*4.3. Absence of Adiabatic Contribution*

Let us first show absence of the adiabatic contribution to the spin pumping in the minimum model. In previous studies, the adiabatic regime of the spin pumping in the minimum model has been studied based on the linear expansion of the Green function in the rotation frequency of *M*(*t*) [23,24], which we referred to as the adiabaticity No. (2). The purpose of the present subsection is to re-examine the adiabatic contribution of the spin pumping from the view point of the adiabaticity No. (3), where we consider a sufficiently slow rotation of *M*(*t*) comparing to the relaxation time, that is, Ω *τ*<sup>−</sup><sup>1</sup> *r* following the procedure by Sinitsyn and Nemenman in Reference [27].

Following the procedure, the adiabatic regime is assessed by dividing the cycle of modulation into time intervals *<sup>δ</sup>t*(≡ T /*N*) and assuming a quick approach of the system to its steady state in each interval. In the steady state of the minimum model, we can expect that the quantum dot is occupied by a single electron whose spin is aligned toward the direction of *<sup>M</sup>*(*t*), and, because of the rotational symmetry of the model, the steady state populations of the quantum dot are invariant under the rotation of *M*(*t*) around the *z*-axis. It indicates that no electron transfer occurs in the adiabatic regime. As a result, we can expect absence of the adiabatic contribution to the spin current generation. We provide an analytical proof of the intuitive observation in Appendix C (see also the original argumen<sup>t</sup> in Section 4 in Reference [58]).

As a result, we need to include the nonadiabatic effect to obtain a finite spin current. It is in marked contrast to the previous example of the energy pumping, where the adiabatic contribution to the energy pumping G*<sup>ν</sup>ad*is finite.

### *4.4. Numerical Evaluation of the Nonadiabatic Spin Pumping*

We now turn to examine nonadiabaticity in spin pumping. For this purpose, we evaluate numerically the instantaneous spin current *J*spin(*t*) and its temporal average *<sup>I</sup>*spin.

To describe the dot–lead coupling, we use the Ohmic spectral density with an exponential cutoff *v*(*ω*) ≡ ∑*k <sup>v</sup>*2*k<sup>δ</sup>*(*<sup>ω</sup>* − *<sup>ω</sup>k*) = *λω* exp[−*<sup>ω</sup>*/*<sup>ω</sup>c*], where *λ* is the coupling strength and *ωc* is the cutoff frequency. For the numerical calculation, we chose 2*M*, the energy difference between the spin-↑ and

