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derson model discussed above, see Fig. 2. This is because in the antiparallel configuration the resultant coupling to ferromagnetic leads does not depend on spin and the sys- tem effectively behaves as a nonmagnetic one. As before, one observes a quantum phase transition at J = J AP c , where now J AP c ≈ 0 . 1858. The only difference is that for | ε$_{d}$ | ̸ = U/ 2 the spectral function displays an asymmetric behavior with respect to ω = 0, see the inset in Fig. 5. 

The situation, however, changes significantly when the magnetizations of the leads switch to the parallel config- uration. The corresponding spectral function for spin- ↑ and spin- ↓ is shown in Fig. 6. Note, that now the spectral function is shown for both positive and negative energies. As before, let us consider first the case of J = 0. Due to an effective exchange field originating from the presence of ferromagnetic electrodes, the spin degeneracy of the dot level is lifted. At zero temperature, the magnitude of the splitting due to exchange field, Δ ε$_{d}$ , can be estimated from the formula 6,7,10 

For the assumed parameters one then finds, Δ ε$_{d}$ ≈ 2 . 15 × 10 − $^{3}$. The exchange field leads generally to the suppression of the Kondo peak, however the reminiscent of the Kondo effect are still visible as relatively small peaks in the d -level spectral function. The position of these peaks is shifted away from the Fermi level – to positive energies for spin- ↓ and to negative energies for spin- ↑ . In fact, the peaks occur for energies comparable to the magnitude of the exchange field. As can be seen in Fig. 6, they develop at ω/T$_{K}$ ≈ 10 2 for spin- ↓ and at ω/T$_{K}$ ≈ − 10 2 for spin- ↑ components of the spectral function. The other peaks in the spectral functions cor- 

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respond to the dot level ε$_{d}$ and its Coulomb counterpart ε$_{d}$ + U . 

When the coupling parameter J increases, the weak Kondo resonances in the spectral function gradually dis- appear for both spin orientations. The physics behind this disappearance remains similar to that described above, i.e. screening of the dot’s spin by the nonmag- netic reservoir exchange-coupled to the dot. Interest- ingly, there is no quantum phase transition in the case of parallel magnetic configuration shown in Fig. 6. 

Let us now assume that there is an external magnetic field B applied to the dot along the z th direction. In the case of antiparallel configuration, the magnetic field destroys both the Kondo resonance and quantum phase transition with changing J . However, when the leads are aligned in parallel, the Kondo effect is already suppressed by the effective exchange field coming from ferromagnetic electrodes, and one may consider the possibility of restor- ing the Kondo peak by applying an external magnetic field which compensates the effects due to exchange field. In Fig. 7 we show the spectral functions for the parallel magnetic configuration in the case of an asymmetric An- derson model, calculated for three different values of the exchange constant J in the presence of external magnetic field B . The insets display behavior of the spectral func- tion associated with the Kondo peaks. When J = 0, see Fig. 7(a) and (b), the full Kondo peak at the Fermi level 

