*2.1. Model Hamiltonian*

We consider a three-terminal hybrid nanodevice: an interaction QD connected to one superconducting lead and two ferromagnetic leads, as shown in Figure 1. The Hamiltonian of the system can be written as [40]

$$H = H\_L + H\_R + H\_{QD} + H\_{T\prime} \tag{1}$$

where

$$H\_{\eta} = \sum\_{k\sigma} \epsilon\_{\eta k\sigma} \epsilon\_{\eta k\sigma}^{\dagger} \epsilon\_{\eta k\sigma} \tag{2}$$

$$H\_{QD} = \sum\_{\sigma} \varepsilon\_d \mathfrak{c}\_{d\sigma}^{\dagger} \mathfrak{c}\_{d\sigma} + \mathcal{U}n\_1 n\_2 + \Gamma\_s (\mathfrak{c}\_{d1}^{\dagger} \mathfrak{c}\_{d2}^{\dagger} + \mathfrak{c}\_{d1} \mathfrak{c}\_{d2}),\tag{3}$$

$$H\_T = \sum\_{\eta k r} \left( V\_{\eta k} \mathbf{c}\_{\eta k x}^\dagger \mathbf{c}\_{d r} + \text{H.c.} \right). \tag{4}$$

Here, *η* = *L*, *R* denotes the left and right leads, while *σ* = 1, 2 represents the spin degree of freedom. In the above equations, *<sup>c</sup>*†*ηkσ* (*<sup>c</sup>ηkσ*) and *c*†*dσ* (*cdσ*) are creation (annihilation) operators of electrons with spin *σ* in the *η*-th ferromagnetic lead and in the QD, respectively. In the dot Hamiltonian *HQD*,  *d* is the energy level of the QD, *nσ* = *<sup>c</sup>*†*dσcd<sup>σ</sup>*, and *U* is the on-site Coulomb repulsion between opposite spin electrons. *HT* depicts the tunneling between the QD and the two ferromagnetic leads, and *<sup>V</sup>η<sup>k</sup>* is the corresponding tunneling matrix element. In general, the tunneling amplitude *<sup>V</sup>η<sup>k</sup>* is assumed to be independent of spin and energy, and thus the effect of spin-polarized tunneling is captured by the spin-dependent tunneling rates, <sup>Γ</sup>*ησ* = 2*π* ∑*k* |*<sup>V</sup>η<sup>k</sup>*|<sup>2</sup>*δ*(*<sup>ω</sup>* − *η<sup>k</sup>σ*).

**Figure 1.** (Color online) Schematic diagram of a quantum dot connected to one superconducting lead and two ferromagnetic leads.

In this paper, since we are only interested in the subgap tunneling, it is natural to consider the limit of an extremely large superconducting gap in the superconducting lead. Therefore, the degree of freedom of the superconducting lead can be integrated out and an effective term can be constructed in the dot Hamiltonian, the third term in Equation (3). The parameter Γ*s* plays the role of describing the superconducting proximity effect on the dot. It is evident that this new proximized term mixes the empty state |0 and the doubly occupied state | ↑↓ in the dot, and results in two new eigenstates with energies, *E*± = *ε* ± *ε*<sup>2</sup> + Γ2*s* (here *ε* =  *d* + *U*/2), which are known as the Andreev bound states. What we are interested in this paper is the effect of Andreev reflection on the electron tunneling through an interacting QD in the Kondo regime.

According to the finite-*U* slave-boson approach, one can introduce four additional auxiliary boson operators, *e*, *p<sup>σ</sup>*, and *d*, which are associated with the empty, singly occupied, and doubly occupied electron states, respectively, of the QD, to discuss the above problem without interparticle couplings in an enlarged space with constraints: The completeness relation [46]

$$\sum\_{\sigma} p\_{\sigma}^{\dagger} p\_{\sigma} + \varepsilon^{\dagger} \varepsilon + d^{\dagger} d = 1,\tag{5}$$

and the particle number conservation condition

$$\mathbf{c}\_{d\sigma}^{\dagger}\mathbf{c}\_{d\sigma} = p\_{\sigma}^{\dagger}p\_{\sigma} + d^{\dagger}d.\tag{6}$$

Within the mean-field scheme, the effective Hamiltonian becomes (please see Appendix A) [46]

$$\begin{split} H &= \sum\_{\sigma} \varepsilon\_{d} c\_{d\sigma}^{\dagger} c\_{d\sigma} + \mathsf{U} d^{\dagger} d + \Gamma\_{\mathrm{s}} (z\_{1}^{\*} z\_{2}^{\*} c\_{d1}^{\dagger} c\_{d2}^{\dagger} + z\_{1} z\_{2} c\_{d1} c\_{d2}) + \sum\_{\eta k\sigma} (V\_{\eta k} c\_{\eta k\sigma}^{\dagger} c\_{d\sigma} z\_{\sigma} + V\_{\eta k}^{\*} c\_{d\sigma}^{\dagger} c\_{\eta k\sigma} z\_{\sigma}^{\*}) \\ &+ \sum\_{\eta k\sigma} \epsilon\_{\eta k\sigma} c\_{\eta k\sigma}^{\dagger} c\_{\eta k\sigma} + \lambda^{1} (\sum\_{\sigma} p\_{\sigma}^{\dagger} p\_{\sigma} + e^{\dagger} e + d^{\dagger} d - 1) + \sum\_{\sigma} \lambda\_{\sigma}^{2} (c\_{d\sigma}^{\dagger} c\_{d\sigma} - p\_{\sigma}^{\dagger} p\_{\sigma} - d^{\dagger} d), \end{split} \tag{7}$$

where three Lagrange multipliers *λ*<sup>1</sup> and *λ*2*σ* are drawn in order to make the constraints valid, and *zσ* is the correctional parameters in the hopping term to recover the many-body effect on tunneling with

$$z\_{\sigma} = (1 - d^{\dagger}d - p\_{\sigma}^{\dagger}p\_{\sigma})^{-1/2}(e^{\dagger}p\_{\sigma} + p\_{\sigma}^{\dagger}d)(1 - e^{\dagger}e - p\_{\sigma}^{\dagger}p\_{\sigma})^{-1/2}.\tag{8}$$
