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was observed experimentally in molecular quantum dots coupled to nonmagnetic leads. 25 

In the present paper we focus our interest on the inter- play of underscreened Kondo effect and itinerant electron ferromagnetism in the leads. To that end we consider a two level quantum dot coupled to a single reservoir of conduction electrons exhibiting spin imbalance. We as- sume that the second Kondo temperature is always much smaller than the experimental temperature, therefore we consider the second lead as a weakly coupled probe only. 

The considered system consists of a two-level quantum dot coupled to a ferromagnetic reservoir, see Fig. 1, and its Hamiltonian is given by 

where the first term describes the noninteracting itinerant electrons in ferromagnetic lead, H$_{FM}$ = ∑ kσ ε$_{kσ}$c † $_{kσ}$c$_{kσ}$ , where c † kσ is the electron creation op- erator with wave number k and spin σ , while ε$_{kσ}$ is the corresponding energy. The second part of the Hamilto- nian describes a two-level quantum dot and is given by 

where n$_{jσ}$ = d † $_{jσ}$d$_{jσ}$ and d † jσ creates a spin- σ electron in the j th level ( j = 1 , 2), ε$_{j}$ denotes the corresponding energy of an electron in the dot. Here, ε$_{1}$ = ε − δ/ 2 and ε$_{2}$ = ε + δ/ 2, with δ = ε$_{2}$ − ε$_{1}$ being the level spac- ing. The on-level (inter-level) Coulomb correlations are denoted by U ( U $^{′}$), respectively. J is the ferromagnetic exchange coupling ( J < 0) with ⃗ S = ⃗ S$_{1}$ + ⃗ S$_{2}$ , where ⃗ S$_{j}$ = 1 2 ∑ σσ ′ d † $_{jσ}$⃗ σ$_{σσ}$ ′ d$_{jσ}$ ′ is the spin operator for elec- trons in the dot level j and ⃗ σ denotes the vector of Pauli spin matrices. The last term of H$_{QD}$ describes the Zee- man splitting with B ( gµ$_{B}$ ≡ 1) being the external mag- netic field applied along the z th direction. 

Finally, the tunnel Hamiltonian is given by 

where t$_{jσ}$ describes the spin-dependent hopping matrix elements between the j th dot level and ferromagnetic lead. The strength of the coupling between the dot and lead can be expressed as Γ$_{jσ}$ = πρ | t$_{jσ}$ | $^{2}$, where ρ is the density of states (DOS) in the lead. In the following, we assume a flat band of width 2 D and use D ≡ 1 as energy unit, as not stated otherwise. Note that by assuming constant DOS the whole spin-dependence has been shifted into the coupling constants. While this as- sumption may simplify the calculations, it does not affect 

## II. THEORETICAL DESCRIPTION 

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the low-energy physics we are interested in. 16,17 To pa- rameterize the couplings, it is convenient to introduce the spin polarization of ferromagnetic lead, p , defined as p = (Γ$_{↑}$ − Γ$_{↓}$) / (Γ$_{↑}$ + Γ$_{↓}$), where we have assumed that each dot level is coupled with the same strength to the lead, Γ$_{jσ}$ = Γ$_{σ}$. Then, the coupling for the spin- ↑ (spin- ↓ ) electrons can be written as Γ$_{↑}$$_{(}$$_{↓}$$_{)}$ = (1 ± p )Γ, where Γ = (Γ$_{↑}$ + Γ$_{↓}$) / 2. For p = 0, the couplings do not de- pend on spin and the system behaves as if coupled to nonmagnetic lead. However, for finite spin polarization, p ̸ = 0, the couplings are spin-dependent, giving rise to an effective exchange field which may spin-split the levels in the dot, suppressing the Kondo resonance. Such behav- ior has been extensively studied theoretically in the case of single-level quantum dots, 14–19,26–29 where the usual S = 1 2 Kondo effect develops, while the effect of ferro- magnetism on the other types of Kondo effect remains to a large extent unexplored. 

In this paper we thus consider the zero-temperature equilibrium transport properties of two-level quantum dots coupled to ferromagnetic lead in the case of under- screened Kondo effect. In order to perform this analysis in most accurate and exact way, we employ the numeri- cal renormalization group (NRG) method. 30,31 The NRG consists in a logarithmic discretization of the conduction band and mapping of the system onto a semi-infinite chain with the quantum dot sitting at the end of the chain. By diagonalizing the Hamiltonian at consecutive sites of the chain and storing the eigenvalues and eigen- vectors of the system, one can accurately calculate the static and dynamic quantities of the system. In partic- ular, numerical results presented here were obtained us- ing the flexible density-matrix numerical renormalization group (DM-NRG) code, which allows to use arbitrary number of Abelian and non-Abelian symmetries. 32–34 In fact, exploiting symmetries as much as possible is crucial in obtaining highly accurate data. Because in the case of ferromagnetic leads the full spin rotational invariance is generally broken, in the case of finite spin polarization we have used the Abelian symmetries for the total charge and spin, while in the case of p = 0 we have exploited the full spin SU (2) symmetry. 

Using the NRG, we have calculated the expecta- 

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