*6.2. Spin-Dependent Prefactors*

In this section, we examine the scattering amplitudes, *t*± and *r*± and the prefactors of the Berry curvatures in Equations (34) and (35). We define these factors as *dn*ˆ = |*<sup>r</sup>*+ − *r*−|<sup>2</sup> − |*<sup>t</sup>*+|<sup>2</sup> + |*t*−|<sup>2</sup> and *d*−*n*<sup>ˆ</sup> = − |*<sup>r</sup>*+ − *r*−|<sup>2</sup> − |*<sup>t</sup>*+|<sup>2</sup> + |*t*−|2. To be compatible with the analysis in the previous subsection, we focus on the geometry such that *β* = *π*/5 and *ν* = *π*/2. For simplicity, we chose symmetric setup of the interferometer, where *J*0*b* = *Jb*1 = *J*<sup>0</sup>*c* = *Jc*1 = *j* and 0 = 1 =  *b* =  *<sup>c</sup>*. Then, *γb* = *γc* = *j*2  *k*−0 . Moreover, in the following calculation we chose  *k* = <sup>−</sup>*j*. First, we show the result of *d*−*n*<sup>ˆ</sup> for 0 = 0.9*j* in Figure 4 with choosing the AB phase *φ* = *π*/3. This function is negatively enhanced near (*<sup>α</sup>*R, *<sup>α</sup>*D)=(*π*/2, 0) and (0, *<sup>π</sup>*/2). In contrast, the factor *dn*ˆ is much smaller as shown in the linear plot for *α*D = 0. If one chose AB phase *φ* = 5*π*/3, *d*−*n*<sup>ˆ</sup> is suppressed and alternatively *dn*ˆ is enhanced near (*<sup>α</sup>*R, *<sup>α</sup>*D)=(*π*/2, 0) and (0, *π*/2) (with changing sign of the data in the left Figure 4). The AB phase *φ* and site energy 0 dependence of *d*±*n*<sup>ˆ</sup> are shown in the left and right of Figure 5, respectively. Therefore, a large contrast of the QAP in two spin directions can be obtained by choosing *φ* = *π*/3 and 0 = 0.9*j*.

**Figure 4.** (**Left**) Contour plot of the function *d*−*n*<sup>ˆ</sup> for 0 = 0.9*j* and *φ* = *π*/3. (**Right**) Line plot of the functions *d*±*n*<sup>ˆ</sup>as a function of *α*Rwith *α*D= 0.

**Figure 5.** (**Left**) AB phase dependence of the function *d*±*n*<sup>ˆ</sup> for 0 = 0.9*j* and *α*R = *π*/2, *α*D = 0. (**Right**) Site energy dependence of the function *d*±*n*<sup>ˆ</sup> for *f* = *π*/3 and *α*R = *π*/2, *α*D = 0.
