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tion value of the dot’s spin as well as the spectral function of the dot, A$_{jj}$ ′ $_{σ}$( ω ) = − 1 2 $_{π}$Im[ G R jj ′ $_{σ}$( ω ) + G R j ′ $_{jσ}$( ω )], where G R jj ′ $_{σ}$( ω ) denotes the Fourier trans- form of the dot retarded Green’s function, G R jj ′ $_{σ}$( t ) = − i Θ( t ) 〈{ d$_{jσ}$ ( t ) , d † j ′ $_{σ}$(0) }〉 . The diagonal elements of the spectral function A$_{jσ}$ ( ω ) ≡ A$_{jjσ}$ ( ω ) are related to the spin-resolved density of states, whereas the off-diagonal elements A$_{jj}$ ′ $_{σ}$( ω ) may be associated with processes of in- jecting and removing an electron at different sites. The symmetrized spin-resolved spectral function of the dot can be found from 

On the other hand, the normalized full spectral function is given by π ∑ σ Γ$_{σ}$ A$_{σ}$ ( ω ) and is directly related to the conductivity of the dot. 

Before presenting numerical results, it is instructive to analyze the eigen-spectrum of the decoupled quan- tum dot Hamiltonian, Eq. (2). The respective eigenstates | Q, S $^{z}$〉 and eigenvalues E$_{Q,S}$ z of H$_{QD}$ are listed in Table I, where Q = ( ∑ jσ n$_{jσ}$ − 2) denotes the charge in the dot and S z is the z th component of the dot spin. By tuning ε the ground state of the dot changes. In particular, in the absence of magnetic field and for ferromagnetic ex- change coupling ( J < 0), when ε > ε$_{−}$$_{2}$$_{,}$$_{−}$$_{1}$ = δ 2 − 3 J 4 , the dot is empty, for ε = ε$_{−}$$_{1}$$_{,}$$_{0}$ = − δ 2 − U ′ − 5 J 4 , the doublet | Q = − 1 , S = 1 $_{2}$〉 and triplet | Q = 0 , S = 1 〉 

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## III. NUMERICAL RESULTS 

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states become degenerate, where now S is the full dot’s spin, whereas for ε = ε$_{0}$$_{,}$$_{1}$ = δ 2 − U ′ − U + 5 J 4 , the states | Q = 0 , S = 1 〉 and | Q = 1 , S = 1 $_{2}$〉 are degenerate. Con- sequently, for ε$_{−}$$_{2}$$_{,}$$_{−}$$_{1}$ > ε > ε$_{−}$$_{1}$$_{,}$$_{0}$ , the ground state of the decoupled dot is the doublet and the system will exhibit usual S = 1 2 Kondo effect, whereas for ε$_{−}$$_{1}$$_{,}$$_{0}$ > ε > ε$_{0}$$_{,}$$_{1}$ the dot is in the triplet state and the system will show an underscreened Kondo effect. In calculations we have taken the following parameters: δ = 0 . 1, J = − 0 . 2, U = U ′ = 0 . 4, and Γ = 0 . 04 (in units of D = 1). One then finds, ε$_{−}$$_{2}$$_{,}$$_{−}$$_{1}$ = 0 . 2, ε$_{−}$$_{1}$$_{,}$$_{0}$ = − 0 . 2 and ε$_{0}$$_{,}$$_{1}$ = 1. 

We have calculated the expectation value of the dot’s spin operators S 2 and S z in case of nonmagnetic and ferromagnetic leads, respectively. The results are sum- marized in Fig. 2. As the level position is lowered by sweeping the gate voltage, the dot is tuned through the Coulomb blockade valleys with electron number N = 1 , 2 , 3 and spin states S = 1 / 2 , 1 , 1 / 2, respectively. When the dot is attached to a ferromagnetic lead, the spin asymmetry in the hybridization results in the spin split- ting of the dot level, thus leading to a spin polarized 

