**Appendix A. Scattering Matrix**

In this Appendix, we argue the scattering problem through the interferometer. First, we inject an electron flux with an energy  =  *k* from the left lead. The wavefunction for *u* ≤ 0 is

$$\left|\psi(\mu)\right\rangle \quad = \left. \varepsilon^{ikna} \left| \chi\_{\text{in}} \right\rangle + \varepsilon^{-ikna} \left| \chi\_{\text{r}} \right\rangle \right. \tag{A1}$$

and the wavefunction for *u* ≥ 1 is

$$\left|\psi(u)\right\rangle \quad = \left. \left. \varepsilon^{ik(u-1)a} \right| \chi\_t \right\rangle , \tag{A2}$$

where |*<sup>χ</sup>*in is the injected wavefunction and |*<sup>χ</sup>*r, |*<sup>χ</sup>*t are the (un-normalized) wavefunctions of reflection and transmission. In particular, at sites *u* = 0, −1,

$$\left|\psi(0)\right\rangle = \left|\chi\_{\rm in}\right\rangle + \left|\chi\_{\rm r}\right\rangle, \qquad \left|\psi(-1)\right\rangle = e^{-i\mathbf{k}\mathbf{r}}\left|\chi\_{\rm in}\right\rangle + e^{i\mathbf{k}\mathbf{r}}\left|\chi\_{\rm r}\right\rangle. \tag{A3}$$

and at sites *u* = 1, 2,

$$\left|\psi(1)\right\rangle = \left|\chi\_{\mathfrak{t}}\right\rangle, \qquad \left|\psi(2)\right\rangle = \mathfrak{e}^{ik\mathfrak{a}} \left|\chi\_{\mathfrak{t}}\right\rangle. \tag{A4}$$

By putting these into Equations (3) and (4), we have

$$\left\{ (\epsilon\_k - y\_0) \left\{ \left| \chi\_{\rm in} \right\rangle + \left| \chi\_{\rm f} \right\rangle \right\} \right\} \quad = \quad \hat{W} \left| \chi\_{\rm f} \right\rangle - j \left\{ e^{-i\mathbf{k}\mathbf{r}} \left| \chi\_{\rm in} \right\rangle + e^{i\mathbf{k}\mathbf{r}} \left| \chi\_{\rm f} \right\rangle \right\} \,, \tag{A5}$$

$$\left| \left( \varepsilon\_{\rm k} - y\_{\rm l} \right) \left| \chi\_{\rm l} \right\rangle \right| \quad = \left| \hat{\mathcal{W}}^{\dagger} \left\{ \left| \chi\_{\rm in} \right\rangle + \left| \chi\_{\rm l} \right\rangle \right\} - j e^{i \mathbf{k} \mathbf{r}} \left| \chi\_{\rm l} \right\rangle. \tag{A6}$$

> Then from Equation (A6),

$$|\chi\_{\rm t}\rangle \quad = \frac{1}{\epsilon\_{\rm k} - \chi\_{\rm 1} + j e^{i\mathbf{k}\mathbf{a}}} \hat{W}^{\dagger} \left\{ |\chi\_{\rm in}\rangle + |\chi\_{\rm r}\rangle \right\},\tag{A7}$$

and putting this into Equation (A5), we have

$$\left(\left(\varepsilon\_{\rm k} - y\_0 + j\varepsilon^{-i\rm kx}\right)\left|\chi\_{\rm in}\right\rangle + \left(\varepsilon\_{\rm k} - y\_0 + j\varepsilon^{i\rm ka}\right)\left|\chi\_{\rm r}\right\rangle = \hat{\mathcal{W}} \frac{1}{\varepsilon\_{\rm k} - y\_1 + j\varepsilon^{i\rm ka}} \hat{\mathcal{W}}^{\dagger}\left\{\left|\chi\_{\rm in}\right\rangle + \left|\chi\_{\rm r}\right\rangle\right\}.\tag{A8}$$

Defining complex parameters *Xu* ≡  *k* − *yu* + *jeika* (*u* = 0, 1) and noting  *k* − *y*0 + *je*−*ika* = *X*0 <sup>−</sup>*iηk*, we solve this equation

$$\begin{array}{rcl} \left| \chi\_{\mathbf{r}} \right\rangle &=& -\left[ \mathbb{I} \mathbf{X}\_{0} \mathbf{X}\_{1} - \hat{\mathsf{W}} \hat{\mathsf{W}}^{\dagger} \right]^{-1} \left[ \mathbb{I} \mathbf{X}\_{0} \mathbf{X}\_{1} - \hat{\mathsf{W}} \hat{\mathsf{W}}^{\dagger} - \mathbb{I} \hat{\mathsf{n}} \eta\_{k} \mathbf{X}\_{1} \right] \left| \chi\_{\mathbf{\hat{n}}} \right\rangle \dots \end{array}$$

Then we have obtained the reflection amplitude matrix

$$\begin{aligned} \dot{\mathcal{T}} & \equiv \ & -\left[\mathbb{I}\mathbf{X}\_{0}\mathbf{X}\_{1} - \hat{\mathsf{W}}\hat{\mathsf{W}}^{\dagger}\right]^{-1}\left[\mathbb{I}\mathbf{X}\_{0}\mathbf{X}\_{1} - \hat{\mathsf{W}}\hat{\mathsf{W}}^{\dagger} - \mathbb{I}\boldsymbol{i}\boldsymbol{\eta}\_{k}\mathbf{X}\_{1}\right] \\ &= & -\mathbb{I} + i\boldsymbol{\eta}\_{k}\mathbf{X}\_{1}\left[\mathbb{I}\mathbf{Y} - \hat{\mathsf{W}}\hat{\mathsf{W}}^{\dagger}\right]^{-1}, \end{aligned} \tag{A9}$$

where we have introduced *Y* ≡ *X*0*X*1. Using this, transmitted state is calculated with Equation (A7),

$$\begin{aligned} \left| \chi\_{\mathbf{t}} \right\rangle &= \begin{array}{c} \frac{1}{X\_1} \hat{\mathsf{W}}^{\dagger} \left\{ \left| \chi\_{\mathbf{in}} \right\rangle + \left| \chi\_{\mathbf{r}} \right\rangle \right\} \\ &= \begin{array}{c} \frac{1}{X\_1} \hat{\mathsf{W}}^{\dagger} \left[ \mathbb{I} - \mathbb{I} + i \eta\_k X\_1 \left[ \mathbb{I} \mathcal{Y} - \hat{\mathsf{W}} \hat{\mathsf{W}}^{\dagger} \right]^{-1} \right] \left| \chi\_{\mathbf{in}} \right\rangle \\ &= \left[ i \eta\_k \hat{\mathsf{W}}^{\dagger} \left[ \mathbb{I} \mathcal{Y} - \hat{\mathsf{W}} \hat{\mathsf{W}}^{\dagger} \right]^{-1} \left| \chi\_{\mathbf{in}} \right\rangle \right. \end{array} \end{aligned}$$

hence the transmission amplitude matrix is

$$\hat{\mathbf{f}}^{\dagger} = \left[ i \eta\_k \hat{\mathbf{W}}^{\dagger} \left[ \mathbb{I} \mathbf{Y} - \mathbb{I} \mathbf{W} \mathbf{W}^{\dagger} \right]^{-1} \right]^{-1}. \tag{A10}$$

We alternatively consider the situation that the electron is injected from the right lead. The wavefunction for *u* ≥ 1 is

$$\left|\psi(n)\right\rangle \quad = \left. e^{-ik(n-1)a} \left|\chi\_{\rm in}^{\prime}\right\rangle + e^{ik(n-1)a} \left|\chi\_{\rm r}^{\prime}\right\rangle \right. \tag{A11}$$

and the wavefunction for *u* ≤ 0 is

$$\left|\psi(n)\right\rangle \quad = \left.e^{-ikna}\right|\chi\_{\mathfrak{t}}^{\prime}\rangle\,,\tag{A12}$$

where |*χ*in is the incoming wavefunction and |*χ*r, |*χ*t are the (un-normalized) wavefunctions of reflection and transmission. At sites *u* = 1, 2,

$$
\langle \Psi(1) \rangle = \left| \chi\_{\rm in}' \right\rangle + \left| \chi\_{\rm r}' \right\rangle, \qquad \left| \psi(2) \right\rangle = e^{-i\mathbf{k}\mathbf{a}} \left| \chi\_{\rm in}' \right\rangle + e^{i\mathbf{k}\mathbf{a}} \left| \chi\_{\rm r}' \right\rangle. \tag{A13}
$$

and at sites *u* = 0, −1,

$$\left|\psi(0)\right\rangle = \left|\chi\_{\mathbf{t}}'\right\rangle, \qquad \left|\psi(-1)\right\rangle = e^{i\mathbf{k}\mathbf{a}} \left|\chi\_{\mathbf{t}}'\right\rangle. \tag{A14}$$

By putting these into Equations (3) and (4), we have

$$\left(\epsilon\_{\rm k} - y\_0\right) \left| \chi\_{\rm t}'\right\rangle \quad = \quad \hat{\mathcal{W}} \left\{ \left| \chi\_{\rm in}'\right\rangle + \left| \chi\_{\rm t}'\right\rangle \right\} - j \epsilon^{\rm ika} \left| \chi\_{\rm t}'\right\rangle,\tag{A15}$$

$$\left\{ (\varepsilon\_{\mathrm{k}} - y\_{1}) \left\{ |\chi\_{\mathrm{in}}'\rangle + |\chi\_{\mathrm{r}}'\rangle \right\} \right. \\ \left. \quad = \quad \left. \mathcal{W}^{\dagger} \left| \chi\_{\mathrm{t}}' \right\rangle - j \left\{ e^{-i\mathbf{k}\mathbf{a}} \left| \chi\_{\mathrm{in}}' \right\rangle + e^{i\mathbf{k}\mathbf{a}} \left| \chi\_{\mathrm{r}}' \right\rangle \right\}. \tag{A16}$$

From Equation (A15),

$$\left|\chi\_{\rm t}^{\prime}\right\rangle\_{\rm t} = \frac{1}{X\_0} \mathcal{W} \left\{ \left|\chi\_{\rm in}^{\prime}\right\rangle + \left|\chi\_{\rm r}^{\prime}\right\rangle \right\},\tag{A17}$$

and putting this into Equation (A16),

$$\begin{aligned} \left(X\_1 - i\eta\_k\right)\left|\chi\_{\rm in}'\right\rangle + X\_1\left|\chi\_{\rm r}'\right\rangle &= \quad \hat{\mathcal{W}}^\dagger \frac{1}{X\_0} \hat{\mathcal{W}}\left\{\left|\chi\_{\rm in}'\right\rangle + \left|\chi\_{\rm r}'\right\rangle\right\}, \\\\ &\tag{A18} \end{aligned} \tag{A18}$$

which is solved as

$$\begin{aligned} \left| \chi\_{\mathbf{r}}' \right\rangle &= \left[ \mathbb{I}Y - \mathcal{W}^{\dagger}\mathcal{W} \right]^{-1} \left\{ - \left( \mathbb{I}Y - \mathcal{W}^{\dagger}\mathcal{W} \right) + i\eta\_{k} \mathcal{X}\_{0} \mathbb{I} \right\} \left| \chi\_{\text{in}}' \right\rangle \\ &= \left\{ - \mathbb{I} + i\eta\_{k} \mathcal{X}\_{0} \left[ \mathbb{I}Y - \mathcal{W}^{\dagger}\mathcal{W} \right]^{-1} \right\} \left| \chi\_{\text{in}}' \right\rangle . \end{aligned}$$

Therefore, the reflection amplitude matrix is

$$\mathcal{H}^{\prime} = -\mathbb{I} + i\eta\_k X\_0 \left[ \mathbb{I}Y - \mathcal{W}^\dagger \mathcal{W} \right]^{-1}. \tag{A19}$$

Putting this into Equation (A17),

$$\begin{aligned} \left| \chi\_{\mathbf{t}}' \right\rangle &= \left. \frac{\dot{\mathcal{W}}}{X\_0} \left\{ \mathbb{I} - \mathbb{I} + i \eta\_k X\_0 \left[ \mathbb{I} \mathbf{Y} - \hat{\mathcal{W}}^\dagger \hat{\mathcal{W}} \right]^{-1} \right\} \left| \chi\_{\mathbf{in}}' \right\rangle \\ &= \left. i \eta\_k \hat{\mathcal{W}} \left[ \mathbb{I} \mathbf{Y} - \hat{\mathcal{W}}^\dagger \hat{\mathcal{W}} \right]^{-1} \left| \chi\_{\mathbf{in}}' \right\rangle \right. \end{aligned}$$

and hence the transmission amplitude matrix is

$$\hat{\mathbf{I}}^{\prime} \quad = \begin{bmatrix} i\eta\_k \boldsymbol{\mathcal{W}} \left[ \mathbb{I} \boldsymbol{Y} - \boldsymbol{\mathcal{W}}^{\dagger} \boldsymbol{\mathcal{W}} \right]^{-1} \text{.} \tag{A20}$$
