# Topological superconductivity in a magnetic-texture coupled Josephson
junction

Ignacio Sardinero ID Department of Theoretical Condensed Matter Physics,
Condensed Matter Physics Center (IFIMAC) and Instituto Nicolás Cabrera,
Universidad Autónoma de Madrid, 28049 Madrid, Spain  Rubén Seoane Souto ID
Department of Theoretical Condensed Matter Physics, Condensed Matter Physics
Center (IFIMAC) and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid,
28049 Madrid, Spain  Division of Solid State Physics and NanoLund, Lund
University, S-221 00 Lund, Sweden  Center for Quantum Devices, Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark  Instituto de
Ciencia de Materiales de Madrid (ICMM), Consejo Superior de Investigaciones
Científicas (CSIC), Sor Juana Inés de la Cruz 3, 28049 Madrid, Spain.  Pablo
Burset ID Department of Theoretical Condensed Matter Physics, Condensed Matter
Physics Center (IFIMAC) and Instituto Nicolás Cabrera, Universidad Autónoma de
Madrid, 28049 Madrid, Spain

###### Abstract

Topological superconductors are appealing building blocks for robust and
reliable quantum information processing. Most platforms for engineering
topological superconductivity rely on a combination of superconductors,
materials with intrinsic strong spin-orbit coupling, and external magnetic
fields, detrimental for superconductivity. We propose a setup where a
conventional Josephson junction is linked via a magnetic-textured barrier.
Antiferromagnetic and ferromagnetic insulators with periodically arranged
domains are compatible with our proposal which does not require intrinsic
spin-orbit or external magnetic fields. We find that the topological phase
depends on the magnitude and period of the barrier magnetization. The
superconducting phase controls the topological transition, which could be
detected as a sharp suppression of the supercurrent across the junction.

Introduction.— Majorana bound states (MBSs) are charge neutral, zero-energy
quasiparticle excitations appearing at the boundaries of topological
superconductors [1, 2, 3, 4, 5, 6]. A pair of MBSs localized at the ends of a
one-dimensional topological superconductor encodes a nonlocal fermionic state
[7]. These states are robust against local perturbations and display non-
Abelian exchange properties [8, 9], making them attractive for fault-tolerant
quantum information processing [10]. The experimental realization of
topological MBSs requires the combination of superconductivity, helical
electrons usually created from spin-orbit coupling, and time-reversal breaking
from magnetism [11]. Over the last decade [12], several material platforms
have been explored based on topological insulators [13, 14], semiconductor
nanowires [15, 16, 17, 5], planar Josephson junctions [18, 19, 20, 21], chains
of magnetic adatoms [22, 23, 24], and, recently, ferromagnetic insulators
combined with other time-reversal symmetry breaking effects [25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38].

An alternative strategy has been recently implemented where the spin-orbit is
synthetically engineered using spatially-varying magnetic fields [39, 40, 41,
42, 43]. In proximitized one-dimensional (1D) systems, the spatial magnetic
modulation can be achieved by interactions [44, 45, 46], adatoms [47, 48, 49,
50, 51, 52, 53, 54, 55], or local magnets [56, 57, 58, 59, 60]. Planar setups
offer more sophisticated magnetic textures in proximity to superconductors
[61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75]. For example,
skyrmion textures on superconductors [76, 77, 78, 79] have been recently
measured [80, 81] and signatures consistent with MBSs where found in
proximitized magnetic monolayers [82, 83, 84]. Proximitized structures need to
be carefully engineered so that the competing magnetic and superconducting
orders coexist. This challenge could be circumvented if the magnetic texture
featuring synthetic spin-orbit interaction is coupled to superconducting
contacts in a Josephson setup [39]. Magnetic textures with spatial variation
across the junction, i.e., from contact to contact [41], have been implemented
[39]. However, for future braiding applications a higher degree of control
over the emerging MBSs could be achieved with textures along the junction
interface, see Fig. 1.

Figure 1: Josephson junction mediated by a magnetic-textured barrier. (a) Two
superconductors are linked by a magnetic-texture barrier with a local
magnetization that changes in space, schematically represented by arrows. (b)
Two possible magnetic texture profiles. (c) Localization of the ground state
wavefunction at the interface in the trivial (red) and topological (blue)
regimes.

Here, we explore such a configuration studying the topological properties of a
two-dimensional (2D) Josephson junction (JJ) coupled through a magnetic-
textured barrier [Fig. 1(a)] with a spatial modulation along the junction
interface [Fig. 1(b)]. We find that the system enters the topological phase
when the superconducting coherence length and the magnetization periodicity
are comparable. The topological regime, characterized by MBSs localized at the
edges of the JJFig. 1(c), can extend up to rather large occupations and is
very sensitive to the phase bias across the junction. Moreover, we show that
the formation and localization of pairs of MBSs has an observable effect on
the junction current-phase relation. The superconducting phase difference is a
substitute of the external magnetic field and helps reaching the topological
regime [18, 19, 85]. Recent experiments have already shown the possibility of
tuning subgap states varying the phase difference between planar Josephson
junctions [20, 21, 86, 87, 88, 89]. Our proposal is thus highly controllable,
does not require external magnetic fields, and bypasses the need for intrinsic
spin-orbit coupling and low-carrier densities.

Model and formalism.— We consider a JJ formed by two conventional singlet
$s$-wave superconductors joined by a magnetic-textured barrier, see Fig. 1. We
model the system using a square-lattice tight-binding Hamiltonian
$H=H_{L}+H_{R}+H_{t}$, with

$\displaystyle H_{L,R}={}$
$\displaystyle\sum_{\sigma=\uparrow,\downarrow}(-t\sum_{\langle
x,x^{\prime}\rangle}c_{i^{\prime}j^{\prime},\sigma}^{\dagger}c_{ij,\sigma}-\sum_{i,j}\mu_{ij}c_{ij,\sigma}^{\dagger}c_{ij,\sigma})$
$\displaystyle+\sum_{i,j}\Delta_{0}\mathrm{e}^{i\phi_{L,R}}c^{\dagger}_{ij,\uparrow}c^{\dagger}_{ij,\downarrow}+\mathrm{h.c.}\,,$
(1)

being the Hamiltonian of the superconducting leads, where $\langle
x,x^{\prime}\rangle$ stands for nearest-neighbors combinations of the
horizontal and vertical indices $i,i^{\prime}$ and $j,j^{\prime}$. The
operator $\hat{c}^{\dagger}_{ij,\sigma}$ ($\hat{c}_{ij,\sigma}$) creates
(annihilates) an electron with spin $\sigma$ on the lattice site $(i,j)$.
Here, $t$ is the nearest-neighbors hopping integral, $\mu_{ij}\equiv\mu$ the
uniform chemical potential of the lattice, $\Delta_{0}>0$ the superconducting
pairing amplitude, and $\phi_{L,R}$ the superconducting phases; we denote
their phase difference as $\phi=\phi_{R}-\phi_{L}$. We consider a finite size
lattice with $N_{x}$ and $N_{y}$ horizontal and vertical sites, respectively,
and only examine symmetric junctions where both superconductors have the same
gap, length $L=N_{x}a/2$ (with $a$ being the lattice distance constant), and
width $N_{y}a$.

The superconductors couple via a magnetic-textured barrier mediating tunneling
between them,

$H_{t}=-\frac{1}{2}\sum_{\sigma,\sigma^{\prime}}\sum_{j}t_{j,\sigma\sigma^{\prime}}\hat{c}^{\dagger}_{L,j,\sigma}\hat{c}_{L+1,j,\sigma^{\prime}}+\mathrm{h.c.}\,.$
(2)

We consider that the magnetization of the barrier has a spatial modulation,
given by the matrix in spin space

$\hat{t}_{j}=t_{0}\hat{\sigma}_{0}+\bm{t}_{j}\cdot\bm{\sigma}.$ (3)

The index $j$ runs along the width of the junction (Fig. 1) and $\bm{\sigma}$
is the vector of Pauli matrices $\hat{\sigma}_{0,1,2,3}$ in spin space. Here,
$t_{0}$ is the uniform amplitude for the spin-conserving hopping term and
$\bm{t}_{j}$ the spatially-varying spin-tunneling part.

Analytic 1D topological model.— To study the bulk topological properties of
the system, we consider a perfect harmonic spatial variation along the spin
$yz$-plane with period $\xi_{m}$ and constant magnitude $t_{\text{m}}$,

$\bm{t}_{j}=t_{m}\left[0,\sin{(2\pi ja/\xi_{m})},\cos{(2\pi
ja/\xi_{m})}\right].$ (4)

We reach a solvable model setting $N_{x}=2$, effectively reducing the system
to two superconducting linear chains coupled along the $y$-direction by Eq. 4,
and assuming $N_{y}\rightarrow\infty$, i.e., applying periodic boundary
conditions to go to the bulk limit. Then, we change into a rotating-frame
basis so that the magnetization orientation always falls along the $z$-axis
[56]. As a result, the two linear superconductors acquire an effective spin-
orbit coupling [44, 57, 56], and are described by the (left, right)
Hamiltonians

$\begin{split}\hat{h}_{L,R}={}&\left(-2t\cos(k_{y}a)-\mu-\dfrac{\pi^{2}}{\xi_{m}^{2}}\right)\hat{\sigma}_{0}\hat{\tau}_{3}\\\
&+\sin(k_{y}a)\dfrac{2\pi
i}{\xi_{m}}\hat{\sigma}_{2}\hat{\tau}_{3}+\Delta_{0}\mathrm{e}^{i\phi_{L,R}}\hat{\sigma}_{2}\hat{\tau}_{2},\end{split}$
(5)

coupled by the tunnel Hamiltonian

$\hat{h}_{t}=(t_{0}\hat{\sigma}_{0}+t_{m}\hat{\sigma}_{3})e^{i\phi/2}\hat{\tau}_{3},$
(6)

with $\hat{\tau}_{1,2,3}$ acting in Nambu (particle-hole) space.

The Hamiltonian of each linear chain $\hat{h}_{L,R}$ is particle-hole and
mirror symmetric, while $\hat{h}_{t}$ breaks time-reversal symmetry.
Therefore, the system is in BDI class [18, 90] and can be characterized by a
$\mathcal{W}\in\mathbb{Z}$ invariant [91], whose parity
$\mathcal{M}=(-1)^{\mathcal{W}}\in\mathbb{Z}_{2}$ can be determined after a
change to the so-called Majorana basis. Rewriting
$H{(k_{y})}=\hat{h}_{L}+\hat{h}_{R}+\hat{h}_{t}$ as a skew-symmetric matrix
$A$ is possible by a unitary transformation, and we thus reach

${}\mathcal{M}=\textrm{sgn}\left(\textrm{Pf}[A(0)]/\textrm{Pf}[A(\pi/a)]\right),$
(7)

with $\textrm{Pf}[A]$ referring to the Pfaffian of $A$. See Supplemental
Material (SM) [92] for more details.

(a)  Figure 2: Topological phase diagram. (a) Lowest 40 energy bands as a
function of the chemical potential for $\xi_{m}\approx 0.8\xi_{s}$ and
$t_{m}/t=0.75$. The coloured background represents the number of pairs of edge
states $\bar{\mathcal{W}}$. Figures (b-d) show maps of $\bar{\mathcal{Q}}$ at
$\mu/t=-3.8$; blue indicates the topologically nontrivial phase. (b,c)
$\bar{\mathcal{Q}}$ with $\xi_{m}=\xi_{s}$ as a function of $t_{0}$ and
$t_{m}$ for (b) $\phi=0$ and (c) $0.9\pi$. (d) $\bar{\mathcal{Q}}$ as a
function of $t_{m}$ and $\xi_{m}$ for $t_{0}/t=1$ and $\phi=0$. The
topological phase boundary for $0.9\pi$ is highlighted as a yellow line. In
all cases, $\Delta_{0}/t=0.2$ and $N_{x}\times N_{y}=16\times 128$.

Topological superconductivity.— Using the analytic model as a guide, we now
focus on the finite size system depicted in Fig. 1 and described by Eqs. 1 and
2. We first consider a harmonic variation of the magnetic texture with
constant amplitude $t_{m}$, Eq. 4, although our findings remain qualitatively
invariant for other periodic magnetization profiles [92]. The barrier locally
polarizes the tunneling electrons inducing an effective exchange field in the
two superconductors. Additionally, we account for a local exchange field close
to the barrier [92].

In the absence of magnetism ($t_{m}=0$), the system features a set of spin-
degenerate Andreev bound states inside the superconducting gap with an energy
that depends on the phase and the transmission of the channel [93]. A uniform
spin polarization along a fixed direction, $\mathbf{t}_{j}=t_{m}(0,0,1)$,
splits these subgap states in two different spin species, eventually closing
the superconducting gap. The system, however, is still in the trivial phase.
We describe the magnetic texture by introducing the spatial variation in Eq.
4, which mixes the two spin components facilitating the formation of equal-
spin triplet pairing close to the junction [94], to the second term of Eq. 3.
We now describe the conditions for the gap reopening indicating a transition
into the topological phase with localized MBSs at the edges of the JJ.

To characterize the topological phase of the finite system we define an
approximate topological invariant $\bar{\mathcal{W}}$, which we compute as the
number of MBSs pairs lying at $E\approx 0$ separated from the subsequent bulk
modes by an effective gap $\delta\leq\Delta_{0}$ [92]. The system features two
Majorana states ($\bar{\mathcal{W}}=1$) when the chemical potential of the 2D
superconductors is $\mu\sim-3.5t$. This regime is shown in Fig. 2(a)
highlighted by a blue background color. Other topological phases with
$\bar{\mathcal{W}}>1$ appear at higher fillings of the superconductors, shown
as a yellow background in Fig. 2(a). We thus focus below on fillings with only
one MBS pair.

We compute $\bar{\mathcal{Q}}=(-1)^{\bar{\mathcal{W}}}\delta/\Delta_{0}$ in
Fig. 2(b-d) to characterize the topological phase and show that the magnetic
tunneling $t_{m}$ required to enter the topological regime (blue regions) is
minimal close to $t\approx t_{0}$, where the junction’s normal transmission is
maximal. In the analytic 1D model, the boundary between topological phases for
$\phi=0$ [white regions in Fig. 2(b-d)] follows the simple condition [92]

$t_{m}^{2}=\left[\mu_{\rm eff}(\xi_{m})\pm t_{0}\right]^{2}+\Delta_{0}^{2},$
(8)

with $\mu_{\rm eff}(\xi_{m})=2t-\mu+\pi^{2}/\xi_{m}^{2}$ [92]. The minimum
parameters to reach the topological phase are thus $t_{m}=\Delta_{0}$ and
$|\mu_{\rm eff}|=t_{0}$, which qualitatively coincides with the minimum of the
blue region of Fig. 2(b).

The phase difference $\phi$ between superconductors provides another way of
controlling the topological phase transition. Tuning $\phi$ from $0$ to $\pi$
reduces the energy of the subgap states so that the phase transition occurs
for lower $t_{m}$ values, with a minimum located at $\phi=\pi$ (mod $2\pi$).
Figure 2(c) shows the phase diagram for $\phi=0.9\pi$ illustrating the
reduction of the minimal $t_{m}$ required to enter the topological phase with
respect to the $\phi=0$ case shown in Fig. 2(b). However, we note that now the
topological gaps are smaller (lighter blue in the topological region) than the
ones found for $\phi=0$. From the 1D model, the minimal required $t_{m}$ value
is given by

$t^{\rm
min}_{m}=\Delta_{0}\sqrt{1+\cos{\phi}+\Delta_{0}^{2}\dfrac{\sin^{2}{\phi}}{2\mu_{\rm{\rm
eff}}(\xi_{m})}},$ (9)

with the limits $t_{m}(\phi=0)=\Delta_{0}$ and $t_{m}(\phi=\pi)=0$ [92].
Comparing the minima in Fig. 2(b) and Fig. 2(c) shows that these results are
in qualitative agreement with the finite-size calculations.

The spatial periodicity of the magnetic texture, $\xi_{m}$, is another
important parameter for reaching the topological phases, see Fig. 2(d). The
local magnetization induced in the superconductors decays with a typical
length-scale given by the superconducting coherence length $\xi_{s}$ [95].
Therefore, for long magnetic periods, $\xi_{s}/\xi_{m}\ll 1$, electrons only
feel the exchange field locally induced by the barrier and the gap collapses
for sufficiently strong $t_{m}$ values. In the opposite regime of small
magnetic periods, $\xi_{s}/\xi_{m}\gg 1$, the barrier magnetization cannot
close the superconducting gap and the system remains in the trivial phase.
Consequently, a robust topological phase requires $\xi_{s}/\xi_{m}\sim 1$
where the topological gap is maximum while the $t_{m}$ required for the phase
transition is minimal [Fig. 2(d)].

Figure 3: Phase-controllable topological order. (a-c) Phase dependence of
energy bands for $t_{m}/\Delta_{0}=1.0$, $2.0$, and $2.5$ in the barrier with
harmonic texture. (d-f) Spatial distribution of the lowest-energy wavefunction
$|\psi_{0}^{\pm}|^{2}$ evaluated along the junction interface ($i=L$). (g-i)
Current-phase relation in the case of the clean (C), harmonic- (H) and domain-
textured (D) junctions. $I_{0}$ is the current maximum for the clean junction
with $t_{m}=0$ and $t_{0}=t$. Arrows indicate the kink at the phase
transition. In all cases, $\mu/t=-3.8$, $\xi_{m}=2\xi_{s}$, and
$\Delta_{0}/t=0.2$.

Phase-controlled topological order.— We now discuss in more detail the effect
of the phase difference between superconductors. As shown in Fig. 2, the
system can transition from the trivial to the topological regime when
increasing the phase difference. Therefore, a finite superconducting phase
difference reduces the energy of the subgap states, making it possible to
transition to the topological regime for smaller $t_{m}$ values. The
superconducting phase is thus a convenient parameter to control topology and,
consequently, the emergence and localization of MBSs. As an illustration, we
compare in Fig. 3 three different configurations: far into the trivial region
(left column); close to the boundary between topological phases from the
trivial region (center); and inside the nontrivial phase (right). In the first
case (left), the modulation of the subgap states with the phase is
insufficient to close the gap. The center column shows a gap closure and
reopening for a finite phase ($\phi\sim\pi/2$), so that a pair of topological
edge states appear when it reopens ($\phi>\pi/2$). Figures 3(d-f) display the
ground state wavefunctions, $|\psi_{0}^{\pm}|^{2}$, showing the formation of
localized edge modes in the topological regime. In the nontrivial case (right
column), the superconducting phase increases the topological gap in the region
away from $\phi=0$ and $\phi=\pi$. At $\phi=\pi$ the gap collapses and
Majoranas delocalize across the full system.

The results presented thus far correspond to a harmonically rotating magnetic
texture along the junction interface [dashed line in Fig. 1(b)]. A more
realistic texture is represented by the solid line in Fig. 1(b) featuring
anti-parallel domains with non-coplanar domain walls described by

$\bm{t}_{j}=t_{m}\lambda_{0}^{-1}\left[0,\tanh\left(\lambda\sin{y_{j}}\right),\tanh\left(\lambda\cos{y_{j}}\right)\right],$
(10)

with $y_{j}=2\pi ja/\xi_{m}$, $\lambda_{0}=\tanh(\lambda)$ and the parameter
$\lambda$ controlling the length of the magnetic domains [92]. Using this
domain magnetization model with the same parameters of Fig. 2 we reproduce the
positions of the crossings in Fig. 3 and the overall structure of the map is
preserved, see Ref. [92]. Consequently, we conclude that the details of the
magnetic texture do not qualitatively affect the topological phase, as long as
$\xi_{s}/\xi_{m}\sim 1$.

Finally, we compute the Josephson current at zero temperature, i.e.,
$I\propto\sum_{\epsilon_{i}<0}\partial_{\phi}\epsilon_{i}(\phi)$, for the
negative eigenvalues $\epsilon_{i}$ of the full Hamiltonian [92]. We show the
current-phase relation (CPR) for the three cases commented above in Fig.
3(g-i). Interestingly, the phase-induced topological transition as a function
of the phase appears as a kink on the CPR, see arrows in Fig. 3(h). This kink
is produced by an avoided crossing of the remaining trivial Andreev bound
states resulting from the topological protection of the newly formed gap, and
it is more pronounced for the domain-texture (red dashed lines) than for the
harmonic case.

Role of the magnetization profile.— The domain-texture model, Eq. 10, allows
us to systematically introduce disorder by randomizing the domain size
($\xi_{m}$), domain magnetization ($t_{m}$), or domain wall helicity. Weak
disorder in $\xi_{m}$ and $t_{m}$ does not disrupt the topological phase
because these quantities preserve the system symmetry [92]. Indeed, disorder
on $\xi_{m}$ makes the topological gap size variable along the interface,
since it is controlled by the magnitude of the effective spin-orbit coupling;
while random domain magnetization values broaden the value of $\phi$ needed
for the phase transition, thus making the avoided crossing in Fig. 3 wider and
the kink less pronounced. Conversely, changes in domain wall helicity alter
the sign of the effective spin-orbit coupling along the wire, leading to
topological phase boundaries where different highly overlapping subgap modes
can enter the topological gap [92].

Conclusions.— In this work, we have shown the onset of topological
superconductivity in a Josephson junction mediated by a spin-textured barrier.
Using a two-dimensional tight-binding model, we identified the conditions for
the emergence of topological Majorana zero modes at the edges of the system.
The presence of the magnetic texture eliminates the need for spin-orbit
coupling or external magnetic fields, while the phase bias across the junction
provides control over the topological phase. We support these results
computing the topological invariant in an analytical 1D model of the junction.
Additionally, we investigate the impact of disorder on the topological phase
transition, finding that only sign changes of the magnetic texture’s helicity
introduce trivial states within the topological gap.

Our work proposes a platform for topological superconductors that do not rely
on intrinsic spin-orbit coupling and external magnetic fields. Materials
including antiferromagnetic insulators with a small out-of-plane magnetization
or ferromagnetic insulators with ordered domains are suitable candidates for
magnetic-textured barriers [96, *Bell_2003_PRB, 97, 98, 99, 100, 101, 102].
The possible detrimental effect from stray fields could be reduced by choosing
a texture with fields pointing away from the superconductors.

###### Acknowledgements.

We thank E.J.H. Lee and A. Levy Yeyati for insightful discussions. I.S. and
P.B. acknowledge support from the Spanish CM “Talento Program” project No.
2019-T1/IND-14088 and the Agencia Estatal de Investigación projects No.
PID2020-117992GA-I00 and No. CNS2022-135950. R.S.S. acknowledges funding from
the Spanish CM “Talento Program” project No. 2022-T1/IND-24070, the European
Union’s Horizon 2020 research and innovation program under the Marie
Sklodowska-Curie Grant Agreement No. 10103324, the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme under Grant Agreement No. 856526 and Nanolund.

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Supplemental material to “Topological superconductivity in a magnetic-texture
coupled Josephson junction”

This Supplementary Material is organized as follows. We introduce the one-
dimensional model and study its topology in Section I. In Section II we
present the domain magnetic texture model and its generalization to include
disorder effects. Next, we define in Section III the approximate topological
invariant valid for the finite-size systems explored in the main text. Section
IV is devoted to study disorder effects on all relevant parameters of the
domain magnetic texture. We then analyze the impact of the band filling on the
topological phase in Section V. Finally, we define the Josephson current in
Section VII.

##  I Analytic 1D topological model

We develop a 1D model that can be solved exactly to obtain analytic insights
on our main results. The model consists of two infinite 1D superconductors
along the $y$ direction, L and R, coupled via a harmonic spin-textured barrier
$V_{LR}(y)$. It is represented by the Hamiltonian

$\check{H}=\begin{pmatrix}H_{L}&V_{LR}(y)\\\
V_{LR}^{\dagger}(y)&H_{R}\end{pmatrix},$ (S 1)

which spans in Nambu, spin, and L-R spaces. Each superconductor is described
as

$H_{L,R}=-\left[\mu+2t\cos(k_{y}a)\right]\hat{\sigma}_{0}\hat{\tau}_{3}+\Delta_{L,R}\mathrm{e}^{i\phi_{L,R}}\hat{\sigma}_{2}\hat{\tau}_{2},$
(S 2)

where the Pauli matrices $\hat{\sigma}_{0,1,2,3}$ and $\hat{\tau}_{0,1,2,3}$
respectively act in spin and Nambu spaces, $\mu$ is the uniform on-site
potential, $t$ the nearest-neighbours hopping, and
$\Delta_{L,R}\mathrm{e}^{i\phi_{L,R}}$ the superconducting pair potential. We
impose that the superconductors are equal,
$\Delta_{L}=\Delta_{R}\equiv\Delta_{0}>0$, and denote $\phi=\phi_{R}-\phi_{L}$
their phase difference. We consider the system to be translation invariant
and, therefore, treat the problem in momentum space with conserved wavenumber
$k_{y}$. The tunneling between the two superconductors is given by

$V_{LR}(y)=\begin{pmatrix}-t_{0}-t_{m}\cos\theta(y)&-t_{m}\sin\theta(y)\\\
t_{m}\sin\theta(y)&-t_{0}+t_{m}\cos\theta(y)\end{pmatrix}\hat{\tau}_{3},\quad\theta(y)=\dfrac{2\pi}{\xi_{m}}y,$
(S 3)

where $t_{m}$ and $\xi_{m}$ are the amplitude and the spatial period of the
magnetic texture, respectively, and $t_{0}$ a spin-independent hopping.

To simplify the Hamiltonian, we locally rotate the electron spin basis so that
it always points in the $z$ direction using the unitary transformation

$U=\left[\cos(\theta/2)\hat{\sigma}_{0}-i\sin(\theta/2)\hat{\sigma}_{2}\right]\hat{\tau}_{0}.$
(S 4)

The rotated Hamiltonian $\tilde{H}=U^{\dagger}HU$ acquires two extra terms
coming from the transformation

$\displaystyle
U^{\dagger}\partial_{y}^{2}U-\partial_{y}^{2}=U^{\dagger}U^{\prime\prime}\partial_{y}^{2}+2U^{\dagger}U^{\prime}\partial_{y},$
(S 5)

with $U^{\prime}=\partial_{y}U$. Substituting
$\partial_{y}^{2}\rightarrow-2t\cos(k_{y}a)$ and
$\partial_{y}\rightarrow-2t\sin(k_{y}a)$ we find

$\tilde{H}_{L,R}=H_{L,R}-\dfrac{\pi^{2}}{2\xi_{m}^{2}}\hat{\sigma}_{0}\hat{\tau}_{3}-\dfrac{2\pi}{\xi_{m}}\sin(k_{y}a)i\hat{\sigma}_{2}\hat{\tau}_{3},$
(S 6)

giving rise to an effective spin-orbit $\alpha_{\rm eff}\equiv\pi/(2\xi_{m})$
and an effective chemical potential $\mu_{\rm
eff}\equiv\mu+\pi^{2}/(2\xi_{m}^{2})$. Now, the Zeeman term is aligned along
the spin $z$ axis and the tunneling term reads as

$\tilde{V}_{LR}(y)=U^{\dagger}{V}_{LR}U=-\left(t_{0}\hat{\sigma}_{0}+t_{m}\hat{\sigma}_{3}\right)\hat{\tau}_{3}.$
(S 7)

###  I.1 Topological classification and topological invariant

We can now classify the Hamiltonian in Eq. S 1 according to its symmetries.
The superconducting terms impose particle-hole symmetry (PHS) but the magnetic
texture breaks time-reversal symmetry (TRS). Therefore, one could in principle
assume that the system has no chiral symmetry and would belong in symmetry
class D [103]. However, aside from the three non-spatial symmetries (TRS, PHS,
chiral), we can also consider a mirror symmetry with respect to the interface.
A combination of time-reversal and mirror symmetries,
$\tilde{\mathcal{T}}=\mathcal{T}\mathcal{M}$, together with PHS, realizes the
chiral symmetry [90]. As a result, Eq. S 1 belongs in the BDI class,
explaining the appearance of phases with more than one MBS per side (yellow
regions of Fig. 2 in the main text).

Since the system is two-dimensional and belongs in class BDI, it is possible
to take it to an off-diagonal form in the chiral basis and use the winding
number $\mathcal{W}$ as topological invariant [104]. To do this, we consider
the three symmetries that the system has, and how each term in the Hamiltonian
transforms under each of them, and write down a possible matrix representation
for each symmetry. The global chiral symmetry can then be written as

$\check{C}=-\hat{\tau}_{2}\hat{u}_{1}\hat{\sigma}_{3},$ (S 8)

with Pauli matrices $\hat{u}_{0,1,2,3}$ acting in the left-right space. As a
result, the Hamiltonian in this basis is

$\check{C}\check{H}(k_{y})\check{C}^{\dagger}=\begin{pmatrix}0&A(k_{y})\\\
A^{T}(-k_{y})&0\end{pmatrix},$ (S 9)

with

$A(k)=\begin{pmatrix}\mu_{\rm eff}+2t\cos{k_{y}}&-\alpha_{\rm
eff}\sin{k_{y}}&t_{0}-t_{m}&-i\Delta_{0}e^{i\phi/2}\\\ \alpha_{\rm
eff}\sin{k_{y}}&\mu_{\rm
eff}+2t\cos{k_{y}}&-i\Delta_{0}e^{i\phi/2}&t_{0}+t_{m}\\\
t_{0}-t_{m}&-i\Delta_{0}e^{-i\phi/2}&\mu_{\rm eff}+2t\cos{k_{y}}&-\alpha_{\rm
eff}\sin{k_{y}}\\\ -i\Delta_{0}e^{-i\phi/2}&t_{0}+t_{m}&\alpha_{\rm
eff}\sin{k_{y}}&\mu_{\rm eff}+2t\cos{k_{y}}\end{pmatrix}.$ (S 10)

The parity of the winding gives the onset of even or odd number of uncoupled
MBSs at each edge of the system. Therefore, the invariant we are interested in
is given by [91, 105]

$\mathcal{M}=\textrm{sgn}\left\\{\dfrac{\textrm{Pf}\left[\check{H}_{sk}(k_{y}=0)\right]}{\textrm{Pf}\left[\check{H}_{sk}(k_{y}=\pi/a)\right]}\right\\}=(-1)^{\mathcal{W}},$
(S 11)

where $\textrm{Pf}\left[\check{H}_{sk}(k_{y})\right]$ stands for the Pfaffian
of $\check{H}_{sk}$, which is the Hamiltonian rotated into a skew-symmetric
form $\check{H}_{sk}^{T}=-\check{H}_{sk}$; namely
$\check{H}_{sk}(k_{y})=U\check{H}(k_{y})U^{\dagger}$, with

$U=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
-i&i\end{pmatrix}\hat{\sigma}_{0}\hat{u}_{0}.$ (S 12)

The topological invariant can thus be recast as

$\mathcal{M}=\textrm{sgn}(c_{0}/c_{\pi})=\textrm{sgn}(c_{0}c_{\pi}),$ (S 13)

with

$c_{0,\pi}=\Delta_{0}^{4}+2\Delta_{0}^{2}(\mu_{\rm
eff}^{0,\pi})^{2}+(t_{0}-t_{m})(t_{0}+t_{m})\cos\phi+\prod\limits_{\begin{subarray}{c}s_{0}=\pm
1\\\ s_{m}=\pm 1\end{subarray}}(\mu_{\rm eff}^{0,\pi}+s_{0}t_{0}+s_{m}t_{m}),$
(S 14)

and

$\mu_{\rm eff}^{0,\pi}=\mu+\pi^{2}/\xi_{m}^{2}\pm 2t.$

###  I.2 Boundaries of the topological phase

Equation S 13 has sign changes representing the system’s topological phase
transitions every time the numerator or the denominator go through a zero.
Setting $\mathcal{M}=0$ we get eight solutions $\pm t_{m}^{0,\pi;\pm}$ with

$\left(t_{m}^{0,\pi;\pm}\right)^{2}=t_{0}^{2}+\mu_{\rm
eff}^{0,\pi}+\Delta_{0}^{2}\cos\phi\pm\sqrt{2\mu_{\rm
eff}^{0,\pi}\left[2t_{0}^{2}-\Delta_{0}^{2}\left(1-\cos\phi\right)\right]-\Delta_{0}^{4}\sin^{2}\phi}.$
(S 15)

Equation S 15 reduces to a very simple condition for $\phi=0$ indicating the
transition into a topological phase:

$t_{m}^{2}>\left[\pm 2t-\mu_{\rm eff}\pm t_{0}\right]^{2}+\Delta_{0}^{2}.$ (S
16)

Equation S 16 describes two circles in the $t-\Delta_{0}$ plane each one with
radius $t_{m}$ and centered around $2|t|=\pm\mu_{\rm eff}-t_{0}$. The system
is in the topological phase for parameters inside the two circles where
$t_{m}$ dominates and Eq. (S 16) is satisfied. Outside, the system is in the
trivial regime. The two circles intersect when $t_{m}>\mu_{\rm eff}$, where
there are topological phases with higher winding number hosting more than a
Majorana pair. The superconducting phase difference $\phi$ enters Eq. S 14 in
the term $(t_{m}^{2}-t_{0}^{2})\cos(\phi)$. Therefore, a phase $\phi=\pi$
exchanges the role of $t_{m}$ and $t_{0}$ in Eq. S 16, while for $\phi=\pi/2$
they play the same role.

###  I.3 Role of the superconducting phase difference

From Eq. S 15 we can deduce the minimal $t_{m}>0$ solution marking the
transition to the topologically nontrivial $\mathcal{W}=1$ phase in which we
are interested. The critical values of $t_{0}$ that minimize that solution are

$t_{0}^{\rm{min}}=\sqrt{\mu_{\text{eff}}^{2}+\Delta_{0}^{2}\sin^{2}\frac{\phi}{2}+\dfrac{\Delta_{0}^{4}}{4\mu_{\text{eff}}^{2}}\sin^{2}\phi}\,.$
(S 17)

Equation S 17 indicates the horizontal position of the minimum of the
parabolic nontrivial region (blue area) in the left panels of Fig. S 1, which
is in good correspondence with the full 2D calculation, see middle panels.

Substituting Eq. S 17 into Eq. S 15 we obtain the corresponding minimal
magnetic tunneling:

$t_{m}^{\text{min}}=\Delta_{0}\sqrt{\cos^{2}\frac{\phi}{2}+\left(\dfrac{\Delta_{0}}{2|\mu_{\text{eff}}|}\right)^{2}\sin^{2}\frac{\phi}{2}}\,.$
(S 18)

The phase difference can thus reduce the minimum $t_{m}$ required to enter the
topological phase for a given value of $t_{0}$. This effect explains the
expansion of the topological phase between Fig. 2(b) for $\phi=0$ and Fig.
2(c) with $\phi\lesssim\pi$ in the main text. For $t_{m}/t\gtrsim 1$ the
superconducting phase does not significantly change the topological boundary,
see, e.g., in Fig. 2(d) of the main text the comparison between the numerical
finite-size calculation (color map) and the analytical result represented by a
yellow line.

Consequently, the analytical model qualitatively reproduces the behavior at
low band fillings of the finite-size numerical calculation described in Fig. 2
of the main text, see also Fig. S 1 below. The finite-size effects only
introduce small variations in the minimal $t_{0}$ and $t_{m}$ values where the
topological transition occurs.

##  II Domain magnetic texture model

The domain model depicted in Fig. 1(b) of the main text consists of an
alternating series of regions with parallel magnetization (domains), like in
an antiferromagnetic texture, separated by smaller regions where the
magnetization rotates from one orientation to its opposite (domain walls). We
again assume a magnetization with constant magnitude $t_{m}$ that rotates in
the $yz$-plane with period $\xi_{m}$, like in Eq. 4 of the main text. Each
period is formed by two domains involving $d$ sites each and two domain walls
with $w$ sites each, that is, $\xi_{m}=2a(d+w)$. The two domains in a period
feature anti-parallel magnetization along the $z$ spin axis, $\pm
t_{m}\hat{z}$, and the magnetization in the domain walls rotates from
$+t_{m}\hat{z}$ to $-t_{m}\hat{z}$. The domain-model magnetic texture can then
be compactly described by [cf. Eq. 10 of the main text]

$\mathbf{t}_{j}=\frac{t_{m}}{\tanh\lambda}\left(0,\tanh\left[\lambda\sin\left(\frac{2\pi
ja}{\xi_{m}}\right)\right],\tanh\left[\lambda\cos\left(\frac{2\pi
ja}{\xi_{m}}\right)\right]\right),$ (S 19)

with the parameter $\lambda$ controlling the length of the magnetic domains
$d$ and domain walls $w$.

We further discretize the model in Eqs. 10 and S 19 to gain more control over
its parameters. To do so, we divide all the indices along the junction into
domains and domain walls. The domains are regions where the magnetization is
constant, and the domain walls as smaller regions ($w\leq d$ in what follows)
where the magnetization rotates speedily. The domain walls thus represent a
region where the exchange interaction between domains with opposite
magnetization forces the spins to rotate between those orientations.

In a junction of width $N_{y}a$ there are $n=\lfloor N_{y}a/\xi_{m}\rfloor$
full periods, with $\lfloor x\rfloor$ being the integer floor of the real
number $x$. There are thus at least $2n$ domains and domain walls on each
width. For simplicity, we assume that the first interface site
$(i,j)=(N_{x}/2,1)$ always starts a domain so that sites belonging to the
$k$-th domain are in the interval

$D_{k}=\left[1+(k-1)(d+w),d+(k-1)(d+w)\right],$ (S 20)

and those in the $k$-th domain wall belong to

$W_{k}=\left[1+d+(k-1)(d+w),k(d+w)\right].$ (S 21)

The magnetization of the domain model is thus given by

$\bm{t}_{j}=\begin{cases}(-1)^{k-1}t_{m}(0,0,1),&j\in D_{k}\\\
t_{m}\left[0,\sin\left(x_{k}+j_{w}\right),\cos\left(x_{k}+j_{w}\right)\right],&j\in
W_{k}\end{cases},$ (S 22)

with

$x_{k}=\frac{1+(-1)^{k}}{2}\pi,\quad\text{and}\quad
j_{w}=\frac{j-kd+(k-1)w}{w+1}\pi.$ (S 23)

We use the discretized model in Eq. S 22 to compute the supercurrent in Fig.
3(g-i) of the main text and the phase diagram in Fig. S 1 below.

###  II.1 Generalized domain model for disorder effects.

To introduce disorder effects we can further generalize the domain model in
Eq. S 22 by allowing each domain and domain wall to have a different size.
Sites belonging to the $k$-th domain of length $d_{k}$ are then in the
interval

$D_{k}=[1+\sum_{l=1}^{k-1}(d_{l}+w_{l}),d_{k}+\sum_{l=1}^{k-1}(d_{l}+w_{l})],$
(S 24)

and those in the $k$-th domain wall of length $w_{k}$ are inside

$W_{k}=[1+d_{k}+\sum_{l=1}^{k-1}(d_{l}+w_{l}),\sum_{l=1}^{k}(d_{l}+w_{l})].$
(S 25)

In the following, we do not randomize the length of domain walls, $w_{k}$,
assuming that in physical setups the exchange interaction that rotates the
spins between neighboring domains is fixed as long as the domains are
sufficiently large.

The magnitude of the magnetization in each domain can also be different and
we, therefore, write

$\bm{t}_{j}=\begin{cases}t_{m}^{k}(0,0,1),&j\in D_{k}\\\
\mathbf{t}_{m,W}^{k}(j),&j\in W_{k}\\\ \end{cases},$ (S 26)

where $t_{m}^{k}$ is the magnetic tunnelling through the $k$-th domain. The
magnitude of the $k$-th domain wall tunneling, $\mathbf{t}_{m,W}^{k}$, must
now vary spatially to adjust to the different magnitudes of the neighboring
domains $k$ and $k+1$. We thus have

$\mathbf{t}_{m,W}^{k}(j)=\left(t_{m}^{k}\frac{j_{e}^{k}-j+1/2}{j_{e}^{k}-j_{s}+1}+t_{m}^{k+1}\frac{j-j_{s}^{k}+1/2}{j_{e}^{k}-j_{s}^{k}+1}\right)\left[0,s_{r}^{k}\sin(x_{k}),\cos(x_{k})\right].$
(S 27)

Here, $j_{s}^{k}=1+d_{k}+\sum_{l=1}^{k-1}(d_{l}+w_{l})$ is the index of the
starting site in the $k$-th domain wall and
$j_{e}^{k}=\sum_{l=1}^{k}(d_{l}+w_{l})$ is that of the ending site. We
introduced the parameter $s_{r}^{k}$ to control the “sign of the rotation”.
For example, in the harmonic model it is set to $1$.

According to these expressions:

  * •

A perfectly antiferromagnetic order of domains is imposed by
$\textrm{sgn}(t_{m}^{k})=-\textrm{sgn}(t_{m}^{k-1})$.

  * •

A “domain flip” disorder is introduced using $\textrm{sgn}({t_{m}^{k}})=n_{\rm
df}^{k}\textrm{sgn}({t_{m}^{k-1}})-(1-n_{\rm
df}^{k})\textrm{sgn}({t_{m}^{k-1}})$, where at the $k$-th domain there is a
chance $\sigma_{\rm df}$ that $n_{\rm df}^{k}=1$ (otherwise 0), in which case
the domains $k$ and $k+1$ are ferromagnetically aligned.

  * •

“Domain wall flip” disorder is set by choosing $s_{r}^{k}$ from a uniform
random distribution in the interval $(-\sigma_{\rm dwf},1-\sigma_{\rm dwf})$.
Therefore, at the $k$-th domain wall the “sign of the rotation” is allowed to
change with probability $\sigma_{\rm dwf}$.

  * •

Disorder in the magnetic amplitude, the domain size, or the electrostatic
potential are introduced by randomly choosing the values of $|t_{m}^{k}|$,
$d_{k}$, or $\mu_{i,j}$ from a Gaussian distribution of mean $\bar{t}_{m}$,
$\bar{d}$, or $\bar{\mu}$ and standard deviation $\sigma_{t_{m}}$,
$\sigma_{\xi_{m}}$, or $\sigma_{\mu}$, respectively.

(a)  Figure S 1: Topological phase diagram as a function of $t_{0}$ and
$t_{m}$ for $\phi=0$ (top panels) and $\phi=0.9\pi$ (bottom panels). The left
column shows the winding number $\mathcal{W}$ for the 1D model with
$\mu_{\text{eff}}/t=-1.85$. In the central column we compute the parity
$\bar{\mathcal{Q}}$ of the approximate topological invariant for a finite size
system with a harmonic texture and parameters $N_{x}=16$, $N_{y}=256$, and
$\mu/t=-3.85$. In the right column we show the domain texture with $N_{x}=32$,
$N_{y}=512$, $w=1$, $d=2$, and $\mu/t=-3.35$.

##  III Approximate topological invariant

The topological invariant, Eq. S 11, is only well defined for infinite
systems. By analogy with the analytical model, Fig. S 1, we now define an
approximate topological invariant to help us distinguish the presence of zero-
energy edge modes in the finite-size system.

We first compute the eigenvalues $\epsilon_{n}$ satisfying
$H\ket{\psi^{\pm}}=\pm\epsilon_{n}\ket{\psi^{\pm}}$ for the eigenvectors
$\ket{\psi^{\pm}}$ of the Hamiltonian $H=H_{L}+H_{t}+H_{R}$ in the main text
(see Eqs. 1 and 2 in the main text). In our description, $\bar{\mathcal{W}}$
quantifies the number of localized MBSs pairs,
$\\{\pm\epsilon_{1},\pm\epsilon_{2},...,\pm\epsilon_{\bar{\mathcal{W}}}\\}$,
lying at $E\approx 0$ well separated from the bulk of states. Consequently,
$\bar{\mathcal{W}}$ resembles the winding number characterizing the
topological phase for an infinite size system.

The parity of this pseudo-winding number,
$\bar{\mathcal{M}}=(-1)^{\bar{\mathcal{W}}}$, determines the even or odd
number of pairs of MBSs at the edges of the system, while the size of the
topological gap is simply $\epsilon_{\bar{\mathcal{W}}+1}$. In the simplest
scenario, with a single eigenstate close to zero energy, we have

$\bar{\mathcal{M}}=(-1)^{\bar{\mathcal{W}}}=\begin{cases}1,&\epsilon_{2}/2<\epsilon_{1}\quad\text{(trivial)}\\\
-1,&\epsilon_{2}/2>\epsilon_{1}\quad\text{(nontrivial)}\end{cases},$ (S 28)

while the gap is given by

$\delta=\begin{cases}\epsilon_{1},&\epsilon_{2}/2<\epsilon_{1}\quad\text{(trivial
gap)}\\\ \epsilon_{2},&\epsilon_{2}/2>\epsilon_{1}\quad\text{(nontrivial
gap)}\end{cases}.$ (S 29)

We can now study the topology of a junction with finite-size superconductors
and compare it to the analytical one-dimensional model in Fig. S 1. In the
left column we show $\mathcal{M}=(-1)^{\mathcal{W}}$ according to Eq. S 13 for
the analytical model. For the finite-size system we compute the parity of the
approximate topological invariant $\bar{\mathcal{M}}$ in Eq. S 28 re-scaled by
the topological gap in Eq. S 29, i.e.,
$\bar{\mathcal{Q}}=\bar{\mathcal{M}}\left(\delta/\Delta_{0}\right)$. The blue
regions in the central and right columns of Fig. S 1 thus represent the
topologically nontrivial phases, with the boundary where the gap closes
appearing in white. It is clear that the phase diagram for the analytical
model is qualitatively the same as for the the finite-size junction with a
harmonic texture (central column of Fig. S 1) or a domain one (right column on
Fig. S 1). The latter features extra low-energy trivial states that can lead
to non-topological gap closings for some parameters. That is the case for the
white lines in the right column of Fig. S 1 that separate two trivial red
regions.

(a)  Figure S 2: (a) Lowest energy bands as a function of $\xi_{m}$ for the
yellow line in panel (b). The map in panel (b) is repeated from Fig. 2(d) of
the main text.

We corroborate the predictions of the approximate topological invariant
studying the energy spectrum. In our calculations for finite-size junctions we
can vary $\xi_{m}$ from $2a$ (purely antiferromagnetic order at the interface)
to $\xi_{m}\to\infty$ (pure ferromagnetic order). We show a representative
result in Fig. S 2(a) illustrating the onset of lowest-energy edge states and
the emergence of zero energy states for $\xi_{m}\sim\xi_{s}$. This calculation
corresponds to a fixed value of $t_{m}$ in Fig. 2(d) of the main text, see
yellow line of Fig. S 2(b).

By contrast, the system is in the trivial regime when these two length scales
are very different. For $\xi_{m}\gg\xi_{s}$ the superconductor feels a
constant exchange field locally that can close the superconducting gap for
sufficiently large $t_{m}$ values. On the other hand, for $\xi_{m}\ll\xi_{s}$
the superconductor averages out the barrier magnetization recovering the
conventional superconducting spectrum.

(a)  Figure S 3: Topological phase diagram for the domain model showing the
energy distribution of eigenvalues (top panels) and the approximate invariant
$\bar{\mathcal{Q}}$ (bottom panels). We show the case without disorder in (a)
to compare it with the effect of disorder in the magnitude $t_{m}$ (b) and the
period $\xi_{m}$ (c) of the magnetic texture. Parameters are $d=6$, $w=2$,
$\mu/t=4$, $\Delta_{0}/t=0.2$, $t_{0}/t=1$, $N_{x}=8$, and $N_{y}=100$.

##  IV Role of disorder

The minimal model in Section I has helped us connect the magnetic spatial
variation along the interface with the emergence of a synthetic spin-orbit
coupling responsible for the nontrivial topology. We have then shown that the
topological phase transition is robust when this idealized model is extended
to a finite-size sequence of domain regions, see Fig. S 1. We now study the
effect of disorder in the domain model to test the stability of the
topological phase, see Section II.

Starting with a perfectly ordered sequence of domains connecting finite-size
superconductors, see Fig. S 3(a), we first consider the possibility of having
spatial irregularities in the period and the magnitude of the magnetic
texture. Disorder effects are thus included by modifying the value of a
specific parameter $p$ using a Gaussian distribution with a standard deviation
$\sigma_{p}$, e.g., $p=t_{m}$ in Fig. S 3(b). Each map point in the bottom
panels of Fig. S 3 then represents a random realization with the indicated
$\sigma_{p}$. On the top panels we show the lowest energy eigenvalues for $20$
random realizations using the parameters indicated by black dots on the
corresponding maps.

(a)  Figure S 4: Lowest energy bands (gray lines) and zero energy mode (blue
line) in the topological phase as a function of the disorder strength in the
magnitude of the magnetization $t_{m}$. Parameters are the same as in Fig. S
3(b) with $\bar{t}_{m}/t=1.2$.

The boundary of the topological phase is sensitive to the amplitude of the
magnetization, $t_{m}$, as discussed above and shown in Fig. S 1 and Fig. 2 of
the main text. Therefore, a disorder on the magnitude of $t_{m}$ blurs the
critical boundary of the topological phase and also reduces the topological
gap inside it, see the white region in the bottom panel of Fig. S 3(b). The
topological gap reduction can be seen in the eigenvalue distribution on the
top panel; for this case, the topological gap is still open. In order to fully
close the gap we need to introduce a disorder strength comparable to the
minimal magnetic amplitude required to enter the topological phase. For
example, Fig. S 4 indicates that the topological gap for $t_{m}/t\sim 1$ is
closed when the variations in amplitude reach $50\%$ of the amplitude without
disorder. Note, however, that the system is still in the nontrivial phase with
zero energy modes indicated by the colored lines in Fig. S 4.

One of our main results is that the topological phase is easier to achieve
when the magnetic and the superconducting coherence lengths are comparable,
i.e., $\xi_{m}\sim\xi_{s}$. By introducing disorder on the domains forming
each period $\xi_{m}$, while keeping the domain wall size fixed, we can
analyze the effect of having domains with different sizes. The phase diagram
in Fig. S 3(c) indicates that the topological region is barely affected by
this type of disorder.

(a)  Figure S 5: Topological phase diagram for the domain model and lowest
energy eigenvalues showcasing disorder effects on (a) the electrostatic
potential, (b) the magnetic orientation of the domains, and (c) the
orientation of the domain walls. Parameters are $d=6$, $w=2$, $\mu/t=4$,
$\Delta_{0}/t=0.2$, $t_{0}/t=1$, $N_{x}=8$, and $N_{y}=100$.

We can also study non-magnetic disorder effects introducing a random variation
on the chemical potential $\mu$. We consider in Fig. S 5(a) a disorder
strength in $\mu$ that can, in principle, reach values that would take the
system out of the topological region, see Fig. 2(a) in the main text. However,
the phase diagram is almost unaffected and the topological gap barely reduced.
The topological phase is thus robust to small changes in the band filling.

Finally, we consider effects that would directly affect the emerging synthetic
spin-orbit coupling at the interface. First, in Fig. S 5(b) we introduce a
random flip in the orientation of each domain, that is, in the sign of $t_{m}$
for each group of parallel spins in a domain, see Eqs. S 22 and S 26. This
disorder, however, does not affect the sign of $t_{m}$ in the domain walls. As
a result, the topological gap is reduced a bit, but the nontrivial phase is
maintained.

By contrast, the opposite situation where the disorder randomly changes the
rotation in the domain walls has a stronger effect on the topological phase.
The map in Fig. S 5(c) is computed imposing domain-wall-flips around 80% of
the times and features many white regions inside the nontrivial area where the
topological gap has closed. This type of disorder is affecting the rotation
direction of the magnetic texture and, therefore, the effective spin-orbit
coupling. When the direction of the magnetization rotation changes the average
spin-orbit is zero locally at one point, setting a topological phase boundary
and leading therefore to additional Andreev bound states inside the gap. If
this happens randomly along the sample with enough frequency many trivial low-
energy states will localize at the junction, leading to a dense density of
states around zero energy that could be a false positive in experimental
setups [106].

In summary, variations in the magnetization magnitude or period and
fluctuations on the exchange field are not enough to drive the system from the
topological to the trivial phase; although they can reduce the topological
gap. By contrast, it is important to maintain the magnetization rotation to
avoid the appearance of undesired low-energy states inside the system.

(a)
Figure S 6: Energy bands as a function of $\mu$ (in units of the bandwidth
$W$) for (a) the finite size system with the same parameters as in Fig. 2 of
the main text and (b) the analytic 1D model, corresponding to $N_{x}=2$.

##  V Band filling

In the main text we focused on chemical potentials in the region
$\mu\sim-3.5t$, which maintain electron-hole symmetry in the two dimensional
bands and feature only one pair of MBS in the topological phase. However, as
we show in Fig. 2(a) of the main text, different topological phases emerge as
a function of $\mu$. We now show in Fig. S 6(a) the full range of phases for a
setup with the same parameters as Fig. 2 of the main text. The lowest energy
states are showcased as red and blue lines for $\epsilon_{1}$ and
$-\epsilon_{1}$, respectively.

Similarly, the analytical 1D model can be approximated to a finite length
chain setting $N_{x}=2$ and a finite $N_{y}$. With two edges in the $y$
direction, we can explore the emergence of topological zero-energy states in
the 1D model. Figure S 6(b) shows that the topological phase is more robust
than in the bulk case, extending for almost every value of $\mu$ for a
sufficiently large $t_{m}/t$ value. For chemical potentials around $\mu\sim 0$
we find topological phases characterized by more than one Majorana state at
each edge in both models (see yellow regions in Fig. S 6).

##  VI Extended magnetic texture

To represent a Josephson junction mediated by a finite-size magnetic texture
we couple the superconducting leads, Eq. 1 in the main text, to a central
region described by

$\displaystyle H_{C}={}$
$\displaystyle-\sum_{\sigma=\uparrow,\downarrow}(t_{0}\sum_{\langle
x,x^{\prime}\rangle}c_{i^{\prime}j^{\prime},\sigma}^{\dagger}c_{ij,\sigma}+\sum_{i,j}\mu_{ij}c_{ij,\sigma}^{\dagger}c_{ij,\sigma})+\sum_{\begin{subarray}{c}i,j\\\
\alpha\beta\end{subarray}}c^{\dagger}_{ij,\alpha}\left(\bm{m}_{j}\cdot\bm{\sigma}\right)_{\alpha,\beta}c^{\dagger}_{ij,\beta},$
(S 30)

where the index $i$ runs from $1$ to $N_{W}$, the number of barrier sites in
the transport direction. Note that we now model the magnetic texture as a
local magnetization with constant magnitude, $|\mathbf{m}_{j}|=m$, as

$\bm{m}_{j}=m\left[\sin{(2\pi ja/\xi_{m})},0,\cos{(2\pi ja/\xi_{m})}\right].$
(S 31)

The Hamiltonian of the coupled system is

$H=\begin{pmatrix}H_{L}&V_{LC}&0\\\ V_{LC}^{\dagger}&H_{C}&V_{RC}^{\dagger}\\\
0&V_{RC}&H_{R}\end{pmatrix},$ (S 32)

where

$V_{XC}=-\frac{t_{0}}{2}\sum_{j,\sigma}\hat{c}^{\dagger}_{x,j,\sigma}\hat{c}_{x+1,j,\sigma}+\mathrm{h.c.}\,,$
(S 33)

with $x=N_{x}/2$, the width of the left superconductor region, when $X=L$ in
$V_{LC}$, and $x=N_{x}/2+N_{W}+1$ when $X=R$ in $V_{RC}$.

In Fig. S 7 we show the topological phase diagram as a function of $m$ and
$\xi_{m}$ for different widths of the central region, $N_{W}=1,\dots,4$. As
the barrier width is increased, the topological gap is reduced due to the
lower transmission. However, the minimum $t_{m}$ to achieve the topological
phase is also minimised, approaching the same value $\Delta_{0}$ as the one
predicted for the bulk minimal model.

(a)  Figure S 7: Topological phase diagram of a long Josephson junction
including $N_{W}$ central magnetic-textured sites between superconductors. The
rest of parameters are $\Delta_{0}/t=0.2$, $t_{0}/t=1$ and $\mu=-3.85t$.

##  VII Current-phase relation

Finally, we describe here the calculation of the Josephson current for a
phase-biased junction shown in Fig. 3 of the main text.

From diagonalization of the full Hamiltonian, Eqs. 1 and 2 in the main text,
we obtain a set of eigenvalues $E_{i}(\phi)$. The corresponding free energy at
a temperature $T$ can be computed as

$F=-T\log(Z)=-8k_{B}T\sum_{i}\log\left[\cosh\left(\dfrac{E_{i}}{2k_{B}T}\right)\right],$
(S 34)

where $Z=\sum_{i}\mathrm{e}^{-E_{i}/{k_{B}T}}$ is the partition function of
the system and $k_{B}$ the Boltzmann constant.

We define the Josephson current [107] as the derivative with respect to the
superconducting phase difference $\phi$, namely,

$I(\phi)=2e\dfrac{dF}{d\phi}=-8e\sum_{i}\tanh\left(\dfrac{E_{i}}{2k_{B}T}\right)\dfrac{dE_{i}}{d\phi}.$
(S 35)

In the limit of zero temperature the supercurrent simply reduces to

$I(\phi)=-8e\sum_{E_{i}<0}\frac{\partial E_{i}}{\partial\phi}.$ (S 36)