# Majorana corner modes and tunable patterns in an altermagnet heterostructure

Yu-Xuan Li  Cheng-Cheng Liu<EMAIL_ADDRESS>Centre for Quantum Physics, Key
Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement
(MOE), School of Physics, Beijing Institute of Technology, Beijing 100081,
China

###### Abstract

The mutual competition and synergy of magnetism and superconductivity provide
us with a very valuable opportunity to access topological superconductivity
and Majorana Fermions. Here, we devise a heterostructure consisting of an
$s$-wave superconductor, a 2D topological insulator and an altermagnet, which
is classified as the third magnet and featured by zero magnetization but spin
polarization in both real and reciprocal spaces. We find that the altermagnet
can induce mass terms at the edges that compete with electron pairing, and
mass domains are formed at the corners of sample, resulting in zero-energy
Majorana corner modes (MCMs). The presence or absence of MCMs can be
engineered by only changing the direction of the Néel vector. Moreover,
uniaxial strain can effectively manipulate the patterns of the MCMs, such as
moving and interchanging MCMs. Experimental realization, remarkable advantages
of our proposal and possible braiding are discussed.

blackIntroduction.—In the past decade, Majorana zero modes (MZMs) have been
extensively researched as building blocks for topological quantum computation
[1, 2, 3, 4, 5, 6]. Topological superconductors have gained tremendous
attention in both experimental and theoretical settings [7, 1, 8, 9, 10, 11]
as a promising platform for the realization of MZMs. The key feature that
differentiates topological superconductors from trivial ones is the presence
of MZMs at the boundaries of the system [12, 13, 14] or the core of the
vortices [15, 16, 17, 18].

The higher-order topological states of matter have extended our understanding
of topological states [19, 20, 21, 22, 23, 24, 25, 26, 27, 28] and provided a
new path for the realization of MZMs [29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. Compared with first-order
conventional topological superconductors, higher-order topological
superconductors (HOTSCs) have a special “bulk-edge” correspondence.
Specifically, the codimension ($d_{c}$) of edge states or end states in
conventional topological superconductors is 1, while the $d_{c}$ of hinge
states or corner states in HOTSCs is equal to or greater than 2. Previous
studies have proposed several heterojunction systems to realize HOTSCs with
MZMs, such as utilizing topological $p$-wave superconductors [36],
unconventional or conventional superconductor/topological insulator
heterojunctions [35, 34, 47] and ferromagnetic heterojunctions [32, 42].
Although progress has been made in the study of HOTSCs, platforms for tunable
MZMs are still difficult to achieve.

Figure 1: (a) Schematic of the proposed altermanget heterostructure. A two-
dimensional topological insulator is sandwiched between an $s$-wave
superconductor and an altermagnets. The emerging MZMs are localized at the
corners. Altermagnets have various spin-momentum locking with the even-parity
wave form, such as $d$-wave, $g$-wave, and $i$-wave. (b) The phase diagram in
the plane of Néel vector polar angle $\theta$ and chemical potential $\mu$.
(c) The real-space wave function distribution with one MZM localized at each
corner. (Inset) Plot of the several eigenvalues near zero energy. The common
parameters are $m_{0}=1.0,t_{x}=A_{x}=A_{y}=1.0,t_{y}=0.5,\Delta_{0}=0.5$, and
$J_{0}=1.0$.

Recently, a new magnetic phase dubbed altermagnets [51, 52] was discovered in
several materials, such as RuO2 [53, 54], Mn5Si3 [55] and KRu4O8 [54], which
breaks the time-reversal symmetry (TRS) despite having zero net magnetization.
Interestingly, altermangets with non relativistic highly anisotropic spin
splitting [54] are used to explore some interesting properties [56, 57, 58,
51, 59, 60, 61, 62, 63], such as finite-momentum Cooper pairing, Andreev
reflection, Hall effect, etc.

A natural question is whether we can use the highly anisotropic spin splitting
and zero magnetization unique to altermangets to design and optimize
topological superconductivity, especially higher-order topological
superconductivity and the hallmark Majorana modes.

In this work, we investigate the combination of altermagnets with
superconductor/topological insulator heterojunctions, as illustrated in Fig.
1(a) and give an affirmative answer to the above question. There are several
forms of spin splitting in altermagnets with different symmetries, such as
$d$-wave, $g$-wave, and $i$-wave [54]. Without loss of generality, we consider
an altermagnets with $d$-wave magnetism as an example, and the other
symmetries have a similar analysis procedure [64]. Our findings demonstrate
that altermagnetism can drive the system into a HOTSC phase, resulting in the
emergence of MZMs at the corners of the system, i.e., Majorana corner modes
(MCMs). By adjusting the chemical potential or varying the direction of the
Néel vector, which are both readily achieved experimentally, we can create or
annihilate a pair of MCMs. Furthermore, combined with uniaxial stress, we can
move and switch MCMs arbitrarily, offering flexible adjustability for further
research.

blackModel.— We begin with a general form of the superconducting mean-field
Hamiltonian $\mathcal{H}=\frac{1}{2}\sum_{\bm{k}}\Psi^{\dagger}_{\bm{k}}H^{\rm
BdG}(\bm{k})\Psi_{\bm{k}}$, where the Nambu spinor basis
$\Psi^{\dagger}_{\bm{k}}=(c^{\dagger}_{\bm{k}\alpha},c_{-\bm{k}\alpha})$ with
$c^{\dagger}_{\bm{k}\alpha}$($c_{\bm{k}\alpha}$) representing the fermion
creation (annihilation) operator for the $\alpha$ degree of freedom. The
corresponding Bogoliubov-de Gennes (BdG) Hamiltonian is given by

$\displaystyle H^{\rm BdG}(\bm{k})$
$\displaystyle=\left(\begin{array}[]{cc}H_{0}(\bm{k})-\mu\Gamma_{00}&-i\Gamma_{20}\Delta_{0}\\\
i\Gamma_{20}\Delta_{0}&\mu\Gamma_{00}-H_{0}^{*}(-\bm{k})\end{array}\right),$
(1) $\displaystyle H_{0}(\bm{k})$
$\displaystyle=M(\bm{k})\Gamma_{03}+A_{x}\sin k_{x}\Gamma_{21}+A_{y}\sin
k_{y}\Gamma_{11}$ $\displaystyle+J(\bm{k})\bm{s}\cdot\bm{n}\otimes\sigma_{x}.$

$\Gamma_{ij}=s_{i}\otimes\sigma_{j}$, where $s_{j}$ and $\sigma_{j}$ act on
the spin $(\uparrow,\downarrow)$ and orbital $(a,b)$ degree of freedom,
respectively. The Néel vector, represented by a unit vector
$\bm{n}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$, can be
rotated experimentally [65]. The first line of the Hamiltonian $H_{0}(\bm{k})$
can describe the conventional topological insulator with a nontrivial
$\mathcal{Z}_{2}$ invariant [66] with $M(\bm{k})=(m_{0}-t_{x}\cos
k_{x}-t_{y}\cos k_{y})$, while the second line denotes the $d$-wave
altermagnetism with $J(\bm{k})=J_{0}(\cos k_{x}-\cos k_{y})$ [54, 52].
Throughout this work we use the lattice constants as units, and take all the
parameters $m_{0},t_{x(y)},A_{x(y)},J_{0}$, and $\Delta_{0}$ positive. When we
ignore the altermagnetism term in the normal states, the Hamiltonian depicts a
two-dimensional topological insulator with time-reversal symmetry (TRS) [67].
The TRS operation is given by $\mathcal{T}=is_{y}\mathcal{K}$, where
$\mathcal{K}$ is the complex conjugate. When the“mass-ring” $M(\bm{k})=0$
[where the $M(\bm{k})$ term changes sign] of the topological insulator
contains an odd number of time-reversal-invariant momentums (TRIMs), the
system has gapless helical edge states protected by the TRS [68].

Figure 2: (a) Quasiparticle spectrum for a cylindrical geometry. In the
presence of altermagnetism and electron pairing, the otherwise helical edge
states open a gap. The inset shows the variation of the edge gap with order
parameter strength $J_{0}$ of altermagnets. The closure of the edge states at
the critical $J_{0}=J_{0}^{c}$ indicates a topological phase transition at the
boundary. (b) Bulk bands along the high symmetry path at $J_{0}=1.0$. The
inset shows the evolution of the bulk gap with $J_{0}$, and no bulk gap
closure is found. (c) Schematic demonstrating the Dirac mass terms of adjacent
edges with opposite signs, forming two domain walls. I, II, III and IV donate
four edges. (d) The real-space energy spectrum of an open-boundary square
structure of size $L_{x}=L_{y}=30$. Eight MZMs emerge (red dots). (e) The
distribution of the MZM wavefunction in real space. (f) The phase diagram in
the plane of the azimuth angle $\theta$ and $J_{0}$. The blue dotted lines
mark the phase boundary. The common parameters are $m_{0}=1.0$,
$t_{x}=t_{y}=A_{x}=A_{y}=2.0$, and $\Delta_{0}=0.5$.

blackZero Energy MCMs.—When proximitized with an $s$-wave superconductor and
an altermagnet, the helical edge states will be gapped by breaking the $U(1)$
charge conservation symmetry and TRS, respectively. The interplay of the
$s$-wave pairing and altermagnet order parameter will drive the system into
the HOTSC phase. We consider the general case and choose the chemical
potential and Néel vector as variable parameters, finding that both have
significant effects on the MCMs, as illustrated in the phase diagram [Fig.
1(b)]. One can find that there are no MCMs when the Néel vector is close to
the in-plane direction. On the contrary, when the Néel vector has a certain
out-of-plane component, it favors the formation of MCMs. In the HOTSC phase,
there are one MCM at each corner, as shown in Fig. 1(c).

To get an intuitive picture of the MCMs, we fix the chemical potential $\mu=0$
and Néel vector along the $z$ direction ($\theta=0$) for simplicity. When
$\Delta_{0}\neq 0,J_{0}=0$, the TRS of the system still holds, but the broken
$U(1)$ symmetry makes the helical edge states open a gap, and the system is a
trivial superconductor [47]. With the altermagnetism turned on and the
increase of $J_{0}$, the edge-state gap along the direction of $x$($y$) will
be closed and reopened again, as shown in Fig. 2(a). However, during the
process, the bulk state is always fully gapped [Fig. 2(b)]. These indicates a
possible higher-order topological phase transition [28], which can be made
more clearly by the edge theory in the next section. After the phase
transition, eight MCMs appear with two MCMs at each corner, which are
characteristic of the HOTSC phase, as shown in Figs. 2(d) and 2(e). We will
see later that the doubling of MCMs here is due to the additional symmetry
brought about by the zero chemical potential. In addition, adjusting the Néel
vector will affect the location of higher-order topological phase transitions.
With the chemical potential $\mu=0$, we analytically obtained the phase
diagram in the plane of the amplitude $J_{0}$ and the polar angle $\theta$ of
the altermagnetic order parameter, as plotted in Fig. 2(f).

blackEdge theory.—The appearance of MCMs after the higher-order topological
phase transition can be intuitively understood by edge theory. For simplicity,
we first consider the case of the chemical potential $\mu=0$ and Néel vector
along the $z$ axis [69]. We expand the Hamiltonian $H^{\rm BdG}(\bm{k})$ in
Eq. (1) around $\bm{\Gamma}=(0,0)$ to the second order,

$\displaystyle H^{\rm BdG}(\bm{k})$
$\displaystyle=(m+\frac{t_{x}}{2}k_{x}^{2}+\frac{t_{y}}{2}k_{y}^{2})\Lambda_{303}+A_{x}k_{x}s_{y}\Lambda_{321}$
(2)
$\displaystyle+A_{y}k_{y}\Lambda_{011}-\frac{J_{0}}{2}(k_{x}^{2}-k_{y}^{2})\Lambda_{331}+\Delta_{0}\Lambda_{220},$

with $\Lambda_{ijk}=\tau_{i}\otimes\Gamma_{jk}$, and $\tau_{i}$ are Pauli
matrices in the particle-hole space. The parameter $m=m_{0}-t_{x}-t_{y}<0$
ensures that the mass ring contains the odd number of TRIMs [68]. To
distinguish the boundaries, we label the four boundaries using I, II, III, and
IV, respectively, as shown in Fig. 2(c). We first focus on edge I. The
momentum $k_{x}$ is replaced by $-i\partial_{x}$, and the Hamiltonian can be
decomposed into two parts $H=H_{0}+H^{\prime}$, which are denoted as
$H_{0}(-i\partial_{x},k_{y})=(m-t_{x}\partial_{x}^{2}/2)\Lambda_{303}-iA_{x}\partial_{x}\Lambda_{321}$
and
$H^{\prime}(-i\partial_{x},k_{y})=A_{y}k_{y}\Lambda_{011}+J_{0}\partial_{x}^{2}/2\Lambda_{331}+\Delta_{0}\Lambda_{220}$,
respectively. We solve the eigenequation
$H_{0}\psi_{\alpha}(x)=E_{\alpha}\psi_{\alpha}(x)$, satisfying the boundary
condition $\psi_{\alpha}(0)=\psi_{\alpha}(+\infty)=0$, and obtain four
$E_{\alpha}=0$ solutions [64]. The perturbation part $H^{\prime}$ is projected
onto the bases $\psi_{\alpha}$ and reads as $H_{\rm
I}=A_{y}k_{y}\eta_{z}-M_{\rm I}\eta_{y}\gamma_{y}+J_{\rm
I}\eta_{x}\gamma_{z}$, where the Pauli matrices $\gamma_{i},\eta_{j}$ are
defined in the four zero-energy-states $\psi_{\alpha}$ space. The mass terms
$J_{\rm I}=J_{0}m/t_{x}$ and $M_{\rm I}=\Delta_{0}$ originate from the
altermagnetism and electron pairing, respectively [64].

The effective Hamiltonian on the other edges (II-IV) can be obtained similarly
[64]. For the convenience of the discussion, we define the edge coordinates
$l$ counterclockwise, and the effective Hamiltonian can be compactly written
as

$H^{\rm
edge}(l)=-iA(l)\eta_{z}\partial_{l}+M(l)\eta_{y}\gamma_{y}+J(l)\eta_{x}\gamma_{z},$
(3)

where for the edge coordinate $l=\left\\{\text{I-IV}\right\\}$,
$A(l)=\left\\{A_{y},A_{x},A_{y},A_{x}\right\\},M(l)=\left\\{-\Delta_{0},\Delta_{0},\Delta_{0},-\Delta_{0}\right\\}$,
and
$J(l)=\left\\{J_{0}m/t_{x},-J_{0}m/t_{y},J_{0}m/t_{x},-J_{0}m/t_{y}\right\\}$,
respectively. From Eq. (3) one can obtain that the Dirac masses originated
from $s$-wave pairing and the altermagnets are commutable, indicating that
they compete with each other on the edges. The boundary energy spectrum of Eq.
(3) reads as $E^{\rm edge}_{\rm
I,III}(k_{y})=\pm\sqrt{(A_{y}k_{y})^{2}+(\mp\Delta_{0}\pm J_{\rm
I(III)})^{2}}$, $E^{\rm edge}_{\rm
II,IV}(k_{x})=\pm\sqrt{(A_{x}k_{x})^{2}+(\pm\Delta_{0}\pm J_{\rm
II(IV)})^{2}}$. One can find that as $J_{0}$ increases, the gap of the edge
state decreases, and at the critical value
$J_{0}^{c}m/t_{x(y)}=\pm\Delta_{0}$, the gap closes that corresponds to the
higher-order topological phase transition and is consistent with the numerical
results of Fig. 2.

Furthermore, we can decompose the Hamiltonian (3) as $H=H^{+}\oplus H^{-}$
with the help of a conserved quantity $\Pi=\eta_{z}\gamma_{x}$. The effective
Hamiltonian for two subspaces with $\Pi=\pm 1$ can be obtained as

$\displaystyle H^{+}$
$\displaystyle=iA(l)\tilde{\eta}_{z}\partial_{l}+\tilde{M}^{+}(l)\tilde{\eta}_{x},$
(4) $\displaystyle H^{-}$
$\displaystyle=iA(l)\tilde{\eta}_{z}\partial_{l}+\tilde{M}^{-}(l)\tilde{\eta}_{x},$

with Pauli matrices $\tilde{\eta}$ acting in the subspace and
$\tilde{M}^{\pm}(l=\text{I-IV})=\\{\pm\Delta_{0}+J_{0}|m|/t_{x},\mp\Delta_{0}-J_{0}|m|/t_{y},\mp\Delta_{0}+J_{0}|m|/t_{x},\pm\Delta_{0}-J_{0}|m|/t_{y}\\}$.
It can be found that once $J_{0}$ exceeds the critical value
$J_{c}=\Delta_{0}t_{0}/|m|$ (considering the isotropic case
$t_{x}=t_{y}=t_{0}$), the signs of the mass terms at the adjacent boundaries
in $H^{\pm}$ are opposite and two mass domain walls are formed at each corner.
The zero-energy MCMs formed by $H^{+}$ and $H^{-}$ are decoupled from each
other, so that two MCMs exist at each corner, as illustrated in Figs.
2(c)–2(e).

Figure 3: (a) The phase diagram for the isosceles-right-triangle geometry. (b)
The energy spectrum of an isosceles triangle structure with two MCMs (red
dots) and the density plot of the two MCMs (c) with V representing the
hypotenuse. $\Delta_{0}$=0.2, $J_{0}$=0.5. (d) The real-space energy spectrum
evolves with electron-pairing amplitude in the isosceles-right-triangle
structure with MCMs marked in red existing in a large range. The common
parameters are $m_{0}=1.0$, $t_{y}=A_{y}=t_{x}=A_{x}=2.0$ and $\mu=0.1$.

blackEffect of chemical potential.—In contrast to the TRS-protected Majorana
Kramers pairs [34], only one MZM can stably exist at a given corner without
TRS. The chemical potential will give to the edge Hamiltonian (3) a new term
$\mu\gamma_{z}$, which couples the $H^{+}$ and $H^{-}$ subblocks. This
coupling causes two MCMs from the two sub-blocks at the same corner to mix and
disappear. However, two such MCMs are intact if they are located at different
corners in spite of nonzero chemical potential. To demonstrate this effect, we
start with an isosceles triangle geometry. We first analyze the situation
where $t_{x}=t_{y}\equiv t_{0}$, and we will analyze the situation where
$t_{x}\neq t_{y}$ later. The altermagnetism causes the Dirac mass
$J(\alpha)=J_{0}m\cos(2\alpha)/t_{0}$ at an edge, where $\alpha$ is the angle
between the edge and the edge I. For an edge with $\alpha=\pi/4$, i.e., edge V
in Fig. 3(c), $J(\alpha)=0$. Similarly we can also get the Dirac mass at edge
V, $\tilde{M}_{\rm V}^{\pm}=\pm\Delta_{0}$ [64]. By carefully analyzing the
signs of the mass terms on the three sides of the sample, we find that when
$\Delta_{0}<J_{0}|m|/t_{0}$, there is a pair of robust MCMs at two vertices of
the triangle sample. In the two subblocks, the signs of the mass terms read
$\text{Sign}[\tilde{M}^{+}(\text{I, V, IV})]=\\{+,+,-\\}$ and
$\text{Sign}[\tilde{M}^{-}(\text{I, V, IV})]=\\{+,-,-\\}$. Note that two
adjacent edges with opposite signs will form a domain wall, so there will be a
MCM at the intersection. It is useful to define the corner position, corner
i-j, which is defined as the intersection of adjacent edge i and edge j. It
can be seen that for $H^{+}$ subblock, there are two MCMs at corner IV-V and
corner IV-I, while for $H^{-}$ subblock, there are two MCMs at corner I-V and
corner I-IV. The nonzero chemical potential will annihilate a pair of MCMs on
the same corner (corner IV-I) with only a pair of stable MCMs at two corners
(corner V-I, corner V-IV). When $\Delta_{0}>J_{0}|m|/t_{0}$, there is no MCM.
These theoretical analyses are also verified by our direct numerical
calculations, as shown in Figs. 3(b)–3(d). We also give the phase diagram in
Fig. 3(a).

blackTunable patterns of MCMs.—We discover that uniaxial strain can
interestingly manipulate the patterns of the MCMs. We first apply a uniaxial
stress to a square structure, say along the $x$-direction such that
$t_{x}>t_{y}$. We find that as the pairing strength increases, the gap of the
edge states along the $k_{y}$ direction of the system is closed and then
reopened, but it is always open along the $k_{x}$ direction, indicating a
higher-order topological phase transition [64]. The numerical results in Figs.
4 (a) and 4(b) confirm the higher-order topological phase with four highly
localized MZMs at each corner. Such results can be understood through edge
theory [64]. When $J_{0}|m|/t_{x}<\Delta_{0}<J_{0}|m|/t_{y}$, we obtain, for
the two subblocks, the signs of the mass terms
$\text{Sign}[\tilde{M}^{+}(l={\rm I-IV})]=\left\\{+,-,-,-\right\\}$ and
$\text{Sign}[\tilde{M}^{-}(l={\rm I-IV})]=\left\\{-,-,+,-\right\\}$,
respectively. For $H^{+}$ sub-block, there are two MCMs at corner I-IV and
corner I-II, while for $H^{-}$ sub-block, there are two MCMs at corner III-II
and corner III-IV. Since the four MCMs are located at four different corners,
the nonzero chemical potential does not work, and they can exist stably, as
shown in Fig. 4 (c). Such an analysis can clearly explain the above numerical
results.

We can also apply a uniaxial stress along another direction, such as the $y$
direction, such that $t_{x}<t_{y}$. We find a higher-order topological phase
transition occurs again with one highly localized MZM at each corner [64].
When $J_{0}|m|/t_{y}<\Delta_{0}<J_{0}|m|/t_{x}$, similarly, for the two
subblocks, the signs of the mass terms read
$\text{Sign}[\tilde{M}^{\pm}(l={\rm I-IV})]=\left\\{+,\mp,+,\pm\right\\}$.
There are two MCMs at corner II-I (IV-III) and corner II-III (IV-I) for
$H^{+}$ ($H^{-}$) subblock. Since the four MCMs are located at four different
corners, they can also exist stably despite the non-zero chemical potential,
as shown in Fig. 4 (d).

It is worth noting that comparing Figs. 4(c) and 4(d), one can find that by
changing the uniaxial stress direction, we can interchange two MCMs at corner
IV-I and corner II-III. Furthermore, we can also freely move a pair of MCMs
spatially to locate at any two vertices of the isosceles structure by applying
uniaxial stress [64]. The intriguing properties with tunable patterns of MCMs
hold promise for the implementation of Majorana braiding [5] and are
particularly useful in the design and fabrication of Majorana systems with
desired properties [70].

Figure 4: Results with nonzero chemical potential $\mu$. (a) The real-space
wave function distribution with one MZM localized at each corner. Inset plots
the several eigenvalues near the zero energy. (b) The real-space energy
spectrum evolves with electron-pairing amplitude in a square structure with
MCMs marked in red existing in a range. (c), (d) Tunable MCM patterns. The
blue hollow (green solid) circles represent the MCMs from the $H^{+}$
($H^{-}$) block. The arrows indicate the uniaxial stress direction. The common
parameters are $m_{0}=1.0$, $t_{y}=A_{y}=1.0$,
$t_{x}=A_{x}=2.0,J_{0}=\Delta_{0}=0.5$, and $\mu=0.1$.

blackDiscussion.—Although altermagnets have been proposed as the third
magnetic phase besides ferromagnets and antiferromagnets for a short time,
there are already abundant altermagnets available, such as RuO2 [54, 53],
KRu4O8 [54] and Mn5Si3 [55], etc. Recently, the spin splitting in altermagnets
has been observed by spin- and angle-resolved photoemission spectroscopy
measurement [71]. On the other hand, superconducting/topological insulator
heterojunctions have been experimentally realized, and induced electron
pairing by proximity effect has been observed in topological insulators [72,
73, 74, 75, 76]. Consequently, it is evident that the required material
ingredients for implementing the proposal are all experimentally accessible.
In the experiment, scanning tunneling microscope can be used to detect MZMs
[77], where the differential conductivity peaks at zero energy due to the
resonant Andreev reflection are quantized at $2e^{2}/h$ [78, 79]. In addition,
the fractional Josephson effect [12] is also strong evidence for the existence
of a zero-energy Majorana mode, and anomalous response to radio frequency
irradiation produced by a $4\pi$-Josephson current could be observed
experimentally [80, 81]. The existence of MZMs can be proved by combining the
above experimental means.

Our proposal has certain remarkable advantages. First, it does not necessitate
the use of unconventional superconductors with small electron coherence
lengths and an external magnetic field which make experimental realization and
confirmation of the existence of MZMs more difficult. Second, by adjusting the
direction of the Néel order parameter of altermagnets, which is readily
experimentally, one can create MCMs. In experiments, current, voltage [82, 83]
and spin-orbit torques [84] can manipulate and detect the direction of the
Néel vector in antiferromagnetic materials. Last but not least, altermagnets
with zero net magnetization have a non-negligible advantage over conventional
magnetic materials when combined with superconductivity [85]. All of these
make experimental realization of the proposal very possible and opening a new
path for the realization of MCMs.

In addition, the patterns of MCMs implemented by our scheme are flexible and
adjustable. For example, for an isosceles triangular structure, a pair of MCMs
can be freely moved to any two vertices of the triangle by adjusting uniaxial
strain, and for a square structure, two MCMs can be interchanged by adjusting
uniaxial strain. These operations are important for braiding MCMs. In
particular, we noticed that the triangular structure hosts a pair of MCMs, so
it is equivalent to 1D topological superconductivity nanowires [13, 14], and
thus can be used to construct $T$ junctions [70] or $Y$ junctions [86, 87, 88]
for braiding MCMs to exhibit the non-Abelian statistics and potential for
topological quantum computation.

The work is supported by the National Key R $\&$ D Program of China (Grant No.
2020YFA0308800) and NSF of China (Grant No. 11922401).

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Supplementary material for “Majorana corner modes and tunable patterns in an
altermagnet heterostructure”
Yu-Xuan Li and Cheng-Cheng Liu

Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum
Architecture and Measurement (MOE), School of Physics, Beijing Institute of
Technology, Beijing 100081, China

The supplementary material contains three parts. The derivation of the edge
Hamiltonian of the square structure when the Neél vector is along any
direction and the phase diagram are given in Section I. The edge Hamiltonian
of the isosceles structure, when the Neél vector is along the $z$-direction,
is given in Section I. The effective boundary Hamiltonian for the
heterostructures of altermagnets with $g$-wave and $i$-wave spin splitting is
derived in Section I. In Section II, the effect of anisotropy and chemical
potential on the MCMs in the square structure is studied. In Section III, we
manipulate the pattern of MCMs by uniaxial stresses. We can interchange MCMs
in a square structure and move the pair of MCMs spatially to locate at any two
vertices of an isosceles structure.

##  I edge theory of Neel vector field along arbitrary direction

In this section, we analyze both the general Neél vector $\bm{n}$ and
$d$-wave-like altermagnetism, as previously explored in Refs. [54, 52]. The
associated bulk Hamiltonian can be expressed as

$\displaystyle H^{\rm BdG}(\bm{k})$
$\displaystyle=\left(\begin{array}[]{cc}H_{0}(\bm{k})-\mu_{0}&-is_{y}\Delta_{0}\\\
is_{y}\Delta_{0}&\mu_{0}-H_{0}^{*}(-\bm{k})\end{array}\right),$ (S-1)
$\displaystyle H_{0}(\bm{k})$ $\displaystyle=(m_{0}-t_{x}\cos k_{x}-t_{y}\cos
k_{y})\sigma_{z}+A_{x}\sin k_{x}s_{y}\sigma_{x}+A_{y}\sin
k_{y}s_{x}\sigma_{x}+\bm{J}(\bm{k})\bm{s}\cdot\bm{n}\sigma_{x},$
$\displaystyle J_{x}(\bm{k})$ $\displaystyle=J^{x}_{0}(\cos k_{x}-\cos
k_{y}),J_{y}(\bm{k})=J^{y}_{0}(\cos k_{x}-\cos
k_{y}),J_{z}(\bm{k})=J^{z}_{0}(\cos k_{x}-\cos k_{y}).$

The bulk Hamiltonian Eq. (S-1) we consider includes Pauli matrices $s_{i}$ and
$\sigma_{j}$, which act on the spin $(\uparrow,\downarrow)$ and orbit $(a,b)$
degrees of freedom, respectively. The unit vector
$\bm{n}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$ represents
the direction of Neél vector, with $\theta$ and $\varphi$ denoting the polar
and azimuthal angles, respectively (Fig. S2(a)). Specifically, the magnitudes
of the components in each direction are
$J_{0}^{x}=J_{0}\sin\theta\cos\varphi$,
$J_{0}^{y}=J_{0}\sin\theta\sin\varphi$, and $J_{0}^{z}=J_{0}\cos\theta$.

Figure S1: (a) The schematic shows the edge coordinate $l$ with the
counterclockwise direction selected as the positive direction. Edges I, II,
III, and IV are labeled for use in the edge theory. (b) Schematic of an
isosceles triangle structure with the hypotenuse marked as V and the rest of
the edges identical to the square structure. (c) Schematic diagram of the
rotated coordinate system and the tangential boundary.

###  I.1 Square geometry

In this subsection, we present in detail the edge theory of the square
geometry (Fig. S1(a)). For simplicity, we set the chemical potential as
$\mu=0$ and expand the Hamiltonian in Eq. (S-1) around $\bm{k}=(0,0)$ to the
second order

$\displaystyle H(\bm{k})$
$\displaystyle=(m+\frac{t_{x}}{2}k_{x}^{2}+\frac{t_{y}}{2}k_{y}^{2})\sigma_{z}\tau_{z}+A_{x}k_{x}s_{y}\sigma_{x}\tau_{z}+A_{y}k_{y}s_{x}\sigma_{x}$
(S-2)
$\displaystyle-\frac{1}{2}J_{0}^{x}(k_{x}^{2}-k_{y}^{2})s_{x}\sigma_{x}\tau_{z}-\frac{1}{2}J_{0}^{y}(k_{x}^{2}-k_{y}^{2})s_{y}\sigma_{x}$
$\displaystyle-\frac{1}{2}J_{0}^{z}(k_{x}^{2}-k_{y}^{2})s_{z}\sigma_{x}\tau_{z}+\Delta_{0}s_{y}\tau_{y}.$

To ensure the topological insulator has helical edge states we set
$m=m_{0}-t_{x}-t_{y}<0$. To facilitate discussing different edges we adopt the
counterclockwise direction as positive and label the four edges of a square
geometry as edge I, edge II, edge III, and edge IV, as shown in Fig. 1(a).

We begin by focusing on the region $x\in[0,+\infty)$ corresponding to edge I.
By decomposing the Hamiltonian into $H=H_{0}+H^{\prime}$ and applying
$k_{x}\rightarrow-i\partial_{x}$, we obtain

$\displaystyle H_{0}(-i\partial_{x})$
$\displaystyle=(m-\frac{t_{x}}{2}\partial_{x}^{2})\sigma_{z}\tau_{z}-iA_{x}\partial_{x}\sigma_{x}s_{y}\tau_{z},$
(S-3) $\displaystyle H^{\prime}(-i\partial_{x},k_{y})$
$\displaystyle=A_{y}k_{y}\sigma_{x}s_{x}+J_{x}(-i\partial_{x})\sigma_{x}s_{x}\tau_{z}+J_{y}(-i\partial_{x})\sigma_{x}s_{y}+J_{z}(-i\partial_{x})\sigma_{x}s_{z}\tau_{z}+\Delta_{0}s_{y}\tau_{y},$

where we ignore the $k_{y}^{2}$ terms.
$J_{\alpha,(\alpha=x,y,z)}(-i\partial_{x})=J_{0}^{\alpha,(\alpha=x,y,z)}\partial_{x}^{2}/2$.
When the pairing and altermagnetism are small compared to the energy gap,
$H^{\prime}$ is regarded as a perturbation. By solving
$H_{0}\psi_{\alpha}(x)=E_{\alpha}\psi_{\alpha}(x)$ for $E_{\alpha}=0$ with the
boundary condition $\psi_{\alpha}(0)=\psi_{\alpha}(+\infty)=0$, we obtain four
zero-energy eigenstates

$\psi_{\alpha}(x)=\mathcal{N}_{x}\sin(\kappa_{1}x)e^{-\kappa_{2}x}e^{ik_{y}y}\chi_{\alpha},$
(S-4)

with normalization constant
$|\mathcal{N}_{x}|^{2}=4|\kappa_{2}(\kappa_{1}^{2}+\kappa_{2}^{2})/\kappa_{1}^{2}|$,
where
$\kappa_{1}=\sqrt{|2m/t_{x}|-(A_{x}^{2}/t_{x}^{2})},\kappa_{2}=A_{x}/t_{x}$.

The spinor component of the eigenstate $\chi_{\alpha}$ satisfies
$\tau_{0}s_{y}\sigma_{y}\chi_{\alpha}=-\chi_{\alpha}$. Furthermore, we denote
the eigenvectors with the eigenvalue equal to $-1$ as $\xi_{\alpha}$, which
satisfy $\tau_{0}s_{y}\sigma_{y}\xi_{\alpha}=-\xi_{\alpha}$

$\displaystyle\xi_{1}$
$\displaystyle=|\tau_{z}=+1\rangle\otimes|\sigma_{y}=+1\rangle\otimes|s_{y}=-1\rangle,$
(S-5) $\displaystyle\xi_{2}$
$\displaystyle=|\tau_{z}=+1\rangle\otimes|\sigma_{y}=-1\rangle\otimes|s_{y}=+1\rangle,$
$\displaystyle\xi_{3}$
$\displaystyle=|\tau_{z}=-1\rangle\otimes|\sigma_{y}=+1\rangle\otimes|s_{y}=-1\rangle,$
$\displaystyle\xi_{4}$
$\displaystyle=|\tau_{z}=-1\rangle\otimes|\sigma_{y}=-1\rangle\otimes|s_{y}=+1\rangle.$

We can obtain $\chi_{i}$ by combining the eigenvectors $\xi_{i}$

$\displaystyle\chi_{1}$
$\displaystyle=(\xi_{3}+\xi_{4})/2=\frac{1}{\sqrt{2}}\left(0,0,0,0,1,0,0,1\right)^{T},$
(S-6) $\displaystyle\chi_{2}$
$\displaystyle=-i(\xi_{3}-\xi_{4})/2=\frac{1}{\sqrt{2}}\left(0,0,0,0,-1,1,0,0\right)^{T},$
$\displaystyle\chi_{3}$
$\displaystyle=(\xi_{1}+\xi_{2})/2=\frac{1}{\sqrt{2}}\left(1,0,0,1,0,0,0,0\right)^{T},$
$\displaystyle\chi_{4}$
$\displaystyle=-i(\xi_{1}-\xi_{2})/2=\frac{1}{\sqrt{2}}\left(0,-1,1,0,0,0,0,0\right)^{T}.$

Using these bases, the matrix elements of $H_{p}$ can be expressed as

$H_{\text{I},\alpha\beta}=\int_{0}^{\infty}dx\psi_{\alpha}^{*}(x)H^{\prime}(-i\partial_{x},k_{y})\psi_{\beta}(x).$
(S-7)

After calculations, we obtain

$H_{\rm I}(k_{y})=A_{y}k_{y}\eta_{z}-M_{\rm I}\gamma_{y}\eta_{y}+J^{z}_{\rm
I}\gamma_{z}\eta_{x}-J^{x}_{\rm I}\gamma_{z}\eta_{z},$ (S-8)

where $\eta_{i}$ and $\gamma_{j}$ are Pauli matrices in bases of
$\psi_{\alpha}$ and the coefficients are

$\displaystyle J^{z}_{\text{I}}$
$\displaystyle=\frac{J_{0}^{z}}{2}\int_{0}^{\infty}dx\psi_{\alpha}^{*}(x)(\partial_{x}^{2})\psi_{\alpha}=\frac{J_{0}^{z}m}{t_{x}},\quad
J^{x}_{\text{I}}=\frac{J_{0}^{x}}{2}\int_{0}^{\infty}dx\psi_{\alpha}^{*}(x)(\partial_{x}^{2})\psi_{\alpha}(x)=\frac{J_{0}^{x}m}{t_{x}},$
(S-9) $\displaystyle M_{\rm I}$
$\displaystyle=\Delta_{0}\int_{0}^{\infty}dx\psi_{\alpha}^{*}(x)\psi_{\alpha}(x)=\Delta_{0}.$

It should be noted that the $y$ component of Neél vector does not impact the
behavior of the helical edge states and the $x$ component of Neél vector does
not induce a gap on the helical edge states of the topological insulator.

For edge III, the occupied space is $x\in(-\infty,0]$. Solving the
eigenequation $H_{0}\psi_{\alpha}(x)=0$ with the boundary condition
$\psi_{\alpha}(0)=\psi_{\alpha}(-\infty)=0$, we obtain the four solutions

$\psi_{\alpha}(x)=\mathcal{N}_{x}\sin(\kappa_{1}x)e^{\kappa_{2}x}e^{ik_{y}y}\chi_{\alpha}.$
(S-10)

Unlike edge I, the spinor parts of the wave function are the eigenstates of
$\tau_{0}s_{y}\sigma_{y}$ with the eigenvalue equal to $+1$, and they are
denoted as

$\displaystyle\xi_{1}$
$\displaystyle=|\tau_{z}=+1\rangle\otimes|\sigma_{y}=-1\rangle\otimes|s_{y}=-1\rangle,$
(S-11) $\displaystyle\xi_{2}$
$\displaystyle=|\tau_{z}=+1\rangle\otimes|\sigma_{y}=+1\rangle\otimes|s_{y}=+1\rangle,$
$\displaystyle\xi_{3}$
$\displaystyle=|\tau_{z}=-1\rangle\otimes|\sigma_{y}=-1\rangle\otimes|s_{y}=-1\rangle,$
$\displaystyle\xi_{4}$
$\displaystyle=|\tau_{z}=-1\rangle\otimes|\sigma_{y}=+1\rangle\otimes|s_{y}=+1\rangle.$

We combine the eigenvectors $\xi_{i}$ to obtain $\chi_{i}$

$\displaystyle\chi_{1}$
$\displaystyle=(\xi_{3}+\xi_{4})/2=\frac{1}{\sqrt{2}}\left(0,0,0,0,-1,0,0,1\right)^{T},$
(S-12) $\displaystyle\chi_{2}$
$\displaystyle=-i(\xi_{3}-\xi_{4})/2=\frac{1}{\sqrt{2}}\left(0,0,0,0,1,1,0,0\right)^{T},$
$\displaystyle\chi_{3}$
$\displaystyle=(\xi_{1}+\xi_{2})/2=\frac{1}{\sqrt{2}}\left(-1,0,0,1,0,0,0,0\right)^{T},$
$\displaystyle\chi_{4}$
$\displaystyle=-i(\xi_{1}-\xi_{2})/2=\frac{1}{\sqrt{2}}\left(0,1,1,0,0,0,0,0\right)^{T}.$

After choosing the basis function as $\psi_{\alpha}$, the matrix elements of
$H^{\prime}$ under these bases are expressed as

$H_{\text{III},\alpha\beta}=\int_{-\infty}^{0}dx\psi_{\alpha}^{*}(x)H^{\prime}(-i\partial_{x},k_{y})\psi_{\beta}(x).$
(S-13)

Through algebraic operations, one can obtain the edge Hamiltonian for edge III

$H_{\rm III}(k_{y})=-A_{y}k_{y}\eta_{z}+M_{\rm
III}\gamma_{y}\eta_{y}+J^{z}_{\rm III}\gamma_{z}\eta_{x}+J^{x}_{\rm
III}\gamma_{z}\eta_{z},$ (S-14)

where $\eta_{i}$ and $\gamma_{j}$ are Pauli matrices in bases of
$\psi_{\alpha}$. And coefficients are

$\displaystyle J^{z}_{\text{III}}$
$\displaystyle=\frac{J_{0}^{z}}{2}\int^{0}_{-\infty}dx\psi_{\alpha}^{*}(x)(\partial_{x}^{2})\psi_{\alpha}=\frac{J_{0}^{z}m}{t_{x}},\quad
J^{x}_{\text{III}}=\frac{J_{0}^{x}}{2}\int^{0}_{-\infty}dx\psi_{\alpha}^{*}(x)(\partial_{x}^{2})\psi_{\alpha}(x)=\frac{J_{0}^{x}m}{t_{x}},$
(S-15) $\displaystyle M_{\rm III}$
$\displaystyle=\Delta_{0}\int^{0}_{-\infty}dx\psi_{\alpha}^{*}(x)\psi_{\alpha}(x)=\Delta_{0}.$

Next, we consider the edges II and IV. The process is similar to the edge I.
For the edge II, we can decompose the Hamiltonian into two parts

$\displaystyle H_{0}(k_{x},-i\partial_{y})$
$\displaystyle=(m-t_{y}\partial_{y}^{2}/2)\tau_{z}\sigma_{z}-iA_{y}s_{x}\sigma_{x}\partial_{y},$
(S-16) $\displaystyle H^{\prime}(k_{x},-i\partial_{y})$
$\displaystyle=A_{x}k_{x}\tau_{z}s_{y}\sigma_{x}+J_{x}(-i\partial_{y})\sigma_{x}s_{s}\tau_{z}+J_{y}(-i\partial_{y})\sigma_{x}s_{y}+J_{z}(-i\partial_{y})\sigma_{x}s_{z}\tau_{z}+\Delta_{0}s_{y}\tau_{y},$

where the $k_{x}^{2}$ terms are ignored.
$J_{\alpha,(\alpha=x,y,z)}(-i\partial_{y})-J_{0}^{\alpha,(\alpha=x,y,z)}\partial_{y}^{2}/2$.
The space $y\in[0,+\infty)$ is occupied and we solve eigenequation
$H_{0}\psi_{\alpha}(y)=E_{\alpha}\psi_{\alpha}(y)$ under the boundary
condition $\psi_{\alpha}(0)=\psi_{\alpha}(+\infty)=0$. We obtain the four
zero-energy solutions

$\psi_{\alpha}(y)=\mathcal{N}_{y}\sin(\kappa_{3}y)e^{-\kappa_{4}y}e^{ik_{x}x}\chi_{\alpha}.$
(S-17)

Here, the normalization parameter is given by
$|\mathcal{N}_{y}|^{2}=4|\kappa_{4}(\kappa_{3}^{2}+\kappa_{4}^{2})/\kappa_{3}^{2}|$,
where
$\kappa_{3}=\sqrt{(2|m|/t_{y})-(A_{y}^{2}/t_{y}^{2})},\kappa_{4}=(A_{y}/t_{y})$.
We obtain the spinor parts of the eigenstates $\psi_{\alpha}$, which satisfy
$\tau_{z}s_{x}\sigma_{y}\chi_{\alpha}=-\chi_{\alpha}$. The eigenstates of
$\tau_{z}s_{x}\sigma_{y}$ with the eigenvalue equal to $-1$ are expressed as

$\displaystyle\xi_{1}$
$\displaystyle=|\tau_{z}=+1\rangle\otimes|\sigma_{y}=+1\rangle\otimes|s_{x}=-1\rangle,$
(S-18) $\displaystyle\xi_{2}$
$\displaystyle=|\tau_{z}=+1\rangle\otimes|\sigma_{y}=-1\rangle\otimes|s_{x}=+1\rangle,$
$\displaystyle\xi_{3}$
$\displaystyle=|\tau_{z}=-1\rangle\otimes|\sigma_{y}=+1\rangle\otimes|s_{x}=+1\rangle,$
$\displaystyle\xi_{4}$
$\displaystyle=|\tau_{z}=-1\rangle\otimes|\sigma_{y}=-1\rangle\otimes|s_{x}=-1\rangle.$

We combine the eigenvectors $\xi_{i}$ to obtain $\chi_{i}$

$\displaystyle\chi_{1}$
$\displaystyle=(\xi_{3}+\xi_{4})/2=\frac{1}{\sqrt{2}}\left(0,0,0,0,-i,0,0,1\right)^{T},$
(S-19) $\displaystyle\chi_{2}$
$\displaystyle=(\xi_{3}-\xi_{4})/2=\frac{1}{\sqrt{2}}\left(0,0,0,0,0,i,1,0\right)^{T},$
$\displaystyle\chi_{3}$
$\displaystyle=(\xi_{1}+\xi_{2})/2=\frac{1}{\sqrt{2}}\left(i,0,0,1,0,0,0,0\right)^{T},$
$\displaystyle\chi_{4}$
$\displaystyle=(\xi_{2}-\xi_{1})/2=\frac{1}{\sqrt{2}}\left(0,-i,1,0,0,0,0,0\right)^{T}.$

The perturbation matrix elements read

$H_{\text{II},\alpha\beta}=\int_{0}^{+\infty}dy\psi_{\alpha}^{*}(y)H^{\prime}(k_{x},-i\partial_{y})\psi_{\beta}(y).$
(S-20)

After some algebraic calculations, we obtain

$H_{\text{II}}(k_{x})=-A_{x}k_{x}\eta_{z}-M_{\text{II}}\gamma_{x}\eta_{y}+J^{z}_{\rm
II}\gamma_{z}\eta_{x}+J^{y}_{\rm II}\gamma_{z}\eta_{z},$ (S-21)

where $\eta_{i}$ and $\tau_{j}$ are Pauli matrices in bases of $\psi_{\alpha}$
and

$\displaystyle J^{z}_{\text{II}}$
$\displaystyle=-\frac{J_{0}^{z}}{2}\int_{0}^{\infty}dy\psi_{\alpha}^{*}(y)(\partial_{y}^{2})\psi_{\alpha}(y)=-\frac{J_{0}^{z}m}{t_{y}},\quad
J^{y}_{\text{II}}=-\frac{J_{0}^{z}}{2}\int_{0}^{\infty}dy\psi_{\alpha}^{*}(y)(\partial_{y}^{2})\psi_{\alpha}(y)=-\frac{J_{0}^{y}m}{t_{y}},$
(S-22) $\displaystyle M_{\text{II}}$
$\displaystyle=\Delta_{0}\int_{0}^{\infty}dy\psi_{\alpha}^{*}(y)\psi_{\alpha}(y)=\Delta_{0}.$

Finally, for edge IV, we need to solve the eigenvalue equation
$H_{0}\psi_{\alpha}(y)=E_{\alpha}\psi_{\alpha}(y)$ under boundary condition
$\psi_{\alpha}(0)=\psi_{\alpha}(-\infty)=0$, we obtain the four zero-energy
solutions

$\psi_{\alpha}(y)=\mathcal{N}_{y}\sin(\kappa_{3}y)e^{\kappa_{4}y}e^{ik_{x}x}\chi_{\alpha}.$
(S-23)

The eigenvectors $\chi_{\alpha}$ satisfy
$\tau_{z}s_{x}\sigma_{y}\chi_{\alpha}=\chi_{\alpha}$. For
$\tau_{z}s_{x}\sigma_{y}$, the eigenvectors with the eigenvalue equal to $+1$
are denoted as

$\displaystyle\xi_{1}$
$\displaystyle=|\tau_{z}=+1\rangle\otimes|\sigma_{y}=-1\rangle\otimes|s_{x}=-1\rangle,$
(S-24) $\displaystyle\xi_{2}$
$\displaystyle=|\tau_{z}=+1\rangle\otimes|\sigma_{y}=+1\rangle\otimes|s_{x}=+1\rangle,$
$\displaystyle\xi_{3}$
$\displaystyle=|\tau_{z}=-1\rangle\otimes|\sigma_{y}=-1\rangle\otimes|s_{x}=+1\rangle,$
$\displaystyle\xi_{4}$
$\displaystyle=|\tau_{z}=-1\rangle\otimes|\sigma_{y}=+1\rangle\otimes|s_{x}=-1\rangle.$

We combine the eigenvectors $\xi_{i}$ to obtain $\chi_{i}$ with

$\displaystyle\chi_{1}$
$\displaystyle=(\xi_{3}+\xi_{4})/2=\frac{1}{\sqrt{2}}\left(0,0,0,0,i,0,0,1\right)^{T},$
(S-25) $\displaystyle\chi_{2}$
$\displaystyle=(\xi_{3}-\xi_{4})/2=\frac{1}{\sqrt{2}}\left(0,0,0,0,0,-i,1,0\right)^{T},$
$\displaystyle\chi_{3}$
$\displaystyle=(\xi_{1}+\xi_{2})/2=\frac{1}{\sqrt{2}}\left(-i,0,0,1,0,0,0,0\right)^{T},$
$\displaystyle\chi_{4}$
$\displaystyle=(\xi_{2}-\xi_{1})/2=\frac{1}{\sqrt{2}}\left(0,i,1,0,0,0,0,0\right)^{T},$

and the perturbation matrix elements read

$H_{\text{IV},\alpha\beta}=\int_{-\infty}^{0}dy\psi_{\alpha}^{*}(y)H^{\prime}(k_{x},-i\partial_{y})\psi_{\beta}(y).$
(S-26)

After some algebraic calculations, we obtain

$H_{\text{IV}}(k_{x})=A_{x}k_{x}\eta_{z}+M_{\text{IV}}\gamma_{x}\eta_{y}+J^{z}_{\rm
IV}\gamma_{z}\eta_{x}-J^{y}_{\rm IV}\gamma_{z}\eta_{z},$ (S-27)

where $\eta_{i}$ and $\tau_{j}$ are Pauli matrices in bases of $\psi_{\alpha}$
and

$\displaystyle J^{z}_{\text{IV}}$
$\displaystyle=-\frac{J_{0}^{z}}{2}\int^{0}_{-\infty}dy\psi_{\alpha}^{*}(y)(\partial_{y}^{2})\psi_{\alpha}(y)=-\frac{J_{0}^{z}m}{t_{y}},\quad
J^{y}_{\text{IV}}=-\frac{J_{0}^{z}}{2}\int^{0}_{-\infty}dy\psi_{\alpha}^{*}(y)(\partial_{y}^{2})\psi_{\alpha}(y)=-\frac{J_{0}^{y}m}{t_{y}},$
(S-28) $\displaystyle M_{\text{IV}}$
$\displaystyle=\Delta_{0}\int_{0}^{\infty}dy\psi_{\alpha}^{*}(y)\psi_{\alpha}(y)=\Delta_{0}.$

In summary, the effective Hamiltonian of the four edges are

$\displaystyle H_{\rm I}(k_{y})=A_{y}k_{y}\eta_{z}-M_{\rm
I}\gamma_{y}\eta_{y}+J^{z}_{\rm I}\gamma_{z}\eta_{x}-J^{x}_{\rm
I}\gamma_{z}\eta_{z},$ (S-29) $\displaystyle
H_{\text{II}}(k_{x})=-A_{x}k_{x}\eta_{z}-M_{\text{II}}\gamma_{x}\eta_{y}+J^{z}_{\rm
II}\gamma_{z}\eta_{x}-J^{y}_{\rm II}\gamma_{z}\eta_{z},$ $\displaystyle H_{\rm
III}(k_{y})=-A_{y}k_{y}\eta_{z}+M_{\rm III}\gamma_{y}\eta_{y}+J^{z}_{\rm
III}\gamma_{z}\eta_{x}+J^{x}_{\rm III}\gamma_{z}\eta_{z},$ $\displaystyle
H_{\text{IV}}(k_{x})=A_{x}k_{x}\eta_{z}+M_{\text{IV}}\gamma_{x}\eta_{y}+J^{z}_{\rm
IV}\gamma_{z}\eta_{x}+J^{y}_{\rm IV}\gamma_{z}\eta_{z}.$

From the equation (S-29) we obtain that only the $z$ component of Neél vector
contributes a Dirac mass for the helical edge states. This indicates that the
$z$ component plays a crucial role in determining the topological properties
of the system. Therefore, we focus on the $z$ component of the Neél vector and
examine its effects on the edge states

$\displaystyle H_{\rm I}(k_{y})=A_{y}k_{y}\eta_{z}-M_{\rm
I}\gamma_{y}\eta_{y}+J^{z}_{\rm I}\gamma_{z}\eta_{x},$ (S-30) $\displaystyle
H_{\text{II}}(k_{x})=-A_{x}k_{x}\eta_{z}-M_{\text{II}}\gamma_{x}\eta_{y}+J^{z}_{\rm
II}\gamma_{z}\eta_{x},$ $\displaystyle H_{\rm
III}(k_{y})=-A_{y}k_{y}\eta_{z}+M_{\rm III}\gamma_{y}\eta_{y}+J^{z}_{\rm
III}\gamma_{z}\eta_{x},$ $\displaystyle
H_{\text{IV}}(k_{x})=A_{x}k_{x}\eta_{z}+M_{\text{IV}}\gamma_{x}\eta_{y}+J^{z}_{\rm
IV}\gamma_{z}\eta_{x}.$

We introduce a unitary transformation

$U=e^{i\frac{\pi}{4}\gamma_{z}}\otimes\mathbf{1}_{2\times 2},$ (S-31)

where $\mathbf{I}_{2\times 2}$ is a 2D identity matrix. Under the unitary
transformation, the effective Hamiltonian for edge II and edge IV is
transformed to

$\displaystyle H_{\rm II}$ $\displaystyle=-A_{x}k_{x}\eta_{z}+M_{\rm
II}\gamma_{y}\eta_{y}+J^{z}_{\rm II}\gamma_{z}\eta_{x},$ (S-32) $\displaystyle
H_{\rm IV}$ $\displaystyle=A_{x}k_{x}\eta_{z}-M_{\rm
IV}\gamma_{y}\eta_{y}+J^{z}_{\rm IV}\gamma_{z}\eta_{x}.$

The effective Hamiltonian of the four edges read

$\displaystyle H_{\rm I}$
$\displaystyle=A_{y}k_{y}\eta_{z}-\Delta_{0}\gamma_{y}\eta_{y}+\frac{J^{z}_{0}m}{t_{x}}\gamma_{z}\eta_{x},$
(S-33) $\displaystyle H_{\rm II}$
$\displaystyle=-A_{x}k_{x}\eta_{z}+\Delta_{0}\gamma_{y}\eta_{y}-\frac{J^{z}_{0}m}{t_{y}}\gamma_{z}\eta_{x},$
$\displaystyle H_{\rm III}$
$\displaystyle=-A_{y}k_{y}\eta_{z}+\Delta_{0}\gamma_{y}\eta_{y}+\frac{J^{z}_{0}m}{t_{x}}\gamma_{z}\eta_{x},$
$\displaystyle H_{\rm IV}$
$\displaystyle=A_{x}k_{x}\eta_{z}-\Delta_{0}\gamma_{y}\eta_{y}-\frac{J^{z}_{0}m}{t_{y}}\gamma_{z}\eta_{x}.$

Taking the counterclockwise direction as positive, we can take an ’edge
coordinate’ $l$ as shown in Fig. 1(a), and the effective Hamiltonian can be
expressed as

$H_{\text{edge},l}=-iA(l)\eta_{z}\partial_{l}+M(l)\gamma_{y}\eta_{y}+J(l)\gamma_{z}\eta_{x},$
(S-34)

with the coefficients $A(l)=\\{A_{y},A_{x},A_{y},A_{x}\\}$,
$M(l)=\left\\{-\Delta_{0},\Delta_{0},\Delta_{0},-\Delta_{0}\right\\}$ and
$J(l)=\left\\{J_{0}^{z}m/t_{x},-J_{0}^{z}m/t_{y}\right.$,
$\left.J_{0}^{z}m/t_{x},-J_{0}^{z}m/t_{y}\right\\}$ for $l=\\{$I, II, III,
IV$\\}$, respectively.

Figure S2: (a) The unit vector $\bm{\hat{n}}$ of the Neél vector with the
polar angle $\theta$ and azimuthal $\varphi$ angles in the spherical
coordinates. (b) Phase diagram concerning the polar $\theta$ and azimuthal
$\varphi$ angles. Here the critical values $\theta_{c}$ exist for the polar
angle. (c) and (d) The variation of $\tilde{M}^{+}$ and $\tilde{M}^{-}$ with
the polar angle of the Neél vector, respectively. The Dirac mass terms
$\tilde{M}^{\pm}$ of the adjacent edges in the orange region have opposite
signs. Common parameters are:
$m_{0}=1.0,t_{x}=t_{y}=A_{x}=A_{y}=2.0,\Delta_{0}=J_{0}=0.5,\mu=0$.

The Hamiltonian $H_{\text{edge},l}$ can be decoupled as
$H_{\text{edge},l}=H_{\text{edge},l}^{+}\oplus H_{\text{edge},l}^{-}$ with the
help of $\Pi=\gamma_{x}\eta_{z}$, where
$H^{\pm}=\langle\phi_{i}^{\pm}|H_{\text{edge},l}|\phi^{\pm}_{j}\rangle$ and
$\phi^{\pm}_{i}$ is the eigenvectors of $\Pi$ with eigenvalue of $\pm 1$. When
the eigenvalue of $\Pi$ equal to $+1$ we obtain

$H_{\text{edge},l}^{+}=iA(l)\tilde{\eta}_{z}\partial_{l}+\tilde{M}^{+}(l)\tilde{\eta}_{x},$
(S-35)

where
$\tilde{M}^{+}(l=\text{I-IV})=\\{\Delta_{0}+J_{0}^{z}|m|/t_{x},-\Delta_{0}-J_{0}^{z}|m|/t_{y},-\Delta_{0}+J_{0}^{z}|m|/t_{x},\Delta_{0}-J_{0}^{z}|m|/t_{y}\\}$.
For the case of $-1$ we obtain

$H_{\text{edge},l}^{-}=iA(l)\tilde{\eta}_{z}\partial_{l}+\tilde{M}^{-}(l)\tilde{\eta}_{x},$
(S-36)

where
$\tilde{M}^{-}(l=\text{I-IV})=\\{-\Delta_{0}+J_{0}^{z}|m|/t_{x},\Delta_{0}-J_{0}^{z}|m|/t_{y},\Delta_{0}+J_{0}^{z}|m|/t_{x},-\Delta_{0}-J_{0}^{z}|m|/t_{y}\\}$.
When $J_{0}^{z}|m|/t_{0}>\Delta_{0}$, the signs of the Dirac masses of the
adjacent edges in $H_{\text{edge},l}^{\pm}$ are opposite (We first consider
the isotropic case with $t_{0}=t_{x}=t_{y}$.) and two domain walls are formed
around each corner. Although the $x$ and $y$ components of Neél vector do not
produce Dirac mass on the edge states, the Dirac mass generated from the $z$
component of Neél vector changes as the polar angle. The variation of Dirac
masses $\tilde{M}^{+}(l)$ and $\tilde{M}^{-}(l)$ with the polar angle is shown
in Figs. S2 (c) and (d). The yellow region indicates the signs of the Dirac
masses on the adjacent edges are opposite. Correspondingly, the phase diagram
in the plane of $(\theta,\varphi)$ is shown in Fig. S2(b). The yellow area
indicates the HOTSC phase while the light blue area indicates the absence of
MCMs in the system.

In the above, we have established that the in-plane component of the Neél
vector does not generate a Dirac mass on the edge states. Now, we focus solely
on exploring the effect of the $z$ component Neél vector with $\theta=0$. We
can obtain the edge Hamiltonian from the general expression in Eq. (S-29) with
only the $z$ component of Neél vector taken into account

$\displaystyle H_{\rm I}$
$\displaystyle=A_{y}k_{y}\eta_{z}-\Delta_{0}\gamma_{y}\eta_{y}+\frac{J^{z}_{0}m}{t_{x}}\gamma_{z}\eta_{x},$
(S-37) $\displaystyle H_{\rm II}$
$\displaystyle=-A_{x}k_{x}\eta_{z}+\Delta_{0}\gamma_{y}\eta_{y}-\frac{J^{z}_{0}m}{t_{y}}\gamma_{z}\eta_{x},$
$\displaystyle H_{\rm III}$
$\displaystyle=-A_{y}k_{y}\eta_{z}+\Delta_{0}\gamma_{y}\eta_{y}+\frac{J^{z}_{0}m}{t_{x}}\gamma_{z}\eta_{x},$
$\displaystyle H_{\rm IV}$
$\displaystyle=A_{x}k_{x}\eta_{z}-\Delta_{0}\gamma_{y}\eta_{y}-\frac{J^{z}_{0}m}{t_{y}}\gamma_{z}\eta_{x}.$

The edges spectrum read

$\displaystyle E_{\rm
I,III}(k_{y})=\pm\sqrt{(A_{y}k_{y})^{2}+(\mp\Delta_{0}\pm\frac{J_{0}^{z}m}{t_{x}})^{2}},$
(S-38) $\displaystyle E_{\rm
II,IV}(k_{x})=\pm\sqrt{(A_{x}k_{x})^{2}+(\pm\Delta_{0}\pm\frac{J_{0}^{z}m}{t_{y}})^{2}}.$

Taking the counterclockwise direction as positive, we can define an ’edge
coordinate’ $l$ as shown in Fig. 1(a), and the effective Hamiltonian can be
expressed as

$H_{\text{edge},l}=iA(l)\eta_{z}\partial_{l}+M(l)\gamma_{y}\eta_{y}+J(l)\gamma_{z}\eta_{x},$
(S-39)

with the coefficients $A(l)=\\{A_{y},A_{x},A_{y},A_{x}\\}$,
$M(l)=\left\\{-\Delta_{0},\Delta_{0},\Delta_{0},-\Delta_{0}\right\\}$ and
$J(l)=\left\\{J_{0}^{z}m/t_{x},-J_{0}^{z}m/t_{y}\right.$,
$\left.J_{0}^{z}m/t_{x},-J_{0}^{z}m/t_{y}\right\\}$ for $l=\\{$I, II, III,
IV$\\}$, respectively. The Hamiltonian $H_{\text{edge},l}$ can be decoupled as
$H_{\text{edge},l}=H_{\text{edge},l}^{+}\oplus H_{\text{edge},l}^{-}$ with the
help of $\Pi=\gamma_{x}\eta_{z}$. For $\gamma_{x}\eta_{z}=+1$ we obtain

$H_{\text{edge},l}^{+}=iA(l)\tilde{\eta}_{z}\partial_{l}+\tilde{M}^{+}(l)\tilde{\eta}_{x},$
(S-40)

where
$\tilde{M}^{+}(l=\text{I-IV})=\\{\Delta_{0}+J_{0}^{z}|m|/t_{x},-\Delta_{0}-J_{0}^{z}|m|/t_{y},-\Delta_{0}+J_{0}^{z}|m|/t_{x},\Delta_{0}-J_{0}^{z}|m|/t_{y}\\}$.
For $\gamma_{x}\eta_{z}=-1$ we obtain

$H_{\text{edge},l}^{-}=iA(l)\tilde{\eta}_{z}\partial_{l}+\tilde{M}^{-}(l)\tilde{\eta}_{x},$
(S-41)

where
$\tilde{M}^{-}(l=\text{I-IV})=\\{-\Delta_{0}+J_{0}^{z}|m|/t_{x},\Delta_{0}-J_{0}^{z}|m|/t_{y},\Delta_{0}+J_{0}^{z}|m|/t_{x},-\Delta_{0}-J_{0}^{z}|m|/t_{y}\\}$.
Without loss of generality, $J_{0}^{z}$, $\Delta_{0}$, $t_{x,y}$ and $A_{x,y}$
are taken as positive. The signs of Dirac masses of all the adjacent edges for
$H_{\text{edge},l}^{\pm}$ are opposite when $J_{0}^{z}|m|/t_{x}>\Delta_{0}$,
which results in a mass domain wall and gives rise to the two MCMs at each
corner [89]. Since the edge Hamiltonian $H^{+}$ and $H^{-}$ are decoupled from
each other with the chemical potential $\mu=0$, two MZMs can still exist at
each corner despite the time-reversal symmetry being broken.

###  I.2 Isosceles triangle geometry

In this subsection, we present the edge theory of the isosceles triangle
geometry (Fig. 1(b)). Specifically, for edges I and IV, the effective
Hamiltonian remains the same as that of the square geometry

$\displaystyle H_{\rm I}$
$\displaystyle=A_{y}k_{y}\eta_{z}-\Delta_{0}\gamma_{y}\eta_{y}+\frac{J^{z}_{0}m}{t_{x}}\gamma_{z}\eta_{x},$
(S-42) $\displaystyle H_{\rm IV}$
$\displaystyle=A_{x}k_{x}\eta_{z}-\Delta_{0}\gamma_{y}\eta_{y}-\frac{J^{z}_{0}m}{t_{y}}\gamma_{z}\eta_{x}.$

Concerning edge V, it should be noted that the Dirac mass term contribution
stemming from the $s$-wave electron pairing is $\Delta_{0}$. However, in the
case of altermagnets, the magnetic order spin splitting $J(\bm{k})$ becomes
zero in the $k_{x}=k_{y}$ direction. As a result, there is no contribution to
the Dirac mass term from the altermagnets on the hypotenuse V of the isosceles
right triangle [34]. Accordingly, we can obtain the effective Hamiltonian on
edge V as follows

$H_{\rm V}=A_{\rm V}k_{\rm V}\eta_{z}-\Delta_{0}\gamma_{y}\eta_{y}.$ (S-43)

Likewise, by utilizing $\Pi=\gamma_{x}\eta_{z}$, one can decouple the
Hamiltonian as $H=H^{+}\oplus H^{-}$. The corresponding mass terms associated
with each subblock are expressed as

$\displaystyle\tilde{M}^{+}(l=\text{I,V,IV})=\left\\{\Delta_{0}+\frac{J_{0}^{z}|m|}{t_{x}},\Delta_{0},\Delta_{0}-\frac{J_{0}^{z}|m|}{t_{y}}\right\\},$
(S-44)
$\displaystyle\tilde{M}^{-}(l=\text{I,V,IV})=\left\\{-\Delta_{0}+\frac{J_{0}^{z}|m|}{t_{x}},-\Delta_{0},-\Delta_{0}-\frac{J_{0}^{z}|m|}{t_{y}}\right\\}.$

The interplay between electron pairing $\Delta_{0}$ and altermagnetism
$J_{0}^{z}$ at the edge can result in the formation of MCMs, which will be
discussed later.

###  I.3 The effective boundary Hamiltonian for the heterostructures of
altermagnets with $g$-wave and $i$-wave spin splitting

For the heterostructures of altermagnets with the spin splitting of other
symmetries, including $g$-wave, and $i$-wave [54], the boundary Hamiltonian
can be obtained by rotating the coordinate system. Within the rotated system,
the boundary Hamiltonian reads as

$H(\alpha)=A_{0}k_{\parallel}\eta_{0}\gamma_{z}+\mathcal{L}(\alpha)\eta_{z}\gamma_{x}+\Delta_{0}\left(\sin\alpha\eta_{y}\gamma_{y}+\cos\alpha\eta_{x}\gamma_{y}\right),$
(S-45)

where

$\left\\{\begin{array}[]{c}\mathcal{L}(\alpha)=J_{\rm eff}\sin(4\alpha)\qquad
g-\text{wave},\\\ \mathcal{L}(\alpha)=J_{\rm eff}\sin(6\alpha)\qquad
i-\text{wave},\end{array}\right.$ (S-46)

and $\gamma_{i},\eta_{j}$ are Pauli matrices, $J_{\rm eff}$ is the effective
strength of the altermagnet on the boundary. In the Eq. (S-45), $\alpha$ is
the angle of rotation of the coordinate axis, which is the direction of the
tangential boundary, as shown in Fig. 1(c). The second and the third terms are
the Dirac mass terms from the altermagnet and $s$-wave electron pairing,
respectively. The edge spectra read

$E(\alpha)=\pm\sqrt{A_{0}^{2}k_{\parallel}^{2}+\left(\mathcal{L}(\alpha)\pm\Delta_{0}\right)^{2}}.$
(S-47)

The two mass terms compete with each other since they are commutative. This
competition leads to a topological phase transition on the boundary, which
creates a mass domain wall capable of binding MCMs in the direction of
$|\mathcal{L}(\alpha)|=\Delta_{0}$.

##  II Effect of chemical potential

In this section, we explore the impact of finite chemical potential on MCMs.
Specifically, in investigating the chemical potential $\mu\neq 0$, we adopt a
similar approach in that chemical potential is treated as a perturbation as
the one used in Section I. We obtain the edge Hamiltonian

$H_{\text{edge},l}=iA(l)\eta_{z}\partial_{l}+M(l)\gamma_{y}\eta_{y}+J(l)\gamma_{z}\eta_{x}-\mu(l)\gamma_{z},$
(S-48)

where the coefficients $A(l)=\\{A_{y},A_{x},A_{y},A_{x}\\}$,
$M(l)=\left\\{-\Delta_{0},\Delta_{0},\Delta_{0},-\Delta_{0}\right\\}$,
$J(l)=\left\\{J_{0}^{z}|m|/t_{x},-J_{0}^{z}|m|/t_{y}\right.$,
$\left.J_{0}^{z}|m|/t_{x},-J_{0}^{z}|m|/t_{y}\right\\}$ and $\mu(l)=\mu_{0}$
for $l=\\{$I, II, III, IV$\\}$, respectively.

For $\mu\neq 0$, there is no such $\Pi$ matrix that can decompose the edge
Hamiltonian in Eq. (S-48) into $H^{+}\oplus H^{-}$. This indicates that a non-
zero chemical potential brings about coupling between the two parts. Two MZMs
localized at the same position would mix and vanish.

Figure S3: Quasiparticle spectrum for a cylinder geometry under different
pairing strengths. For (a) $\Delta_{0}=0.1$. (b) $\Delta_{0}=0.5$. (c)
$\Delta_{0}=0.6$. (d) Bulk spectrum and edge spectrum for the open boundary
conditions along the $x$ direction with the critical electron pairing
$\Delta_{0}=0.5$. (e) The density plot displays the corner localized
probability distribution of the four MCMs. The insert shows real space
eigenvalues for $\Delta_{0}=0.6$. (f) Variation of real space spectrum with
pairing intensity $\Delta_{0}$ with the MCMs labeled in red. The common
parameters are: $m_{0}=1.0,t_{y}=A_{y}=1.0,t_{x}=A_{x}=2.0,J_{0}=0.5,\mu=0.1$.
Figure S4: Quasiparticle spectrum for a cylinder geometry under different
pairing strengths. For (a) $\Delta_{0}=0.3$. (b) $\Delta_{0}=0.5$. (c)
$\Delta_{0}=0.6$. (d) Bulk spectrum and edge spectrum for the open boundary
conditions along the $x$ direction with the critical electron pairing
$\Delta_{0}=0.5$. (e)The real space eigenvalues for $\Delta_{0}=0.6$. (f) The
density plot displays the corner localized probability distribution of the
four zero-energy MCMs. The common parameters are:
$m_{0}=1.0,t_{y}=A_{y}=2.0,t_{x}=A_{x}=1.0,J_{0}=0.5,\mu=0.1$.

Assuming $t_{x}>t_{y}$ and selecting a finite chemical potential, we observe
that as the electron pairing becomes larger, the gap of the edge states along
the $k_{y}$ direction is closed and reopened with the gap of the bulk state
and the gap of the edge states along the $k_{x}$ direction always open, as
depicted in Figs. S3(a)-(d), which indicates a higher-order topological phase
transition. In fact, according to Eq. (S-38), it can be concluded that once
anisotropy occurs ($t_{x}\neq t_{y}$), the edge states $E(k_{x(y)})=0$ in the
$x$ and $y$ directions have different critical $\Delta_{0}^{c}$. This
observation is also in line with the energy spectrum of the cylinder geometry
obtained via numerical calculations. A similar pattern is likewise observed in
the real space spectrum for the variation of electron pairing $\Delta_{0}$ (as
shown in Fig. S3(f)), where one MCM per corner localization takes place after
a higher-order topological phase transition, and these states subsequently are
split again as the electron pairing continues to increase.

For the case where $t_{x}<t_{y}$, we observe that as the strength of electron
pairing increases, a higher-order topological phase transition occurs but with
the gap of the edge states along the $k_{y}$ direction close and reopen, as
depicted in Figs. S4(a)-(d). The hallmark MZMs emerge at the corners of the
system, as shown in Figs. S4(e) and (f).

##  III Manipulation of MCMs

###  III.1 Square geometry

In this subsection, we will first manipulate the MCMs in square geometry by
using uniaxial stress. It is useful to define the corner position, corner i-j,
which is defined as the intersection of the adjacent edge i and edge j. From
the picture of mass terms in edge theory, we can know that the order between
$J_{0}^{z}|m|/t_{x}$, $J_{0}^{z}|m|/t_{y}$, and $\Delta_{0}$ is important. In
fact, there are six sequences, but only four are not equivalent, as shown in
the following four cases. The chemical potential will couples the $H^{+}$ and
$H^{-}$ subblocks. This coupling causes two MCMs from the two subblocks at the
same corner to mix and annihilate. However, such two MCMs are intact if they
are located at different corners.

Figure S5: Schematic diagram (a) shows the $H^{-}$ subblock generating MCMs at
corner III-IV and corner III-II. (b) shows $H^{+}$ subblock producing MCMs at
corner I-II and corner I-IV. (c) The MCMs generated by $H^{+}$ and $H^{-}$ are
spatially separated, so that stable MCMs can emerge after the non-zero
chemical potential couples $H^{+}$ and $H^{-}$.

  * •

Case (1): $\mu\neq 0,J_{0}^{z}|m|/t_{x}<\Delta_{0}<J_{0}^{z}|m|/t_{y}$.

Firstly, we consider the case where $t_{x}>t_{y}$ and $\mu\neq 0$. When the
electron pairing satisfies $J_{0}^{z}|m|/t_{x}<\Delta_{0}<J_{0}^{z}|m|/t_{y}$,
one can obtain the corresponding signs of the mass terms from Eq. (S-40) and
Eq. (S-41)

$\text{Sign}(\tilde{M}^{+})=\left\\{+,-,-,-\right\\},\qquad\text{Sign}(\tilde{M}^{-})=\left\\{-,-,+,-\right\\}.$
(S-49)

The Eq. (S-49) indicates that the MCMs caused by $H^{+}$ emerge at the corner
I-IV and corner I-II, as illustrated in Fig. S5(b). On the other hand, the
MCMs at corner III-IV and corner III-II are due to $H^{-}$, as depicted in
Fig. S5(a). While $H^{+}$ and $H^{-}$ are coupled in the presence of a nonzero
chemical potential, the final four MCMs are still stable since they are
located at distinct corners. Consequently, when
$J_{0}^{z}|m|/t_{x}<\Delta_{0}<J_{0}^{z}|m|/t_{y}$, one stable MZM is
localized at each corner, as shown in Fig. S5(c). This analysis is in
agreement with the numerical results presented in Section II.

Figure S6: Schematic diagram (a) shows $H^{-}$ subblock generating MCMs at
corner IV-I and corner IV-III. (b) shows $H^{+}$ subblock giving birth to MCMs
at corner II-IV and corner II-III. (c) The MCMs generated by $H^{+}$ and
$H^{-}$ are spatially separated, so that stable MCMs can emerge despite the
non-zero chemical potential.

  * •

Case (2) : $\mu\neq 0,J_{0}^{z}|m|/t_{y}<\Delta_{0}<J_{0}^{z}|m|/t_{x}$.

The values of $t_{x}$ and $t_{y}$ can be controlled experimentally through the
application of uniaxial stress. Now we consider
$J_{0}|m|/t_{y}<\Delta_{0}<J_{0}|m|/t_{x}$, the signs of Dirac masses of the
edge Hamiltonian $H^{\pm}$ are expressed as

$\text{Sign}(\tilde{M}^{+})=\left\\{+,-,+,+\right\\},\qquad\text{Sign}(\tilde{M}^{-})=\left\\{+,+,+,-\right\\}.$
(S-50)

One can obtain that $H^{+}$ leads to the formation of MCMs at corner II-I and
corner II-II, as shown in Fig. S6(b). In contrast, the MCMs induced by the
$H^{-}$ term appear at corner IV-I and corner IV-III, as illustrated in Fig.
S6(a). As previously noted, since the MCMs generated by the two blocks occupy
distinct corners, they do not overlap and can form stable MCMs even when the
two subspaces are coupled with $\mu\neq 0$.

Figure S7: Schematic diagram (a) shows $H^{-}$ subblock generating MCMs at
corner III-IV and corner I-II. (b) shows $H^{+}$ subblock bounding MCMs at the
same corners. (c) Since the MCMs generated by $H^{+}$ and $H^{-}$ are at the
same position, the non-zero chemical potential coupling will cause the MCMs to
mix and vanish.

  * •

Case (3) : $\mu\neq 0,J_{0}^{z}|m|/t_{x}\neq J_{0}^{z}|m|/t_{y}<\Delta_{0}$.

We now examine the scenario where the electron pairing satisfies
$J_{0}|m|/t_{x}\neq J_{0}|m|/t_{y}<\Delta_{0}$. The signs of Dirac masses on
the edges are expressed as

$\text{Sign}(\tilde{M}^{+})=\left\\{+,-,-,+\right\\},\qquad\text{Sign}(\tilde{M}^{-})=\left\\{-,+,+,-\right\\}.$
(S-51)

Equation (S-51) reveals that the MCMs generated by the two subblocks $H^{\pm}$
emerge at the same location, as illustrated in Fig. S7. Consequently, the
finite chemical potential causes $H^{+}$ and $H^{-}$ coupled, and the
otherwise pair of MCMs will mix and disappear.

Figure S8: (a) (b) The four MCMs at four corners from $H^{-}$ subblock and
$H^{+}$ subblock, respectively. Since the MCMs generated by $H^{+}$ and
$H^{-}$ are at the same position, the non-zero chemical potential coupling
will cause the MCMs to mix and vanish.

  * •

Case (4) : $\mu\neq 0,\Delta_{0}<J_{0}^{z}|m|/t_{x}\neq J_{0}^{z}|m|/t_{y}$.

Finally, we consider the case where $\Delta_{0}<J_{0}|m|/t_{x}\neq
J_{0}|m|/t_{y}$. The signs of the Dirac masses term can be expressed as

$\text{Sign}(\tilde{M}^{+})=\left\\{+,-,+,-\right\\},\qquad\text{Sign}(\tilde{M}^{-})=\left\\{+,-,+,-\right\\}.$
(S-52)

We find that the MCMs contributed by $H^{\pm}$ subblocks are located at the
same corners. The non-zero chemical potential coupling $H^{+}$ and $H^{-}$
makes them mix and vanish, as shown in Fig. S8.

Figure S9: Tunable patterns and interchange of MCMs. (a) When
$J_{0}^{z}|m|/t_{x}<\Delta_{0}<J_{0}^{z}|m|/t_{y}$, two MCMs (blue hollow
circles) emerge located at the left two corners from $H^{+}$, whereas two MCMs
(green solid circles) emerge located at the right two corners from $H^{-}$.
(b) When $J_{0}^{z}|m|/t_{x}<\Delta_{0}<J_{0}^{z}|m|/t_{y}$, two MCMs (blue
hollow circles) emerge located at the lower two corners from $H^{+}$, whereas
two MCMs (green solid circles) emerge located at the upper two corners from
$H^{-}$. The arrow indicates the uniaxial stress direction.

Interestingly, we find that it is possible to manipulate MZMs by adjusting the
relative magnitudes of $t_{x}$ and $t_{y}$ via uniaxial stress. As shown in
Figs. S9(a) and (b), the arrows indicate the direction of the uniaxial stress.
We can interchange a pair of MCMs by applying uniaxial stresses in different
directions.

###  III.2 Isosceles right triangle geometry

Similar to the above subsection, we investigate how MCMs evolve in an
isosceles right triangle geometry.

  * •

Case (1): $\mu\neq
0,0<\Delta_{0}<J_{0}^{z}|m|/t_{x},\Delta_{0}<J_{0}^{z}|m|/t_{y}$.

According to Eq. (S-44), the signs of the Dirac mass terms on the triangle
edge can be expressed as follows in the case

$\text{Sign}\left[\tilde{M}^{+}(l=\text{I,V,IV})\right]=\left\\{+,+,-\right\\},\qquad\text{Sign}\left[\tilde{M}^{-}(l=\text{I,V,IV})\right]=\left\\{+,-,-\right\\}.$
(S-53)

From Eq. (S-53) we obtain that the MCMs produced by $H^{+}$ appear at the
corner IV-V and corner I-IV, as depicted in Fig. S10(b). Similarly, for
$H^{-}$, the MCMs emerge at the corner I-IV and corner I-V, as illustrated in
Fig. S10(a).

Figure S10: Schematic diagram (a) shows $H^{-}$ subblock generating MCMs at
corner I-IV and corner I-V. (b) shows $H^{+}$ subblock producing MCMs at
corner IV-I and corner IV-V. (c) The non-zero chemical potential couples
$H^{+}$ and $H^{-}$, the MCMs in the same position are mixed and disappear.

When a finite chemical potential is present, $H^{+}$ and $H^{-}$ can couple
with each other, causing the MCMs located at the same spatial location to mix
and vanish, while MCMs at different locations can remain exist. Therefore,
only two MCMs can exist stably, as illustrated in Fig. S10(c), which is in
agreement with the numerical results presented in the main text.

  * •

Case (2): $\mu\neq 0,0<J_{0}^{z}|m|/t_{x}<J_{0}^{z}|m|/t_{y}<\Delta_{0}$.

The signs of the Dirac masses at the boundary can be expressed as follows in
the case

$\text{Sign}\left[\tilde{M}^{+}(l=\text{I,V,IV})\right]=\left\\{+,+,+\right\\},\qquad\text{Sign}\left[\tilde{M}^{-}(l=\text{I,V,IV})\right]=\left\\{-,-,-\right\\}.$
(S-54)

As can be obtained from Eq. (S-54), there is no domain wall, which means that
MCMs cannot emerge.

Figure S11: Schematic (a) indicates that the $H^{-}$ subblock does not
contribute to the MCMs. (b) shows $H^{+}$ subblock giving birth to MCMs at
corner IV-I and corner IV-V. (c) The non-zero chemical potential couples
$H^{+}$ and $H^{-}$. Figure S12:  Schematic (a) indicates that the $H^{-}$
subblock generating MCMs at corner V-IV and corner V-I. (b) shows that the
$H^{+}$ subblock does not generate MCMs. (c) The non-zero chemical potential
couples $H^{+}$ and $H^{-}$.

  * •

Case (3): $\mu\neq 0,J_{0}|m|/t_{x}<\Delta_{0}<J_{0}|m|/t_{y}$.

The signs of the Dirac masses on the edges in the case are expressed as

$\text{Sign}\left[\tilde{M}^{+}(l=\text{I,V,IV})\right]=\left\\{+,+,-\right\\},\qquad\text{Sign}\left[\tilde{M}^{-}(l=\text{I,V,IV})\right]=\left\\{-,-,-\right\\}.$
(S-55)

Upon analyzing the signs of the mass terms, one can observe that all domain
walls are generated by the $H^{+}$ subblock. The MCMs emerge and are stable at
corner IV-I and corner IV-V, as illustrated in Fig. S11.

  * •

Case (4): $\mu\neq 0,J_{0}|m|/t_{y}<\Delta_{0}<J_{0}|m|/t_{x}$.

In this scenario, the signs of the Dirac masses can be expressed as follows:

$\text{Sign}\left[\tilde{M}^{+}(l=\text{I,V,IV})\right]=\left\\{+,+,+\right\\},\qquad\text{Sign}\left[\tilde{M}^{-}(l=\text{I,V,IV})\right]=\left\\{+,-,-\right\\}.$
(S-56)

According to Eq. (S-56), the MCMs are generated solely by $H^{-}$, as
illustrated in Fig. S12. Hence, non-zero chemical potential does not work with
two stable MCMs finally.

In summary, the movement of a pair of MCMs in space is achieved by uniaxial
stress in the isosceles triangle structure.