3 both Abelian and non-Abelian symmetries. 35,36 In calcu- lations we have thus used the U (1) symmetry for the z th component of the total spin, the U (1) symmetry for the charge in the first channel, and the SU (2) symmetry for the charge in the second channel. Furthermore, in calcu- lations we have taken the discretization parameter Λ = 2 and kept 3000 states at each iteration step. Using the NRG we can calculate the spectral function of the dot, A$_{σ}$ ( ω ) = − 1 $_{π}$Im G R $_{dσ}$( ω ), where G R $_{dσ}$( ω ) de- notes the Fourier transform of the dot retarded Green’s function, G R $_{dσ}$( t ) = − i Θ( t ) 〈{ d$_{σ}$ ( t ) , d † $_{σ}$(0) }〉 . On the other hand, the spectral function can be directly related to the spin-resolved linear conductance G$_{σ}$ by the following for- mula spin-dependent couplings we also introduce the spin po- larization of ferromagnetic leads, p = (Γ$_{↑}$ − Γ$_{↓}$) / (Γ$_{↑}$+Γ$_{↓}$). The couplings can be then written in a compact form as, Γ$_{↑}$$_{(}$$_{↓}$$_{)}$ = (1 ± p )Γ, where Γ = (Γ$_{↑}$ + Γ$_{↓}$) / 2. Assuming symmetric coupling strength of the dot to the leads, the resultant coupling in the antiparallel configuration is the same for the spin-up and spin-down electrons, Γ AP ↑ ( ↓ ) = Γ. On the other hand, in the parallel configuration, the couplings are then different for the two spin directions, Γ P ↑ ( ↓ ) = (1 ± p )Γ, which effectively leads to spin splitting of the dot level and, when this is the case, the Kondo res- onance may become suppressed because of broken spin degeneracy. 6,7 Finally, the exchange Hamiltonian describing the cou- pling between the dot and the second (nonmagnetic) reservoir is given by where f ( ω ) is the Fermi distribution function and the to- tal conductance is given by, G = G$_{↑}$ + G$_{↓}$ . At zero tem- perature, the spin dependent conductance for the paral- lel configuration is given by, G P ↑ ( ↓ ) = e 2 h (1 ± p ) π Γ A P ↑ ( ↓ $_{)}$, while for the antiparallel configuration one gets, G AP ↑ ( ↓ ) = e 2 h (1 − p $^{2}$) π Γ A AP ↑ ( ↓ ) = G $^{AP}$/ 2, where A P / AP σ is the zero- temperature spectral function of the d -level operator in respective magnetic configuration, taken at ω = 0. where ⃗ S = 1 2 ∑ σσ ′ d † $_{σ}$⃗ σ$_{σσ}$ ′ d$_{σ}$ ′ is the spin in the dot, J de- notes the exchange coupling constant and ⃗ σ is a vector of Pauli spin matrices. We note that in addition to the ex- change scattering of electrons [described by Eq. (5)] there could be also potential scattering. However, in this work we are mainly interested in the low energy physics, where the Kondo effect emerges, so the potential scattering may be neglected, as it does not lead to any Kondo-type cor- relations. To analyze the equilibrium transport properties of the considered system, we employ the numerical renormal- ization group method 33 – nonperturbative, very powerful and essentially exact numerical method to address quan- tum impurity problems. 34 The NRG consists in a loga- rithmic discretization of the conduction band and map- ping of the system onto a semi-infinite chain with the impurity (quantum dot) sitting at the end of the chain. By diagonalizing the Hamiltonian at consecutive sites of the chain and storing the eigenvalues and eigenvectors of the system, one can calculate the static and dynamic quantities of the system. In the case of model consid- ered in this paper, the Hamiltonian is mapped onto two semi-infinite chains, where the first chain corresponds to ferromagnetic leads tunnel-coupled to the dot, while the second one to nonmagnetic reservoir exchange-coupled to the dot. Because such two-channel calculations are usually very demanding numerically, it is crucial to ex- ploit as many symmetries of the system’s Hamiltonian as possible. Especially, using the SU (2) symmetry de- creases the size of Hilbert space and thus increases con- siderably the accuracy of calculations. In particular, to efficiently perform the analysis, we have used the flexible density-matrix numerical renormalization group (DM- NRG) code, which can tackle with arbitrary number of In the following we present numerical results on the equilibrium spectral function and linear conductance, when the quantum dot is in the Kondo regime. We will distinguish between two different situations; symmetric ( ε$_{d}$ = − U/ 2) and asymmetric ( ε$_{d}$ = − U/ 2) Anderson models. The origin of such a distinction stems from the way in which ferromagnetic leads act on the quantum dot. More specifically, in the asymmetric Anderson model fer- romagnetism of the leads gives rise to a spin splitting of the Kondo resonance in the parallel configuration, while in the symmetric model no such a splitting appears (as- suming that the dot is coupled with the same strength to the left and right leads). 6,7,10 In other words, an effective exchange field, due to coupling to magnetic leads, acts on the dot in the former case, while such a field vanishes in the latter case. The effective field is directly related to the difference in the coupling strengths of the dot and ferromagnetic leads for the two spin orientations. Since the coupling in the spin-up channel is larger than that in the spin-down one, energy of the spin-up (spin-down) electron in the dot decreases (increases) by Δ ε$_{d}$/ 2. Con- sequently, the spin-dependent coupling acts as an effec- tive magnetic field, leading to spin-splitting Δ ε$_{d}$ of the dot level. 6,7,10 III. NUMERICAL RESULTS B. Method G$_{σ}$ = e 2 h 4Γ$_{L}$$_{σ}$Γ$_{R}$$_{σ}$ Γ$_{L}$$_{σ}$ + Γ$_{R}$$_{σ}$ ∫ dωπA$_{σ}$ ( ω ) ( − ∂f ( ω ) ∂ω ) , (6) H$_{exch}$ = J 2 ∑ σσ ′ ∑ k ⃗ Sa † $_{kσ}$⃗ σ$_{σσ}$ ′ a$_{kσ}$ ′ , (5)