TABLE I: The respective eigenstates | Q, S $^{z}$〉 and eigenvalues E Q,S z of the decoupled quantum dot Hamiltonian. Here, Q = ( ∑$_{jσ}$ n jσ − 2) and S z are the charge and z th component of the spin in the dot, while | χ 1 χ 2 〉 denotes the local states with χ j = 0 , ↑ , ↓ , d for zero, spin-up, spin-down and two electrons in the level j . FIG. 2: (color online) The expectation value of the dot spin operator S 2 in presence of nonmagnetic (a) and the expecta- tion value of dot spin operator S z for ferromagnetic (b) lead as a function of the average level position ε = ( ε 1 + ε 2 ) / 2. The solid line corresponds to Γ = 0 . 04, while the dashed line corresponds to Γ = 0 . 015 (in units of D = 1). The expecta- tion values separately for the two dot levels are also shown. The parameters are δ = 0 . 1, J = − 0 . 2, U = U ′ = 0 . 4, B = 0, and in the ferromagnetic case p = 0 . 4. The shadowed regions indicate Coulomb valleys with different electron number in the dot, where N = ∑$_{jσ}$ n jσ . n | Q, S $^{z}$〉 E Q,S z 1 | − 2 , 0 〉 = | 00 〉 0 2 | − 1 , − 1 $_{2}$〉 = | ↓ 0 〉 ε − δ 2 + 3 J 4 − B 2 3 | − 1 , − 1 $_{2}$〉 = | 0 ↓〉 ε + δ 2 + 3 J 4 − B 2 4 | − 1 , 1 $_{2}$〉 = | ↑ 0 〉 ε − δ 2 + 3 J 4 + B 2 5 | − 1 , 1 $_{2}$〉 = | 0 ↑〉 ε + δ 2 + 3 J 4 + B 2 6 | 0 , 0 〉 = | d0 〉 2 ε − δ + U 7 | 0 , 0 〉 = | 0d 〉 2 ε + δ + U 8 | 0 , 0 〉 = 1 √ $_{2}$( | ↑↓〉 − | ↓↑〉 ) 2 ε + U ′ 9 | 0 , − 1 〉 = | ↓↓〉 2 ε + U ′ + 2 J − B 10 | 0 , 0 〉 = 1 √ $_{2}$( | ↑↓〉 + | ↓↑〉 ) 2 ε + U ′ + 2 J 11 | 0 , 1 〉 = | ↑↑〉 2 ε + U ′ + 2 J + B 12 | 1 , − 1 $_{2}$〉 = | d ↓〉 3 ε − δ 2 + 3 J 4 + U + 2 U ′ − B 2 13 | 1 , − 1 $_{2}$〉 = | ↓ d 〉 3 ε + δ 2 + 3 J 4 + U + 2 U ′ − B 2 14 | 1 , 1 $_{2}$〉 = | d ↑〉 3 ε − δ 2 + 3 J 4 + U + 2 U ′ + B 2 15 | 1 , 1 $_{2}$〉 = | ↑ d 〉 3 ε + δ 2 + 3 J 4 + U + 2 U ′ + B 2 16 | 2 , 0 〉 = | dd 〉 4 ε + 2 U + 4 U ′ 0.0 0.4 0.8 1.2 1.6 2.0 p=0 -1.6 -1.2 -0.8 -0.4 0.0 0.4 -0.8 -0.4 0.0 0.4 0.8 N= N=1 N=2 N=3 p=0.4 N=4 b a 〈 S 2 〉 〈 S z 〉 ε/D 〈 S 2 $_{1}$〉 〈 S 2 $_{2}$〉 〈 S $^{2}$〉 〈 S z 〉 〈 S z $_{2}$〉 〈 S z $_{1}$〉 3 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 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$_{−}$$_{1}$$_{,}$$_{0}$ = − 0 . 2 and ε$_{0}$$_{,}$$_{1}$ = 1. We have calculated the expectation value of the dot’s spin operators S 2 and S z in case of nonmagnetic and ferromagnetic leads, respectively. The results are sum- marized in Fig. 2. As the level position is lowered by sweeping the gate voltage, the dot is tuned through the Coulomb blockade valleys with electron number N = 1 , 2 , 3 and spin states S = 1 / 2 , 1 , 1 / 2, respectively. When the dot is attached to a ferromagnetic lead, the spin asymmetry in the hybridization results in the spin split- ting of the dot level, thus leading to a spin polarized Before presenting numerical results, it is instructive to analyze the eigen-spectrum of the decoupled quan- tum dot Hamiltonian, Eq. (2). The respective eigenstates | Q, S $^{z}$〉 and eigenvalues E$_{Q,S}$ z of H$_{QD}$ are listed in Table I, where Q = ( ∑ jσ n$_{jσ}$ − 2) denotes the charge in the dot and S z is the z th component of the dot spin. By tuning ε the ground state of the dot changes. In particular, in the absence of magnetic field and for ferromagnetic ex- change coupling ( J < 0), when ε > ε$_{−}$$_{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 〉 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. III. NUMERICAL RESULTS A$_{σ}$ ( ω ) = ∑ jj ′ A$_{jj}$ ′ $_{σ}$( ω ) . (4)