6 FIG. 6: (color online) The spin-up (a), spin-down (b) and the full (c) spectral function in the underscreened Kondo regime for ε = − 0 . 4 in the presence of external magnetic field B , as indicated in the figure. The parameters are the same as in Fig. 2 with Γ = 0 . 04 and p = 0 . 4. The inset in (c) displays the dependence of the spectral function associated with the Kondo peak. FIG. 5: (color online) The spin-up (a), spin-down (b) and the full (c) spectral function in the S = 1 2 Kondo regime for ε = 0 in the presence of external magnetic field B , as indicated in the figure. The parameters are the same as in Fig. 2 with Γ = 0 . 04 and p = 0 . 4. The inset in (c) shows the zoom- out full spectral function. The Kondo effect is restored when B = 0 . 009215. although it is possible to restore the resonance peak at ω = 0, the full restoration of the underscreened Kondo effect is not possible, as the height of the peak is below the unitary limit. sition. For finite spin polarization of the leads, p = 0, the Kondo phenomenon becomes suppressed due to an effective exchange field originating from the presence of ferromagnetic leads, and so is the critical behavior. In the transport regime where the system exhibits un- derscreened Kondo effect and for p = 0, we have found that the polarization of the dot abruptly changes sign in the middle of the S = 1 Coulomb blockade region. This is associated with virtual tunneling processes (either electron-like or hole-like) that give the dominant contri- bution to the renormalization of the dot levels. Furthermore, we have also analyzed the effect of an external magnetic field B applied to the dot. It has been shown that an appropriately tuned B can restore the Kondo effect in the case of the S = 1 2 Coulomb valley. The underscreened Kondo effect, on the other hand, ex- hibits an extremely high sensitivity towards even a tiny In this paper we have analyzed the equilibrium trans- port properties of a two-level quantum dot asymmetri- cally coupled to ferromagnetic leads. We have shown that by tuning the position of the dot levels, the ground state of the system changes from a Fermi liquid in the S = 1 2 Coulomb valley into a Fermi liquid plus residual spin-a-half in the S = 1 Coulomb valley, the latter being the example of underscreened Kondo effect. The bound- ary between these two regime is a quantum phase tran- IV. CONCLUSIONS -0.0012 -0.0006 0.0000 0.0006 0.0012 0.0 0.4 0.8 1.2 1.6 -0.4 0.0 0.4 0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 B  = 0 B  = 0.0088 B  = 0.009 B  = 0.009215 0.0 0.2 0.4 0.6 0.8 c b a ω/D π Γ ↑ A p =0 . 4 ↑ π Γ ↓ A p =0 . 4 ↓ π ∑ σ Γ σ A p =0 . 4 σ 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 0.0 0.4 0.8 1.2 1.6 -0.0004 0.0000 0.0004 0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 B  = 0 B  = 0.007 B  = 0.0072991186 B  = 0.0072991187 B  = 0.007299112 B  = 0.0072992 B  = 0.0073 0.0 0.2 0.4 0.6 0.8 c b a ω/D π Γ ↑ A p =0 . 4 ↑ π Γ ↓ A p =0 . 4 ↓ π ∑ σ Γ σ A p =0 . 4 σ