*3.3. Adiabatic and Nonadiabatic Contributions*

Taking the Riemann sum on G*ν*1 and G*ν*2 by setting *N* → ∞ and *δt* → 0, we find that they reduce to the dynamical and geometrical phases, respectively. For instance, we obtain the energy transfer with the environment R with setting *ν* = *R* as [56]

$$\mathcal{G}\_1^{\mathbb{R}} = \int\_0^{\mathcal{T}} dt' \frac{\Gamma\_L \Gamma\_R(n\_L(t') - n\_R(t'))}{K},\tag{25}$$

with *K* ≡ ∑*<sup>ν</sup>*=*L*,*<sup>R</sup>* <sup>Γ</sup>*ν*(<sup>1</sup> + <sup>2</sup>*nν*(*t*)), and

$$\mathcal{G}\_2^R = \int \int dT\_L dT\_R \left\{ \frac{2\Gamma\_L \Gamma\_R (\Gamma\_L + \Gamma\_R)}{K^3} \frac{dn\_R}{dT\_R} \frac{dn\_L}{dT\_L} \right\},\tag{26}$$

which coincide with the ones in Reference [28] and imply that the sum of G*ν*1 and G*ν*2 corresponds to the adiabatic contribution.

Considering this point, we find that the nonadiabatic contribution is described with a new extra term in <sup>Δ</sup>*q<sup>ν</sup>* added to the adiabatic contribution in the form,

$$
\langle \Delta q^{\nu} \rangle = \mathcal{G}\_{ad}^{\nu} + \mathcal{G}\_{nad^{\nu}}^{\nu} \tag{27}
$$

with G*<sup>ν</sup>ad* = G*ν*1 + G*ν*2 and G*<sup>ν</sup>nad* = G*ν*3 . This is consistent with the expression of G*ν*3 , which shows that, when *ρ*00(0) = *ρ<sup>s</sup>*,<sup>1</sup> and the absolute value of Λ*iδt* is sufficiently large, we can neglect G*ν*3 . The former condition corresponds to the adiabatic approximation in Reference [28], where the population of the relevant system instantaneously approaches the steady state for the temperature setting at an initial time. The term G*ν*3 shows that the nonadiabatic contribution to the transferred quantity explicitly depends on the initial condition of the relevant system, *ρ*00(0). Moreover, expanding Equation (13) about *δt* up to the first order, we find that the nonadiabatic effect described in G*ν*3 shows a correction to both G*ν*1 and G*ν*2 . In the following, we present a numerical evaluation of the formulas obtained.

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