0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 1.0 G AP [(1 − p 2 )2 e 2 /h ] J 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 0.0 0.2 0.4 0.6 0.8  J=0  J=0.18  J=0.21  J=0.218  J=0.2204  J=0.221  J=0.22138  J=0.2218  J=0.223  J=0.228  J=0.24  J=0.26 0.0 0.1 0.2 0.3 0.4 0.5  J=0  J=0.2  J=0.22  J=0.226  J=0.228  J=0.2286  J=0.22878  J=0.229  J=0.23  J=0.232  J=0.24  J=0.26 0.0 0.1 0.2 0.3  J=0  J=0.18  J=0.21  J=0.218  J=0.2204  J=0.221  J=0.22138  J=0.2218  J=0.223  J=0.228  J=0.24  J=0.26 c b a A AP σ /A 0 A P ↑ /A 0 A P ↓ /A 0 ω/T$_{K}$ FIG. 3: The linear conductance as a function of exchange cou- pling constant J for the antiparallel magnetic configuration. The conductance was determined from the spectral function shown in Fig. 2(a). The parameters are the same as in Fig. 2 and J is in units of D = 1. FIG. 2: (color online) The spectral function of the d -level operator in the antiparallel (a) and parallel (b,c) magnetic configurations for the symmetric Anderson model and for dif- ferent values of the exchange coupling J . The parameters are: ε d = − 0 . 05, U = 0 . 1, Γ = 0 . 0077, p = 0 . 4, and T = 0. The Kondo temperature is defined as a half-width of the spec- tral function for J = 0 and p = 0, T K = 2 . 5 × 10 − $^{4}$, while A 0 = ∑$_{σ}$ A σ ( ω = 0) for J = 0 and p = 0. All the parameters are given in the units of D ≡ 1. A. Symmetric Anderson model For the symmetric Anderson model we assume the fol- lowing parameters (in the units of D ), ε$_{d}$ = − 0 . 05 and U = 0 . 1. The zero-temperature spin-dependent spectral function A$_{σ}$ , normalized to A$_{0}$ , with A$_{0}$ = ∑ $_{σ}$A$_{σ}$ ( ω = 0) taken for J = 0 and p = 0, is shown in Fig. 2 for both an- tiparallel (a) and parallel (b,c) magnetic configurations (note the logarithmic energy scale), and for indicated val- ues of the exchange coupling parameter J . The spectral function is plotted as a function of ω/T$_{K}$ , where T$_{K}$ is the The situation changes when the electron in the dot is additionally exchange coupled to the nonmagnetic reser- voir. When the coupling is antiferromagnetic and the coupling parameter J increases, the width of the Kondo peak becomes gradually narrower and narrower. The hight of the peak, however, remains unchanged, as can be clearly seen in Fig. 2 for some small values of the ex- change coupling constant. In order to see this behavior Let us consider first the situation with vanishing ex- change coupling of the dot to the nonmagnetic reservoir, J = 0. For ω < T$_{K}$ , a Kondo peak develops in the dot spectral function due to screening of the dot’s spin by conduction electrons of ferromagnetic leads, which leads to the formation of a non-local spin singlet. The height of the Kondo peak is independent of spin in the antipar- allel configuration and depends on spin in the parallel one. Apart from this, a Hubbard peak corresponding to ε$_{d}$ + U is visible in the spectral function shown in Fig. 2. This behavior of the Kondo phenomenon in the presence of ferromagnetic leads is in agreement with that found by other methods, for instance by the equation of mo- tion for the Green functions 37 and also by the real-time diagrammatic technique. 38 Kondo temperature defined as a half-width of the d -level spectral function for J = 0 and p = 0, T$_{K}$ = 2 . 5 × 10 − $^{4}$. It can be seen that in the antiparallel configuration the spectral function is independent of the spin orientation [Fig. 2(a)], A AP ↑ = A AP ↓ , while it depends on electron spin in the parallel magnetic configuration, A P ↑ = A P $_{↓}$, see Fig. 2(b) and (c). Note, that for symmetric Ander- son model the spectral function is symmetric with respect to ω = 0, therefore here it is shown only for positive en- ergies, i.e. for energies above the Fermi level. Moreover, note also that the spectral functions are normalized to that for the corresponding paramagnetic limit ( p = 0 and J = 0), so A$_{↑}$ ( ω = 0) + A$_{↓}$ ( ω = 0) = A$_{0}$ in the parallel configuration. 4