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Nonequilibrium self-energies, Ng approach and heat current of a nanodevice for small bias voltage and temperature

Using non-equilibrium renormalized perturbation theory to second order in the renormalized Coulomb repulsion, we calculate the lesser $Σ^<$ and and greater $Σ^>$ self-energies of the impurity Anderson model, which describes the current through a quantum dot, in the general asymmetric case. While in general a numerical integration is required to evaluate the perturbative result, we derive an analytical approximation for small frequency $ω$, bias voltage $V$ and temperature $T$ which is exact to total second order in these quantities. The approximation is valid when the corresponding energies $\hbar ω$, $eV$ and $k_B T$ are small compared to $k_B T_K$, where $T_K$ is the Kondo temperature. The result of the numerical integration is compared with the analytical one and with Ng approximation, in which $Σ^<$ and $Σ^>$ are assumed proportional to the retarded self-energy $Σ^r$ times an average Fermi function. While it fails at $T=0$ for $\hbar|ω| \lesssim eV$ we find that the Ng approximation is excellent for $k_B T > eV/2$ and improves for asymmetric coupling to the leads. Even at $T=0$, the effect of the Ng approximation on the total occupation at the dot is very small. The dependence on $ω$ and $V$ are discussed in comparison with a Ward identity that is fulfilled by the three approaches. We also calculate the heat currents between the dot and any of the leads at finite bias voltage. One of the heat currents changes sign with the applied bias voltage at finite temperature.