**Appendix B. Scattering Eigenstates**

In this appendix, we show the details of the calculations of the scattering eigenstates discussed in Section 3. First, we evaluate

$$\begin{aligned} \mathcal{W}\mathcal{W}^{\dagger} &= (\gamma\_b \mathcal{U}\_b + \gamma\_c \mathcal{U}\_c)(\gamma\_b \mathcal{U}\_b^{\dagger} + \gamma\_c \mathcal{U}\_c^{\dagger}) \\ &= (\gamma\_b^2 + \gamma\_c^2) \mathbb{I} + \gamma\_b \gamma\_c (\mathcal{U} + \mathcal{U}^{\dagger}), \end{aligned} \tag{A21}$$

with *u*ˆ ≡ *U* ˆ *bU* ˆ † *c* representing the total development around the interferometer in the order 0 → *b* → 1 → *c* → 0. We study following matrix

$$\begin{split} \hat{\mathfrak{a}} &= \ & \left\{ \begin{aligned} & \mathfrak{a} = e^{-i\phi} \left( \mathbb{I}\delta + i\mathfrak{r}\cdot\hat{\mathfrak{e}} \right) \left( \mathbb{I}\delta' - i\mathfrak{r}'\cdot\hat{\mathfrak{e}} \right) \\ & \quad = & e^{-i\phi} \left\{ \mathbb{I}\delta\delta' - i\delta\mathfrak{r}'\cdot\hat{\mathfrak{e}} + i\delta'\mathfrak{r}\cdot\hat{\mathfrak{e}} + (\mathfrak{r}\cdot\hat{\mathfrak{e}})(\mathfrak{r}'\cdot\hat{\mathfrak{e}}) \right\} \\ & \quad = & e^{-i\phi} \left\{ \mathbb{I}\delta\delta' + i(\delta'\mathfrak{r} - \delta\mathfrak{r}')\cdot\hat{\mathfrak{e}} + \mathbb{I}\mathfrak{r}\cdot\mathfrak{r}' + i(\mathfrak{r}\times\mathfrak{r}')\cdot\hat{\mathfrak{e}} \right\}. \end{split}$$

> We introduce a unit vector defined by

$$\mathfrak{n} \equiv \mathcal{N} \left\{ \delta' \mathfrak{r} - \delta \mathfrak{r}' + \mathfrak{r} \times \mathfrak{r}' \right\} = (n\_{x\_{\prime}} n\_{y\_{\prime}} n\_{z}),\tag{A22}$$

where N is a normalization constant. We define *z*-direction (in spin space) in parallel to *δτ* − *δτ*, namely

$$\mathcal{N}\left\{\delta'\mathfrak{r}-\delta\mathfrak{r}'\right\} \equiv \ n\_{\tilde{z}}\mathfrak{z}\_{\prime} \tag{A23}$$

with a unit vector *z*ˆ in *z*-direction. Since *τ* × *τ* is orthogonal to *δτ* − *δτ*, we set

$$\mathcal{N}\left\{\mathfrak{r}\times\mathfrak{r}^{\prime}\right\}\quad\equiv\quad n\_{\mathfrak{x}}\mathfrak{x}+n\_{\mathfrak{y}}\mathfrak{y},\tag{A24}$$

with unit vectors in *x*, *y*-directions. Solving Equation (A23) for *<sup>τ</sup>*, we have

$$
\pi^{\prime} \quad = \quad \frac{\delta^{\prime}}{\delta}\pi - \frac{n\_z}{\delta \mathcal{N}}\hat{\mathfrak{z}}\_{\prime} \tag{A25}
$$

and hence

*τ* × *τ* = − *nz δ*N (*τ* × *<sup>z</sup>*<sup>ˆ</sup>). (A26)

Using this, we obtain for *τ* = (*<sup>τ</sup>x*, *<sup>τ</sup>y*, *<sup>τ</sup>z*),

$$m\_{\mathbf{x}} = -\pounds \cdot \mathcal{N} \left\{ \mathbf{r} \times \mathbf{r}' \right\} = -\frac{n\_z}{\delta} \tau\_y. \tag{A27}$$

Similarly, we also have *ny* = *nzδτx*. Normalization condition requires

$$1 \quad = \quad \left(-\frac{n\_z}{\delta}\tau\_y\right)^2 + \left(\frac{n\_z}{\delta}\tau\_x\right)^2 + n\_z^2 = \frac{1-\tau\_z^2}{\delta^2}n\_{z'}^2\tag{A28}$$

hence we determine

$$m\_z = \frac{\delta}{\sqrt{1-\tau\_z^2}}\tag{A29}$$

and

$$
\hat{\mathfrak{n}}\_{\perp} = \frac{1}{\sqrt{1 - \tau\_z^2}} (-\tau\_{\mathfrak{y}\prime} \tau\_{\mathfrak{x}\prime} \delta). \tag{A30}
$$

Therefore,

$$\begin{split} \hat{\mathfrak{a}} &= \ \mathfrak{e}^{-i\phi} \left\{ \mathbb{I} \left( \delta \delta' + \mathfrak{r} \cdot \mathfrak{r}' \right) + i \frac{1}{\mathcal{N}} \mathfrak{a} \cdot \mathfrak{r} \right\} \\ &\equiv \ \mathfrak{e}^{-i\phi} \left\{ \mathbb{I} \cos \omega + i \sin \omega \mathfrak{r} \cdot \mathfrak{r} \right\} , \end{split} \tag{A31}$$

where the real parameter *ω* is determined from cos *ω* ≡ *δδ* + *τ* · *<sup>τ</sup>*. The unitarity condition of *u*ˆ can be checked by noting

$$\begin{split} \frac{1}{\mathcal{N}^2} &= \quad \left| \delta' \mathbf{\dot{r}} - \delta \mathbf{\dot{r}}' + \mathbf{\tau} \times \mathbf{\dot{\tau}'} \right|^2 \\ &= \quad \left| \delta' \mathbf{\dot{r}} - \delta \mathbf{\dot{r}}' \right|^2 + (\mathbf{\dot{r}} \times \mathbf{\dot{\tau}'}) \cdot (\mathbf{\dot{r}} \times \mathbf{\dot{\tau}'}) \\ &= \quad \delta^2 + \delta'^2 - 2\delta \delta' \mathbf{\dot{\tau}} \cdot \mathbf{\dot{\tau}'} + (1 - \delta^2)(1 - \delta'^2) - (\mathbf{\dot{\tau}} \cdot \mathbf{\dot{\tau}'})^2 \\ &= \quad 1 - (\delta \delta' + \mathbf{\dot{\tau}} \cdot \mathbf{\dot{\tau}'})^2 = 1 - \cos^2 \omega, \tag{A32} \end{split} \tag{A32}$$

where we used the relation (*τ* × *τ*) · (*τ* × *τ*) = |*τ*|<sup>2</sup> |*τ*|<sup>2</sup> − (*τ* · *τ*)2. Now, the operator *u*ˆ + *u*ˆ† is calculated as

$$\begin{array}{rcl} \hat{\mathfrak{u}} + \hat{\mathfrak{u}}^{\dagger} &=& 2\cos\phi\cos\omega\mathbb{I} + 2\sin\phi\sin\omega\mathbb{i} \cdot \hat{\mathfrak{v}}, \end{array} \tag{A33}$$

hence

$$\begin{split} \boldsymbol{\mathcal{W}} \boldsymbol{\mathcal{W}}^{\dagger} &= \, \, \vert \, \gamma\_{\flat}^{2} + \gamma\_{\varepsilon}^{2} \rangle \mathbb{I} + 2 \gamma\_{\varnothing} \gamma\_{\varepsilon} (\cos \phi \cos \omega \mathbb{I} + \sin \phi \sin \omega \mathbb{i} \cdot \boldsymbol{\mathcal{\sigma}}) \\ &\equiv \, \, \, \, \, \, \, \, \, \mathrm{A} \mathbb{I} + \mathbf{B} \cdot \boldsymbol{\mathcal{\sigma}}. \end{split} \tag{A34}$$

We defined

$$A\_{\quad} = \,\_1\gamma\_b^2 + \gamma\_c^2 + 2\gamma\_b\gamma\_c\cos\phi\cos\omega\_{\,\,t} \tag{A35}$$

$$\mathbf{B} \quad = \ \ 2\gamma\_b \gamma\_c \sin\phi \sin\omega \mathbf{\hat{n}}.\tag{A36}$$

Alternatively, we evaluate

$$\begin{split} \hat{\mathcal{W}}^{\dagger} \hat{\mathcal{W}} &= \, \left( \gamma\_{b} \hat{\mathcal{U}}\_{b}^{\dagger} + \gamma\_{c} \hat{\mathcal{U}}\_{c}^{\dagger} \right) (\gamma\_{b} \hat{\mathcal{U}}\_{b} + \gamma\_{c} \hat{\mathcal{U}}\_{c}) \\ &= \, \left( \gamma\_{b}^{2} + \gamma\_{c}^{2} \right) \mathbb{I} + \gamma\_{b} \gamma\_{c} (\hat{\mathcal{U}} + \hat{\mathcal{U}}^{\dagger}), \end{split} \tag{A37}$$

where *u*ˆ ≡ *U*ˆ †*c U*ˆ *b*, which represents the total development around the interferometer in the order 1 → *c* → 0 → *b* → 1. With similar procedure done for *u*ˆ, we have

$$\begin{split} \mathfrak{u}' &= \ \mathfrak{e}^{-i\phi} \left\{ \mathbb{I}\delta\delta' + i(\delta'\mathfrak{r} - \delta\mathfrak{r}') \cdot \mathfrak{e} + \mathbb{I}\mathfrak{r}' \cdot \mathfrak{r} + i(\mathfrak{r}' \times \mathfrak{r}) \cdot \mathfrak{e} \right\} \\ &= \ \mathfrak{e}^{-i\phi} \left\{ \mathbb{I}\cos\omega + i\sin\omega \mathfrak{u}' \cdot \mathfrak{e} \right\}, \end{split} \tag{A38}$$

where

$$\hat{\mathfrak{n}}' = \frac{1}{\sqrt{1-\tau\_z^2}}(\tau\_{\mathcal{Y}'} - \tau\_{\mathcal{X}}\delta),\tag{A39}$$

and with the same *ω* as before. Now, by calculating the factor *u*ˆ + *u*<sup>ˆ</sup>†, we obtain

$$\begin{aligned} \mathcal{W}^\dagger \mathcal{W} &= \begin{aligned} (\gamma\_b^2 + \gamma\_c^2) \mathbb{I} + 2\gamma\_b \gamma\_c (\cos\phi \cos\omega \mathbb{I} + \sin\phi \sin\omega \mathbb{i}' \cdot \hat{\sigma}) \\ &\equiv \begin{array}{c} A \mathbb{I} + \mathbb{B}' \cdot \hat{\sigma}, \end{array} \end{aligned} \tag{A40}$$

where

$$\mathbf{B}^{\prime} \quad = \ \ 2\gamma\_b \gamma\_c \sin\phi \sin\omega \mathbf{n}^{\prime}.\tag{A41}$$

As shown in the main text, we introduce two sets of normalized eigenstates of the operators *n*ˆ · *σ*ˆ and ˆ*n* · *σ*ˆ , |±*n*<sup>ˆ</sup> and |±*n*<sup>ˆ</sup>, which obey

$$\begin{aligned} \left| \hat{\mathcal{W}} \hat{\mathcal{W}}^{\dagger} \right| \pm \left| \mathfrak{H} \right> &= \left| \lambda \pm \left| \pm \mathfrak{H} \right> ,\\ \left| \hat{\mathcal{W}}^{\dagger} \hat{\mathcal{W}} \right| \pm \mathfrak{H}' \rangle &= \left| \lambda \pm \left| \pm \mathfrak{H}' \right> , \end{aligned} \tag{A42}$$

with the same eigenvalues as Equation (19).

> It can be shown that these two sets of eigenstates {|±*n*<sup>ˆ</sup>, |±*n*<sup>ˆ</sup>} are related with each other by

$$\left| \pm \hbar' \right\rangle \quad = \begin{array}{c} \frac{1}{\sqrt{\lambda\_{\pm}}} \hat{\mathcal{W}}^{\dagger} \left| \pm \hbar \right\rangle \end{array} \tag{A43}$$

with double-sign correspondence. This can be checked by the eigen-equation

$$
\begin{split}
\langle \hat{\mathcal{W}}^{\dagger} \hat{\mathcal{W}} \,|\pm \hbar' \rangle &= \begin{array}{c} \frac{1}{\sqrt{\lambda\_{\pm}}} \hat{\mathcal{W}}^{\dagger} (\hat{\mathcal{W}} \hat{\mathcal{W}}^{\dagger}) \,|\pm \hbar \rangle \\ &= \frac{1}{\sqrt{\lambda\_{\pm}}} \hat{\mathcal{W}}^{\dagger} \lambda\_{\pm} \,|\pm \hbar \rangle \\ &= \, \, \lambda\_{\pm} \,|\pm \hbar' \rangle , \end{split} \tag{A44}
$$

and the normalization condition

$$
\begin{split}
\langle \pm \mathfrak{H}' | \pm \mathfrak{H}' \rangle &= \frac{1}{\lambda \pm} \langle \pm \mathfrak{H} | \hat{W} \hat{W}^{\dagger} | \pm \mathfrak{H} \rangle \\ &= \langle \pm \mathfrak{H} | \pm \mathfrak{H} \rangle = 1.
\end{split}
\tag{A45}
$$

Similarly, we can also prove the relation

$$\left| \pm \hbar \right\rangle \quad = \begin{array}{c} \frac{1}{\sqrt{\lambda\_{\pm}}} \hat{\mathcal{W}} \left| \pm \hbar' \right\rangle. \tag{A46}$$

We have the spectral decomposition of the matrices *W* ˆ † and *W*ˆ by

$$
\hat{\mathcal{W}}^{\dagger} \quad = \sqrt{\lambda\_{+}} \left| \hat{\mathfrak{n}}' \right\rangle \left\langle \hat{\mathfrak{n}} \right| + \sqrt{\lambda\_{-}} \left| -\hat{\mathfrak{n}}' \right\rangle \left\langle -\hat{\mathfrak{n}} \right|, \tag{A47}
$$

$$
\hat{\mathcal{W}}\_{-} = \begin{array}{c c} \sqrt{\lambda\_{+}} \left| \mathfrak{H} \right> \left< \mathfrak{H}' \right| + \sqrt{\lambda\_{-}} \left| -\mathfrak{H} \right> \left< -\mathfrak{H}' \right|, \end{array} \tag{A48}
$$

where the second relation is obtained by taking Hermite conjugate of the first relation.

Now, let us turn to discuss the scattering wavefunctions using these eigenstates. For the left incoming states, we choose |*<sup>χ</sup>*in = |±*n*<sup>ˆ</sup>, then the transmitted states are

$$\begin{aligned} \left| \chi\_{\mathbf{t}, \pm} \right\rangle &= \left| \hat{\mathbf{t}} \right| \pm \hbar \rangle \\ &= \left| i \eta\_k \hat{\mathbf{W}}^{\dagger} \left[ \Pi \mathbf{Y} - \hat{\mathbf{W}} \hat{\mathbf{W}}^{\dagger} \right]^{-1} \right| \pm \hbar \rangle \\ &= \left| i \eta\_k \hat{\mathbf{W}}^{\dagger} \left[ \Pi \mathbf{Y} - \Pi \lambda\_{\pm} \right]^{-1} \right| \pm \hbar \rangle \\ &= \left| \frac{i \eta\_k}{Y - \lambda\_{\pm}} \hat{\mathbf{W}}^{\dagger} \left| \pm \hbar \right\rangle \\ &= \left| \frac{i \eta\_k}{Y - \lambda\_{\pm}} \sqrt{\lambda\_{\pm}} \mid \pm \hbar' \right\rangle \\ &= \left| \pm \mid \pm \hbar' \right\rangle, \end{aligned} \tag{A49}$$

where we used Equations (5) and (A47) and defined two transmission amplitudes,

$$t\_{\pm} \equiv \begin{array}{c} i\eta\_{k}\sqrt{\lambda\_{\pm}}\\ \underline{Y}-\lambda\_{\pm} \end{array} \tag{A50}$$

Using the orthogonality of the eigenstates, the transmission amplitude matrix ˆ*t* is expressed as in Equation (24). Similarly, the reflected states are

$$<\langle \chi\_{\mathbf{r}, \pm} \rangle \quad = \quad \hat{r} \left| \pm \hbar \right\rangle = r\_{\pm} \left| \pm \hbar \right\rangle \,\,,\tag{A51}$$

where the reflection amplitudes are

$$r\_{\pm} \equiv -1 + \frac{i\eta\_k X\_1}{Y - \lambda\_{\mp}}.\tag{A52}$$

The reflection amplitude matrix *r*ˆ is diagonal and is given in Equation (27). Similarly, for the right incoming states, the transmitted states are

$$\left|\chi\_{\mathfrak{t},\pm}^{\prime}\right\rangle \quad = \quad \hat{\mathfrak{t}}^{\prime} \left|\pm \mathfrak{n}^{\prime}\right\rangle = \mathfrak{t}\_{\pm} \left|\pm \mathfrak{n}\right\rangle \,,\tag{A53}$$

and hence the transmission amplitude matrix ˆ*t* is given in Equation (25). The reflected states are

$$\left|\chi\_{\mathbf{r},\pm}^{\prime}\right\rangle\_{\mathbf{r}} = \left|\mathbf{r}^{\prime}\right| \left|\pm \mathbf{\hat{n}}^{\prime}\right\rangle = r\_{\pm}^{\prime} \left|\pm \mathbf{\hat{n}}^{\prime}\right\rangle,\tag{A54}$$

where we defined

$$r'\_{\pm} \equiv -1 + \frac{i\eta\_k X\_0}{Y - \lambda\_{\pm}},\tag{A55}$$

and hence the reflection amplitude matrix *r*ˆ is given in Equation (28).

### **Appendix C. Unitarity of the Scattering Matrix**

The scattering matrix needs to satisfy the unitarity condition, such that

$$
\hat{\mathfrak{l}}^{\dagger}\hat{\mathfrak{f}} + \hat{\mathfrak{r}}^{\dagger}\hat{\mathfrak{r}}^{\dagger} = \stackrel{\tag{A56}}{\text{I.}} \tag{A56}
$$

Using the results of Equations (24) and (27),

$$\left|\hat{\mathfrak{H}}^{\dagger}\hat{\mathfrak{H}} + \hat{\mathfrak{r}}^{\dagger}\hat{\mathfrak{r}}\right| = \left| |t\_{+}|^{2} \left| \hat{\mathfrak{H}} \right\rangle \left\langle \hat{\mathfrak{n}} \right| + \left| t\_{-} \right|^{2} \left| -\hat{\mathfrak{n}} \right\rangle \left\langle -\hat{\mathfrak{n}} \right| + \left| r\_{+} \right|^{2} \left| \hat{\mathfrak{n}} \right\rangle \left\langle \hat{\mathfrak{n}} \right| + \left| r\_{-} \right|^{2} \left| -\hat{\mathfrak{n}} \right\rangle \left\langle -\hat{\mathfrak{n}} \right|.$$

Therefore, if |*t*±|<sup>2</sup> + |*r*±|<sup>2</sup> = 1, using the completeness relation of |±*n*<sup>ˆ</sup>, the unitarity is confirmed. Let us check this:

$$\begin{split} &|\boldsymbol{t}\_{\pm}|^{2} + |\boldsymbol{r}\_{\pm}|^{2} = \left| \frac{i\eta\_{k}\sqrt{\lambda\_{\pm}}}{Y - \lambda\_{\pm}} \right|^{2} + \left| -1 + \frac{i\eta\_{k}X\_{1}}{Y - \lambda\_{\pm}} \right|^{2} \\ &= \frac{\eta\_{k}^{2}\lambda\_{\pm}}{\left| Y - \lambda\_{\pm} \right|^{2}} + 1 + \frac{i\eta\_{k}X\_{1}^{\*}}{Y^{\*} - \lambda\_{\pm}} - \frac{i\eta\_{k}X\_{1}}{Y - \lambda\_{\pm}} + \frac{\eta\_{k}^{2}\left| X\_{1} \right|^{2}}{\left| Y - \lambda\_{\pm} \right|^{2}} \\ &= \left| 1 + i\eta\_{k}\frac{X\_{1}^{\*}(Y - \lambda\_{\pm}) - X\_{1}(Y^{\*} - \lambda\_{\pm})}{\left| Y - \lambda\_{\pm} \right|^{2}} + \frac{\eta\_{k}^{2}(\lambda\_{\pm} + |X\_{1}|^{2})}{\left| Y - \lambda\_{\pm} \right|^{2}} \\ &= \left| 1 + \frac{\eta\_{k}}{\left| Y - \lambda\_{\pm} \right|^{2}} \right| \left[ i\left\{ X\_{1}^{\*}(Y - \lambda\_{\pm}) - X\_{1}(Y^{\*} - \lambda\_{\pm}) \right\} + \eta\_{k}(\lambda\_{\pm} + |X\_{1}|^{2}) \right]. \end{split}$$

The factor in the square bracket is

$$\begin{aligned} \left[\bullet\right] &=&-i(X\_1^\*-X\_1)\lambda\_\pm + i(X\_0\left|X\_1\right|^2 - X\_0^\*\left|X\_1\right|^2) + \eta\_k(\lambda\_\pm + \left|X\_1\right|^2) \\ &=&\left[-i(-i\eta\_k) + \eta\_k\right]\lambda\_\pm + \left[i(i\eta\_k) + \eta\_k\right]\left|X\_1\right|^2 = 0,\end{aligned}$$

hence |*t*±|<sup>2</sup> + |*r*±|<sup>2</sup> = 1 is confirmed.

### **Appendix D. Derivatives of the Scattering Amplitude Matrices**

In this Appendix, we evaluate the Berry curvature, Equation (33), with two control parameters, *g*1, *g*2, which only modify the scattering eigenvalues *λ*± and corresponding eigenvectors |±*n*<sup>ˆ</sup>, |±*n*<sup>ˆ</sup>. We need to calculate the derivatives of the scattering amplitude matrices by a control parameter (*g* = *g*1, *g*2), *∂r*<sup>ˆ</sup> *∂g* and *∂*<sup>ˆ</sup>*t ∂g* . Since scattering amplitude matrices are expressed with the eigenstates as shown in Equations (24) and (27), we first evaluate the first order derivative of the basis states

$$
\frac{\partial}{\partial \mathbf{g}} \left( \begin{array}{c} \vert \dot{\mathfrak{H}} \rangle \\\vert - \dot{\mathfrak{H}} \rangle \end{array} \right)\_{\mathbf{i}} = \left( \begin{array}{c} a\_{\mathcal{S}} & b\_{\mathcal{S}} \\\ \tilde{b}\_{\mathcal{S}} & \tilde{a}\_{\mathcal{S}} \end{array} \right) \left( \begin{array}{c} \vert \dot{\mathfrak{H}} \rangle \\\ \vert - \dot{\mathfrak{H}} \rangle \end{array} \right)\_{\mathbf{i}} \tag{A57}
$$

$$
\frac{\partial}{\partial \mathbf{g}} \left( \begin{array}{c} \left| \mathfrak{H}' \right> \\ \left| -\mathfrak{H}' \right> \end{array} \right) \quad = \quad \left( \begin{array}{c} a\_{\mathcal{S}'} & b\_{\mathcal{S}'} \\ \tilde{b}\_{\mathcal{S}'} & \tilde{a}\_{\mathcal{S}'} \end{array} \right) \left( \begin{array}{c} \left| \mathfrak{H}' \right> \\ \left| -\mathfrak{H}' \right> \end{array} \right) . \tag{A58}$$

Since the basis states are normalized,

$$\frac{\partial}{\partial \boldsymbol{\mathfrak{g}}} \left< \boldsymbol{\mathfrak{h}} \middle| \boldsymbol{\mathfrak{h}} \right> \quad = \quad \left( \frac{\partial}{\partial \boldsymbol{\mathfrak{g}}} \middle| \boldsymbol{\mathfrak{h}} \right) \left| \boldsymbol{\mathfrak{h}} \right> + \left< \boldsymbol{\mathfrak{h}} \middle| \begin{array}{c} \boldsymbol{\mathfrak{h}} \middle| \boldsymbol{\mathfrak{h}} \end{array} \right> = a\_{\mathcal{S}}^{\*} + a\_{\mathcal{S}} = 0,\tag{A59}$$

and *ag* should be pure imaginary. Similarly, *<sup>a</sup>*˜*g*, *ag* and *a*˜*g* are also pure imaginary. Using the orthogonality condition, we have the relation

$$\frac{\partial}{\partial \mathbf{g}} \left< \hat{\mathfrak{n}} \right| - \hat{\mathfrak{n}} \right>\_{\mathbf{r}} = \left( \frac{\partial \left< \hat{\mathfrak{n}} \right|}{\partial \mathbf{g}} \right) \left| - \hat{\mathfrak{n}} \right> + \left< \hat{\mathfrak{n}} \right| \frac{\left< \hat{\mathfrak{n}} \right> - \hat{\mathfrak{n}}}{\partial \mathbf{g}} = b\_{\mathfrak{g}}^{\*} + \tilde{b}\_{\mathfrak{g}} = 0,\tag{A60}$$

and *b*∗*g* + ˜ *bg* = 0.

To have the formula for *ag*, *bg* and ˜ *bg*, we consider a unit vector *n*ˆ ≡ (*nx*, *ny*, *nz*) = (sin Θ cos Φ, sin Θ sin Φ, cos <sup>Θ</sup>), with angles Θ and Φ. Since the operator ˆ*n* · *σ*ˆ in the matrix form,

$$
\hat{\mathfrak{H}} \cdot \hat{\mathfrak{G}}^{\ \ \ \ \ \hat{\mathfrak{G}}} \equiv \begin{pmatrix}
\cos \Theta & \sin \Theta e^{-i\Phi} \\
\sin \Theta e^{i\Phi} & -\cos \Theta
\end{pmatrix} \tag{A61}
$$

has two eigenvalues *λ* = ±1, the eigenvector in a form (*<sup>c</sup>*1, *<sup>c</sup>*2)*<sup>t</sup>* satisfies for *λ* = 1, (cos Θ − <sup>1</sup>)*<sup>c</sup>*1 + (sin <sup>Θ</sup>*e*<sup>−</sup>*i*<sup>Φ</sup>)*<sup>c</sup>*2 = 0 and normalization condition, and hence

$$|\mathfrak{n}\rangle\_{\mathfrak{n}} = \left(\frac{n\_x - i n\_y}{\sqrt{2(1 - n\_z)}}, \frac{\sqrt{1 - n\_z}}{\sqrt{2}}\right)^t,\tag{A62}$$

and for *λ* = −1, (cos Θ + <sup>1</sup>)*<sup>c</sup>*1 + (sin <sup>Θ</sup>*e*<sup>−</sup>*i*<sup>Φ</sup>)*<sup>c</sup>*2 = 0, hence

$$| -n \rangle \quad = \left( \frac{-n\_x + i n\_y}{\sqrt{2(1 + n\_z)}}, \frac{\sqrt{1 + n\_z}}{\sqrt{2}} \right)^t. \tag{A63}$$

Differentiation with *g* gives

$$\frac{\partial}{\partial \mathbf{g}} \left| \mathbf{n} \right> \quad = \quad \left( \frac{\frac{\partial \mathbf{n}\_x}{\partial \mathbf{g}} - i \frac{\partial \mathbf{n}\_y}{\partial \mathbf{g}}}{\sqrt{2(1 - n\_z)}} + \frac{(n\_x - i n\_y) \frac{\partial n\_z}{\partial \mathbf{g}}}{2 \sqrt{2} (1 - n\_z)^{3/2}} - \frac{\frac{\partial n\_x}{\partial \mathbf{g}}}{2 \sqrt{2} \sqrt{1 - n\_z}} \right)^{\mathbf{f}},\tag{A64}$$

and

$$\frac{\partial}{\partial g} \left| -n \right> \quad = \quad \left( \frac{-\frac{\partial n\_x}{\partial g} + i \frac{\partial n\_y}{\partial \xi}}{\sqrt{2(1 - n\_z)}} + \frac{(n\_x - i n\_y) \frac{\partial n\_z}{\partial \xi}}{2\sqrt{2}(1 + n\_z)^{3/2}} , \frac{\frac{\partial n\_x}{\partial \xi}}{2\sqrt{2}\sqrt{1 + n\_z}} \right)^t. \tag{A65}$$

Therefore,

$$a\_{\mathcal{S}} \equiv \langle n | \frac{\partial}{\partial \mathcal{g}} | n \rangle = \frac{i}{2(1 - n\_z)} \left\{ n\_y \frac{\partial n\_x}{\partial \mathcal{g}} - n\_x \frac{\partial n\_y}{\partial \mathcal{g}} \right\}\_{\text{plane}} \tag{A66}$$

$$\bar{a}\_{\mathcal{S}} \equiv \langle -\mathfrak{n} | \frac{\partial}{\partial \mathcal{g}} | -\mathfrak{n} \rangle = \frac{i}{2(1+n\_z)} \left\{ n\_y \frac{\partial n\_x}{\partial \mathcal{g}} - n\_x \frac{\partial n\_y}{\partial \mathcal{g}} \right\},\tag{A67}$$

$$\begin{split} \boldsymbol{b}\_{\mathcal{S}} & \equiv & \langle -\mathfrak{n} | \frac{\partial}{\partial \mathcal{g}} | \mathfrak{n} \rangle \\ &= & -\frac{1}{2\sqrt{1 - n\_z^2}} \left\{ i n\_{\mathcal{Y}} \frac{\partial n\_x}{\partial \mathcal{g}} - i n\_x \frac{\partial n\_y}{\partial \mathcal{g}} + \frac{\partial n\_z}{\partial \mathcal{g}} \right\}, \end{split} \tag{A68}$$

$$\begin{split} \left| \bar{b}\_{\mathcal{S}} \right| &\equiv \quad \langle n \vert \frac{\partial}{\partial \mathcal{g}} \vert -n \rangle = -b\_{\mathcal{S}}^{\*} \\ &= \quad \frac{1}{2\sqrt{1 - n\_z^2}} \left\{ -in\_{\mathcal{Y}} \frac{\partial n\_x}{\partial \mathcal{g}} + in\_x \frac{\partial n\_y}{\partial \mathcal{g}} + \frac{\partial n\_z}{\partial \mathcal{g}} \right\}. \end{split} \tag{A69}$$

We also evaluate *ag* , *a*˜*g* , *bg* and ˜ *bg* similarly. From Equations (A30) and (A39), we found that following relations hold: *ag* = *ag*, *a*˜*g* = *a*˜*g* and *bg* = *b*∗*g* and ˜ *bg* = <sup>−</sup>*bg*.

Using these relations, we evaluate the derivatives of the scattering amplitudes:

$$\begin{split} \frac{\partial \hat{\boldsymbol{\tau}}}{\partial \boldsymbol{\xi}} &= \quad \frac{\partial \boldsymbol{r}\_{+}}{\partial \boldsymbol{g}} \left| \dot{\boldsymbol{\mu}} \right> \left< \dot{\boldsymbol{\mu}} \right| + \boldsymbol{r}\_{+} \frac{\partial \left| \dot{\boldsymbol{\mu}} \right>}{\partial \boldsymbol{g}} \left< \dot{\boldsymbol{\mu}} \right| + \boldsymbol{r}\_{+} \left| \dot{\boldsymbol{\mu}} \right> \frac{\partial \left< \dot{\boldsymbol{\mu}} \right|}{\partial \boldsymbol{g}} \\ &+ \frac{\partial \boldsymbol{r}\_{-}}{\partial \boldsymbol{g}} \left| - \dot{\boldsymbol{\mu}} \right> \left< -\dot{\boldsymbol{\mu}} \right| + \boldsymbol{r}\_{-} \frac{\partial \left| - \dot{\boldsymbol{\mu}} \right>}{\partial \boldsymbol{g}} \left< -\dot{\boldsymbol{\mu}} \right| + \boldsymbol{r}\_{-} \left| - \dot{\boldsymbol{\mu}} \right| \frac{\partial \left< -\dot{\boldsymbol{\mu}} \right|}{\partial \boldsymbol{g}} \\ &\equiv \quad \left( \left| \dot{\boldsymbol{\mu}} \right>, \left| - \dot{\boldsymbol{\mu}} \right> \right) \dot{\boldsymbol{r}}\_{\mathcal{S}} \left( \begin{array}{c} \left| \dot{\boldsymbol{\mu}} \right| \\ \left< -\dot{\boldsymbol{\mu}} \right| \end{array} \right), \tag{A70} \tag{A70} \end{split} \tag{A70}$$

with defining a matrix

$$\mathfrak{f}\_{\mathfrak{F}\_{\mathfrak{F}}} = \left( \begin{array}{c} \frac{\partial r\_{+}}{\partial \mathfrak{F}} \\ (r\_{+} - r\_{-}) b\_{\mathfrak{F}} \end{array} \begin{array}{c} (r\_{+} - r\_{-}) b\_{\mathfrak{F}}^{\*} \\ \frac{\partial r\_{-}}{\partial \mathfrak{g}} \end{array} \right) . \tag{A71}$$

Similarly,

$$\begin{split} \frac{\partial \dot{\boldsymbol{\mu}}^{\prime}}{\partial \boldsymbol{\mathcal{S}}} &= \quad \frac{\partial t\_{+}}{\partial \boldsymbol{\mathcal{S}}} \left| \dot{\boldsymbol{\mathfrak{n}}} \right\rangle \left\langle \dot{\boldsymbol{\mathfrak{n}}}^{\prime} \right| + t\_{+} \frac{\partial \left| \dot{\boldsymbol{\mathfrak{n}}} \right\rangle}{\partial \boldsymbol{\mathfrak{g}}^{\prime}} \left\langle \dot{\boldsymbol{\mathfrak{n}}}^{\prime} \right| + t\_{-} \left| \dot{\boldsymbol{\mathfrak{n}}} \right\rangle \frac{\partial \left\langle \dot{\boldsymbol{\mathfrak{n}}}^{\prime} \right|}{\partial \boldsymbol{\mathfrak{g}}} \\ &+ \quad \frac{\partial t\_{-}}{\partial \boldsymbol{\mathfrak{g}}} \left| -\dot{\boldsymbol{\mathfrak{n}}} \right\rangle \left\langle -\dot{\boldsymbol{\mathfrak{n}}}^{\prime} \right| + t\_{-} \frac{\partial \left| -\dot{\boldsymbol{\mathfrak{n}}} \right\rangle}{\partial \boldsymbol{\mathfrak{g}}} \left\langle -\dot{\boldsymbol{\mathfrak{n}}}^{\prime} \right| + t\_{-} \left| -\dot{\boldsymbol{\mathfrak{n}}} \right\rangle \frac{\partial \left\langle -\dot{\boldsymbol{\mathfrak{n}}}^{\prime} \right|}{\partial \boldsymbol{\mathfrak{g}}} \\ &\equiv \quad \left( \left| \dot{\boldsymbol{\mathfrak{n}}} \right\rangle, \left| -\dot{\boldsymbol{\mathfrak{n}}} \right) \rangle \dot{\boldsymbol{\mathfrak{f}}}^{\prime}\_{\mathcal{S}} \left( \left. \frac{\left\langle \dot{\boldsymbol{\mathfrak{n}}}^{\prime} \right|}{\left. \left. -\dot{\boldsymbol{\mathfrak{n}}}^{\prime} \right|} \right\rangle \right) . \end{split} \tag{A72}$$

with defining a matrix

$$\hat{\mathbf{f}}'\_{\mathcal{S}} = \begin{pmatrix} \frac{\partial \mathbf{t}\_{+}}{\partial \mathcal{S}} & \mathbf{t}\_{+} \mathbf{b}\_{\mathcal{S}} - \mathbf{t}\_{-} \mathbf{b}\_{\mathcal{S}}^{\*} \\ \mathbf{t}\_{+} \mathbf{b}\_{\mathcal{S}} - \mathbf{t}\_{-} \mathbf{b}\_{\mathcal{S}}^{\*} & \frac{\partial \mathbf{t}\_{-}}{\partial \mathcal{S}} \end{pmatrix} . \tag{A73}$$

Obviously, it is convenient to take the scattering eigenstates |±*n*<sup>ˆ</sup> to the spin axis |*σ* = ±1 in the Berry curvatures, which is written as

$$\begin{array}{rcl} \Pi\_{\mathsf{H}/-\mathsf{H}}(\mathsf{g}\_{1},\mathsf{g}\_{2}) & = & \frac{1}{\pi} \heartsuit \left\{ \left( \mathfrak{f}\_{\mathsf{S}2} \mathfrak{f}\_{\mathsf{S}1}^{\mathsf{t}} + \mathfrak{f}\_{\mathsf{S}2}^{\mathsf{t}} \mathfrak{f}\_{\mathsf{S}1}^{\mathsf{t}} \right)\_{\langle 1,1 \rangle/\langle 2,2 \rangle} \right\}. \end{array} \tag{A74}$$

Then the diagonal components of *<sup>r</sup>*<sup>ˆ</sup>*g*2 *r*ˆ†*g*1is

$$\left(\hat{r}\_{\mathcal{S}2}\hat{r}\_{\mathcal{S}1}^{\dagger}\right)\_{(1,1)} = \quad \frac{\partial r\_{+}}{\partial \mathcal{g}\_{2}}\frac{\partial r\_{+}^{\*}}{\partial \mathcal{g}\_{1}} + |r\_{+} - r\_{-}|^{2} \, b\_{\mathcal{S}2}^{\*} b\_{\mathcal{S}1'} \tag{A75}$$

$$\left(\hat{r}\_{\mathbb{S}^2}\hat{r}\_{\mathbb{S}^1}^\dagger\right)\_{(2,2)} = \begin{cases} \frac{\partial r\_-}{\partial \mathcal{g}\_2} \frac{\partial r\_-^\*}{\partial \mathcal{g}\_1} + |r\_+ - r\_-|^2 b\_{\mathbb{S}^2} b\_{\mathbb{S}^1}^\* \\\\ \end{cases} \tag{A76}$$

and the diagonal components of ˆ*tg*2 ˆ*t*†*g*1 is

$$\left(\hat{I}\_{\mathfrak{J}\_2}^{\prime}\hat{I}\_{\mathfrak{J}\_1}^{\prime\dagger}\right)\_{(1,1)} = \underbrace{\partial t\_+}\_{\partial\mathfrak{J}\_2}\frac{\partial t\_+^\*}{\partial\mathfrak{J}\_1} + (t\_+b\_{\mathfrak{J}\_2} - t\_-b\_{\mathfrak{J}\_2}^\*)(t\_+^\*b\_{\mathfrak{J}\_1}^\* - t\_-^\*b\_{\mathfrak{J}\_1}),\tag{A77}$$

$$
\begin{pmatrix} \hat{\mathfrak{f}}'\_{\mathfrak{J}^2} \hat{\mathfrak{l}}^{\dagger}\_{\mathfrak{J}} \end{pmatrix}\_{(2,2)} = \begin{array}{c} \frac{\partial \mathfrak{t} - \partial \mathfrak{t}^\*}{\partial \mathfrak{J}^2} \frac{\partial \mathfrak{t}^\*}{\partial \mathfrak{J}} + (\mathfrak{t} + b\_{\mathfrak{J}^2} - \mathfrak{t} - b\_{\mathfrak{J}^2}^\*) (\mathfrak{t}^\*\_{+} b\_{\mathfrak{J}^1}^\* - \mathfrak{t}^\*\_{-} b\_{\mathfrak{J}^1}) . \end{array} \tag{A78}$$

As stated before, we restrict the type of control parameters, *g*, that only change the eigenvalues *λ*± in the transmission amplitudes, *r*± and *t*± and corresponding eigenstates, and we take

$$
\frac{\partial r\_{\pm}}{\partial \mathbf{g}} = \frac{\partial \lambda\_{\pm}}{\partial \mathbf{g}} \frac{\partial r\_{\pm}}{\partial \lambda\_{\pm}}, \qquad \frac{\partial t\_{\pm}}{\partial \mathbf{g}} = \frac{\partial \lambda\_{\pm}}{\partial \mathbf{g}} \frac{\partial t\_{\pm}}{\partial \lambda\_{\pm}}.\tag{A79}
$$

Then the factors in the Berry curvature, Equations (A75) and (A77),

$$
\frac{\partial r\_{\pm}}{\partial \mathbf{g}\_{2}} \frac{\partial r\_{\pm}^{\*}}{\partial \mathbf{g}\_{1}} \quad \left. \quad \frac{\partial \lambda\_{\pm}}{\partial \mathbf{g}\_{2}} \frac{\partial \lambda\_{\pm}}{\partial \mathbf{g}\_{1}} \right| \frac{\partial r\_{\pm}}{\partial \lambda\_{\pm}} \right|^{2}, \tag{A80}
$$

$$
\frac{\partial t\_{\pm}}{\partial \mathbf{g}\_{2}} \frac{\partial t\_{\pm}^{\*}}{\partial \mathbf{g}\_{1}} \quad \left. \quad - \right. \frac{\partial \lambda\_{\pm}}{\partial \mathbf{g}\_{2}} \frac{\partial \lambda\_{\pm}}{\partial \mathbf{g}\_{1}} \left| \frac{\partial t\_{\pm}}{\partial \lambda\_{\pm}} \right|^{2} \,, \tag{A81}
$$

are real and are not contributing to the pumping.

Then, the Berry curvature for the spin |*n*<sup>ˆ</sup> is

$$\begin{split} \Pi\_{\mathsf{fl}}(\mathcal{g}\_{1}, \mathcal{g}\_{2}) &= \ \frac{1}{\pi i} \Im \left\{ \left| r\_{+} - r\_{-} \right|^{2} b\_{\mathcal{G}\_{2}}^{\*} b\_{\mathcal{G}\_{1}} + \left( t\_{+} b\_{\mathcal{G}\_{2}} - t\_{-} b\_{\mathcal{G}\_{2}}^{\*} \right) \left( t\_{+}^{\*} b\_{\mathcal{G}\_{1}}^{\*} - t\_{-}^{\*} b\_{\mathcal{G}\_{1}} \right) \right\} \\ &= \ \frac{1}{2 \pi i} (\left| r\_{+} - r\_{-} \right|^{2} - \left| t\_{+} \right|^{2} + \left| t\_{-} \right|^{2}) \left( b\_{\mathcal{G}\_{2}}^{\*} b\_{\mathcal{G}\_{1}} - b\_{\mathcal{G}\_{2}} b\_{\mathcal{G}\_{1}}^{\*} \right) . \end{split} \tag{A82}$$

Using Equation (A68) and the relation

$$\frac{\partial n\_z}{\partial \mathcal{g}} \quad = \quad -\frac{1}{n\_z} \left( n\_x \frac{\partial n\_x}{\partial \mathcal{g}} + n\_y \frac{\partial n\_y}{\partial \mathcal{g}} \right), \tag{A83}$$

derived from *nz* = /1 − *n*2*x* − *n*2*y*, the factor in the last bracket of Equation (A82) is manipulated to

$$b\_{\mathcal{S}^2}^\* b\_{\mathcal{S}^1} - b\_{\mathcal{S}^2} b\_{\mathcal{S}^1}^\* = \frac{\mathrm{i}}{2n\_z} \left( \frac{\partial n\_x}{\partial \mathcal{g}\_2} \frac{\partial n\_y}{\partial \mathcal{g}\_1} - \frac{\partial n\_x}{\partial \mathcal{g}\_1} \frac{\partial n\_y}{\partial \mathcal{g}\_2} \right) \equiv 2\pi \mathrm{i} \mathcal{C}\_{\mathbb{S}^{1, \mathcal{G}2}}.\tag{A84}$$

From Equation (A30), the derivatives of the elements of *n*ˆ by some control parameter *g* are calculated as

$$\frac{\partial n\_x}{\partial g} = -\frac{1}{\sqrt{1-\tau\_z^2}} \left\{ \frac{\partial \tau\_y}{\partial g} + \frac{\tau\_y \tau\_z}{1-\tau\_z^2} \frac{\partial \tau\_z}{\partial g} \right\},\tag{A85}$$

$$\frac{\partial n\_{\mathbf{y}}}{\partial \mathbf{g}} = \frac{1}{\sqrt{1 - \tau\_z^2}} \left\{ \frac{\partial \tau\_\mathbf{x}}{\partial \mathbf{g}} + \frac{\tau\_\mathbf{x} \tau\_\mathbf{z}}{1 - \tau\_z^2} \frac{\partial \tau\_\mathbf{z}}{\partial \mathbf{g}} \right\},\tag{A86}$$

therefore, the factor *Cg*1,*g*2 in Equation (A84) is obtained as Equation (36) and the Berry curvatures are given as Equations (34) and (35).

### **Appendix E. Formulation of Diamond-Shape Interferometer**

This section explains the foundation of the Schrödinger Equations (3) and (4). At the four sites in the interferometer, the Schrödinger equation is

$$\left(\left(\epsilon-\epsilon\_{\rm u}\right)\left|\psi(\boldsymbol{u})\right>\right) = \left.-\sum\_{\boldsymbol{v}}\hat{\mathcal{U}}\_{\rm uv}\left|\psi(\boldsymbol{v})\right>\right.\tag{A87}$$

Explicitly, at sites *u* = 0, 1, *b*, *c*:

$$\left(\left(\varepsilon-\epsilon\_{0}\right)\left|\psi(0)\right>\right) = \left.-\left(\left<\mathbb{I}\_{0b}\left|\psi(b)\right> + \left<\mathbb{I}\_{0c}\left|\psi(c)\right>\right) - j\left|\psi(-1)\right>\right)\right.\tag{A88}$$

$$\left(\epsilon-\epsilon\_1\right)\left|\psi(1)\right\rangle = \left.-\left(\bar{\mathcal{U}}\_{b1}^\dagger \left|\psi(b)\right\rangle + \bar{\mathcal{U}}\_{c1}^\dagger \left|\psi(c)\right\rangle\right) - j\left|\psi(2)\right\rangle,\tag{A89}$$

$$\left(\left(\epsilon-\epsilon\_{b}\right)\left|\psi(b)\right>\right) = \left.-\left(\bar{l}\bar{l}\_{0b}^{\dagger}\left|\psi(0)\right> + \bar{l}\bar{l}\_{b1}\left|\psi(1)\right>\right),\tag{A90}$$

$$\left(\left(\varepsilon-\mathfrak{e}\_{\mathfrak{c}}\right)\left|\psi(\mathfrak{c})\right\rangle\right)=\left.-\left(\mathcal{U}\_{0\mathfrak{c}}^{\sharp}\left|\psi(0)\right\rangle+\mathcal{U}\_{\mathfrak{c}1}\left|\psi(1)\right\rangle\right).\tag{A91}$$

Using Equations (A90) and (A91),

$$\left|\psi(b)\right\rangle \quad = \left. -\frac{1}{\epsilon - \epsilon\_b} \left[ \hat{\mathcal{U}}\_{0b}^{\dagger} \left| \psi(0) \right\rangle + \hat{\mathcal{U}}\_{b1} \left| \psi(1) \right\rangle \right] \tag{A92}$$

$$\left|\psi(\varepsilon)\right\rangle \quad = \quad -\frac{1}{\varepsilon - \varepsilon\_{\varepsilon}} \left[\mathcal{U}\_{0\varepsilon}^{\dagger}\left|\psi(0)\right\rangle + \mathcal{U}\_{\varepsilon 1}\left|\psi(1)\right\rangle\right].\tag{A93}$$

By putting these into Equation (A88),

$$
\begin{split}
\left< \left( \epsilon - \epsilon\_{0} \right) \left| \psi(0) \right> &= \left. -\tilde{U}\_{0b} \left( -\frac{1}{\epsilon - \epsilon\_{b}} \right) \left[ \tilde{U}\_{0b}^{\dagger} \left| \psi(0) \right> + \tilde{U}\_{b1} \left| \psi(1) \right> \right] \\ &- \Omega\_{0c} \left( -\frac{1}{\epsilon - \epsilon\_{c}} \right) \left[ \mathcal{U}\_{0c}^{\dagger} \left| \psi(0) \right> + \tilde{\Omega}\_{c1} \left| \psi(1) \right> \right] - j \left| \psi(-1) \right> \\ &= \left( \frac{\bar{J}\_{0b} \bar{l}\_{b0}}{\epsilon - \epsilon\_{b}} + \frac{\bar{J}\_{0c} \bar{l}\_{c0}}{\epsilon - \epsilon\_{c}} \right) \left| \psi(0) \right> + \left[ \frac{\bar{\mathcal{U}}\_{0b} \bar{l}\_{b1}}{\epsilon - \epsilon\_{b}} + \frac{\bar{\mathcal{U}}\_{0c} \bar{\mathcal{U}}\_{c1}}{\epsilon - \epsilon\_{c}} \right] \left| \psi(1) \right> - j \left| \psi(-1) \right>.
\end{split}
$$

Then we define real variables

$$
\gamma\_{uvw} \quad \equiv \begin{array}{c} \text{J}\_{uv}\text{J}\_{uv} \\ \text{\(\varepsilon-\varepsilon\_v\)} \end{array} \tag{A94}
$$

and introducing a 2 × 2 matrix

$$\begin{array}{rcl} \mathcal{W} & \equiv & \frac{\mathcal{Q}\_{0b}\mathcal{Q}\_{b1}}{\mathfrak{c} - \mathfrak{c}\_{b}} + \frac{\mathcal{Q}\_{0c}\mathcal{Q}\_{c1}}{\mathfrak{c} - \mathfrak{c}\_{c}}\\ & = & \gamma\_{0b1}\hat{\mathcal{Q}}\_{0b}\hat{\mathcal{Q}}\_{b1} + \gamma\_{0c1}\hat{\mathcal{Q}}\_{0c}\hat{\mathcal{Q}}\_{c1} \end{array} \tag{A95}$$

we obtain the relation equivalent to Equation (3)

$$\left| \left( \epsilon - y\_0 \right) \left| \psi(0) \right> \right| \quad = \quad \left| \Psi \right| \left| \psi(1) \right> - j \left| \psi(-1) \right> , \tag{A96}$$

where we defined renormalized site energy at *u* = 0, *y*0 ≡ 0 + *γ*0*b*0 + *γ*0*c*0. By putting Equation (A92) in Equation (A89),

( − 1)|*ψ*(1) = *U*˜ †*b*1 1  −  *b U*˜ †0*b* |*ψ*(0) + *U*˜ *b*1 <sup>|</sup>*ψ*(1) +*U* ˜ † *c*1 1  −  *c U*˜ †0*c* |*ψ*(0) + *U*˜ *c*1 <sup>|</sup>*ψ*(1) − *j* |*ψ*(2) = *J*1*b Jb*0  −  *b U*ˆ †*b*1*U*<sup>ˆ</sup> †0*b* + *J*<sup>1</sup>*c Jc*0  −  *c U*ˆ †*c*1*U*<sup>ˆ</sup> †0*c* |*ψ*(0) + *J*1*b Jb*1  −  *b* + *J*<sup>1</sup>*c Jc*1  −  *c* |*ψ*(1) − *j* |*ψ*(2).

Hence, we have the equation equivalent to Equation (4)

$$\left| \left( \varepsilon - y\_1 \right) \left| \psi(1) \right> \right| \quad = \quad \hat{\mathcal{W}}^\dagger \left| \psi(0) \right> - j \left| \psi(2) \right> , \tag{A97}$$

where we introduced renormalized site energy at *u* = 1, *y*1 ≡ 1 + *γ*1*b*1 + *γ*1*c*1. We defined *γb* ≡ *γ*0*b*1, *γc* ≡ *γ*0*c*1 and *U* ˆ *b* ≡ *U* ˆ 0*bU* ˆ *b*1, *U* ˆ *c* ≡ *U* ˆ <sup>0</sup>*cU* ˆ *c*1.
