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ground state. Precisely in the middle of the N = 2 Coulomb blockade valley, i.e. for ε = ( ε$_{−}$$_{1}$$_{,}$$_{0}$ + ε$_{0}$$_{,}$$_{1}$ ) / 2 = − U ′ − U 2 = − 0 . 6, the spin polarization changes sign, as to the right (left) hand side from that point the electron (hole) like virtual processes dominate in the renormaliza- tion of the level position. 

Note that the abrupt jump of the dot polarization is due to the fact that the underscreened Kondo model is extremely susceptible to even a tiny magnetic field. The reason is that the ground state of the system consists of a Fermi liquid and a decoupled residual spin S = 1 2 and that residual spin at zero temperature can be polar- ized by any infinitesimal magnetic field. In the case of a weakly coupled second lead there is a second stage of the Kondo screening at T ′ K when the remaining spin de- gree of freedom is quenched by the weakly coupled lead’s electrons. 35 In that case the sudden step in the polar- ization turns into a very steep crossover with a width of ∼ max( T ′ $_{K}$, T ), where T is the experimental temperature. 

Since the dot spin polarization is rather difficult to detect experimentally, we have computed the level- resolved single particle spectral density 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 corresponding retarded Green’s function. This quan- tity contains the information about the transport prop- erties of the dot. Given that the dot is strongly cou- pled to one of the leads, the conductivity through the setup at voltage bias eV is given by dI/dV ∼ e 2 h π ∑ jj ′ σ Γ$_{σ}$ A$_{jj}$ ′ $_{σ}$( ω = eV ). The results for the level-resolved normalized spectral 

functions π Γ A p =0 jj ′ = π ∑ σ Γ$_{σ}$ A p =0 jj ′ σ in the case of non- magnetic lead ( p = 0) are shown in Fig. 3. The spectral functions are plotted as a function of energy ω and aver- age level position ε . To resolved the low-energy behavior of spectral functions logarithmic scale for ω is used. Fur- thermore, only the Coulomb blockade valleys with N = 1 and N = 2 electrons in the dot are shown. The behav- ior of spectral functions for blockade valley with N = 3 electrons is similar to that with a single electron due to the particle-hole symmetry. 

First of all, we note that the two Coulomb blockade regimes with N = 1 and N = 2 are characterized by fun- damentally different fixed points. In the N = 1 regime ( − 0 . 2 < ε < 0 . 2) the effective low-energy model is a Fermi liquid with fully screened spin S = 1 2 ( S = 1 2 Kondo model). On the other hand, in the two-electron regime ( − 1 < ε < − 0 . 2) the effective model is a singu- lar Fermi liquid, 24 i.e. a normal Fermi liquid plus a free spin- 1 2 with weak residual ferromagnetic coupling to the conduction channel. These two distinct ground states of the system are separated by a quantum phase transi- tion, 36,37 which occurs when sweeping the gate voltage through the boundary of N = 1 ( N = 3) and N = 2 Coulomb valleys. The quantum phase transition is of Kosterlitz-Thouless type, 37 with exponentially decreas- ing Kondo temperature when approaching the transition from the normal Fermi liquid side. The different be- havior associated with two distinct fixed points can be 

observed in the dependence of the level-resolved spec- tral functions. In the N = 1 Coulomb valley the diag- onal elements of the spectral function exceed the uni- tary value of π Γ, while the off-diagonal elements become negative and their absolute value is also larger than π Γ. 

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