**Appendix A**

In this appendix, we briefly show how to obtain the effective Hamiltonian Equation (7). Within the formulation of the finite-*U* slave boson approach, the original QD electron operators are replaced by the new varibales as follows: *n*1*n*2 → *d*†*d*, *cdσ* → *zσ fd<sup>σ</sup>*, and *c*† *<sup>d</sup>σcd<sup>σ</sup>* → *f* † *dσ fd<sup>σ</sup>*. The additional degrees of freedom simplify the Coulomb interaction *Un*1*n*2 as *Ud*†*d*, but introduce vectors not describing physically real states. Then, two constraint conditions have to be imposed to eliminate the unphysical part of the enlarged Hilbert space |Φ

$$\begin{cases} (\sum p\_{\sigma}^{\dagger} p\_{\sigma} + \mathfrak{e}^{\dagger} \mathfrak{e} + d^{\dagger} d - 1) |\Phi\rangle = 0, \\ (f\_{d\sigma}^{\dagger} f\_{d\sigma} - p\_{\sigma}^{\dagger} p\_{\sigma} - d^{\dagger} d) |\Phi\rangle = 0. \end{cases} \tag{A1}$$

Therefore, the subspace of the enlarged Hilbert space defined by Equation (A1) is equivalent to the original Hilbert space. Applying Dirac's formulation of constrained dynamics, one should then introduce two *q*-number Lagrange multipliers *λ*<sup>1</sup> and *λ*<sup>2</sup> *σ*, corresponding to the two constraints in Equation (A1), to the Heisenberg equation of motion with respect to the effective Hamiltonian

$$
\tilde{H} \equiv H + \lambda^1 \left( \sum\_{\sigma} p\_{\sigma}^{\dagger} p\_{\sigma} + e^{\dagger} e + d^{\dagger} d - 1 \right) + \sum\_{\sigma} \lambda\_{\sigma}^2 (f\_{d\sigma}^{\dagger} f\_{d\sigma} - p\_{\sigma}^{\dagger} p\_{\sigma} - d^{\dagger} d), \tag{A2}
$$

and consequently any dynamical observable *A*ˆ satisfies the standard equation of motion as a state equation

$$i\hbar \frac{d\check{A}}{dt} |\Phi\rangle = [\hat{A}, \check{H}] |\Phi\rangle. \tag{A3}$$

The next key point of the SBMF approach is to replace all the slave-boson operators and the Lagrange multipliers by their average values according to nonequilibrium steady states (NESS), which can be still expressed in this paper by *e*, *p<sup>σ</sup>*, *d*, *λ*1, and *λ*<sup>2</sup> *σ*. As a result, we can indeed obtain the mean-field expression of the effective Hamiltonian Equation (7) (please note that we still use the notation *cdσ* instead of *fdσ* in the effective Hamiltonian in the main text for the sake of convenience).
