# Chiral topological superconductivity in Josephson junction

Shao-Kai Jian  Condensed Matter Theory Center, Department of Physics,
University of Maryland, College Park, Maryland 20742, USA  Shuai Yin  School
of Physics, Sun Yat-Sen University, Guangzhou 510275, China

###### Abstract

We consider a heterostructure of semiconductor layers sandwiched between two
superconductors, forming a two-dimensional Josephson junction. Applying a
Zeeman field perpendicular to the junction can render a topological
superconducting phase with the chiral Majorana edge mode. We show that the
phase difference between two superconductors can efficiently reduce the
magnetic field required to achieve the chiral topological superconductivity,
providing an experimentally feasible setup to realize chiral Majorana edge
modes. We also construct a lattice Hamiltonian of the setup to demonstrate the
chiral Majorana edge mode and the Majorana bound state localized in vortices.

##  I Introduction

Since the discovery of topological insulators, intensive theoretical and
experimental studies on various kinds of symmetry protected topological order
have arisen Kitaev2009 ; Ludwig2010 , among which the pursuit of Majorana
fermions and Majorana zero modes is one of the most interesting and important
issues. Owing to its non-Ablian statistics, the Majorana zero mode is of great
potential to implement topological qubits in fault-tolerant topological
quantum computations Kitaev2003 ; Alicea2011 ; Halperin2012 ; DasSarma2015 ;
Franz2015 . The Majorana zero mode can be realized in 5/2 fractional quantum
Hall states Stern2010 ; Read2000 , vortices of spinless $p+ip$ superconductors
Kitaev2006  and superconductor-semiconductor heterostructures Fu2008 ; Sau2010
; Alicea2010 ; DasSarma2010  in two dimensions, the edge of a ferromagnetic
atomic chain on a superconductor Kitaev2001 ; Yazdani2013 ; Simon2013 , and
the planar Josephson junction Stern2017 ; Marcus2019 ; Yacoby2019 ; Mayer2019
, etc., leading to ongoing interest among both condensed matter physics and
quantum computation communities.

On the other hand, one-dimensional chiral Majorana fermion—the simplest
solution satisfying the real version of the Dirac equation originally proposed
by Ettore Majorana in 1937 Majorana1937 —also attracts lots of attentions in
recent years. Realization of the chiral Majorana fermions in real world is not
only of great theoretical importance, but more significantly also opens a new
avenue for topological quantum computation Lian2018 . The chiral Majorana
fermion can be realized at the edge of the two-dimensional chiral topological
superconductor in class D Ludwig2010 . The chiral topological superconductor
is a real analog of an integer quantum Hall insulator, where the number of
chiral Majorana edge modes reflects the BdG Chern number of the occupied
bands. A series of theoretical proposals arises including the superconductor-
semiconductor heterostructure Sau2010 , and superconductor-quantum anomalous
Hall insulator heterostructure Qi2010 , and so on. Although various
experimental works have tried to realize the later proposal He2017 , debates
remain about whether true chiral Majorana edge modes are observed Wen2018 ;
Sau2018 ; Chang2019 ; Wang2019 .

In this paper, we propose a different strategy to realize the chiral Majorana
edge mode by taking advantage of the phase difference in the Josephson
junction. As shown in Fig. 1, we consider a few layers of semiconductor
sandwiched between two superconductors, forming a two-dimensional Josephson
junction with a superconducting phase difference $\chi$. In the presence of a
Zeeman field perpendicular to the junction, the phase difference $\chi$
provides a useful knob to tune the topological transition from a trivial
superconductor to a chiral topological superconductor with a Majorana edge
mode. In stead of the external magnetic field, the Zeeman energy can also be
obtained by using a ferromagnetic semiconductor in our setup. Namely, we can
consider the superconductor/ferromagnetic semiconductor/superconductor
heterostructure without applying the external magnetic field. The advantage of
this setup is that (i) the critical Zeeman field to achieve the topological
transition is efficiently reduced by the phase difference $\chi$ without
destroying the superconducting order and (ii) the transition can be realized
without carefully gating the system. The continuous topological transition is
described by two-dimensional Majorana cone at $\Gamma$ point, the center point
of Brillouin zone. We also construct a lattice model to show the chiral
Majorana edge mode and the Majorana zero bound states within the vortices—the
two essential signatures of chiral topological superconductors.

The paper is organized as follows. In Section II we describe the Josephson
junction setup and obtain the phase diagram using the scattering theory. A
remarkable result is the critical Zeeman energy near the phase difference
$\pi$ is in the order of $\Delta^{2}/\mu$, where $\mu$ and $\Delta$ are the
chemical potential and the superconducting gap respectively, which is a
significant improvement compared to previous proposals Sau2010 . In Section
III, we reveal how the topological phase transition to the topological
superconducting phase is achieved by showing that the gap closing point is
described by a real version of the Dirac Hamiltonian that changes BdG Chern
number. The calculation illuminates the robustness of our mechanism to realize
the topological transition. In Section IV, we show the gap of the topological
superconducting phase is sizable, i.e., is in the same order of the
proximitized superconducting gap, over nearly the full phase diagram. In
Section V, a microscopic lattice model of our setup is constructed to
demonstrate the phase diagram, the nontrivial chiral Majorana edge states and
the Majorana zero modes in the topological phase. Finally, we summarize the
results in various aspects and also make a brief comparison with the previous
heterostructure setup to show the advantage of our proposal in Section VI.

Figure 1: (a) A schematic plot of the setup. A few layers of semiconductor
thin film is sandwiched between two superconductors (S). The heterostructure
form a Josephson junction. The order parameters of two superconductors are
given by $\Delta e^{\pm i\chi/2}$. A out-of-plane magnetic field denoted by
$B_{z}$ is also applied to tune the topological superconductivity. As we
discussed in the main text, the Zeeman energy can be obtained by using a
ferromagnetic semiconductor without applying an external magnetic field. (b)
The proximitized SC state forms a domain wall in the semiconductor layers. The
width of the domain wall and of the semiconductor layers are denoted by $L$
and $W$, respectively.

##  II Josephson junction setup and the phase diagram

We consider a two-dimensional Josephson junction shown in Fig. 1, where a few
layers of semiconductor are sandwiched between two superconductors with a
superconducting phase difference $\chi$. Due to the proximity effect, the
superconducting orders will penetrate into the semiconductor layers and form a
domain wall in the semiconductor layers. One may wonder what the domain wall
means since the superconducting order parameter is not a discrete variable.
More precisely, we are considering the situation that the width of the
semiconductor is larger than twice of the superconducting proximity length,
such that in the middle of the semiconductor, the proximitized superconducting
gap is zero and the semiconductor is in the normal state. We refer to the
region with normal states the domain wall. As shown in Fig. 1, the width of
the domain wall and the semiconductor layers are $L$ and $W$, respectively.
The superconducting order in the semiconductor layers is modeled by

$\displaystyle\Delta(z)=\Delta e^{-i\chi/2}\Theta(-z-L/2)+\Delta
e^{i\chi/2}\Theta(z-L/2),$ (1)

where $\Delta$ ($\chi$) is the magnitude (the phase difference) of the
superconductivity, and $\Theta$ is the step function, i.e., $\Theta(x)=0$ for
$x<0$ and $\Theta(x)=1$ for $x\geq 0$. $z$ denotes the direction perpendicular
to the junction. We assume that inside the domain wall $|z|\leq L/2$ the
superconducting order is zero. Because the proximity length of
superconductivity is in general quite large, and the proximitized
superconducting order in semiconductors opens up an energy gap, the amplitude
of wavefunctions inside the semiconductor will concentrate near the domain
wall, one can focus on the superconducting-normal state interface at $\pm L/2$
and take $W/L\rightarrow\infty$.

The BdG Hamiltonian of the semiconductor layers is
$H=\Psi^{\dagger}H_{BdG}\Psi$, where
$\Psi=(\psi_{\uparrow},\psi_{\downarrow},\psi^{\dagger}_{\downarrow},-\psi^{\dagger}_{\uparrow})$,
and

$\displaystyle H_{BdG}({\bf k},z)=$ $\displaystyle\Big{(}\frac{{\bf
k}^{2}-\partial_{z}^{2}}{2m}-\mu\Big{)}\tau^{z}+H_{SOC}({\bf k})\tau^{z}$ (2)
$\displaystyle+(\Delta(z)\tau^{+}+H.c.)+E_{Z}\sigma^{z},$

where ${\bf k}=(k_{x},k_{y})$ denotes the momenta in the junction plane,
$\tau$ ($\sigma$) is the Pauli matrix acting on the Nambu (spin) space and
$\tau^{\pm}\equiv\frac{1}{2}(\tau^{x}\pm i\tau^{y})$. $E_{Z}$ denotes the
Zeeman energy induced by the external magnetic field perpendicular to the
junction (along $z$ direction) or by using ferromagnetic semiconductor with
out-of-plane magnetic orders. $H_{SOC}$ is the Dresselhaus spin-orbit coupling
which will be discussed later.

The phase transition between the trivial and topological superconductors is
characterized by a gap closing at $\bf{k}=0$ and consequently a change of the
BdG Chern number. So, to look for the topological transition, we calculate the
junction spectrum in the $\bf{k}=0$ subspace by the scattering theory
Beenakker1991 . As the spin is conserved during the scattering, we can focus
on $\sigma^{z}=1$ subspace. The transmission amplitude through the domain wall
$-L/2<z<L/2$ is

$\displaystyle T=\left(\begin{array}[]{cccc}e^{ik^{e}L}&0\\\
0&e^{-ik^{h}L}\end{array}\right),$ (5)

where $k^{e/h}=\sqrt{2m(\mu\pm(\epsilon-E_{Z}))}$ is the wave-vector of the
normal state wavefunction and $\epsilon$ is the eigen-energy. By matching the
wavefunction at the superconducting-normal state interface at $z=\pm L/2$ (see
Appendix A for more details), the scattering matrix is obtained to be $S_{\pm
L/2}=e^{\pm i\chi/4\tau^{z}}Se^{\mp i\chi/4\tau^{z}}$, where

$\displaystyle
S=e^{-i\phi_{S}}\left(\begin{array}[]{cccc}re^{i\phi_{N}}&\sqrt{1-r^{2}}\\\
\sqrt{1-r^{2}}&-re^{-i\phi_{N}}\end{array}\right),$ (8)

where $r$ is the normal reflection amplitude, and $\phi_{S,N}$ are two phases
given in the Appendix A. Using the scattering and transmission matrix, the
spectrum of the bound state is determined by Beenakker1991
$\det(1-S_{-L/2}TS_{+L/2}T)=0$, i.e.,

$\displaystyle\cos[(k^{e}-k^{h})L-2\phi_{S}]$
$\displaystyle\quad\quad\quad=(1-r^{2})\cos\chi+r^{2}\cos[(k^{e}+k^{h})L+2\phi_{N}].$
(9)

Assuming a weak superconducting pairing $E_{Z}\lesssim\Delta\ll\mu$, the zero
mode is given by

$\displaystyle\arccos\frac{E_{Z}}{\Delta}=\frac{\tilde{\chi}}{2}+\frac{\pi}{2}\frac{E_{Z}}{E_{T}}+n\pi,$
(10)

where $n\in\mathbb{Z}$, $\tilde{\chi}$ is a function of $\chi$,

$\displaystyle\tilde{\chi}(\chi)=\arccos\Big{[}(1-r^{2})\cos\chi+r^{2}\cos
2k_{F}L\Big{]},$ (11)

and $E_{T}\equiv(\pi/2)v_{F}/L$ is the Thouless energy of the junction. Here
$k_{F}=\sqrt{2m\mu}$ and $v_{F}=\sqrt{2\mu/m}$. Note the domain wall width is
in the order of the lattice constant, thus, the Thouless energy is in the
order of the chemical potential, $E_{T}\sim\mu$, which is the largest energy
scale.

Figure 2: The phase diagram of the Josephson junction as a function of the
junction phase difference and the Zeeman energy. $C$ denotes the BdG Chern
number, where $C=0$ ($C=1$) implies the trivial (topological)
superconductivity. The dashed (solid) line is the phase boundary without
(with) normal reflection.

In the weak superconducting pairing limit, $\Delta\ll\mu$, the normal
reflection Beenakker1991  and the effect from Thouless energy, i.e. the second
term on the right-hand side of Eq. (10), can be neglected. Then phase diagram
is simply given by $\frac{E_{Z}}{\Delta}=|\cos\frac{\chi}{2}|$, as shown by
the dashed line in Fig. 2. It is remarkable that at $\chi=\pi$, the critical
Zeeman energy is reduced to zero, namely, an infinitesimal Zeeman field can
turn the system into the chiral topological superconducting phase. Including
corrections from the normal reflection to the lowest nontrivial order of
$\Delta/\mu$, $r=\Delta/(2\mu)+\mathcal{O}(\Delta^{2}/\mu^{2})$, the phase
diagram is modified to the solid line in Fig. 2. Especially, at $\chi=\pi$,
the correction is given by $\tilde{\chi}(\chi=\pi)=\pi-\Delta/\mu\cos k_{F}L$,
so the critical Zeeman energy at $\chi=\pi$ is modified to be
$\Delta^{2}/\mu$, which is still small at weak pairing limit, making it an
experimentally achievable way to realize the chiral topological
superconductivity.

##  III Topological phase transition

It is well known that spin-orbit coupling is crucial in the realization of
topological phases. Direct semiconductors with an inversion-asymmetric zinc
blende structure (point group $T_{d}$), such as GaAs, InSb, CdTe etc, have
considerable size of spin-orbit coupling and are common in making quantum
structures WinklerBook . Moreover, many of these materials have a similar band
structure with the smallest gap at the $\Gamma$ point. To give a concrete
example of spin-orbit couplings, we consider a heterostructure consisting of a
few layers of such semiconductors grown in (001) direction. Because of the
phase difference in the Josephson junction, the heterostructure breaks
inversion symmetry. However, such a structure inversion asymmetry cannot
induce Rashba spin-orbit coupling, since the system respects the composite
$S_{4}$ and time reversal symmetry. As a result, the symmetry allowed spin-
orbit coupling is the Dresselhaus term WinklerBook ; Dresselhaus1954 ,
$H_{SOC}({\bf k})=\alpha(k_{x}\sigma^{x}-k_{y}\sigma^{y})$, where $\alpha$ is
the strength of spin-orbit coupling.

In the following, we will show that the topological phase transition is
described by a two-dimensional Majorana cone—a real version of the Dirac
fermion. To simplify the question, we set the width of domain wall $L$ to be
zero. At the $\bf k=0$ subspace of the normal semiconductor without the
proximitized superconducting order and the external magnetic field, a
continuous kinetic term of the one Kramers pair near the Fermi points,
$\psi_{1,\uparrow}$ and $\psi_{2,\downarrow}$, is described by one-dimensional
Dirac Hamiltonian $-iv_{F}\partial_{z}\sigma^{z}$ (see Appendix B for more
details). Adding the proximitized SC order and the Zeeman field, the BdG
Hamiltonian in Nambu basis
$\Psi=(\psi_{1,\uparrow},\psi_{2,\downarrow},\psi_{1,\uparrow}^{\dagger},\psi_{2,\downarrow}^{\dagger})^{T}$
is given by (note the basis difference in Eq. (2)), $\mathcal{H}_{BdG}({\bf
k},z)=\mathcal{H}_{0}(z)+\mathcal{H}_{1}({\bf k})$, where

$\displaystyle\mathcal{H}_{0}(z)=-iv_{F}\partial_{z}\sigma^{z}-\Delta^{\prime}\sigma^{y}\tau^{y}-\Delta^{\prime\prime}(z)\sigma^{y}\tau^{x}+E_{Z}\sigma^{z}\tau^{z},$
(12) $\displaystyle\mathcal{H}_{1}({\bf
k})=\frac{k_{x}^{2}+k_{y}^{2}}{2m}\tau^{z}+\alpha(k_{x}\sigma^{x}-k_{y}\sigma^{y}\tau^{z}),$
(13)

and $\Delta^{\prime}=\Delta\cos(\chi/2)$ and
$\Delta^{\prime\prime}(z)=\Delta\sin(\chi/2)\text{sgn}(z)$ refer to the real
and imaginary part of the superconducting order, respectively. We have
separated the Hamiltonian into an unperturbed part $\mathcal{H}_{0}$ and a
perturbation $\mathcal{H}_{1}$, respectively.

In Eq. (III), because of the domain wall formed in $\Delta^{\prime\prime}(z)$
Jackiw1976 , we get two bound states with eigen-energy
$E_{\pm}=\pm(E_{Z}-\Delta\cos\chi/2)$ that are related by particle-hole
transformation,

$\displaystyle\psi_{+}(z)=f(z)(1,0,0,-1)^{T},\quad\psi_{-}(z)=f(z)(0,1,-1,0)^{T},$

where
$f(z)=(\mathcal{N})^{-1/2}\exp[-\int_{0}^{z}dz^{\prime}\frac{\Delta^{\prime\prime}(z^{\prime})}{v_{F}}]$,
and $\mathcal{N}$ is the normalization factor.

Now we consider the perturbation Eq. (13) to the bound state $\psi_{\pm}$.
Using the first-order perturbation theory, it is straightforward to get the
effective Hamiltonian in the bound states subspace $(\psi_{+},\psi_{-})^{T}$,

$\displaystyle\mathcal{H}_{\text{eff}}=\left(\begin{array}[]{cccc}\delta
m&-\alpha(k_{x}+ik_{y})\\\ -\alpha(k_{x}-ik_{y})&-\delta m\end{array}\right),$
(17)

where the mass is $\delta m=E_{Z}-\Delta\cos\chi/2$. At critical point $\delta
m=0$, the Hamiltonian describes a massless Majorana cone, corresponding to the
phase transition from a trivial to a topological superconductor.

Similar analysis can be applied to the other Kramers pair
$\psi_{1,\downarrow}$ and $\psi_{2,\uparrow}$, which shows that the critical
point occurs at $E_{Z}/\Delta=-\cos\chi/2$. Combined with the above results,
the phase boundary is $E_{Z}/\Delta=|\cos\chi/2|$, (we assume $E_{Z}>0$),
consistent with the results from the scattering theory.

The way that the topological transition is realized in Eq. (17) is very robust
in the following sense. So long as there exists some mechanism to achieve a
gap closing at the $\Gamma$ point, the symmetry allowed spin-orbit coupling is
sufficient to render a nontrivial topological transition because the gap
closing point is effectively described by a Dirac Hamiltonian. The tunability
of the domain wall bound state in $\bf k=0$ subspace provides a useful
mechanism to realize the gap closing. We also want to emphasize the fact that
the symmetry in our setup only allows that the Dresselhaus spin-orbit coupling
is not essential. The presence of other kinds of spin-orbit couplings, such as
the Rashba spin-orbit coupling, will not affect the topological phase
transition in general.

Figure 3: (a) The phase diagram as a function of $(\chi,E_{Z}/\Delta)$. $C=0$
($C=1$) is the Chern number indicating the normal SC (the chiral TSC). The
solid line is the phase boundary obtained from the lattice Hamiltonian Eq.
(21). The parameters are given by $W=25$, $L=2$, $t=t^{\prime}=1$, $\mu=-3.2$,
$\alpha=0.6$, and $\Delta=0.5$. The dashed line is determined by Eq. (10) and
the dotted line indicates gapless points. The spectra at $k_{y}=0$ of the
Josephson junction at the three red points are shown in (b-d).
$\Delta^{\prime}=\Delta\cos(\chi/2)$ denotes the smallest gap away from the
$\Gamma$ point.

##  IV Topological gap

The spectrum at $\bf k=0$ is given by $\epsilon=\pm(\Delta\cos\chi/2\pm
E_{Z})$, thus away from the transition point, the gap is still of the same
order. One does not have to concerned about the gap near the $\Gamma$ point.
Here, we analyze the topological gap at generic ${\bf k}$ away from the
$\Gamma$ point, especially, at $k\gg E_{Z}/\alpha$,
$k=\sqrt{k_{x}^{2}+k_{y}^{2}}$. Using the scattering theory (see Appendix C
for more details), the spectrum at $k\gg E_{Z}/\alpha$ is determined by

$\displaystyle\arccos\frac{\epsilon}{\Delta}=\frac{(k_{\sigma}^{e}-k_{\sigma}^{h})L}{2}+\frac{\chi}{2},$
(18)

where $k_{\sigma}^{e/h}=\sqrt{2m(\mu-E_{k\sigma}\pm\epsilon)}$,
$E_{k\sigma}=k^{2}/2m-\sigma\alpha k$, with $\sigma=\pm 1$. In the weak
superconducting pairing limit, $\Delta\ll\mu$, the spectrum is

$\displaystyle\frac{\epsilon}{\Delta}=\cos\Big{[}\frac{\chi}{2}+\frac{k_{F}L}{2\sqrt{1-E_{k\sigma}/\mu}}\frac{\Delta}{\mu}\times\frac{\epsilon}{\Delta}\Big{]}.$
(19)

We can see that the dependence on $k$ is suppressed by $\Delta/\mu$ for
$E_{Z}/\alpha\ll k\ll k_{F}$. As $k$ increases such that $E_{k\sigma}\sim\mu$,
the prefactor of $\Delta/\mu$ diverges. Nevertheless, notice that the bound
state exists only if $k_{\sigma}^{e/h}$ is a real number, i.e.,
$E_{k\sigma}\leq\mu-\epsilon$. So when $E_{k\sigma}=\mu-\epsilon$, the
spectrum changes to

$\displaystyle\frac{\epsilon}{\Delta}=\cos\Big{[}\frac{\chi}{2}+\frac{k_{F}L}{2}\sqrt{\frac{\Delta}{\mu}}\times\sqrt{\frac{\epsilon}{\Delta}}\Big{]}.$
(20)

Combining the results of Eqs. (19) and (20), we conclude that the topological
gap is $\epsilon\sim\Delta\cos\frac{\chi}{2}$ up to a small correction of
$\Delta/\mu$. This gap is also verified in the lattice model introduced in the
next section. Notice that at $\chi=\pi$, the system has a small gap, so the
better place to get the chiral topological superconductor is in between
$\chi=0$ and $\pi$, where the Zeeman field is small while the gap is sizable.

Figure 4: (a,b) The spectrum of the system in $k_{x}$ direction (open boundary
condition in the $y$ direction) in the topological superconducting phase and
the trivial phase, respectively. The parameters are given by $W=5$, $L=2$,
$t=t^{\prime}=1$, $\mu=-3.2$, $\alpha=1$, $\Delta=1$, $\chi=0.6\pi$. (c) The
spectrum of the system with two vortices. The disk (square) points correspond
to the topological (trivial) phase. The parameters are given by $W=6$, $L=2$,
$t=t^{\prime}=1$, $\mu=-3.2$, $\alpha=1$, $\Delta=1$, $\chi=0.6\pi$. (d) The
local density of the ground state wavefunction in the topological phase. Two
Majorana zero modes appear in the vortices. The topological (trivial) phase
corresponds to $E_{Z}=0.8\Delta$ ($E_{Z}=0.1\Delta$).

##  V Tight binding model

To demonstrate the topological phase, we construct a tight binding Hamiltonian
to model the setup in Fig. 1. The lattice model is given by
$H=H_{0}+H_{SOC}+H_{Z}+H_{SC}$, where

$\displaystyle H_{0}=-t\sum_{j_{z}=1}^{W}\sum_{{\bf
i};\hat{\mu}=\hat{x},\hat{y}}(c_{{\bf i},j_{z}}^{\dagger}c_{{\bf
i}+\hat{\mu},j_{z}}+H.c.)-t^{\prime}\sum_{j_{z}=1}^{W-1}\sum_{{\bf i}}(c_{{\bf
i},j_{z}}^{\dagger}c_{{\bf i},j_{z}+1}+H.c.)-\mu\sum_{j_{z}=1}^{W}\sum_{{\bf
i}}c_{{\bf i},j_{z}}^{\dagger}c_{{\bf i},j_{z}},$ (21) $\displaystyle
H_{{SOC}}=i\alpha\sum_{j_{z}=1}^{W}\sum_{{\bf
i};\hat{\mu}=\hat{x},\hat{y}}[(c_{{\bf i},j_{z}}^{\dagger}\sigma^{x}c_{{\bf
i}+\hat{x},j_{z}}-c_{{\bf i},j_{z}}^{\dagger}\sigma^{y}c_{{\bf
i}+\hat{y},j_{z}})-H.c.],$ (22) $\displaystyle
H_{Z}=E_{Z}\sum_{j_{z}=1}^{W}\sum_{{\bf i}}c_{{\bf
i},j_{z}}^{\dagger}\sigma^{z}c_{{\bf i},j_{z}},$ (23) $\displaystyle
H_{SC}=\sum_{{\bf i}}\Big{[}\sum_{j_{z}=1}^{[W/2]-[L/2]}\Delta
e^{-i\chi/2}c_{{\bf i},j_{z}}(i\sigma^{y})c_{{\bf
i},j_{z}}+\sum_{j_{z}=W+1-([W/2]-[L/2])}^{W}\Delta e^{i\chi/2}c_{{\bf
i},j_{z}}(i\sigma^{y})c_{{\bf i},j_{z}}\Big{]}+H.c..$ (24)

Here, $\bf i$ denotes sites in square lattice in the $xy$ plane, and $j_{z}$
is the layer index along the $z$ direction. The number of semiconductor layer
is $W$. $t$ ($t^{\prime}$) is the in-plane (out-of-plane or $z$-direction)
nearest-neighbor hopping amplitude. $\mu$ is the chemical potential, $\alpha$
is the strength of in-plane spin-orbit coupling, and $\Delta$ ($\chi$) is the
amplitude (phase difference) of the proximatized superconducting orders that
are nonzero at $1\leq j_{z}\leq[W/2]-[L/2]$ and $W+1-([W/2]-[L/2])\leq
j_{z}\leq W$. $[x]$ denote the largest integer that is smaller than $x$.

The phase diagram of the lattice model is shown in Fig. 3. The solid line in
Fig. 3 is the phase boundary obtained from the lattice model, while the dashed
line is determined by Eq. (10). One can see that they match each other quite
well. The spectrum of the junction at different phases indicated by the red
points in Fig. 3 are plotted in Figs. 3(b-d), where one can see that by tuning
the phase difference and/or the Zeeman energy, the energy gap will close at
phase boundary and then reopen in the topological superconducting phase. It is
also important to note that the smallest gap away from the $\Gamma$ point is
given by $\Delta^{\prime}=\Delta\cos\frac{\chi}{2}$, which is consistent with
the result from the scattering theory in the previous section.

To explore the chiral edge state of topological superconducting phase, we use
open boundary condition in the $y$ direction, and plot the energy spectrum as
a function of $k_{x}$ as shown in Figs. 4 and 4 for the topological phase and
the trivial phase, respectively. The two chiral Majorana states along $+y$ and
$-y$ edges are clearly shown in Fig. 4 in the topological superconducting
phase, although there is no edge state at the normal superconducting phase
shown in Fig. 4.

Moreover, when vortices are created via external magnetic field, the chiral
topological superconductor is expected to host Majorana zero mode localized in
the vortex core. In Fig. 4, we observe the Majorana zero mode in the
topological phase with two vortices, while in the trivial phase there is no
Majorana zero mode. The local density of ground state wavefunction in the
topological phase is plotted in Fig. 4, corresponding to the localized
Majorana zero mode at the vortex core.

##  VI Conclusions

In this paper, we show that a two-dimensional Josephson junction of the
superconductor-semiconductor-superconductor sandwich heterostructure can
provide a useful setup to realize the chiral topological superconductor with
chiral Majorana edge modes. The topological superconducting phase appears in a
large portion of the phase diagram spanned by the Zeeman field and the
superconducting phase difference of the Josephson junction. Compared to the
previous setup Sau2010 , although the device in our proposal might be a little
more complicated as the superconductor layers cover both the top and the
bottom of the semiconductor, the benefit is that the Zeeman field is reduced
within the critical strength and gating is no longer necessary which is
considered as the main challenges of previous setup. For the simplest setup of
a Rashba semiconductor proximitized with a superconductor, we can model it by
the following BdG Hamiltonian

$\displaystyle H_{BdG}=\left(-t(\cos k_{x}+\cos k_{y})-\mu\right)\tau^{z}$
$\displaystyle+\alpha(\sin k_{x}\sigma^{y}\tau^{z}-\sin
k_{y}\sigma^{x})+E_{Z}\sigma^{z}\tau^{z}+\Delta\sigma^{y}\tau^{x},$ (25)

where $\sigma$ ($\tau$) acts on the spin (Nambu) space. $t$ is the hopping
amplitude, $\mu$ is the chemical potential, $\alpha$ is the spin-orbital
coupling strength, $E_{Z}$ denotes the Zeeman energy, and $\Delta$ is the
proximitized superconducting gap. This setup is able to induce a gap closing
at the $\Gamma$ point and render a topological transition into the chiral
topological superconductivity. It is easy to realize the transition occurs at
$E_{Z}=\sqrt{(2t+\mu)^{2}+\Delta^{2}}$. So, to realize the topological phase
in this setup, the Zeeman energy cannot be less than the superconducting gap,
which is typically hard to achieve. Besides that, to really lower the critical
Zeeman energy, one needs to tune the chemical potential. For our setup,
however, as demonstrated in Fig. 2, the topological phase can be obtained by a
Zeeman energy that is much smaller the superconducting gap and the chemical
potential does not play any essential role. These advantages facilitate
possible experimental reaches.

We mainly consider the narrow Josephson junction limit, i.e., $\Delta\ll
E_{T}$. A wide Josephson junction with $\Delta\gg E_{T}$ can also achieve the
desired topological transition. In this case, there appears multiple bound
states in the Josephson junction, and the topological phase transition occurs
at

$\displaystyle\frac{\tilde{\chi}}{2}\pm\frac{\pi}{2}\frac{E_{Z}}{E_{T}}=\left(n+\frac{1}{2}\right)\pi.$
(26)

One may have concerns that the external magnetic field might not be efficient
when the semiconductor is covered from both sides by the superconductors. It
is useful to consider type-II superconductors in the setup so that the
magnetic field can penetrate through the full device by creating vortices. And
in particular, we have demonstrated by the lattice Hamiltonian that the vortex
is not only harmless to the topological phase, but also useful in generating
Majorana zero modes, as shown in Fig. 4 and 4. Moreover, instead of using the
external magnetic field, it is also possible to consider doping magnetic
elements or using the ferromagnetic semiconductors to render the Zeeman energy
in the Josephson junction. The flexibility of achieving the topological phase
transition and the mature fabrication technique of quasi-two dimensional
semiconductor devices make our proposal a promising setup to realize the
chiral topological superconductor.

Possible experimental signatures include the half-integer longitudinal
conductance plateau Wang2015 , which is implemented in previous experiments
He2017 ; Chang2019 , and zero bias peak in the tunneling conductance of
Majorana zero modes trapped in vortices, which we have explicitly shown in
Fig. 4.

##  VII Acknowledgement

We thank Hong Yao, Sankar Das Sarma, Ady Stern, Zhongbo Yan, Yingyi Huang,
Fuchun Zhang, and Ruirui Du for helpful discussions. S.-K.J. is supported by
the Simons Foundation via the It From Qubit Collaboration. S.Y. is supported
by the National Natural Science Foundation of China (Grant No. 41030090).

##  Appendix A Spectrum in Josephson junction

Due to the proximity effect, the superconducting (SC) order will penetrate
into semiconductor layers. Because the two superconductors have different SC
order (most importantly, the different SC phase), there is a domain wall in
the semiconductor layers. Assuming the size of the domain wall is $L$ as shown
in Fig. 1(b), the SC order in the semiconductor layers is given by

$\displaystyle\Delta(z)=\Delta e^{-i\chi/2}\Theta(-z-L/2)+\Delta
e^{i\chi/2}\Theta(z-L/2),$ (27)

where $\Delta$ is the magnitude of superconductivity, and $\chi$ denotes the
phase difference between two superconductors. $\Theta$ is the step function.
Note the difference between the width of semiconductor layers denoted by $W$,
and the width of domain wall denoted by $L$, as shown in Fig. 5. Because the
proximity length of SC order is quite large, and the proximitized SC order in
semiconductors opens up a finite energy gap, the amplitude of wavefunctions
will concentrate near the domain wall. As a result, one focus on the interface
at $\pm L/2$ and take $W/L\rightarrow\infty$.

The BdG Hamiltonian in the semiconductor layers is
$H=\Psi^{\dagger}\mathcal{H}_{BdG}\Psi$, where
$\Psi=(\psi_{\uparrow},\psi_{\downarrow},\psi^{\dagger}_{\downarrow},-\psi^{\dagger}_{\uparrow})$,
and

$\displaystyle
H_{BdG}(k_{x},k_{y},z)=\Big{(}\frac{k_{x}^{2}+k_{y}^{2}-\partial_{z}^{2}}{2m}-\mu\Big{)}\tau^{z}+H_{SOC}(k_{x},k_{y})\tau^{z}+(\Delta(z)\tau^{+}+H.c.)+E_{Z}\sigma^{z},$
(28)

where $\tau$ ($\sigma$) is the Pauli matrix acting on the Nambu (spin) space
and $\tau^{\pm}=\frac{1}{2}(\tau^{x}\pm i\tau^{y})$. $E_{Z}$ denotes the
Zeeman energy. Note that the Zeeman field is along $z$ direction. $H_{SOC}$ is
the Dresselhaus spin-orbit coupling which will be discussed in the next
Sector.

Owing to the translational symmetry in $xy$-plane, $k_{x}$ and $k_{y}$ are
good quantum numbers. Let’s first focus on $k_{x}=k_{y}=0$ subspace in this
Sector to determine the phase diagram, and then in later Sector on generic
$k_{x},k_{y}$ to explore the topological gap. The Hamiltonian in the
$k_{x}=k_{y}=0$ subspace reduces to

$\displaystyle
H_{BdG}(0,0,z)=\Big{(}-\frac{\partial_{z}^{2}}{2m}-\mu\Big{)}\tau^{z}+(\Delta(z)\tau^{+}+H.c.)+E_{Z}\sigma^{z}.$
(29)

For the normal state, i.e., $|z|<L/2$, the eigenstates with eigen-energy
$\epsilon$ are

$\displaystyle\psi_{e\sigma}^{\pm}(z)=\frac{1}{\sqrt{k_{\sigma}^{e}}}\left(\begin{array}[]{cccc}1\\\
0\end{array}\right)\otimes|\sigma\rangle e^{\pm
ik_{\sigma}^{e}z},\quad\psi_{h\sigma}^{\pm}(z)=\frac{1}{\sqrt{k_{\sigma}^{h}}}\left(\begin{array}[]{cccc}0\\\
1\end{array}\right)\otimes|\sigma\rangle e^{\pm ik_{\sigma}^{h}z},$ (34)

where $\sigma=\pm 1$, $|+1\rangle=(1,0)^{T}$ and $|-1\rangle=(0,1)^{T}$. The
wavevector is $k_{\sigma}^{e/h}=\sqrt{2m(\mu\pm(\epsilon-\sigma E_{Z}))}$. For
the proximitized SC state at $|z|>L/2$, the eigenstates with eigen-energy
$\epsilon$ is give by

$\displaystyle\phi_{e\sigma}^{\pm}(z)=\frac{1}{\sqrt{2q_{\sigma}^{e}}[(\epsilon-\sigma
E_{Z})^{2}/\Delta^{2}-1]^{1/4}}\left(\begin{array}[]{cccc}e^{i\eta_{\sigma}^{e}/2}\\\
e^{-i\eta_{\sigma}^{e}/2}\end{array}\right)\otimes|\sigma\rangle e^{\pm
iq_{\sigma}^{e}z},$ (37)

and $\phi_{h\sigma}^{\pm}$ is given by replacing the index $e$ to $h$ in Eq.
(37). Here $\eta_{\sigma}^{e/h}=\frac{\chi}{2}\pm\eta_{\sigma}$, and
$\eta_{\sigma}=\arccos\frac{\epsilon-\sigma E_{Z}}{\Delta}$. Notice that for
$t>1$, $\arccos t=-i\ln(t+\sqrt{t^{2}-1})$. The wavevector is
$q_{\sigma}^{e/h}=[2m(\mu\pm\sqrt{(\epsilon-\sigma
E_{Z})^{2}-\Delta^{2}})]^{1/2}$, and ${\rm Re}q_{\sigma}^{e/h}>0$, ${\rm
Im}q_{\sigma}^{e}>0$, ${\rm Im}q_{\sigma}^{h}<0$. The wavefunctions are
normalized to have same current.

We are ready to evaluate the scattering matrix. To simplify the problem, it is
easy to observe that the spin is conserved during the scattering. So let’s
focus on $\sigma=1$ and neglect the index $\sigma$. By matching the
wavefunction Eq. (34) and Eq. (37) at the superconducting-normal state
interface at $z=\pm L/2$, the scattering matrix is $S_{\pm L/2}=e^{\pm
i\chi/4\tau^{z}}Se^{\mp i\chi/4\tau^{z}}$, where

$\displaystyle
S=e^{-i\phi_{S}}\left(\begin{array}[]{cccc}re^{i\phi_{N}}&\sqrt{1-r^{2}}\\\
\sqrt{1-r^{2}}&-re^{-i\phi_{N}}\end{array}\right),$ (40)

in which $r=r_{n}/r_{d}$ is normal reflection amplitude. $r_{n}$ and $r_{d}$
are positive numbers, and

$\displaystyle
r_{d}e^{i\phi_{S}}=-e^{-i\eta}(k^{h}-q^{e})(k^{e}-q^{h})+e^{i\eta}(k^{e}+q^{e})(k^{h}+q^{h}),$
(41) $\displaystyle
r_{n}e^{i\phi_{N}}=-e^{-i\eta}(k^{h}-q^{e})(k^{e}+q^{h})+e^{i\eta}(k^{e}-q^{e})(k^{h}+q^{h}).$
(42)

These functions determine the two phases $\phi_{S}$ and $\phi_{N}$ in Eq. (8)
in the main text.

The transmission amplitude is simply

$\displaystyle T=\left(\begin{array}[]{cccc}e^{ik^{e}L}&0\\\
0&e^{-ik^{h}L}\end{array}\right).$ (45)

Using the scattering and transmission matrix, the spectrum of the bound state
is determined by $\det(1-S_{-L/2}TS_{+L/2}T)=0$, i.e.,

$\displaystyle\cos[(k^{e}-k^{h})L-2\phi_{S}]=(1-r^{2})\cos\chi+r^{2}\cos[(k^{e}+k^{h})L+2\phi_{N}].$
(46)

To proceed, we assume weak pairing $E_{Z}\lesssim\Delta\ll\mu$, then we have
$r=\Delta/(2\mu)+\mathcal{O}(\Delta^{2}/\mu^{2})$, $\phi_{S}=\eta$, and
$\phi_{N}=0$. The zero mode is given by

$\displaystyle\arccos\frac{E_{Z}}{\Delta}=\frac{\tilde{\chi}}{2}+\frac{\pi}{2}\frac{E_{Z}}{E_{T}}+n\pi,$
(47)

where $\tilde{\chi}$ is a function of $\chi$:

$\displaystyle\tilde{\chi}(\chi)=\arccos\Big{(}\big{[}1-\frac{\Delta^{2}}{(2\mu)^{2}}\big{]}\cos\chi+\frac{\Delta^{2}}{(2\mu)^{2}}\cos
2k_{F}L\Big{)},$ (48)

and $E_{T}\equiv(\pi/2)v_{F}/L$ is the Thouless energy of the junction. Here
$k_{F}=\sqrt{2m\mu}$ and $v_{F}=\sqrt{2\mu/m}$. Note the domain wall width is
of order of lattice constant, as a result, the Thouless energy is of order of
chemical potential, $E_{T}\sim\mu$, which is the largest energy scale in the
question.

Figure 5: Schematic plot of Josephson junction.

As we will show in the next Section, the appearance of zero mode implies a gap
closing that separates trivial and topological superconducting phases. At the
zeroth order of $\Delta/\mu$, there is no normal reflection and
$\tilde{\chi}={\chi}$, the phase diagram is simply given by
$\frac{E_{Z}}{\Delta}=|\cos\frac{\chi}{2}|$, as shown by the dashed line in
Fig. 2. It is interesting to see that at $\chi=\pi$, an infinitesimal magnetic
field can turn the system into a chiral topological superconducting phase.

Including correction from the normal reflection to the first order of
$\Delta/\mu$, the relation between $\chi$ and $\tilde{\chi}$ is given in Eq.
(48) and the phase diagram is corrected to the solid line in Fig. 2. At
$\chi=0$ and $\chi=\pi$, the correction is in the linear order of
$\Delta/\mu$, i.e., $\tilde{\chi}(\chi=0)=\Delta/\mu\sin k_{F}L$, and
$\tilde{\chi}(\chi=\pi)=\pi-\Delta/\mu\cos k_{F}L$, respectively. While at
other generic values of $\chi$, the correction is in the quadratic order of
$\Delta/\mu$, i.e., $\tilde{\chi}(\chi)=\chi+(\Delta/\mu)^{2}(\cos\chi-\cos
2k_{F}L)/\sin\chi$. It is useful to note that when the normal reflection is
considered, the critical Zeeman energy at $\chi=\pi$ is given by
$\Delta^{2}/\mu$, which is a still a small number at weak pairing.

##  Appendix B Spin-orbit coupling and topological phase transition

It is well known that spin-orbit coupling is crucial in the realization of
topological phases. Direct semiconductors with an inversion-asymmetric zinc
blende structure (point group $T_{d}$), such as GaAs, InSb, CdTe etc, have
considerable size of spin-orbit coupling and are common in making quantum
structures. Moreover, many of these materials have a similar band structure
with the smallest gap at $\Gamma$ point. To give a concrete example of spin-
orbit couplings, we consider a heterostructure consisting of a few layers of
such semiconductors grown in (001) direction. Because of the phase difference
in the Josephson junction, the heterostructure breaks inversion symmetry.
However, such a structure inversion asymmetry cannot induce Rashba spin-orbit
coupling, since the system is invariant under the composite $S_{4}$ and time
reversal operations. As a result, the symmetry allowed spin-orbit coupling is
the Dresselhouse term

$\displaystyle H_{SOC}=\alpha(k_{x}\sigma^{x}-k_{y}\sigma^{y}),$ (49)

where $\alpha$ is the strength of spin-orbit coupling.

In the following, we will show that the gap closing in the pervious Section is
described by two-dimensional Majorana cone. To simplify the question, we set
the width of domain wall $L$ to be zero, and neglect the normal reflection. In
the $k_{x}=k_{y}=0$ subspace of the semiconductor (without proximitized SC
order), the dispersion along $k_{z}$ direction is shown in Fig. 6. We make a
continuous model near the Fermi point, i.e., $\psi_{1,2}$, for right and left
movers. Because the proximitized SC order couples the Kramers pair, we first
analyze the Kramers pair $\psi_{1,\uparrow}$ and $\psi_{2,\downarrow}$ with
kinetic term given by one-dimensional Dirac Hamiltonian
$\mathcal{H}_{0}=-iv_{F}\partial_{z}\sigma^{z}$.

Owing to the proximitized SC order, the BdG Hamiltonian in Nambu space
$\Psi=(\psi_{1,\uparrow,z},\psi_{2,\downarrow,z},\psi_{1,\uparrow,z}^{\dagger},\psi_{2,\downarrow,z}^{\dagger})^{T}$
is given by (note the basis difference in Eq. (28)),
$\mathcal{H}_{BdG}(k_{x},k_{y},z)=H_{0}(z)+H_{1}(k_{x},k_{y})$, where

$\displaystyle
H_{0}(z)=-iv_{F}\partial_{z}\sigma^{z}-\Delta^{\prime}(z)\sigma^{y}\tau^{y}-\Delta^{\prime\prime}(z)\sigma^{y}\tau^{x}+E_{Z}\sigma^{z}\tau^{z},$
(50) $\displaystyle
H_{1}(k_{x},k_{y})=\frac{k_{x}^{2}+k_{y}^{2}}{2m}\tau^{z}+\alpha(k_{x}\sigma^{x}-k_{y}\sigma^{y}\tau^{z}),$
(51)

and $\Delta^{\prime}(z)=\Delta\cos(\chi/2)$ and
$\Delta^{\prime\prime}(z)=\Delta\sin(\chi/2)\text{sgn}(z)$ refer to the real
and imaginary part of SC order given in Eq. (27). Here, we have separated the
Hamiltonian into two parts, $H_{0}$ and $H_{1}$, which will be treated as
unperturbed part and perturbation, respectively.

Figure 6:  The dispersion of $k_{x}=k_{y}=0$ subspace along $k_{z}$ direction.
Because in $k_{x}=k_{y}=0$ subspace the SOC vanishes, the band is doubly
degenerate. We denote left/right mover by $\psi_{1/2}$, respectively.

In $H_{0}$, because of the domain wall formed in $\Delta^{\prime\prime}(z)$,
similar to domain wall in one-dimensional Dirac Hamiltonian, we get two
related bound states that are related by particle-hole transformation,

$\displaystyle\psi_{+}(z)=f(z)(1,0,0,-1)^{T},\quad\psi_{-}(z)=f(z)(0,1,-1,0)^{T},$
(52)

where
$f(z)=(\mathcal{N})^{-1/2}\exp[-\int_{0}^{z}dz^{\prime}\frac{\Delta^{\prime\prime}(z^{\prime})}{v_{F}}]$,
and $\mathcal{N}$ is the normalization factor. The eigen-energies of these
bound states are $E_{\pm}=\pm(E_{Z}-\Delta\cos\chi/2)$, and the zero-mode
appears at $E_{Z}/\Delta=\cos\chi/2$.

Now we consider the perturbation $H_{1}$ to the bound state $\psi_{\pm}$.
Using first-order perturbation theory, it is straightforward to get the
effective Hamiltonian in the bound states subspace $(\psi_{+},\psi_{-})^{T}$,

$\displaystyle\mathcal{H}_{\text{eff}}=\left(\begin{array}[]{cccc}\delta
m&-\alpha(k_{x}+ik_{y})\\\ -\alpha(k_{x}-ik_{y})&-\delta m\end{array}\right),$
(55)

where the mass is $\delta m=E_{Z}-\Delta\cos\chi/2$. At critical point $\delta
m=0$, the Hamiltonian describes a Majorana cone, indicating a topological
transition. Namely, the gap closing realizes a transition from a trivial to a
topological superconductor.

Similar analysis can be applied to the other Kramers pair
$\psi_{1,\downarrow}$ and $\psi_{2,\uparrow}$ in Fig. 6, and the transition
appears at $E_{Z}/\Delta=-\cos\chi/2$. Combined with above results, the phase
boundary is $E_{Z}/\Delta=|\cos\chi/2|$, consistent with the results from the
previous Section.

##  Appendix C Topological gap

In previous sections, we have analyzed the spectrum at $k_{x}=k_{y}=0$. Away
from the transition point, the gap at $k_{x}=k_{y}=0$ is in the order of
$\Delta$. Here, we analyze the topological gap at generic $k_{x},k_{y}$,
especially, when $k_{x},k_{y}\gg E_{Z}/\alpha$. Since we are interested in
$\alpha k\gg E_{Z}$, $k=\sqrt{k_{x}^{2}+k_{y}^{2}}$, we neglect the Zeeman
energy, and the Hamiltonian is

$\displaystyle
H_{BdG}(E_{Z}=0)=\Big{(}\frac{k_{x}^{2}+k_{y}^{2}-\partial_{z}^{2}}{2m}-\mu+H_{SOC}(k_{x},k_{y})\Big{)}\tau^{z}+(\Delta(z)\tau^{+}+H.c.).$
(56)

where the proximitized SC order is again given by Eq. (27). In the following,
we will use the same notation in Section A. For the normal state in $|z|<L/2$,
the eigenstates with eigen-energy $\epsilon$ are given by

$\displaystyle\psi_{e\sigma}^{\pm}(z)=\frac{1}{\sqrt{k_{\sigma}^{e}}}\left(\begin{array}[]{cccc}1\\\
0\end{array}\right)\otimes|\sigma\rangle e^{\pm
ik_{\sigma}^{e}z},\quad\psi_{h\sigma}^{\pm}(z)=\frac{1}{\sqrt{k_{\sigma}^{h}}}\left(\begin{array}[]{cccc}0\\\
1\end{array}\right)\otimes|\sigma\rangle e^{\pm ik_{\sigma}^{h}z},$ (61)

where $\sigma=\pm 1$,
$|+1\rangle=1/\sqrt{2}(e^{i\phi_{k}/2},e^{-i\phi_{k}/2})^{T}$ and
$|-1\rangle=1/\sqrt{2}(e^{i\phi_{k}/2},-e^{-i\phi_{k}/2})^{T}$,
$\phi_{k}=\arctan k_{y}/k_{x}$. The wavevector is
$k_{\sigma}^{e/h}=\sqrt{2m(\mu-E_{k\sigma}\pm\epsilon)}$, where
$E_{k\sigma}=k^{2}/2m-\sigma\alpha k$. For the proximitized SC state in
$|z|<-L/2$, the eigenstates with eigen-energy $\epsilon$ is give by

$\displaystyle\phi_{e\sigma}^{\pm}(z)=\frac{1}{\sqrt{2q_{\sigma}^{e}}[\epsilon^{2}/\Delta^{2}-1]^{1/4}}\left(\begin{array}[]{cccc}e^{i\eta^{e}/2}\\\
e^{-i\eta^{e}/2}\end{array}\right)\otimes|\sigma\rangle e^{\pm
iq_{\sigma}^{e}z},$ (64)

and $\phi_{h\sigma}^{\pm}$ is given by replacing the index $e$ to $h$ in Eq.
(64). Here $\eta^{e/h}=\frac{\chi}{2}\pm\eta$, and
$\eta=\arccos\frac{\epsilon}{\Delta}$. The wavevector is
$q_{\sigma}^{e/h}=[2m(\mu-
E_{k\sigma}\pm\sqrt{\epsilon^{2}-\Delta^{2}})]^{1/2}$, and ${\rm
Re}q_{\sigma}^{e/h}>0$, ${\rm Im}q_{\sigma}^{e}>0$, ${\rm
Im}q_{\sigma}^{h}<0$.

It is easy to observe that the spin is conserved during the scattering. By
matching the wavefunction Eq. (61) and Eq. (64) at the superconducting-normal
state interface $z=\pm L/2$, we can obtain the scattering matrix, and then get
the bound state spectrum. The spectrum is determined by

$\displaystyle\arccos\frac{\epsilon}{\Delta}=\frac{(k_{\sigma}^{e}-k_{\sigma}^{h})L}{2}+\frac{\chi}{2}.$
(65)

Here we have neglected the normal reflection which is a small correction in
the weak pairing. To proceed, we again assume the weak pairing $\Delta\ll\mu$,
and get

$\displaystyle\frac{\epsilon}{\Delta}=\cos\Big{[}\frac{\chi}{2}+\frac{k_{F}L}{2\sqrt{1-E_{k\sigma}/\mu}}\frac{\Delta}{\mu}\times\frac{\epsilon}{\Delta}\Big{]}.$
(66)

The dependence on $k$ is suppressed by $\Delta/\mu$. Thus, to the lowest
order, the gap is given by $\epsilon=\Delta\cos\chi/2$.

However, as $k$ increases such that $E_{k\sigma}\sim\mu$, above approximation
may break down. Nevertheless, notice that the bound state exists when
$k_{\sigma}^{e/h}$ is a real number, i.e., $E_{k\sigma}\leq\mu-\epsilon$. At
$E_{k\sigma}=\mu-\epsilon$, we have

$\displaystyle\frac{\epsilon}{\Delta}=\cos\Big{[}\frac{\chi}{2}+\frac{k_{F}L}{2}\sqrt{\frac{\Delta}{\mu}}\times\sqrt{\frac{\epsilon}{\Delta}}\Big{]},$
(67)

where the suppression is sublinear in $\Delta/\mu$. At generic momentum, the
topological gap is $\epsilon=\Delta\cos\frac{\chi}{2}$ up to a small
correction of $\Delta/\mu$.

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