††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Multi-locational Majorana Zero Modes

Yutaro Nagae  Department of Applied Physics, Nagoya University, Nagoya
464-8603, Japan  Andreas P. Schnyder  Max-Planck-Institut für
Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany  Yukio
Tanaka  Department of Applied Physics, Nagoya University, Nagoya 464-8603,
Japan  Yasuhiro Asano  Department of Applied Physics, Hokkaido University,
Sapporo 060-8628, Japan  Satoshi Ikegaya  Department of Applied Physics,
Nagoya University, Nagoya 464-8603, Japan  Institute for Advanced Research,
Nagoya University, Nagoya 464-8601, Japan

###### Abstract

We show the appearance of an unconventional Majorana zero mode whose wave
function splits into multiple parts located at different ends of different
one-dimensional topological superconductors, hereinafter referred to as a
multi-locational Majorana zero mode. Specifically, we discuss the multi-
locational Majorana zero modes in a three-terminal Josephson junction
consisting of topological superconductors, which forms an elemental qubit of
fault-tolerant topological quantum computers. We also demonstrate anomalously
long-ranged nonlocal resonant transport phenomena caused by the multi-
locational Majorana zero mode.

††preprint: APS/123-QED

Introduction.—Majorana zero modes (MZMs) of topological superconductors (TSs)
have been a central issue in condensed matter physics [1, 2, 3, 4, 5, 6, 7, 8,
9]. This research field has made remarkable progress with finding various
forms of the MZMs, such as Majorana end modes in one-dimensional TSs [10],
chiral MZMs in TSs with broken time-reversal symmetry [11, 12, 13], helical
MZMs in time-reversal invariant TSs [14, 15, 16, 17], dispersionless MZMs in
nodal TSs [18, 19, 20, 21, 22], and Majorana hinge/corner modes in higher-
order TSs [23, 24, 25, 26, 27, 28, 29, 30, 31].

On the basis of the bulk-boundary correspondence, which is an essential
concept in the physics of topological condensed matters, a MZM is usually
localized at _one_ end/surface of _one_ TS [2, 3, 4, 32, 33, 34]. In this
Letter, nevertheless, we predict the appearance of an unconventional MZM whose
wave function splits into _multiple_ parts located at _different_ ends of
_different_ TSs, where the distance between the ends hosting the MZM can
exceed the decay length of the MZM substantially (see Fig. 1). Hereinafter, we
refer to this particular zero mode, which exhibits simultaneously the Majorana
nature and the nonlocal nature, as a multi-locational Majorana zero mode
(MMM). Specifically, we discuss the appearance of the MMM in a three-terminal
Josephson junction (TJJ) consisting of one-dimensional TSs (hereinafter,
referred to as a topological TJJ).

Another important topic in the field of MZMs is the transport anomalies in
junctions consisting of topological superconductors. So far, the following
transport signatures originating from MZMs have been investigated: zero-bias
conductance quantization in normal-metal–superconductor junctions [35, 36, 37,
38, 39, 40, 41, 42], fractional current-phase relationship in Josephson
junctions [43, 44, 45, 46], anomalous transport in multi-terminal devices [47,
53, 48, 55, 56, 57, 54, 59, 51, 52, 58, 49, 50, 60, 61, 62, 63, 64, 65], and
quantum teleportation in devices including capacitors [66, 67, 68]. In this
Letter, we demonstrate that the MMM causes striking nonlocal transport
phenomena, which manifests clearly the coexistence of the Majorana nature and
the nonlocal nature.

The topological TJJ forms an elemental qubit of fault-tolerant topological
quantum computers [69, 70, 71, 72, 73, 74, 75, 76, 77, 78]. At present,
topological TJJs have not yet been realized experimentally. However, on the
way to realizing the topological quantum computations, which is a central goal
of the topological condensed matter physics, future experiments will certainly
build the topological TJJs. The observations of the characteristic transport
phenomena due to the MMM provide smoking-gun evidences for the realizations of
fully functional topological TJJs.

Figure 1: (a) Schematic image of the topological TJJ. (b) Energy eigenvalues
of the topological TJJ, where all TSs have the same length, $L=800\approx
4\xi$. (c) Amplitudes of the MMM in each TS. (d) Inverse participation ratio
of the MMM in $\alpha=1$.

Multi-locational Majorana zero modes.—At first, we consider a low-energy
effective Hamiltonian describing the couplings between MZMs in a topological
TJJ. As illustrated in Fig. 1(a), in the absence of any couplings, one MZM
appears at each end of the three TSs, where we denote the Majorana operator
for the outer (inner) MZM of the $\alpha$-th TS with
$\gamma_{o(i),\alpha}=\gamma^{\dagger}_{o(i),\alpha}$. In the TJJ, the
couplings among these MZMs are described by

$\displaystyle\begin{split}&H=\frac{1}{2}\bm{\gamma}^{\mathrm{T}}\mathcal{H}\bm{\gamma},\quad\mathcal{H}=\left(\begin{array}[]{cc}0_{3\times
3}&i\mathcal{H}_{T}\\\ -i\mathcal{H}_{T}&\mathcal{H}_{J}\end{array}\right),\\\
&\mathcal{H}_{T}=\mathrm{diag}[E_{T,1},E_{T,2},E_{T,3}],\\\
&\mathcal{H}_{J}=\left(\begin{array}[]{ccc}0&iE_{J,12}&iE_{J,13}\\\
-iE_{J,12}&0&iE_{J,23}\\\
-iE_{J,13}&-iE_{J,23}&0\end{array}\right),\end{split}$ (1)

where
$\bm{\gamma}=[\gamma_{o,1},\gamma_{o,2},\gamma_{o,3},\gamma_{i,1},\gamma_{i,2},\gamma_{i,3}]^{\mathrm{T}}$.
The $3\times 3$ null matrix is represented by $0_{3\times 3}$. The couplings
between the outer and inner MZMs of the $\alpha$-th TS are characterized by
$E_{T,\alpha}\propto\exp(-2L_{\alpha}/\xi)$ [71, 79], where $L_{\alpha}$ and
$\xi$ represent the length of the $\alpha$-th TS and the decay length of the
MZM, respectively. The couplings among the inner MZMs at the Josephson
junction is described by $\mathcal{H}_{J}$ [10], where
$E_{J,\alpha\beta}=C_{\alpha\beta}\sin[(\phi_{\alpha}-\phi_{\beta})/2]$, with
$C_{\alpha\beta}=C_{\beta\alpha}$ and $\phi_{\alpha}$ representing the
superconducting phase of the $\alpha$-th TS. For simplicity, we ignore long-
range couplings, i.e., $\gamma_{o,\alpha}$ couples only with
$\gamma_{i,\alpha}$.

Since originally there were three MZMs at the Josephson junction, $H_{J}$
becomes a $3\times 3$ skew-symmetric matrix, which always has an unpaired
zero-energy eigenvalue. The two remaining finite eigenvalues are given by $\pm
E_{J}$ with $E_{J}=\sqrt{E_{J,12}^{2}+E_{J,23}^{2}+E_{J,31}^{2}}$ [76, 77]. By
applying a unitary transformation to diagonalize $\mathcal{H}_{J}$, as also
discussed in the Supplemental Material (SM) [80], we rewrite the Hamiltonian
as

$\displaystyle\begin{split}&H=\frac{1}{2}\tilde{\bm{\gamma}}^{\dagger}\left(\begin{array}[]{cc}\mathcal{H}_{\mathrm{eff}}&\mathcal{V}\\\
\mathcal{V}^{\dagger}&\mathcal{E}_{J}\end{array}\right)\tilde{\bm{\gamma}},\\\
&\mathcal{H}_{\mathrm{eff}}=\left(\begin{array}[]{cccc}0&0&0&iv_{1}\\\
0&0&0&iv_{2}\\\ 0&0&0&iv_{3}\\\
-iv_{1}&-iv_{2}&-iv_{3}&0\end{array}\right),\\\
&\mathcal{V}=\left(\begin{array}[]{cc}iv_{1,+}&iv_{1,-}\\\
iv_{2,+}&iv_{2,-}\\\ iv_{3,+}&iv_{3,-}\\\
0&0\end{array}\right),\quad\mathcal{E}_{J}=\left(\begin{array}[]{cc}E_{J}&0\\\
0&-E_{J}\end{array}\right),\end{split}$ (2)

where
$\tilde{\bm{\gamma}}=[\gamma_{o,1},\gamma_{o,2},\gamma_{o,3},\gamma_{0,J},\gamma_{+,J},\gamma_{-,J}]^{T}$,
with $\gamma_{0,J}$ and $\gamma_{\pm,J}$ corresponding to the zero-energy and
the finite-energy eigenstates of $\mathcal{H}_{J}$, respectively.
Specifically, we obtain
$v_{\alpha}=(E_{T,\alpha}/2)\sum_{\beta,\gamma}\epsilon^{\alpha\beta\gamma}(E_{J,\beta\gamma}/E_{J})$,
where $\epsilon^{\alpha\beta\gamma}$ is the Levi-Civita symbol. The components
in $\mathcal{V}$ are also of the order of $E_{T,\alpha}$ [80]. Thus, in the
limit of $E_{T}\ll E_{J}$ with
$E_{T}=\sqrt{E_{T,1}^{2}+E_{T,2}^{2}+E_{T,3}^{2}}$, the low-energy excitation
of $|E|\ll E_{J}$ is approximately described by [81, 82],

$\displaystyle\begin{split}&H_{\mathrm{eff}}=\frac{1}{2}\bm{\gamma}_{\mathrm{eff}}^{T}\mathcal{H}_{\mathrm{eff}}\bm{\gamma}_{\mathrm{eff}},\\\
&\bm{\gamma}_{\mathrm{eff}}=[\gamma_{o,1},\gamma_{o,2},\gamma_{o,3},\gamma_{0,J}]^{T},\end{split}$
(3)

where the contributions from the finite-energy eigenstates at the Josephson
junction (i.e., $\gamma_{\pm,J}$) are ignored. The detailed derivation of
$H_{\mathrm{eff}}$ is presented in the SM [80]. The effective low-energy
Hamiltonian $H_{\mathrm{eff}}$ has doubly degenerate zero modes:

$\displaystyle\gamma_{n}=\sum_{\alpha=1\text{-}3}c_{\alpha}^{n}\gamma_{o,\alpha}\quad(n=1,2),$
(4)

where the coefficients $c_{\alpha}^{n}$ are real numbers satisfying

$\displaystyle\sum_{\alpha}v_{\alpha}c_{\alpha}^{n}=0,\quad\sum_{\alpha}c_{\alpha}^{n}c_{\alpha}^{m}=\delta_{m,n}.$
(5)

Importantly, since the coefficients $c_{\alpha}^{n}$ are _real_ numbers, the
Majorana relation holds, i.e., $\gamma_{n}=\gamma_{n}^{\dagger}$. Moreover,
$\gamma_{n}$ is described by the superposition of the outer end MZMs (i.e.,
$\gamma_{o,\alpha}$). Namely, the wave function of $\gamma_{n}$ has large
amplitudes at the outer edges of different TSs. This unconventional zero mode
exhibiting both Majorana nature and the nonlocal nature is what we refer to as
the MMM. Due to the degeneracy at zero energy, there is ambiguity in the
coefficients $c_{\alpha}^{n}$. However, the quantity of

$\displaystyle\begin{split}&P_{\alpha}=\sum_{n=1,2}|c_{\alpha}^{n}|^{2}=\sum_{\beta\neq\alpha}\frac{v^{2}_{\beta}}{E_{v}^{2}},\\\
&E_{v}=\sqrt{v_{1}^{2}+v_{2}^{2}+v_{3}^{3}},\end{split}$ (6)

which characterizes the amplitudes of the MZMs in the $\alpha$-th TS, is
independent of the choice of $c_{\alpha}^{n}$. As long as
$v_{\alpha=1,2,3}\neq 0$, we obtain $P_{\alpha}<1$ for all TSs. Namely, _two_
MMMs are distributed over the ends of _three_ different TSs. For instance, we
obtain $P_{1}=P_{2}=P_{3}=2/3$ with $v_{1}=v_{2}=v_{3}$, which means that the
_two_ MMMs are equality distributed over the ends of _three_ different TSs.
When one of $v_{\alpha}$ is zero, e.g. $v_{1}=v_{2}\neq 0$ and $v_{3}=0$, we
obtain $P_{1}=P_{2}=1/2$ and $P_{3}=1$. In this case, _one_ MMM is distributed
over the outer edges of _two_ TSs, namely the first and second one, while the
third TS hosts a conventional MZM at its outer edge.

Here we numerically reproduce the appearance of the MMM by using the tight-
binding Bogoliubov-de Gennes (BdG) Hamiltonian of a spin-less $p$-wave
superconductor (i.e., a Kitaev chain [10]). We note that the low-energy
physics of various one-dimensional TSs [83, 84], such as superconducting
semiconductor nanowires [85, 86, 87, 88], magnetic atom chains deposited on
superconductors [89, 90, 91, 92], and planer topological Josephson junctions
[93, 94, 95, 96, 97, 98, 99, 100, 101], are described effectively by the
Kitaev chain [40, 89, 101]. The BdG Hamiltonian reads
$H_{\mathrm{BdG}}=\sum_{\alpha=1\text{-}3}H_{\alpha}+H_{d}$ with

$\displaystyle H_{\alpha}=$
$\displaystyle\sum_{x_{\alpha}=1}^{L_{\alpha}-1}(-tc^{\dagger}_{x_{\alpha}+1}c_{x_{\alpha}}+\mathrm{h.c.})-\sum_{x_{\alpha}=-\infty}^{L_{\alpha}}\mu
c^{\dagger}_{x_{\alpha}}c_{x_{\alpha}}$
$\displaystyle+\frac{1}{2}\sum_{x_{\alpha}=1}^{L_{\alpha}-1}(i\Delta
e^{i\phi_{\alpha}}c^{\dagger}_{x_{\alpha}+1}c^{\dagger}_{x_{\alpha}}+\mathrm{h.c.}),$
(7) $\displaystyle H_{d}=$
$\displaystyle\sum_{\alpha=1\text{-}3}(-t_{d,\alpha}c^{\dagger}_{d}c_{L_{\alpha}}+\mathrm{h.c.})-\mu_{d}c^{\dagger}_{d}c_{d},$

where $c_{x_{\alpha}}$ ($c^{\dagger}_{x_{\alpha}}$) is the annihilation
(creation) operator of an electron at a site $x_{\alpha}$ of the $\alpha$-th
branch, $t$ is the nearest-neighbor hopping integral, $\mu$ denotes the
chemical potential, and $\Delta$ represents the pair potential in the
superconducting segment. The three TSs are connected through a single normal
site, where the coupling is described by $H_{d}$. In the following
calculations, we fix the parameters as $\mu=\mu_{d}=-0.8t$, $t_{d,\alpha}=t$,
and $\Delta=0.01t$, where the decay length of MZMs is evaluated by
$\xi=1/\mathrm{artanh}(\Delta/2t)\approx 200$. For simplicity, we assume that
all TSs have a same length $L_{1}=L_{2}=L_{3}=L$. In Fig. 1(b), we show the
energy eigenvalues obtained by diagonalizing $H_{\mathrm{BdG}}$, where
$L=800\approx 4\xi$. The superconducting phases are chosen as
$(\phi_{1},\phi_{2},\phi_{3})=(0,2\pi/3,4\pi/3)$, which corresponds to
$v_{1}=v_{2}=v_{3}$ for the effective Hamiltonian $H_{\mathrm{eff}}$. We
observe two zero-energy states, marked by the arrows, corresponding to two
MMMs. The first excited states are approximately located at $E=\pm E_{v}$,
which characterizes the necessary energy resolution to observe the MMMs. The
energy $E_{v}$ is proportional to
$E_{T,\alpha}\approx\Delta\cos(k_{F}L)e^{-2L_{\alpha}/\xi}$ with
$k_{F}=\arccos(\mu/2t)$ and thus oscillates and decays by increasing $L$ [79].
We note that these conditions might be achievable in experiments with
artificial TSs [85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99,
100, 101], where the coupling strength $E_{T,\alpha}$ as well as the decay
length of the MZM can be controlled, for example, by changing the applied
Zeeman fields [79]. In Fig. 1(c), we show the amplitude of the wave function
of the zero energy states in each TS, i.e.,
$\sum_{n=1,2}|\psi_{n}(x_{\alpha})|^{2}$ with $\psi_{n}(x_{\alpha})$
representing the wave function of the $n$-th zero-energy states. The amplitude
of the wave function increases at the outer edge of each TS and is equivalent
for all TSs, which agrees with $P_{1}=P_{2}=P_{3}=2/3$ obtained by the
effective theory based on $H_{\mathrm{eff}}$. In Fig. 1(d), we show the
inverse participation ratio [102] for each TS, defined by
$\mathrm{IPR}_{\alpha}=\sum_{n=1,2}\sum_{x_{\alpha}=1}^{L_{\alpha}}|\psi_{n}(x_{\alpha})|^{4}$.
By increasing the length of TS (i.e., $L$), $\mathrm{IPR}_{\alpha}$ becomes a
constant, which means that the wave function of the zero-energy states is well
localized at the outer edges of the TS and is not distribute in the entire
system. As a result, we numerically verified the appearance of the MMM in the
tight-binding model.

Transport anomalies.—We study transport anomalies due to the MMM. At first, we
capture the essence of transport properties of MMM by an analytical
calculation using the coupling Hamiltonian in Eq. (24). Then, we numerically
reproduce the analytical results by employing recursive Green’s function
techniques [103, 104], where the TSs are described by the Kitaev chains [10].
To study the transport properties of the MMM, we attach a normal-metal lead to
the outer end of each TS. We refer to the lead attached to the $\alpha$-th TS
as the $\alpha$-th lead. In general, for the wide-bandwidth normal-metal
leads, the scattering matrix at energy $E$ are represented by [47, 51]

$\displaystyle\begin{split}S(E)&=\left(\begin{array}[]{cc}s^{ee}&s^{eh}\\\
s^{he}&s^{hh}\end{array}\right)\\\ &=1-2\pi
iW^{\dagger}\left[E-\mathcal{H}+i\pi WW^{\dagger}\right]^{-1}W,\end{split}$
(8)

where $(s^{\eta\zeta})_{\alpha\beta}=s^{\eta\zeta}_{\alpha\beta}$ with
$s^{ee}_{\alpha\beta}$ and $s^{he}_{\alpha\beta}$ ($s^{eh}_{\alpha\beta}$ and
$s^{hh}_{\alpha\beta}$) being the scattering coefficients from the electron
(hole) in the $\beta$-th lead to the electron and hole in the $\alpha$-th
lead, respectively. In our case, the coupling matrix $W$ is given by [47]

$\displaystyle W=\left(\begin{array}[]{cc}w&-w\\\ 0_{3\times 3}&0_{3\times
3}\end{array}\right),\quad w=\left(\begin{array}[]{ccc}w_{1}&0&0\\\
0&w_{2}&0\\\ 0&0&w_{3}\end{array}\right),$ (14)

where the basis for the normal-metal leads is chosen as
$\bm{c}=[c_{e,1},c_{e,2},c_{e,3},c_{h,1},c_{h,2},c_{h,3}]^{\mathrm{T}}$ with
$c_{e(h),\alpha}$ representing a propagating electron (hole) in the
$\alpha$-th lead. $w_{\alpha}$ characterizes the couplings between the
electron/hole in the $\alpha$-th lead and the MZM at the outer end of the
$\alpha$-th TS. Without loss of generality, we set $w_{\alpha}$ to be real
numbers [47]. When $w_{\alpha}^{2}\ll E_{T}\ll E_{J}$, the scattering
coefficients at zero energy is obtained by

$\displaystyle\begin{split}&s^{ee}_{\alpha\alpha}=\frac{v_{\alpha}^{2}w_{\beta}^{2}w_{\gamma}^{2}}{A},\quad
s^{he}_{\alpha\alpha}=-\frac{w_{\alpha}^{2}(v_{\beta}^{2}w_{\gamma}^{2}+w_{\beta}^{2}v_{\gamma}^{2})}{A},\\\
&s^{ee}_{\alpha\beta}=-s^{he}_{\alpha\beta}=\frac{v_{\alpha}v_{\beta}w_{\alpha}w_{\beta}w^{2}_{\gamma}}{A},\\\
&A=v_{1}^{2}w_{2}^{2}w_{3}^{2}+v_{2}^{2}w_{3}^{2}w_{1}^{2}+v_{3}^{2}w_{1}^{2}w_{2}^{2},\end{split}$
(15)

where $\alpha\neq\beta$, $\beta\neq\gamma$, and $\gamma\neq\alpha$. The
detailed derivation for the scattering coefficients are shown in the SM [80].
The scattering coefficients show highly intertwined properties: the local
Andreev reflection probability in the $\alpha$-th lead is related with the
normal reflection probabilities in the different leads as
$|s^{he}_{\alpha\alpha}|=|s^{ee}_{\beta\beta}|+|s^{ee}_{\gamma\gamma}|$;
moreover, all nonlocal scattering probabilities between the $\alpha$-th and
$\beta$-th leads are equivalent (i.e.,
$|s^{ee}_{\alpha\beta}|=|s^{he}_{\alpha\beta}|=|s^{ee}_{\beta\alpha}|=|s^{he}_{\beta\alpha}|$).
When $v_{1}=v_{2}=v_{3}$ and $w_{1}=w_{2}=w_{3}$, for instance, we obtain
$|s^{he}_{\alpha\alpha}|^{2}=4/9$ and
$|s^{ee}_{\alpha\alpha}|^{2}=|s^{ee}_{\alpha\beta}|^{2}=|s^{he}_{\beta\alpha}|^{2}=1/9$
for $\alpha\neq\beta$. When $v_{1}=v_{2}$ and $v_{3}=0$ with
$w_{1}=w_{2}=w_{3}$, we obtain
$|s^{ee}_{\alpha\beta}|^{2}=|s^{he}_{\alpha\beta}|^{2}=1/4$ for
$\alpha,\beta=1,2$. Such highly intertwined scatterings with respect to the
multiple leads are obviously due to the MMM.

We numerically reproduce the anomalous transport of the MMM by using the
tight-binding BdG Hamiltonian. We describe the normal leads located at
$x_{\alpha}<1$ by setting $\Delta=0$ in $H_{\alpha}$ and denote the hopping
integral at the interface of the $\alpha$-th lead and the $\alpha$-th TS
(i.e., the hopping integral between $x_{\alpha}=0$ and $x_{\alpha}=1$) by
$t^{\prime}_{\alpha}$. We numerically compute the scattering coefficients
using the recursive Green’s function techniques [103, 104]. In Fig. 2(a), we
show the local Andreev reflection probability of $|s^{he}_{11}|^{2}$ and the
nonlocal Andreev reflection probability of $|s^{he}_{12}|^{2}$ at zero energy
as a function of the superconductor length $L$. We consider a low-transparency
junction with $t_{\alpha}^{\prime}=0.1t$, which corresponds to
$w_{1}=w_{2}=w_{3}$. The superconducting phases are chosen as
$(\phi_{1},\phi_{2},\phi_{3})=(0,2\pi/3,4\pi/3)$, which corresponds to
$v_{1}=v_{2}=v_{3}$. We clearly find the plateau at $|s^{he}_{11}|^{2}=4/9$
and $|s^{he}_{12}|^{2}=1/9$ in a broad range of $L$, which agrees with Eq.
(15). We also confirm that other scattering amplitudes also exhibit the
plateaus consistent with Eq. (15). For longer $L$ exceeding $7\xi$, the ideal
plateau is affected by the finite energy states at $E=\pm E_{v}\propto
e^{-2L_{\alpha}/\xi}$. For $L$ shorter than $5\xi$, we see the deviation from
the ideal plateau owing to more complex couplings neglected in
$H_{\mathrm{eff}}$. Nevertheless, it has been shown that typical nonlocal
scatterings through superconductors decay exponentially with the distance
between the leads on a scale fixed by $\xi$ [47, 48, 49, 50, 51, 52]. On the
other hand, in the topological TJJ, the plateau at $|s^{he}_{12}|^{2}=1/9$ is
maintained even when the distance between the two leads (i.e., $2L$) exceeds
$10\xi$ (the leads are separated by two TSs). As a result, the robust plateau
in Fig. 2(a) proves the anomalously long-ranged nonlocal transport mediated by
the MMM.

Figure 2: (a) Nonlocal Andreev reflection probability $|s^{he}_{12}|^{2}$ and
nocal Andreev reflection probability $|s^{he}_{11}|^{2}$ at zero energy as a
function of the superconductor length. (b) Differential conductance as a
function of the bias voltage.

We describe how the characteristic scatterings of the MMM affect the charge
transport. For this purpose, we apply the same bias voltage $V$ to all normal-
metal leads, while all TSs are grounded. We assume significantly low
transparency at the interfaces between the leads and TSs, such that the bias
voltages applied to the leads drop only at the interfaces. We also assume
$(\phi_{1},\phi_{2},\phi_{3})=(0,0,\pi)$, such that the Josephson currents
between the TSs are absent; this assumption yields $v_{3}=0$, and thus the MMM
is distributed in the first and second TSs. Within the
Blonder–Tinkham–Klapwijik (BTK) formalism, the differential conductance
$G_{\alpha}(eV)=dI_{\alpha}/dV$ at zero-temperature is given by [105, 106]

$\displaystyle
G_{\alpha}(eV)=\frac{e^{2}}{h}\sum_{\beta}\left[\delta_{\alpha\beta}-|s^{ee}_{\alpha\beta}|^{2}+|s^{he}_{\alpha\beta}|^{2}\right]_{E=eV},$
(16)

where $I_{\alpha}$ is the time-averaged current in the $\alpha$-th lead. We
note that our calculations based on the BTK formalism is qualitatively
justified for bias voltages well below the superconducting gap. The
differential conductance at zero bias voltages with low transparency
interfaces (i.e., $w_{\alpha}^{2}\ll E_{v}\ll E_{J}$) is given by

$\displaystyle\begin{split}&G_{1(2)}(0)=\frac{2e^{2}}{h}\frac{v_{2(1)}^{2}w_{1(2)}^{2}}{v_{1}^{2}w_{2}^{2}+v_{2}^{2}w_{1}^{2}},\quad
G_{3}(0)=\frac{2e^{2}}{h}.\end{split}$ (17)

The differential conductance of the third lead exhibits the zero-bias
conductance quantization [35, 36, 37, 38, 39]. On the other hand, $G_{1}(0)$
and $G_{2}(0)$ depend on the transparency of the interfaces and are not
quantized. Nevertheless, a remarkable relation holds:

$\displaystyle G_{1}(0)+G_{2}(0)=\frac{2e^{2}}{h}.$ (18)

Namely, the sum of zero-bias differential conductance in the different leads
is perfectly quantized. The observation of the nonlocally quantized
conductance can be a smoking-gun evidence for the appearance of the MMM. In
Fig. 2(c), we show the differential conductance obtained numerically by using
the tight-binding model. We choose $L=800\sim 4\xi$,
$(\phi_{1},\phi_{2},\phi_{3})=(0,0,\pi)$ and
$(t_{1}^{\prime},t_{2}^{\prime},t_{3}^{\prime})=(0.07t,0.1t,0.07t)$. While
$G_{1}\neq G_{2}$ owing to $t_{1}^{\prime}\neq t_{2}^{\prime}$, we see
$G_{1}+G_{2}$ at zero-bias voltage is quantized to $2e^{2}/h$. Therefore, we
numerically demonstrate the sum quantization of the conductance due to the
MMM. We also find a second peak at the finite bias voltage $eV\approx E_{v}$.
We remark that the energy $E_{v}$, which characterizes the necessary
resolutions in experiments, can be controlled, for instance, by changing
applied Zeeman fields of the artificial TSs [79].

Even though the current shot noise is more difficult to measure experimentally
than the differential conductance, it also manifests the profound nature of
the intertwined scatterings of the MMM. We specifically discuss the zero-
frequency noise power defined by

$\displaystyle P_{\alpha\beta}=\int^{\infty}_{-\infty}\overline{\delta
I_{\alpha}(0)\delta I_{\beta}(t)}dt,$ (19)

where $\delta I_{\alpha}(t)=I_{\alpha}(t)-I_{\alpha}$ denotes the deviation of
the current at time $t$ from the time averaged current. Within the BTK
formalism, the zero-frequency noise power at zero-temperature is given by
[107, 108]

$\displaystyle\begin{split}&P_{\alpha\beta}=\frac{e^{2}}{h}\int^{eV}_{0}\mathcal{P}_{\alpha\beta}(E)dE,\\\
&\mathcal{P}_{\alpha\beta}(E)=\delta_{\alpha\beta}\sum_{\eta=e,h}p^{\eta\eta}_{\alpha\alpha}-\sum_{\eta,\zeta}\sigma_{\eta}\sigma_{\zeta}p^{\eta\zeta}_{\alpha\beta}p^{\zeta\eta}_{\beta\alpha},\\\
&p^{\eta\zeta}_{\alpha\beta}=\sum_{\gamma=1,2,3}s^{\eta
e}_{\alpha\gamma}(s^{\zeta e}_{\beta\gamma})^{\ast},\end{split}$ (20)

where $\sigma_{\eta}=1$ ($-1$) for $\eta=e$ ($h$). For significantly low bias
voltages with low-transparency interfaces, such as $eV\ll w_{\alpha}^{2}\ll
E_{v}\ll E_{J}$, the zero-frequency noise power is given by

$\displaystyle\begin{split}&P_{11}=P_{22}=-P_{12}=-P_{21}=\frac{2h}{eV}I_{1}I_{2},\\\
&P_{33}=P_{3\alpha}=P_{\alpha 3}=0,\quad(\alpha\neq 3),\end{split}$ (21)

where we approximate
$P_{\alpha\beta}\approx(e^{3}V/h)\mathcal{P}_{\alpha\beta}(0)$ and
$I_{\alpha}\approx G_{\alpha\alpha}(0)V$, which is justified in the linear
response regime of $eV$. We find that the particular relation,

$\displaystyle 2|P_{12}|=P_{11}+P_{22},$ (22)

is satisfied. The cross-correlator in any stochastic processes is bounded by
the auto-correlator as $2|P_{12}|\leq P_{11}+P_{22}$. Thus, the maximized
cross-correlator of $|P_{12}|$ implies that the MMM causes the perfect
correlation between the first and second leads. In addition, the total noise
power satisfies $\sum_{\alpha,\beta}P_{\alpha\beta}=0$, which implies that the
total charge current flowing into the TSs is noiseless due to the resonant
transmissions of the conventional MZM at the third TS and the MMM splitting
into the first and second TSs. Especially, the relation of

$\displaystyle P_{11}+P_{22}+P_{12}+P_{21}=0$ (23)

is closely connected with the nonlocally quantized conductance in Eq. (18);
both are the physical consequence of the highly intertwined resonant
scatterings caused by the MMM. In the SM [80], we also demonstrate the
anomalous current noise [i.e. Eq. (22) and Eq. (23)] using the tight-binding
model.

Discussion.—In this paper, we discuss the charge currents only when the
Josephson currents are absent, i.e., the superconducting phases are fixed at
$(\phi_{1},\phi_{2},\phi_{3})=(0,0,\pi)$. Studying other cases, for example,
the coexistence of charge currents due to the bias voltages and Josephson
currents [56, 57, 58], remains as an important future task. Nevertheless, we
still expect drastic transport anomalies because as shown in Eq. (4), the
topological TJJ hosts the MMM insensitive to the superconducting phases.

We have shown that MMM in the TJJ is created when an odd number of MZMs are
brought into contact and coupled to each other. Moreover, our analysis is
based on an effective Hamiltonian, which focuses only on the couplings among
the MZMs and is irrelevant to the details of the parent TS. Thus, we expect
that our strategy for creating a MMM can be extended to other systems, such as
higher-order TSs exhibiting high controllability for the number and location
of MZMs [28, 29, 30, 31]. Inspired by the various studies on the conventional
MZMs, studying the Cooper pair splittings due to the nonlocal Andreev
reflections [47, 48, 49, 50, 51, 52], the nonlocal Josephson effects forming
Cooper quartets [60, 61, 62], the quantum teleportation [66, 67, 68], and the
full-counting statistics [63, 64, 65] in the presence of MMMs would be
intriguing topics for future works. We hope that our work will stimulate
further investigations into the systems hosting MMMs.

In summary, we demonstrate the appearance of the MMM whose wave function
splits into the multiple parts localized at different edges of different TSs.
Thanks to the coexistence of the Majorana property and the nonlocal property,
the MMM causes the drastic nonlocal resonant transport phenomena. We expect
that our proposal can be used to prove the realization of the fully functional
topological TJJ, which forms an elemental qubit for fault-tolerant topological
quantum computers.

###### Acknowledgements.

S.I. is supported by the Grant-in-Aid for JSPS Fellows (JSPS KAKENHI Grant No.
JP22KJ1507).

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Supplemental Material for “Multi-locational Majorana Zero Modes”
Yutaro Nagae1, Andreas P. Schnyder2, Yukio Tanaka1, Yasuhiro Asano3, and
Satoshi Ikegaya1,4

1Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
2Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569
Stuttgart, Germany
3Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
4Institute for Advanced Research, Nagoya University, Nagoya 464-8601, Japan

##  I Low-energy effective Hamiltonian

In this section, we derive the low-energy effective Hamiltonian for the three-
terminal Josephson junctions consisting of the one-dimensional topological
superconductors (TSs), which is given by Eq. (3) in the main text. We start
with the Hamiltonian in Eq. (1) in the main text, which describes the
couplings between Majorana zero modes (MZMs) in the three-terminal Josephson
junction:

$\displaystyle\begin{split}&H=\frac{1}{2}\bm{\gamma}^{\mathrm{T}}\mathcal{H}\bm{\gamma},\\\
&\bm{\gamma}=[\gamma_{o,1},\gamma_{o,2},\gamma_{o,3},\gamma_{i,1},\gamma_{i,2},\gamma_{i,3}]^{\mathrm{T}},\quad\mathcal{H}=\left(\begin{array}[]{cc}0_{3\times
3}&i\mathcal{H}_{T}\\\ -i\mathcal{H}_{T}&\mathcal{H}_{J}\end{array}\right),\\\
&\mathcal{H}_{T}=\mathrm{diag}[E_{T,1},E_{T,2},E_{T,3}],\quad\mathcal{H}_{J}=\left(\begin{array}[]{ccc}0&iE_{J,12}&iE_{J,13}\\\
-iE_{J,12}&0&iE_{J,23}\\\
-iE_{J,13}&-iE_{J,23}&0\end{array}\right).\end{split}$ (24)

where $\gamma_{o(i),\alpha}=\gamma^{\dagger}_{o(i),\alpha}$ is an outer
(inner) MZM of the $\alpha$-th TS. In the absence of any couplings, as also
illustrated in Fig. 3(a), one MZM appears at each end of the three TSs. The
coupling constant between the outer and inner MZMs in the $\alpha$-th TS is
given by $E_{T,\alpha}$. The coupling between the inner MZMs of the
$\alpha$-th and $\beta$-th TSs at the Josephson junction is represented by
$E_{J,\alpha\beta}=-E_{J,\beta\alpha}$. The $n\times m$ null matrix is given
by $0_{n\times m}$. At first, we define a $3\times 3$ unitary matrix that
diagonalizes $\mathcal{H}_{J}$:

$\displaystyle\begin{split}&\mathcal{U}_{J}^{\dagger}\mathcal{H}_{J}\mathcal{U}_{J}=\left(\begin{array}[]{cc}0&0_{1\times
2}\\\ 0_{2\times 1}&\mathcal{E}_{J}\end{array}\right),\\\
&\mathcal{E}_{J}=\left(\begin{array}[]{cc}E_{J}&0\\\
0&-E_{J}\end{array}\right),\quad
E_{J}=\sqrt{E_{J,12}^{2}+E_{J,23}^{2}+E_{J,31}^{2}},\end{split}$ (25)

with

$\displaystyle\mathcal{U}_{J}=\left(\bm{V}_{0},\bm{V}_{+},\bm{V}_{-}\right),$
(26)

where $\bm{V}_{0}$ and $\bm{V}_{\pm}$ are the three-element vectors satisfying

$\displaystyle\mathcal{H}_{J}\bm{V}_{0}=0,\qquad\mathcal{H}_{J}\bm{V}_{\pm}=\pm
E_{J}\bm{V}_{\pm},$ (27)

respectively. Specifically, $\bm{V}_{0}$ is given by

$\displaystyle\bm{V}_{0}=\frac{1}{E_{J}}\left(\begin{array}[]{cl}E_{J,23}\\\
-E_{J,13}\\\ E_{J,12}\end{array}\right).$ (31)

In addition, we define the field operators of

$\displaystyle\left(\begin{array}[]{cl}\gamma_{0,J}\\\ \gamma_{+,J}\\\
\gamma_{-,J}\end{array}\right)=\mathcal{U}_{J}^{\dagger}\left(\begin{array}[]{cl}\gamma_{i,1}\\\
\gamma_{i,2}\\\ \gamma_{i,3}\end{array}\right)$ (38)

where $\gamma_{0,J}$ and $\gamma_{\pm,J}$ corresponding to the field operators
for the zero-energy eigenstate and the finite-energy eigenstates,
respectively. We note that $\gamma_{0,J}$ still satisfies the Majorana
relation, $\gamma_{0,J}=\gamma_{0,J}^{\dagger}$. By using the $6\times 6$
unitary matrix of

$\displaystyle\mathcal{U}=\left(\begin{array}[]{cc}1_{3}&0_{3\times 3}\\\
0_{3\times 3}&\mathcal{U}_{J}\end{array}\right),$ (41)

with $1_{n}$ being the $n\times n$ identity matrix, we rewrite the Hamiltonian
as

$\displaystyle
H=\frac{1}{2}\bm{\gamma}^{\mathrm{T}}\mathcal{U}\mathcal{U}^{\dagger}\mathcal{H}\mathcal{U}\mathcal{U}^{\dagger}\bm{\gamma}=\frac{1}{2}\tilde{\bm{\gamma}}^{\dagger}\tilde{\mathcal{H}}\tilde{\bm{\gamma}},$
(42)

with

$\displaystyle\tilde{\bm{\gamma}}=\mathcal{U}^{\dagger}\bm{\gamma}=[\gamma_{o,1},\gamma_{o,2},\gamma_{o,3},\gamma_{0,J},\gamma_{+,J},\gamma_{-,J}]^{T},$
(43)

and

$\displaystyle\tilde{\mathcal{H}}=\mathcal{U}^{\dagger}\mathcal{H}\mathcal{U}=\left(\begin{array}[]{cc}0_{3\times
3}&i\mathcal{H}_{T}\mathcal{U}_{J}\\\
-i\mathcal{U}_{J}^{\dagger}\mathcal{H}_{T}&\mathcal{U}_{J}^{\dagger}\mathcal{H}_{J}\mathcal{U}_{J}\end{array}\right).$
(46)

For later convenience, we rewrite the Hamiltonian as

$\displaystyle\begin{split}&H=\frac{1}{2}\tilde{\bm{\gamma}}^{\dagger}\left(\begin{array}[]{cc}\mathcal{H}_{0}&\mathcal{V}\\\
\mathcal{V}^{\dagger}&\mathcal{E}_{J}\end{array}\right)\tilde{\bm{\gamma}},\\\
&\mathcal{H}_{0}=\left(\begin{array}[]{cccc}0&0&0&iv_{1}\\\ 0&0&0&iv_{2}\\\
0&0&0&iv_{3}\\\
-iv_{1}&-iv_{2}&-iv_{3}&0\end{array}\right),\quad\mathcal{V}=\left(\begin{array}[]{cc}iv_{1,+}&iv_{1,-}\\\
iv_{2,+}&iv_{2,-}\\\ iv_{3,+}&iv_{3,-}\\\ 0&0\end{array}\right),\end{split}$
(47)

where

$\displaystyle\left(\begin{array}[]{cl}v_{1}\\\ v_{2}\\\
v_{3}\end{array}\right)=\mathcal{H}_{T}\bm{V}_{0}=\frac{1}{E_{J}}\left(\begin{array}[]{cl}E_{T,1}E_{J,23}\\\
-E_{T,2}E_{J,13}\\\ E_{T,3}E_{J,12}\end{array}\right),$ (54)

and

$\displaystyle\left(\begin{array}[]{cl}v_{1,\pm}\\\ v_{2,\pm}\\\
v_{3,\pm}\end{array}\right)=\mathcal{H}_{T}\bm{V}_{\pm}.$ (58)

The Hamiltonian in Eq. (47) is equivalent to that in Eq. (2) of the main text.

Figure 3: Schematic image for the process creating Multi-locational Majorana
zero modes.

To construct the low-energy effective Hamiltonian, we consider the resolvent
corresponding to $\tilde{\mathcal{H}}$ [81, 82]:

$\displaystyle\mathcal{G}(E)=(E-\tilde{\mathcal{H}})^{-1}=\left(\begin{array}[]{cc}\mathcal{G}_{\mathrm{mzm}}&\mathcal{G}_{\mathcal{V}}\\\
\mathcal{G}_{\mathcal{V}^{\dagger}}&\mathcal{G}_{J}\end{array}\right).$ (61)

The block component related to the MZMs of

$\displaystyle\bm{\gamma}_{\mathrm{eff}}=[\gamma_{o,1},\gamma_{o,2},\gamma_{o,3},\gamma_{0,J}]^{T}$
(62)

is obtained as

$\displaystyle\begin{split}&\mathcal{G}_{\mathrm{mzm}}(E)=[E-\mathcal{H}_{\mathrm{mzm}}(E)]^{-1},\\\
&\mathcal{H}_{\mathrm{mzm}}(E)=\mathcal{H}_{0}+\mathcal{V}(E-\mathcal{E}_{J})^{-1}\mathcal{V}^{\dagger}.\end{split}$
(63)

When we focus on the low-energy excitation of $|E|\ll E_{J}$, we can make the
replacement of

$\displaystyle\mathcal{V}(E-\mathcal{E}_{J})^{-1}\mathcal{V}^{\dagger}\approx-\frac{1}{E_{J}}\mathcal{V}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\mathcal{V}^{\dagger},$ (66)

which yields the effective Hamiltonian describing the low-energy excitation of
$|E|\ll E_{J}$ [81, 82]:

$\displaystyle\mathcal{H}_{\mathrm{eff}}=\mathcal{H}_{0}+\delta\mathcal{H},\qquad\delta\mathcal{H}=-\frac{1}{E_{J}}\mathcal{V}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\mathcal{V}^{\dagger}.$ (69)

The finite matrix elements in $\mathcal{H}_{0}$ are of the order of $E_{T}$,
while finite matrix elements in $\delta\mathcal{H}$ are of the order of
$E_{T}^{2}/E_{J}$, where $E_{T}=\sqrt{E_{T,1}^{2}+E_{T,2}^{2}+E_{T,3}^{2}}$.
Therefore, in the limit of $E_{T}\ll E_{J}$, we can neglect the insignificant
terms of $O(E^{2}_{T}/E_{J})$ and obtain

$\displaystyle\mathcal{H}_{\mathrm{eff}}\approx\mathcal{H}_{0}.$ (70)

Eventually, the effective Hamiltonian describing low-energy excitation of
$|E|\ll E_{J}$ in the limit of $E_{T}\ll E_{J}$ is obtained by

$\displaystyle\begin{split}&H_{\mathrm{eff}}=\frac{1}{2}\bm{\gamma}_{\mathrm{eff}}^{T}\mathcal{H}_{\mathrm{eff}}\bm{\gamma}_{\mathrm{eff}},\quad\mathcal{H}_{\mathrm{eff}}=\left(\begin{array}[]{cccc}0&0&0&iv_{1}\\\
0&0&0&iv_{2}\\\ 0&0&0&iv_{3}\\\
-iv_{1}&-iv_{2}&-iv_{3}&0\end{array}\right),\end{split}$ (71)

which is equivalent to the Hamiltonian in Eq. (3) of the main text. As also
illustrated in Fig. 3(b), we focus on the couplings between four MZMs: three
outer MZMs and one MZM remained at the Josephson junction.

The effective Hamiltonian $H_{\mathrm{eff}}$ is diagonalized as,

$\displaystyle\mathcal{U}_{\mathrm{eff}}^{\dagger}\mathcal{H}_{\mathrm{eff}}\mathcal{U}_{\mathrm{eff}}=\mathcal{E}_{\mathrm{eff}}=\mathrm{diag}[0,0,E_{v},-E_{v}],\qquad
E_{v}=\sqrt{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}},$ (72)

with

$\displaystyle\begin{split}&\mathcal{U}_{\mathrm{eff}}=\left(\bm{u}_{0,1},\bm{u}_{0,2},\bm{u}_{+},\bm{u}_{-}\right),\\\
&\bm{u}_{0,1}=\left(\begin{array}[]{cl}\sin^{2}{\varphi}(1-\cos\theta)+\cos\theta\\\
-\sin\varphi\cos\varphi(1-\cos\theta)\\\ -\cos\varphi\sin\theta\\\
0\end{array}\right),\quad\bm{u}_{0,2}=\left(\begin{array}[]{cl}-\sin\varphi\cos\varphi(1-\cos\theta)\\\
\cos^{2}{\varphi}(1-\cos\theta)+\cos\theta\\\ -\sin\varphi\sin\theta\\\
0\end{array}\right),\quad\bm{u}_{\pm}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cl}\cos\varphi\sin\theta\\\
\sin\varphi\sin\theta\\\ \cos\theta\\\ \pm i\end{array}\right),\end{split}$
(73)

where

$\displaystyle\left(\begin{array}[]{cl}v_{1}\\\ v_{2}\\\
v_{3}\end{array}\right)=E_{v}\left(\begin{array}[]{cl}\cos\varphi\sin\theta\\\
\sin\varphi\sin\theta\\\ \cos\theta\end{array}\right).$ (80)

As a result, we obtain the two zero-energy states:

$\displaystyle\gamma_{n}=\bm{u}_{0,n}^{\dagger}\bm{\gamma}_{\mathrm{eff}}=\sum_{\alpha=1\text{-}3}c_{\alpha}^{n}\gamma_{o,\alpha}\quad(n=1,2),$
(81)

where we represent
$\bm{u}_{0,n}=\left(c_{1}^{n},c_{2}^{n},c_{3}^{n},0\right)^{\mathrm{T}}$.
Importantly, $\gamma_{n}$ is described by the superposition of the outer end
MZMs (i.e., $\gamma_{o,\alpha}$). Moreover, since $c_{\alpha}^{n}$ are the
real number, $\gamma_{n}$ obeys $\gamma_{n}=\gamma_{n}^{\dagger}$. Therefore,
$\gamma_{n}$ describes the multi-locational Majorana zero modes (MMMs) as also
illustrated in Fig. 3(c). Strictly speaking, owing to the degeneracy at zero
energy, there is ambiguity in the representation of the zero-energy
eigenstates. Nevertheless, the quantity of

$\displaystyle\begin{split}&P_{\alpha}=\sum_{n=1,2}|c_{\alpha}^{n}|^{2}=\sum_{\beta\neq\alpha}\frac{v^{2}_{\beta}}{E_{v}^{2}},\end{split}$
(82)

which characterizes the amplitudes of the MZMs in the $\alpha$-th TS, is
irrelevant to the representation of $c_{\alpha}^{n}$.

##  II Scattering coefficients

In this section, we calculate the scattering coefficients in the presence of
the MMM. We start with the scattering matrix at the energy $E$, which is given
in Eq. (8) of the main text [47, 48, 51]:

$\displaystyle\begin{split}&S(E)=\left(\begin{array}[]{cc}s^{ee}&s^{eh}\\\
s^{he}&s^{hh}\end{array}\right)=1_{6}-2\pi
iW^{\dagger}\left[E-\mathcal{H}+i\pi WW^{\dagger}\right]^{-1}W,\\\
&W=\left(\begin{array}[]{cc}w&-w\\\ 0_{3\times 3}&0_{3\times
3}\end{array}\right),\quad w=\left(\begin{array}[]{ccc}w_{1}&0&0\\\
0&w_{2}&0\\\ 0&0&w_{3}\end{array}\right),\end{split}$ (83)

where the real number of $w_{\alpha}$ characterizes the coupling between the
electron/hole in the $\alpha$-th lead and the MZM at the outer end of the
$\alpha$-th TS. By using the $6\times 6$ unitary matrix in Eq. (41), we can
rewrite the scattering matrix as

$\displaystyle\begin{split}S(E)&=1_{6}-2\pi
iW^{\dagger}\mathcal{U}\left[E-\mathcal{U}^{\dagger}\mathcal{H}\mathcal{U}+i\pi\mathcal{U}^{\dagger}WW^{\dagger}\mathcal{U}\right]^{-1}\mathcal{U}^{\dagger}W\\\
&=1_{6}-2\pi iW^{\dagger}\left[E-\mathcal{\tilde{H}}+i\pi
WW^{\dagger}\right]^{-1}W\\\ &=1_{6}-2\pi
iW^{\dagger}\left(\begin{array}[]{cc}E-\mathcal{H}_{\mathrm{eff}}+i\pi\tilde{W}\tilde{W}^{\dagger}&-\mathcal{V}\\\
-\mathcal{V}^{\dagger}&E-\mathcal{E}_{J}\end{array}\right)^{-1}W\\\
&=1_{6}-2\pi
iW^{\dagger}\left(\begin{array}[]{cc}\tilde{\mathcal{G}}_{\mathrm{mzm}}&\tilde{\mathcal{G}}_{\mathcal{V}}\\\
\tilde{\mathcal{G}}_{\mathcal{V}^{\dagger}}&\tilde{\mathcal{G}}_{J}\end{array}\right)W\\\
&=1_{6}-2\pi
i\tilde{W}^{\dagger}\tilde{\mathcal{G}}_{\mathrm{mzm}}\tilde{W},\end{split}$
(84)

where we use $\mathcal{U}^{\dagger}W=W$ and define,

$\displaystyle\begin{split}&\left(\begin{array}[]{cc}\tilde{\mathcal{G}}_{\mathrm{mzm}}&\tilde{\mathcal{G}}_{\mathcal{V}}\\\
\tilde{\mathcal{G}}_{\mathcal{V}^{\dagger}}&\tilde{\mathcal{G}}_{J}\end{array}\right)=\left(\begin{array}[]{cc}E-\mathcal{H}_{\mathrm{eff}}+i\pi\tilde{W}\tilde{W}^{\dagger}&-\mathcal{V}\\\
-\mathcal{V}^{\dagger}&E-\mathcal{E}_{J}\end{array}\right)^{-1},\\\
&\tilde{W}=\left(\tilde{w},-\tilde{w}\right),\quad\tilde{w}=\left(\begin{array}[]{cl}w\\\
0_{1\times 3}\end{array}\right).\end{split}$ (85)

We specifically obtain

$\displaystyle\begin{split}&\tilde{\mathcal{G}}_{\mathrm{mzm}}=\left[E-\mathcal{H}_{\mathrm{eff}}+i\pi\tilde{W}\tilde{W}^{\dagger}-\mathcal{V}(E-\mathcal{E}_{J})^{-1}\mathcal{V}^{\dagger}\right]^{-1}=\tilde{\mathcal{G}}_{\mathrm{eff}}\left[1-\mathcal{V}(E-\mathcal{E}_{J})^{-1}\mathcal{V}^{\dagger}\tilde{\mathcal{G}}_{\mathrm{eff}}\right]^{-1},\\\
&\tilde{\mathcal{G}}_{\mathrm{eff}}=\left[E-\mathcal{H}_{\mathrm{eff}}+i\pi\tilde{W}\tilde{W}^{\dagger}\right]^{-1}=\left[E-\mathcal{H}_{\mathrm{eff}}+i\tilde{\Gamma}\right]^{-1},\\\
&\tilde{\Gamma}=\mathrm{diag}[\tilde{\Gamma}_{1},\tilde{\Gamma}_{2},\tilde{\Gamma}_{3},0],\quad\tilde{\Gamma}_{\alpha}=2\pi
w_{\alpha}^{2}.\end{split}$ (86)

Similarly to the discussion in Sec. I, in the limit of
$E,E_{T},\tilde{\Gamma}_{\alpha}\ll E_{J}$, we can make the replacement of

$\displaystyle\tilde{\mathcal{G}}_{\mathrm{mzm}}\approx\tilde{\mathcal{G}}_{\mathrm{eff}},$
(87)

where the contributions from the finite-energy eigenstates at the Josephson
junction, described by $\gamma_{\pm,J}$, are ignored. Therefore, in the
presence of the MMM, the low-energy scattering matrix is evaluated by

$\displaystyle S(E)=1_{6}-2\pi
i\tilde{W}^{\dagger}\tilde{\mathcal{G}}_{\mathrm{eff}}\tilde{W}=1_{6}-2\pi
i\tilde{W}^{\dagger}\left[E-\mathcal{H}_{\mathrm{eff}}+i\tilde{\Gamma}\right]^{-1}\tilde{W}.$
(88)

From Eq. (88), we obtain the scattering coefficients at zero energy as

$\displaystyle\begin{split}&s^{ee}_{\alpha\alpha}=\frac{v_{\alpha}^{2}w_{\beta}^{2}w_{\gamma}^{2}}{v_{1}^{2}w_{2}^{2}w_{3}^{2}+v_{2}^{2}w_{3}^{2}w_{1}^{2}+v_{3}^{2}w_{1}^{2}w_{2}^{2}},\quad
s^{he}_{\alpha\alpha}=-\frac{w_{\alpha}^{2}(v_{\beta}^{2}w_{\gamma}^{2}+w_{\beta}^{2}v_{\gamma}^{2})}{v_{1}^{2}w_{2}^{2}w_{3}^{2}+v_{2}^{2}w_{3}^{2}w_{1}^{2}+v_{3}^{2}w_{1}^{2}w_{2}^{2}},\\\
&s^{ee}_{\alpha\beta}=-s^{he}_{\alpha\beta}=\frac{v_{\alpha}v_{\beta}w_{\alpha}w_{\beta}w^{2}_{\gamma}}{v_{1}^{2}w_{2}^{2}w_{3}^{2}+v_{2}^{2}w_{3}^{2}w_{1}^{2}+v_{3}^{2}w_{1}^{2}w_{2}^{2}},\quad(\alpha\neq\beta,\;\beta\neq\gamma,\;\gamma\neq\alpha)\end{split}$
(89)

which is equivalent to Eq. (10) in the main text.

##  III Numerical results on zero-frequency noise power

In this section, we show the numerical results on the zero-frequency noise
power calculated by using the tight-binding model, where the scattering
coefficients are computed using the lattice Green’s function techniques [103,
104]. We evaluate the zero-frequency noise power at zero temperature by [107,
108]

$\displaystyle\begin{split}&P_{\alpha\beta}=\frac{e^{2}}{h}\int^{eV}_{0}\mathcal{P}_{\alpha\beta}(E)dE,\\\
&\mathcal{P}_{\alpha\beta}(E)=\delta_{\alpha\beta}\sum_{\eta=e,h}p^{\eta\eta}_{\alpha\alpha}-\sum_{\eta,\zeta}\sigma_{\eta}\sigma_{\zeta}p^{\eta\zeta}_{\alpha\beta}p^{\zeta\eta}_{\beta\alpha},\quad
p^{\eta\zeta}_{\alpha}=\sum_{\gamma=1,2,3}s^{\eta e}_{\alpha\gamma}(s^{\zeta
e}_{\beta\gamma})^{\ast},\end{split}$ (90)

where $\sigma_{\eta}=1$ ($-1$) for $\eta=e$ ($h$), which is equivalent to Eq.
(15) of the main text. In Fig. 4, we show the zero-frequency noise power
$P_{\alpha\beta}$ for $\alpha,\beta=1,2$ as a function of the bias voltage.
The parameters are chosen to be the same as those in Fig. 2(b) of the main
text. Specifically, we plot

$\displaystyle
F_{\alpha\beta}(eV)=\frac{P_{\alpha\beta}(eV)}{I_{1}(eV)+I_{2}(eV)},$ (91)

where $I_{\alpha}(eV)$ is the time-averaged current in the $\alpha$-th lead
wire. At the zero-bias voltage limit, we clearly find the relations of

$\displaystyle 2|P_{12}|=P_{11}+P_{22},\qquad P_{11}+P_{22}+P_{12}+P_{21}=0,$
(92)

which agree with Eq. (17) and Eq. (18) in the main text, respectively. As a
result, we numerically confirm the anomalous zero-frequency noise power due to
the nonlocal resonant scatterings of the MMM.

Figure 4: Zero-frequency noise power normalized by the charge current as a
function of the bias voltage. The solid line represents
$F_{T}=F_{11}+F_{22}+F_{12}+F_{21}$.