Quantum Reflection and Transmission in Ring Systems with Double Y-Junctions: Occurrence of Perfect Reflection

TL;DR

This paper develops a new formalism for analyzing quantum scattering in ring systems with double Y-junctions, revealing conditions for perfect reflection that differ from traditional one-dimensional systems.

Contribution

It introduces a path integral-based approach for quantum scattering in ring systems with double Y-junctions and uncovers conditions for perfect reflection in anti-symmetric configurations.

Findings

Perfect reflection can occur in anti-symmetric ring systems.

New compact formulas for probability amplitudes are derived.

Quantum reflection and transmission depend on junction symmetry.

Abstract

We consider the scattering problems of a quantum particle in a system with a single Y-junction and in ring systems with double Y-junctions. We provide new formalism for such quantum mechanical problems. Based on a path integral approach, we find compact formulas for probability amplitudes in the ring systems. We also discuss quantum reflection and transmission in the ring systems under scale-invariant junction conditions. It is remarkable that perfect reflection can occur in an anti-symmetric ring system, in contrast with the one-dimensional quantum systems having singular nodes of degree 2.

Equations

$$i\hslash \frac{\partial }{\partial t}{\Phi }_i\left(t,x_i\right)=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x_i^2}{\Phi }_i\left(t,x_i\right),$$

$$j_1(t,\xi )+j_2(t,\xi )+j_3(t,\xi )=0,$$

$$j_i(t,x_i):=-\frac{i\hslash }{2m}\left\{{\Phi }_i^{\ast }(t,x_i){\Phi }_i^{\prime }(t,x_i)-{{\Phi }_i^{\ast }}^{\prime }(t,x_i){\Phi }_i(t,x_i)\right\},$$

$$j_i(t,\xi ):=\underset{x_i\to \xi }{\lim }{j}_i(t,x_i).$$

$${\Psi }^{\prime †}\Psi -{\Psi }^{†}{\Psi }^{\prime }=0,$$

$$\Psi:=\left(\begin{array}[]{c}\Phi_{1}(t,\xi)\\ \Phi_{2}(t,\xi)\\ \Phi_{3}(t,\xi)\end{array}\right),\quad\Psi^{\prime}:=\left(\begin{array}[]{c}\Phi^{\prime}_{1}(t,\xi)\\ \Phi^{\prime}_{2}(t,\xi)\\ \Phi^{\prime}_{3}(t,\xi)\end{array}\right).$$

$$\left|\Psi -i{L}_0{\Psi }^{\prime }\right|=\left|\Psi +i{L}_0{\Psi }^{\prime }\right|,$$

$$\left(U-I_3\right)\Psi +i{L}_0\left(U+I_3\right){\Psi }^{\prime }=0,$$

$$W^{\dagger}UW=D:=\left(\begin{array}[]{ccc}e^{i\theta_{1}}&0&0\\ 0&e^{i\theta_{2}}&0\\ 0&0&e^{i\theta_{3}}\\ \end{array}\right)$$

$$W=e^{i\eta }{e}^{i\alpha {\lambda }_3}{e}^{i\beta {\lambda }_2}{e}^{i\gamma {\lambda }_3}{e}^{i\delta {\lambda }_5}{e}^{ia{\lambda }_3}{e}^{ib{\lambda }_2}{e}^{ic{\lambda }_3}{e}^{id{\lambda }_8}$$

$$U=WD{W}^{†}=\mathcal{V}D{\mathcal{V}}^{†},$$

$$\mathcal{V}=e^{i\alpha {\lambda }_3}{e}^{i\beta {\lambda }_2}{e}^{i\gamma {\lambda }_3}{e}^{i\delta {\lambda }_5}{e}^{ia{\lambda }_3}{e}^{ib{\lambda }_2}.$$

$${\Phi }_i\left(t,x_i\right)=e^{-i\frac{\mathcal{E}}{\hslash }t}\left({\varphi }_i{e}^{ik{x}_i}+{\psi }_i{e}^{-ik{x}_i}\right),$$

$$\Psi=e^{-i\frac{\cal E}{\hbar}t}\left\{e^{ik\xi}\left(\begin{array}[]{c}\phi_{1}\\ \phi_{2}\\ \phi_{3}\end{array}\right)+e^{-ik\xi}\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\\ \psi_{3}\end{array}\right)\right\},$$

$$\Psi^{\prime}=e^{-i\frac{\cal E}{\hbar}t}\left\{ike^{ik\xi}\left(\begin{array}[]{c}\phi_{1}\\ \phi_{2}\\ \phi_{3}\end{array}\right)-ike^{-ik\xi}\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\\ \psi_{3}\end{array}\right)\right\},$$

$$\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\\ \psi_{3}\end{array}\right)=S(\xi)\left(\begin{array}[]{c}\phi_{1}\\ \phi_{2}\\ \phi_{3}\end{array}\right).$$

$$S(\xi )=S^{(\mathrm{in})}(\xi ):=e^{2ik\xi }\mathcal{V}{S}_0^{(\mathrm{in})}{\mathcal{V}}^{†},$$

$$S_{0}^{\rm(in)}:=\left(\begin{array}[]{ccc}\frac{ikL_{1}+1}{ikL_{1}-1}&0&0\\ 0&\frac{ikL_{2}+1}{ikL_{2}-1}&0\\ 0&0&\frac{ikL_{3}+1}{ikL_{3}-1}\\ \end{array}\right),$$

$$L_i:=L_0\cot \frac{{\theta }_i}{2}.$$

$$S(\xi )=S^{(\mathrm{out})}(\xi ):=e^{-2ik\xi }\mathcal{V}{S}_0^{(\mathrm{out})}{\mathcal{V}}^{†},$$

$$S_{0}^{\rm(out)}:=\left(\begin{array}[]{ccc}\frac{ikL_{1}-1}{ikL_{1}+1}&0&0\\ 0&\frac{ikL_{2}-1}{ikL_{2}+1}&0\\ 0&0&\frac{ikL_{3}-1}{ikL_{3}+1}\\ \end{array}\right).$$

$${\Phi }_1(t,x_1)$$

$${\Phi }_2(t,x_2)$$

$${\Phi }_3(t,x_3)$$

$${\Phi }_4(t,x_4)$$

$$\left(\begin{array}[]{c}A\\ B\\ D\end{array}\right)=S_{1}^{\rm(in)}(\xi_{1})\left(\begin{array}[]{c}1\\ C\\ E\end{array}\right),$$

$$\left(\begin{array}[]{c}F\\ C\\ E\end{array}\right)=S_{2}^{\rm(out)}(\xi_{2})\left(\begin{array}[]{c}0\\ B\\ D\end{array}\right),$$

$$F=F_2+F_4+F_6+\cdots .$$

$$S^{\rm(in)}_{1}(\xi_{1}):=\left(\begin{array}[]{ccc}s_{11}&s_{12}&s_{13}\\ s_{21}&s_{22}&s_{23}\\ s_{31}&s_{32}&s_{33}\\ \end{array}\right),$$

$$S^{\rm(out)}_{2}(\xi_{2})=\left(\begin{array}[]{ccc}\tilde{s}_{11}&\tilde{s}_{12}&\tilde{s}_{13}\\ \tilde{s}_{21}&\tilde{s}_{22}&\tilde{s}_{23}\\ \tilde{s}_{31}&\tilde{s}_{32}&\tilde{s}_{33}\\ \end{array}\right),$$

$$F_2={\tilde{s}}_{12}{s}_{21}+{\tilde{s}}_{13}{s}_{31}=\sum _{i=2,3}{\tilde{s}}_{1i}{s}_{i1}.$$

$$F_4$$

Full text

1 Introduction

It is truly interesting that the variety of junction conditions appears when the number of space dimension is reduced in quantum mechanics. This feature becomes apparent in one-dimensional quantum systems. For example, when we consider a node (i.e., a point interaction) on a one-dimensional quantum wire, the two adjacent one-dimensional spaces are completely separated by the node. This situation leads to various non-trivial junction conditions. The characteristics of the non-trivial junction conditions in one-dimensional quantum systems have comprehensively been studied by mathematical works [1, 2, 3, 4].

We shed light on quantum problems in a system of a Y-junction. The Y-junction is composed of three one-dimensional quantum wires. These wires intersect at one point. The intersection of degree 3 has a point interaction parametrized by U(3) [1, 2, 3, 4]. Such a system was originally investigated by pioneer works (e.g., [5, 6]) with a simplified $S$-matrix, which was inspired by realized nano-rings. Some mathematical features of transmission of a quantum particle in the system with a Y-junction were investigated in [7] and also in the context of a star graph etc [8, 9, 10, 11]. However, the thorough investigation of the system has not yet been completed.

In the present work, we provide new formalism for the quantum mechanical problems in the system with a single Y-junction and ring systems with double Y-junctions. In particular, by focusing on scale-invariant junction conditions, we investigate quantum reflection and transmission in such systems. This paper is organized as follows. In §2, we formulate the system with a single Y-junction. We also formulate the ring systems with double Y-junctions in §3. In §4, we discuss quantum reflection and transmission on the ring systems under the scale-invariant junction conditions. Finally, we provide a summary in §5.

2 Formulation of a system with a single Y-junction

2.1 The Schrödinger equation

We consider a quantum system with a single Y-junction, in which three one-dimensional quantum wires intersect at one point. Let us assume that the three axes are given by $x_{1}$, $x_{2}$ and $x_{3}$ and directed to the node as shown in Fig. 1(a) (inward axes). We also assume that the node locates at $x_{i}=\xi\ (i=1,2,3)$. Note that the angle between any two axes has no effect on the physical states. On each wire ($x_{i}<\xi$), a quantum particle with mass $m$ obeys the Schrödinger equation

[TABLE]

where $\Phi_{i}$ denotes the wave function on the $x_{i}$-axis. Thus we assume a free particle on the wire.

2.2 Junction conditions

Let us discuss the expression of a junction condition. The junction condition at the node is provided by the conservation of the probability current

[TABLE]

where the probability current $j_{i}(t,x_{i})$ on the $x_{i}$-axis is given by

[TABLE]

and we implicitly assume the limit

[TABLE]

Here the prime $(^{\prime})$ denotes the partial differentiation with respect to the spatial coordinate. Equation (2) can be expressed by

[TABLE]

where

[TABLE]

Equation (5) is equivalently expressed as [4]

[TABLE]

where $L_{0}$ $(\in\mathbb{R})$ is an arbitrary non-vanishing constant with dimension of length (see A for the role of $L_{0}$). Hence $\Psi-iL_{0}\Psi^{\prime}$ is connected to $\Psi+iL_{0}\Psi^{\prime}$ via a unitary transformation. Thus, we obtain the junction condition [4]

[TABLE]

where $I_{3}$ is the $3\times 3$ identity matrix, and $U$ is a $3\times 3$ unitary matrix, i.e., $U\in{\rm U}(3)$. Therefore the junction condition is characterized by the unitary matrix $U$.

Next, we discuss parametrization of the unitary matrix $U$. For this discussion, it is useful to recall that any unitary matrix $U$ can be diagonalized by a unitary matrix $W$ as

[TABLE]

where $\theta_{i}\in\mathbb{R}\ (i=1,2,3)$. Since ${\rm U}(3)={\rm U}(1)\times{\rm SU}(3)$ holds locally [12], the unitary matrix $W$ can always be expressed by an element $V\in$ SU(3) multiplied by a complex factor $e^{i\eta}$, i.e., $W=e^{i\eta}V$, where $\eta\in\mathbb{R}$. When we adopt the Euler angle parametrization [13] for $V$, we have

[TABLE]

where $\lambda_{1},\lambda_{2},\cdots,\lambda_{8}$ are the Gell-Mann Matrices (see B for the definition), and $\alpha,\beta,\gamma,\delta,a,b,c,d\in\mathbb{R}$. Then we derive

[TABLE]

where

[TABLE]

Thus the unitary matrix $U$, which provides the junction condition, is specified by the nine real parameters: $\theta_{1}$, $\theta_{2}$, $\theta_{3}$, $\alpha$, $\beta$, $\gamma$, $\delta$, $a$ and $b$.

2.3 The scattering matrices

Let us consider the scattering of a single mode with wave number $k$ by a Y-junction. We assume that incoming waves and outgoing waves are provided by $\phi_{i}e^{ikx_{i}}$ and $\psi_{i}e^{-ikx_{i}}$ $(i=1,2,3)$, respectively, as shown in Fig. 1(a). Here $\phi_{i},\psi_{i}\in\mathbb{C}$. Then, we have

[TABLE]

and

[TABLE]

[TABLE]

where ${\cal E}:=\hbar^{2}k^{2}/2m$. Substituting Eqs. (14) and (15) into Eq. (8) and using Eq. (11), we can rewrite the expression (8) into the form

[TABLE]

Based on the inward axes as shown in Fig. 1(a), we derive the $S$-matrix as

[TABLE]

where

[TABLE]

and

[TABLE]

Here the superscript “(in)” denotes the result derived on the inward axes. Note that the diagonal component $S_{ii}$ $(i=1,2,3)$ represents the probability amplitude for the reflection from $x_{i}$-axis to $x_{i}$-axis, while the non-diagonal component $S_{ij}$ ($i\neq j$) represents the probability amplitude for the transmission from $x_{j}$-axis to $x_{i}$-axis. The $S$-matrix (17) is not symmetric in general, because the junction conditions do not necessarily satisfy the time-reversal symmetry (see C for the condition of the time-reversal symmetry).

It should also be noted that when we take the axes in the opposite direction as shown in Fig. 1(b) (outward axes), we have to replace the wave number $k$ with $-k$. In this case, we derive the $S$-matrix in the form

[TABLE]

[TABLE]

Here the superscript “(out)” denotes the result derived on the outward axes. This expression is also useful to discuss a ring system with double Y-junctions in the next section.

3 Formulations of a ring system with double Y-junctions

3.1 Preliminary

We consider a ring system made of double Y-junctions as shown in Fig. 2. In this system, the Y-junction on the left has the inward axes of $x_{1}$, $x_{2}$ and $x_{3}$, while the Y-junction on the right has the outward axes of $x_{2}$, $x_{3}$ and $x_{4}$. The nodes are given by $x_{1}=x_{2}=x_{3}=\xi_{1}$ and $x_{2}=x_{3}=x_{4}=\xi_{2}$, where $\xi_{1}>\xi_{2}$. We also assume that the node on the Y-junction $(x_{1}x_{2}x_{3})$ has the same parameters as the node on the Y-junction $(x_{4}x_{2}x_{3})$ symmetrically, as shown in Fig. 3(a). We call this type of a ring a symmetric ring. (We also discuss an anti-symmetric ring below.) When we consider a scattering problem, we take the wave functions on the wires as

[TABLE]

where $A,B,C,D,E,F\in\mathbb{C}$. Applying these expressions to Eq. (16), we have the relations

[TABLE]

[TABLE]

where $S_{1}$ denotes the $S$-matrix on the Y-junction $(x_{1}x_{2}x_{3})$, and $S_{2}$ denotes the $S$-matrix on the Y-junction $(x_{4}x_{2}x_{3})$. By solving Eqs. (26) and (27), we can derive the coefficients $A,B,C,D,E$ and $F$ as functions of the components of $S_{1}^{\rm(in)}(\xi_{1})$ and $S_{2}^{\rm(out)}(\xi_{2})$ (see D).

3.2 Path integral description

Let us derive the solution for the coefficients $A,B,C,D,E$ and $F$ in Eqs. (22)-(25) based on a path integral approach. Our approach is similar to the method of a path decomposition expansion developed in [14] (see [10, 15, 16] for general treatments). The path integral makes physical interpretation clear.

First, we obtain the solution for $F$. We now consider all possible paths from $x_{1}$ to $x_{4}$. Each path can be characterized by the number of scattering at the nodes. Let us assume that $F_{n}$ denotes the contribution from the path having $n$ times scattering to $F$. Here $n=2\ell_{1}\ (\ell_{1}=1,2,3,\cdots)$ is only permitted. Thus, we have

[TABLE]

For example, $F_{2}$ is composed of two paths: ($x_{1}\rightarrow x_{2}\rightarrow x_{4}$) and ($x_{1}\rightarrow x_{3}\rightarrow x_{4}$). Hence, when we write the components of $S^{\rm(in)}(\xi_{1})$ and $S^{\rm(out)}(\xi_{2})$ as

[TABLE]

[TABLE]

we have

[TABLE]

Similarly, we derive

[TABLE]

When we define $2\times 2$ matrices

[TABLE]

[TABLE]

then we derive

[TABLE]

Therefore, we obtain the amplitude $F$ as

[TABLE]

where $I_{2}$ is the $2\times 2$ identity matrix.

Next, we obtain the solution for $A$. We consider all possible paths from $x_{1}$ to $x_{1}$. Let us assume that $A_{n}$ denotes the contribution from the path having $n$ times scattering to $A$. Here $n=2\ell_{2}+1\ (\ell_{2}=0,1,2,\cdots)$. Then we have

[TABLE]

For example, $A_{1}$ is composed of the one path: ($x_{1}\rightarrow x_{1}$), and $A_{3}$ is composed of four paths: ($x_{1}\rightarrow x_{2}\rightarrow x_{2}\rightarrow x_{1}$), ($x_{1}\rightarrow x_{2}\rightarrow x_{3}\rightarrow x_{1}$), ($x_{1}\rightarrow x_{3}\rightarrow x_{2}\rightarrow x_{1}$) and ($x_{1}\rightarrow x_{3}\rightarrow x_{3}\rightarrow x_{1}$). Thus we derive

[TABLE]

Therefore we obtain

[TABLE]

The coefficient $A$ is expressed by the sum of a series of $s\tilde{s}$ in the same manner as $F$.

Finally, in a similar way, we obtain the other coefficients $B,C,D$ and $E$ as

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Thus the coefficients $B,C,D$ and $E$ are also expressed by the sum of a series of $s\tilde{s}$.

The above expressions can be simplified. Since $\left|s_{ij}\right|<1$ and $\left|\tilde{s}_{ij}\right|<1$ ($i,j=2,3$), we have

[TABLE]

where $O_{2}$ is the $2\times 2$ zero matrix. Hence the series of $s\tilde{s}$ converges as

[TABLE]

Therefore we can rewrite the above results as

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By straightforward calculations, we can confirm that the above results coincide with the results derived by solving Eqs. (26) and (27) algebraically (see D). Using Eqs. (75) and (80), we can calculate the probability for reflection $|A|^{2}$ and that for transmission $|F|^{2}$.

4 Systems under scale-invariant junction conditions

In this section, we restrict our attention to the cases of scale-invariant junction conditions for explicit examples.

4.1 A system with a single Y-junction

We consider a system with a single Y-junction. Under the scale-invariant junction conditions, the eigen values of $U$ take $+1$ or $-1$, that is, $\theta_{i}$ takes [math] or $\pi$ (see E for the details). Then, the diagonal matrix $D$ is given by

[TABLE]

where

[TABLE]

Substituting Eqs. (11) and (81) into Eq. (8), we find that the junction condition is divided into two conditions

[TABLE]

[TABLE]

Thus the dimensional parameter $L_{0}$ is dropped in the junction conditions. Furthermore, from Eq. (19), we have

[TABLE]

[TABLE]

Hence we derive

[TABLE]

It should be noted that the $k$-dependence of $S_{0}^{\rm(in)}$ and $S_{0}^{\rm(out)}$ disappears under the scale-invariant junction conditions. Thus, we obtain the $S$-matrix

[TABLE]

Since the three parameters $\theta_{i}\ (i=1,2,3)$ are fixed, six parameters remain in the $S$-matrices. From Eqs. (88) and (89), we find the probability for reflection from the $x_{i}$-axis to the $x_{i}$-axis

[TABLE]

and the probability for transmission from the $x_{i}$-axis to the $x_{j}$-axis

[TABLE]

In these expressions, $k$ does not appear. Therefore, when we consider the scale-invariant junction conditions, the probabilities for reflection and transmission become constant with respect to $k$.

4.2 A symmetric ring system with double Y-junctions

We consider a symmetric ring system with double Y-junctions described in Sec. 3.1 (see also Fig. 3(a)). From Eqs. (88) and (89), we have the $S$ matrix at $\xi_{1}$ and that at $\xi_{2}$ as

[TABLE]

[TABLE]

We drop the superscripts “(in)” and “(out)” in what follows. From these equations, we find the relations

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From Eqs. (94) and (98), we can derive

[TABLE]

and

[TABLE]

Furthermore, we obtain

[TABLE]

Thus an eigen value and an eigen vector of $s\tilde{s}$ are given by $e^{2ik(\xi_{1}-\xi_{2})}\left|s_{11}\right|^{2}$ and \displaystyle\left(\begin{array}[]{c}s_{21}\\ s_{31}\end{array}\right), respectively. Substituting Eq. (101) into Eqs. (68)-(72) and (53), we derive the simplified results

[TABLE]

Consequently, from Eqs. (102) and (107), we find that the perfect transmission ($A=0$) occurs if and only if the condition

[TABLE]

holds except the trivial case of $s_{11}=0$. On the other hand, the perfect reflection ($F=0$) does not occur except the trivial case of $|s_{11}|=1$.

4.3 An anti-symmetric ring system with double Y-junctions

We also discuss an anti-symmetric ring system in which the $x_{2}$-components in the $S$-matrix are replaced with the $x_{3}$-components and vice versa on the Y-junction $(x_{4}x_{2}x_{3})$ on the right, as shown in Fig. 3(b). That is, we replace $S_{2}(\xi_{2})$ with $\overline{S}_{2}(\xi_{2})$, where

[TABLE]

Here $P$ is defined by

[TABLE]

Under the scale-invariant junction conditions, from Eq. (98), we derive

[TABLE]

Thus, we have

[TABLE]

and

[TABLE]

Substituting the last result into Eqs. (75)-(80), we obtain

[TABLE]

where

[TABLE]

From Eq. (117), we can find a condition for perfect transmission ($A=0$). From $A=0$, we derive

[TABLE]

Hence, if the condition

[TABLE]

holds, then the perfect transmission occurs repeatedly as $k$ increases. Furthermore, from Eq. (122), we find that the perfect reflection ($F=0$) occurs if and only if the condition

[TABLE]

holds except the trivial cases of $s_{21}=0$ or $s_{31}=0$. It should be emphasized that this phenomenon never happen when we consider one-dimensional quantum systems with double point interactions of degree 2 (see [17, 18]).

5 Summary

We have newly formulated the system with a single Y-junction and the ring systems with double Y-junctions. For the ring systems, we found the compact formulas for probability amplitudes based on the path integral approach. We have also discussed quantum reflection and transmission on the ring systems. Restricting our attention to the ring systems under the scale-invariant junction conditions, we found the conditions for perfect transmission and/or perfect reflection. It is remarkable that perfect reflection can occur in the case of the anti-symmetric ring system, in contrast to the one-dimensional quantum systems having singular nodes of degree 2. This phenomena might be related to the possible switching of the supercurrent through a ring-shaped Josephson junction, which was investigated by [19]. This probable relation would be investigated elsewhere. General cases beyond the scale-invariant junction conditions should also be investigated. This will be provided in the future works.

Appendix A The property of $L_{0}$ in the junction condition

As shown in [4], the parameter $L_{0}$ does not provide any additional freedom for all possible junction conditions characterized by the unitary matrix $U\in U(3)$. To make the present paper self-contained, we explicitly show this fact in the case of $U(3)$, following the discussion of $U(2)$ in [4]. When we use $W$ satisfying Eq. (9), Eq. (8) can be written as

[TABLE]

where

[TABLE]

Equation (128) is reduced to

[TABLE]

We consider an infinitesimal transformation of $L_{0}$ as

[TABLE]

Under this transformation, if we consider infinitesimal transformations of $\theta_{j}$ as

[TABLE]

then Eq. (130) becomes invariant. Hence, the change of $L_{0}$ can always be absorbed by the changes of parameters $(\theta_{1},\theta_{2},\theta_{3})$. In other words, the parameters $(L_{0},\theta_{1},\theta_{2},\theta_{3})$ give the same junction condition as that of $(\overline{L}_{0},\overline{\theta}_{1},\overline{\theta}_{2},\overline{\theta}_{3})$. Thus, the transformation given by (131) and (132) can be considered to be a kind of a gauge transformation. Therefore, we can regard the parameter $L_{0}$ as a gauge freedom.

Appendix B Gell-Mann Matrices

The Gell-Mann Matrices are defined as

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

These eight matrices give bases in SU(3).

Appendix C The condition of the time-reversal symmetry

We discuss the condition of the time-reversal symmetry for the unitary matrix $U$. When we consider the time-reversal transformation

[TABLE]

we find the transformation of the wave function from Eq. (1),

[TABLE]

Then we also have

[TABLE]

Under the time-reversal transformation, the junction condition (8) is transformed as

[TABLE]

where we assumed the transformation $U\rightarrow\overline{U}$. The complex conjugate of Eq. (8) multiplied by $(-U^{T})$ from the left-hand side, where $T$ denotes transposition, becomes

[TABLE]

Here we have used the property that $U$ is unitary. Comparing Eq. (144) with (145), we derive

[TABLE]

Thus, the time-reversal symmetry requires the condition

[TABLE]

Therefore, if $U$ is a symmetric matrix, then the boundary condition satisfies the time-reversal symmetry as well as the Schrödinger equation (1) does.

Next, we discuss the parameter space which satisfies the condition (147) for the time-reversal symmetry. From Eqs. (11) and (147), we derive the condition

[TABLE]

where ${\cal V}$ is given by Eq. (12). In this case, the S-matrix also becomes symmetric, i.e., $S=S^{T}$. Note that while $i\lambda_{2}$ and $i\lambda_{5}$ are real, $i\lambda_{3}$ is imaginary. From the requirements that $e^{i\alpha\lambda_{3}}$, $e^{i\gamma\lambda_{3}}$, and $e^{ia\lambda_{3}}$ should be real, we find the conditions for the time-reversal symmetry,

[TABLE]

Therefore, when the above conditions (149)-(151) hold, the Y-junction satisfies the time-reversal symmetry, in which the six remaining parameters $\beta,\delta,b,\theta_{1},\theta_{2},\theta_{3}$ are still free.

Finally, we relate our expression of the S-matrix with the symmetric S-matrix in the previous works [5, 6]. If we adopt the parameters

[TABLE]

then we obtain

[TABLE]

It should be noted that this matrix also corresponds to a scale-invariant junction condition. When $0\leq b\leq\frac{\pi}{4}$, we introduce new variables

[TABLE]

Then we find that the S-matrix is reduced to

[TABLE]

This is the same expression as the S-matrix in [5, 6].

Appendix D The solution of the amplitude

$A$, $B$, $C$, $D$, $E$ and $F$

By solving Eqs. (26) and (27) algebraically, we obtain the solution for $A,B,C,D,E$ and $F$ as

[TABLE]

where

[TABLE]

Appendix E Scale-invariant junction conditions

We discuss the Weyl scaling transformation of the wave function (see also [4]), which is given by

[TABLE]

where $\lambda$ is a positive constant, and ${\cal A}$ is a normalization factor. Note that the condition for the normalization of the probability is provided by

[TABLE]

in a system with a single Y-junction (Fig. 1(a)). This normalization condition becomes, under the Weyl scaling transformation,

[TABLE]

Hence, we derive

[TABLE]

We also have

[TABLE]

From these results, we find that the junction condition (8) is transformed, under the Weyl scaling transformation, to

[TABLE]

It follows that the junction condition is not invariant in general under the Weyl scaling transformation. If we impose the Weyl scaling invariance, then we obtain

[TABLE]

The last equation means that the eigen values of $U$ take $+1$ or $-1$. In this case, we call the junction condition scale invariant.