**2. Model System**

In the leads *u* ≤ −1, *u* ≥ 2, we set  *u* = 0 and *W*ˆ *uv* = <sup>−</sup>*j*I, where I is the two-dimensional unit matrix and the real hopping parameter *j* is only nonzero for nearest-neighbor pair of *u*, *v*. With a standard treatment on the tight-binding Hamiltonian, we obtain the eigen-energy  *k* ≡ −2*j* cos(*ka*) and its corresponding eigen-function, |*ψ*(*u*) ∝ *eikau* |*χ*, where *k* is a real wave-number parameter, *a* (> 0) is the lattice constant and |*χ* is a certain state vector.

The system of interferometer is represented between *u* = 0 and *u* = 1 sites and we choose 0 = *y*0 and 1 = *y*1 and *W*ˆ 01 ≡ *W*ˆ and *W*ˆ 10 = *W*ˆ †. The microscopic derivations of *y*0, *y*1 and *W*ˆ for a diamond-shaped interferometer are demonstrated in Section 5. Then the Schrödinger equations at sites *u* = 0, 1 read

$$y\_0 \left| \psi(0) \right\rangle + \hat{W} \left| \psi(1) \right\rangle - j \left| \psi(-1) \right\rangle \quad = \left. \epsilon \left| \psi(0) \right\rangle \right\rangle,\tag{3}$$

$$
\left< \left. y\_1 \left| \psi(1) \right> + \dot{\mathcal{W}}^\dagger \left| \psi(0) \right> - j \left| \psi(2) \right> \right. \\
\left. \left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left| \psi(1) \right> \right. \\
\left. = \left. \epsilon \left$$

The reflection and transmission amplitude matrices for the electron flux with an energy  =  *k* injected from the left lead is

$$\hat{\boldsymbol{\tau}}^{\dagger} \quad = \; \left[ -\mathbb{I} + i\eta\_{k} \mathbf{X}\_{1} \left[ \mathbb{I} \mathbf{Y} - \hat{\boldsymbol{\mathcal{W}}} \hat{\boldsymbol{\mathcal{W}}}^{\dagger} \right]^{-1} \right] \tag{5}$$

$$\hat{\mathbf{I}}^{\dagger} \quad = \quad i\eta\_k \hat{\mathbf{W}}^{\dagger} \left[ \mathbb{I} \mathbf{Y} - \hat{\mathbf{W}} \hat{\mathbf{W}}^{\dagger} \right]^{-1} \text{.} \tag{6}$$

where *Y* ≡ *X*0*X*1 with complex parameters *Xu* ≡  *k* − *yu* + *jeika* (*u* = 0, 1) and we introduced a parameter of energy dimension *ηk* ≡ 2*j* sin(*ka*). The reflection and transmission amplitude matrices for the electron flux injected from the right lead is

$$\begin{array}{rcl}\mathfrak{H}' &=& -\mathbb{I} + i\eta\_{k}\boldsymbol{X}\_{0}\left[\mathbb{I}\boldsymbol{Y} - \mathcal{W}^{\dagger}\boldsymbol{W}\right]^{-1}, \end{array} \tag{7}$$

$$\dot{\mathcal{H}}^{\prime} \quad = \quad i\eta\_k \mathcal{W} \left[ \mathbb{I}Y - \mathcal{W}^{\dagger} \mathcal{W} \right]^{-1} . \tag{8}$$

The details of the derivation of these formulae are given in Appendix A. In the next section, the obtained scattering amplitude matrices are diagonalized and the formulae of the scattering amplitude eigenvalues are given. Then in Section 4, the Berry curvatures for two spin eigenstates, Equations (34) and (35), is given, which allow calculation of QAP spin per cycle.

### **3. Diagonalization of Hopping Operator** *W* **ˆ** *W* **ˆ †**

In this section, we diagonalize the product of hopping operators *W* ˆ and *W* ˆ † appearing in the scattering amplitude matrices derived in the previous section. Then we obtain the scattering eigenstates through an interferometer. This is an extension of the discussion in Reference [3]. We consider an interferometer in *<sup>x</sup>*-*y* plane made of two one-dimensional arms, *b* and *c*, represented by real coupling parameters *γb*, *γc* and 2 × 2 unitary matrices, *U* ˆ *b* and *U* ˆ *c*, showing propagation from the site 0 to 1 via the arms *b* and *c*, respectively. We assume following general expressions characterizing the effect of AB phase and Rashba or Dresselhaus SOI:

$$
\hat{\mathcal{U}}\_b \quad = \quad \varepsilon^{-i\phi\_1} \left( \mathbb{I}\delta + i\boldsymbol{\pi} \cdot \boldsymbol{\mathcal{O}} \right),
\tag{9}
$$

$$\hat{\mathcal{U}}\_{\mathfrak{c}} \; \; \; \; \; \; \; \; \mathcal{E} \; \; \; \; \mathcal{U} \; \; \mathcal{S} \; \; \; \mathcal{U} \; \; \; \mathcal{U} \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \mathcal{U} \; \; \; \; \; \; \; \mathcal{U} \; \; \; \; \; \; \; \; \; \mathcal{U} \; \; \; \; \; \; \; \; \;$$

where *σ*ˆ is the vector of Pauli spin matrices. *φ* ≡ *φ*1 + *φ*2 = <sup>2</sup>*π*(*HS*)/Φ0 is the AB phase with the magnetic field *H* in the *z* direction, the area of the interferometer *S*, and a magnetic flux quantum Φ0. Unitarity condition requires the real parameters, *δ*, *δ* and real three-dimensional vectors *τ*, *τ* to obey *δ*2 + |*τ*|<sup>2</sup> = *δ*<sup>2</sup> + |*τ*|<sup>2</sup> = 1. The hopping matrix *W*ˆ is given by

$$
\hat{\mathcal{W}}\_{-} = \gamma\_b \hat{\mathcal{U}}\_b + \gamma\_c \hat{\mathcal{U}}\_c. \tag{11}
$$

As shown in Appendix B, the matrix factor appearing in the scattering amplitudes for the electron flux injected from the left lead, Equation (5), is

$$
\mathcal{W}\mathcal{W}^\dagger \equiv \ A\mathbb{I} + \mathbf{B} \cdot \mathfrak{G} \,\,\,\,\,\tag{12}
$$

where

$$A \quad = \quad \gamma\_b^2 + \gamma\_c^2 + 2\gamma\_b\gamma\_c\cos\phi\cos\omega,\tag{13}$$

$$\mathcal{B}\_{\;\;\;\;\phi} = \,\,\, 2\gamma\_b \gamma\_c \sin\phi \sin\omega \,\,\, \text{\,\,in}\,\,\,\tag{14}$$

The real parameter *ω* is determined from cos *ω* ≡ *δδ* + *τ* · *τ* and the unit vector *n*ˆ is defined by

$$\mathfrak{H}\_{\omega} = \frac{1}{\sqrt{1 - \mathfrak{r}\_z^2}} (-\mathfrak{r}\_{y\_\prime} \,\, \mathfrak{r}\_{x\_\prime} \delta). \tag{15}$$

We then introduce two normalized eigenstates of the operator ˆ*n* · *σ*ˆ , |*n*<sup>ˆ</sup> and |−*n*<sup>ˆ</sup> such that

$$
\left|\mathfrak{h}\cdot\mathfrak{d}^\*\left|\mathfrak{h}\right>\right.\tag{16}
$$

*n* ˆ · *σ* ˆ |−*n*<sup>ˆ</sup> = − |−*n*<sup>ˆ</sup>. (17)

Clearly, these are also the eigenstates of the operator *W* ˆ *W* ˆ † such that

$$
\left| \hat{\mathcal{W}} \hat{\mathcal{W}}^{\dagger} \left| \pm \hat{\mathfrak{n}} \right> \right. \\
\left. \quad = \quad \lambda\_{\pm} \left| \pm \hat{\mathfrak{n}} \right> , \tag{18}
$$

with the eigenvalues

$$
\lambda\_{\pm} = \gamma\_b^2 + \gamma\_c^2 + 2\gamma\_b\gamma\_c\cos(\phi \mp \omega). \tag{19}
$$

These eigenvalues are positive since *λ*± = ±*n*<sup>ˆ</sup>| *W*ˆ *W*ˆ † |±*n*<sup>ˆ</sup> = *W*ˆ † |±*n*<sup>ˆ</sup> 2 ≥ 0.

To study the scattering eigenstates for the electron flux injected from the right, Equation (7), we evaluate *W* ˆ †*W*<sup>ˆ</sup> with similar procedure as above,

$$
\mathcal{W}^\dagger \mathcal{W} \quad \equiv \quad A\mathbb{I} + \mathcal{B}' \cdot \hat{\sigma}, \tag{20}
$$

where

$$\mathbf{B}^{\prime} \quad = \ 2\gamma\_b \gamma\_{\hat{c}} \sin \phi \sin \omega \ \hat{n}^{\prime},\tag{21}$$

and corresponding unit vector

$$\hat{\mathfrak{n}}' = \frac{1}{\sqrt{1 - \tau\_z^2}} (\tau\_{y\prime} - \tau\_{x\prime}\delta). \tag{22}$$

Then we introduce two normalized eigenstates of the operator ˆ*n* · *σ*ˆ , |±*n*<sup>ˆ</sup>, which obey

$$
\left| \hat{\mathcal{W}}^{\dagger} \hat{\mathcal{W}} \left| \pm \mathfrak{H}' \right> \right. \\
\left. \quad = \quad \lambda\_{\pm} \left| \pm \mathfrak{H}' \right> , \tag{23}
$$

with the same eigenvalues as Equation (18).

The elements of the scattering matrix are now explicitly evaluated with the obtained scattering eigenstates. As detailed in Appendix B, we can show that the transmission amplitude matrices are

$$\left|\hat{\mathbf{f}}\right| = \left|\mathbf{t} + \left|\hat{\mathfrak{n}}'\right>\left<\mathfrak{n}\right| + \left|\mathbf{t} - \left|-\hat{\mathfrak{n}}'\right>\left<-\mathfrak{n}\right|\right>,\tag{24}$$

$$\left|\hat{\mathfrak{f}}\right\rangle\_{-}=\left|t+\left|\mathfrak{h}\right\rangle\left\langle\mathfrak{h}'\right|+\left|t-\left|-\mathfrak{h}\right\rangle\left\langle-\mathfrak{h}'\right|,\tag{25}$$

where we defined two transmission amplitudes,

$$t\_{\pm} \quad \equiv \quad \frac{i\eta\_k \sqrt{\lambda\_{\pm}}}{Y - \lambda\_{\pm}}. \tag{26}$$

Similarly, the reflection amplitude matrices are given by

$$\left|\hat{r}\right| = \left|r\_{+}\left|\hat{\mathfrak{n}}\right>\left<\hat{\mathfrak{n}}\right| + r\_{-}\left|-\hat{\mathfrak{n}}\right>\left<-\hat{\mathfrak{n}}\right|,\tag{27}$$

$$
\hat{r}' \quad = \quad r'\_+ \left| \hat{\mathfrak{n}}' \right> \left< \hat{\mathfrak{n}}' \right| + r'\_- \left| -\hat{\mathfrak{n}}' \right> \left< -\hat{\mathfrak{n}}' \right|, \tag{28}
$$

where the reflection amplitudes are

$$r\_{\pm} \quad \equiv \quad -1 + \frac{i\eta\_k X\_1}{Y - \lambda\_{\pm}} ,\tag{29}$$

$$r'\_{\pm} \equiv -1 + \frac{i\eta\_k X\_0}{Y - \lambda\_{\mp}}.\tag{30}$$

The unitarity condition of the scattering amplitude matrices, ˆ*t*† ˆ*t* + *r*ˆ†*r*<sup>ˆ</sup> = 1, is confirmed in Appendix C. The unitarity condition ˆ*t*† ˆ*t* + *r*<sup>ˆ</sup>†*r*<sup>ˆ</sup> = 1 can also be checked.
