly per- formed at finite temperatures. It is known that this system is characterized by very low energy scales in some parame- ter regimes (in particular in the case of the two-stage Kondo effect$^{27,28,35}$), thus in order to properly describe its transport properties it is imperative to consider thermal effects and to calculate the conductance at finite temperatures using a non - perturbative method. This is the goal of the present work. 

This paper is structured as follows. In Sec. II the model and the numerical renormalization group method are presented. The thermal effects are first studied in the non-interacting model in Sec. III, where we show that at finite temperatures the resonance line shape still has the Fano form to a good ap- proximation, albeit with temperature-dependent parameters. In Sec. IV we then focus on the Fano-Kondo effect in the case where only the directly embedded quantum dot is interacting and experiences the Kondo screening; in this case the reso- nance line shape itself is appreciably modified by the com- bined Fano interference and Kondo effect. In Sec. V we finally study the fully interacting case, where completely different behavior is found. Finally, in Sec.VI the recent experimental results are examined and compared to theory, leading to the conclusion that the observed line-shape (an anti-resonance) is indicative of the occurrence of the two-stage Kondo effect. 

The Hamiltonian under study takes the following form: 

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where n d = ∑ σ d † σ d σ and n a = ∑ σ a † σ a σ . The operators d † σ and a † σ are creation operators for an electron with spin σ on the embedded quantum dot d and on the side-coupled dot a , see Fig. 1. The on-site electron-electron repulsion is denoted by U α , where α stands for d or a . The on-site energies ϵ α of the dots are shifted by U α / 2 in order to define the parameters 

which measure the detuning from the particle-hole symmetric case which corresponds to δ a = δ d = 0 . The coupling be- tween the dots is described by the interdot tunnel coupling t . The dot d is assumed to couple to the source and drain elec- trodes in a symmetric way, thus it hybridizes with the even- parity combination of electrons from both leads 16,36 with the strength 

which will be assumed to be constant for all energies within the conduction band of half-width D (equivalently, the den- 

## II. MODEL AND METHOD 

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sity of states in the conduction band, ρ , is assumed to be con- stant, ρ = 1 / 2 D ). Throughout this work, Γ d will be fixed to Γ d /D = 0 . 06 . 

For the correct description of correlated regimes in clus- ters of quantum dots 30,37–51,51–53 it is necessary to use non- perturbative techniques that take properly into account not only the local correlation effects on the impurity sites, but also the charge fluctuations and inter-site spin-spin coupling i n- duced by the exchange interaction. One such technique is the numerical renormalization group (NRG)$^{54–69}$which involves three steps: 1) logarithmic discretization of the conduction band, 2) mapping onto a one-dimensional tight-binding chain with exponentially decreasing hopping constants, and 3) itera- tive diagonalization of the resulting Hamiltonian. The results presented in this work have been calculated using the NRG discretization parameter Λ = 4 with N z = 8 equidistant val- ues of the twist parameter z using an improved discretization scheme$^{54,70}$, with the truncation cutoff set at 10 ω N . The ex- pectation values are calculated in the standard way, while the conductance curves at finite temperatures are computed usin g the Meir-Wingreen formula from the spectral data$^{61,63,71}$: 

where G 0 = 2 e $^{2}$/h is the conductance quantum (i.e., the conductance corresponding to full transmission), f ( ω ) = [1 + exp( βω )] − 1 is the Fermi function where β = 1 /k B T and the chemical potential has been fixed at zero energy, and A d ( ω, T ) is the spectral function on the impurity site d . 

It should be noted that there is a significant conceptual dif- ference as regards the usage of the Fano formula in the context of scattering experiments in atomic physics and the present context of nanoscopic transport. In atomic physics, the res- onance is observed in the scattering cross-section as a func- tion of the energy of the incoming particle while the reso- nance energy is fixed. In transport experiments, the resonan ce is observed in the conductance which relates to the scatter- ing of electrons at the fixed energy of the Fermi level (this is strictly true only at zero temperature; at finite temperatur es, electrons within an energy window of order k B T participate in the transport) and the independent variable is the energy of the side-coupled dot which is swept using gate voltages. In spite of the seemingly symmetric roles of parameters ϵ and ϵ r in the expressions in Eqs. (1) and (2), there is a notable difference as regards the role of the background channel. The background scattering phase shift namely depends on the vari- able parameter in the first case, while it is essentially cons tant in the second case (at zero temperature). There may be some small variation of the background scattering phase shift when the parameters of the side-coupled dot are changed, since the two impurities are coupled and there will be hybridization ef- fects induced by the side-coupled dot on the embedded dot. Nevertheless, this effect is rather small. We therefore con- clude that any significant deviation from the universal Fano line-shape profile may only result from electron correlatio n effects or from thermal effects. 

Furthermore, it should be remarked that the approach used in this work does not take into account that at high tempera- 

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