2 see Fig. 1. It is assumed that the external leads are made of the same ferromagnetic material and their magnetiza- tions are collinear, so that the system can be either in the parallel or antiparallel magnetic configuration. The total Hamiltonian is given by Here, H$_{FM}$ describes the ferromagnetic leads, H$_{FM}$ = ∑ rkσ ε$_{rkσ}$c † $_{rkσ}$c$_{rkσ}$ , where c † rkσ is the electron creation operator with wave number k , spin σ in the left ( r = L) or right ( r = R) lead, and ε$_{rkσ}$ is the corresponding energy. The second part, H$_{NM}$ , corresponds to a nonmagnetic electron reservoir and is given by, H$_{NM}$ = ∑ kσ ε$_{k}$a † $_{kσ}$a$_{kσ}$ , with a † kσ being the respective creation operator and ε$_{k}$ is the single-particle energy. The quantum dot is described by the Anderson Hamiltonian, where d † σ creates a spin- σ electron, ε$_{d}$ denotes the en- ergy of an electron in the dot, and U describes the Coulomb correlations between two electrons occupying the dot. The last term corresponds to external magnetic field B applied along the z th direction ( gµ$_{B}$ ≡ 1) and S$_{z}$ = 1 $_{2}$( d † $_{↑}$d$_{↑}$ − d † $_{↓}$d$_{↓}$ ). The tunnel Hamiltonian is given by terplay between the effects due to ferromagnetism of the leads and exchange coupling to the additional nonmag- netic reservoir. Depending on the strength of the tunnel c oupling t and exchange coupling J , the dot’s spin can be screened either by electrons in the ferromagnetic leads or by electrons in the nonmagnetic reservoir. By analyz- ing the equilibrium spectral functions and the conduc- tance through the dot, we show that in the antiparallel magnetic configuration, the system undergoes a quan- tum phase transition with increasing exchange coupling J , where the conductance drops from the maximum value to zero. For a certain critical value of J , J AP c , both elec- tron channels try to screen the dot’s spin and the con- ductance approaches a half of the quantum conductance. In the parallel configuration, on the other hand, the con- ductance is generally suppressed, irrespective of the ex- change constant J , due to effective spin splitting of the dot level caused by the exchange field coming from fer- romagnetic leads. 6,7 We show that the Kondo effect can be restored by applying a properly tuned external mag- netic field B for J below the corresponding critical point, J < ˜ J P c . Furthermore, the quantum critical regime can also be recovered, which however requires a fine-tuning in the parameter space of J and B . where t$_{rσ}$ describes the spin-dependent hopping matrix elements between the dot and ferromagnetic leads. The coupling to magnetic leads can be described by Γ$_{rσ}$ = πρ$_{r}$ | t$_{rσ}$ | $^{2}$, where ρ$_{r}$ ≡ ρ is the density of states in the lead r . We have thus shifted the whole spin-dependence into the coupling constants and assumed a flat band of width 2 D , 6,7 where D ≡ 1 is set as the energy unit, if not stated otherwise. By means of a unitary transformation in the left-right basis, 31,32 one can map the problem of tunneling through quantum dot coupled to the left and right leads into a problem where the dot is effectively coupled to a single lead with a new coupling constant, Γ$_{σ}$ = Γ$_{L}$$_{σ}$ + Γ$_{R}$$_{σ}$. This can be done by introducing the following symmetric operators, α$_{kσ}$ = ˜ t$_{L}$$_{σ}$c$_{L}$$_{kσ}$ + ˜ t$_{R}$$_{σ}$c$_{R}$$_{kσ}$ , with dimensionless coefficients ˜ t$_{rσ}$ = t$_{rσ}$/ √ t 2 L σ + t 2 R $_{σ}$. Then, the tunneling Hamiltonian can be written as The paper is organized as follows. In section II we present the model as well as briefly describe the NRG method together with some details of calculations. In turn, in section III we present numerical results for sym- metric and asymmetric Anderson models in both parallel and antiparallel magnetic configurations of the system. Finally, we conclude in section IV. The considered system consists of a single-level quan- tum dot tunnel-coupled to left and right ferromagnetic leads and exchange-coupled to a nonmagnetic reservoir, One can see that now the dot is tunnel-coupled to only one effective electron reservoir, H$_{FM}$ = ∑ kσ ε$_{kσ}$α † $_{kσ}$α$_{kσ}$ , with new spin-dependent coupling constant Γ$_{σ}$. The other parts of the system Hamiltonian, Eq. (1), are not affected by this transformation. To parameterize the H$_{tun}$ = ∑ kσ √ Γ$_{σ}$ πρ ( d † $_{σ}$α$_{kσ}$ + α † $_{kσ}$d$_{σ}$ ) . (4) A. Model II. THEORETICAL DESCRIPTION H$_{tun}$ = ∑ rkσ t$_{rσ}$ ( d † $_{σ}$c$_{rkσ}$ + c † $_{rkσ}$d$_{σ}$ ) , (3) H$_{QD}$ = ∑ σ ε$_{d}$d † $_{σ}$d$_{σ}$ + Ud † $_{↑}$d$_{↑}$ d † $_{↓}$d$_{↓}$ + BS$_{z}$, (2) H = H$_{FM}$ + H$_{NM}$ + H$_{QD}$ + H$_{tun}$ + H$_{exch}$. (1) FIG. 1: (color online) The schematic of a quantum dot (QD) tunnel-coupled to external ferromagnetic leads and exchange- coupled to a nonmagnetic electron reservoir. The spin- dependent coupling to the left (right) lead is described by Γ L σ (Γ R σ ), while J denotes the exchange coupling constant. The magnetizations of the leads can form either parallel or antiparallel magnetic configuration, as indicated. QD L R J