∎

11institutetext: Department of Applied Mathematical and Physical Sciences,
NTUA, 15780, Athens, Greece 22institutetext: Institut für Theoretische Physik,
ETH, 8093, Zürich, Switzerland

We propose a novel platform for the creation and manipulation of Majorana zero
modes consisting of a ferromagnetic metallic wire placed between conventional
superconductors which are in proximity to ferromagnetic insulators. Our device
relies on the interplay of applied supercurrents and exchange fields emerging
from the ferromagnetic insulators. We assert that the proposed
superconductor/ferromagnet heterostructures exhibit enhanced controllability,
since topological superconductivity can be tuned apart from gate voltages
applied on the ferromagnetic wire, also by manipulating the applied
supercurrents and/or the magnetisation of the ferromagnetic insulators.

# Platform for controllable Majorana zero modes using
superconductor/ferromagnet heterostructures

Giorgos Livanasaddr1  Nikolaos Vanasaddr1  Manfred Sigristaddr2  Georgios
Varelogiannisaddr1

##  1 Introduction

Topological quantum computation is the most promising answer to the quantum
decoherence problem hindering the development of quantum computers Nayak .
Topologically protected qubits formed by non-Abelian anyons would allow to
store and process quantum information in decoherence-resilient schemes.
Majorana zero modes (MZMs) are arguably the most natural realization of such
non-Abelian anyons. Indeed, MZMs are expected to emerge at the edges of 1D
Kitaev  and at the vortices of 2D Ivanov  topological p-wave superconductors
but unfortunately p-wave superconductivity appears to be scarce in nature.
However, a topological p-wave superconducting state has been demonstrated to
emerge in heterostructures comprising topological insulators and conventional
superconductors Kane , where proximity induced singlet superconductivity pairs
the helical states confined at the boundaries of the topological insulator
realising an effectively spinless superconducting state. Following this
seminal work numerous proposals have been put forward involving
superconductor-semicondu
ctor heterostructures SauA ; Alicea ; Linder , non-centrosymmetric
superconductors Ghosh , surfaces of heavy metals Potter , the quantum
anomalous Hall state Qi  and 2D electron gases Hell . Key elements for
realising effectively spinless superconductivity in all the aforementioned
proposals, is spin momentum locking and a Zeeman energy term, thereby many
schemes for realising MZMs rely on effective spin orbit coupling emerging from
the application of laser beams in ultracold atoms superfluids Zhu ,
nanomagnets in superconducting wires Kjaergaard  or carbon nanoribbons Loss ,
magnetic tunnel junctions in proximized 2D electron gases Matos , from the
helical ordering of magnetic adatoms embedded in conventional superconductors
Choy ; Pientka ; Stano ; Vazifeh ; Schecter  as well as from the curved
geometry of carbon nanotubes SauB  and semiconducting wires Ojanen . Moreover,
in most of the above setups for engineering MZMs, the required Zeeman energy
is provided by the application of an external magnetic field.
Superconductor/ferromagnet heterostructures (SC-FM) support the emergence of
topological superconductivity without the requirement of an external magnetic
field, as has been already demonstrated for ferromagnetic metal/superconductor
heterostructures with spin active interfaces Zhang ; Takei  and
ferromagnetically aligned magnetic adatoms chains embedded in conventional
superconductors with spin orbit coupling Perge . However, no related devices
exhibiting enhanced controllability of the MZMs, necessary for the future
design of braiding operations, has been proposed so far.
Here we propose such a platform for engineering controllable TSC and MZMs
based on superconductor/ferromagnet heterostructures (SC-FM) and application
of supercurrents and gate voltages. The heterostructures consist of a
ferromagnetic metallic wire (FM) placed among conventional superconductors
(SCs) which are in proximity to ferromagnetic insulators (FIs)
(Fig. 1). The general underlying mechanism has already been presented in our
previous publication Livanas  and consists of two stages. In the first stage,
triplet p-wave correlations are induced in the SCs, due to the coexistence of
conventional superconductivity with applied supercurrents and exchange fields
emerging from the FIs. As we show below, the d-vector Ueda  of the induced
p-wave correlations is parallel to polarization of the exchange field of the
FIs. In the second stage, these p-wave correlations mediate into the FM where
an analogous to Kitaev’s chain spinless superconducting state is realised
provided the magnetisation of the wire and the d-vector of the p-wave
correlations are misaligned. We demonstrate that our device platform exhibits
enhanced controllability through the combination of externally applied
supercurrents and the manipulation of the exchange fields resulting from the
FIs that can be achieved by the use of currents Avci  or applied gate voltages
Chiba ; Chiba2 . We remark that although supercurrents were involved in other
proposals for engineering MZMs Romito ; Heimes ; Dmytruk ; Lesser ; Rontynen ;
Melo , in our device they do not constitute simply an alternative to magnetic
fields, but instead they can also be exploited to manipulate the phase of the
induced TSC, a property crucial for designing feasible braiding schemes.

##  2 Induced p-wave superconductivity

Initially, we describe how the coexistence of conventional superconductivity,
supercurrents and exchange fields induces triplet p-wave correlations with a
d-vector parallel to the polarization of the exchange fields. As we have
already pointed out in Livanas  these four fields or order parameters
constitute a quartet Quartets  meaning that the presence of any three of them
induces the missing fourth one. To this end we employ the following
Hamiltonian representing a 1D SC,

$\displaystyle\quad
H=\int\Psi^{{\dagger}}\left([\frac{\partial_{x}^{2}}{2m}\right.+\mu]\tau_{z}+\tau_{z}\bm{h}\cdot\tilde{\bm{\sigma}}+\Delta\tau_{y}\sigma_{y}\biggl{)}\Psi\,$
(1)

where
$\Psi^{{\dagger}}=(\psi_{\uparrow}^{{\dagger}},\psi_{\downarrow}^{{\dagger}},\psi_{\uparrow},\psi_{\downarrow})$
the extended Nambu spinor and $\bm{\tau}$ and
$\tilde{\bm{\sigma}}=(\sigma_{x},\tau_{z}\sigma_{y},\sigma_{z})$ are the Pauli
matrices acting on particle-hole and spin space, respectively, $\bm{h}$ the
exchange field and $\Delta=|\Delta|e^{iJx\tau_{z}}$ the singlet
superconducting field. Under the gauge transformation $\Psi(x)\rightarrow
e^{-iJ/2x\tau_{z}}\Psi(x)$ kinetic term $\frac{\partial_{x}^{2}}{2m}$
transforms to
$\frac{\partial_{x}^{2}}{2m}+i\frac{J}{2m}\partial_{x}-\frac{J^{2}}{8m}\tau_{z}$
where $i\frac{J}{2m}\partial_{x}$ is a current term and
$-\frac{J^{2}}{8m}\tau_{z}$ can be absorbed in the chemical potential $\mu$.
Thus, Hamiltonian Eq. 1 takes the form $H=\int
dx\Psi^{{\dagger}}\biggl{(}[\frac{\partial_{x}^{2}}{2m}-\mu]\tau_{z}+i\frac{J}{2m}\partial_{x}+\tau_{z}\bm{h}\cdot\tilde{\bm{\sigma}}+|\Delta|\tau_{y}\sigma_{y}\biggl{)}\Psi$.
For periodic boundary conditions, a Fourier transformation to momentum $k$
space leads to the energy bands of the wire
$E_{s,\pm}(k)=-sh+\frac{J}{2m}k\pm\sqrt{\left(\frac{k^{2}}{2m}-\mu\right)^{2}+\Delta^{2}}$,
where $s=\pm$, with $+$ and $-$ corresponding to different spin configurations
depending on the orientation of the $\bm{h}$.

For translationally invariant systems p-wave triplet correlations derive from
equation

$\displaystyle\quad<\Delta_{\bm{p}}>\propto\frac{1}{\beta}\Sigma_{i\omega_{n}}Tr\\{k\tau_{x}\bm{d}\cdot\tilde{\bm{\sigma}}i\sigma_{y}\hat{G}(k,i\omega_{n})\\}\,,$
(2)

where $\hat{G}(k,i\omega_{n})$ the Matsubara Green’s function with
$\omega_{n}=(2n+1)\pi/\beta$ the fermion frequencies and $\beta\propto 1/T$
the inverse of temperature. The above expression simplifies to

$\displaystyle\quad<\Delta_{\bm{p}}>\propto\sum_{k}\sum_{m}$ $\displaystyle
n_{F}(E_{m})$ (3)
$\displaystyle[U(k)^{{\dagger}}k\tau_{x}\bm{d}\cdot\tilde{\bm{\sigma}}i\sigma_{y}U(k)]_{mm}\,,$

where $m$ the energy band index, $U(k)$ the transformation matrix which
diagonalises the Hamiltonian matrix $\hat{H}(k)$ while $[]_{mm}$ denotes the
$m$ diagonal term of the corresponding matrix. Based on the above analysis it
is straightforward to derive the expression

$\displaystyle<\Delta_{p,\bm{d}\parallel\bm{h}}>\propto\sum_{k}k\frac{\Delta}{\sqrt{\Delta^{2}+(\frac{k^{2}}{2m}-\mu)^{2}}}\cdot$
$\displaystyle\biggl{[}n_{F}(E_{s,-})-n_{F}(E_{s^{\prime},+})-[n_{F}(E_{s^{\prime},-})-n_{F}(E_{s,+})\biggl{]}$
(4)

Figure 1:  a) Device for engineering controllable MZMs based on
superconductor-ferromagnet heterostructures. A ferromagnetic metallic wire
(FM) is placed between two conventional superconductors (SCs) where
supercurrents (blue arrows) are applied in an anti-parallel configuration.
Exchange fields are induced in the SCs due to proximity with the ferromagnetic
insulators (FIs). The optimal configuration is when the magnetisations of the
FIs (black arrows) is perpendicular to that of the FM (white arrow) and anti-
parallel with each other. The application of supercurrent $J$ and exchange
field $h_{x}$ in the SC induces an equal spin triplet pairing component (blue
dots) which mediates in the FM resulting in the emergence of MZMs
$\gamma_{a}=(\psi^{{\dagger}}+\psi)$ (green dot) and
$\gamma_{b}=i(\psi^{{\dagger}}-\psi)$ (yellow dot) at the edges of the wire.
b) The energy bands of a system with an applied current $J$ and a $h_{x}$
exchange field. The singlet superconducting field $\Delta$ pairs electron from
different bands. The application of $J$ and $h_{x}$ results in an imbalance of
singlet pairing between the two spin configurations, and therefore an equal
spin triplet pairing component is induced. However, the induced triplet
superconducting field still pairs electrons from different bands. c) The
induced equal spin triplet superconducting correlations $\Delta_{p}$ mediate
in the FM where an exchange field $h_{z}$ is present. In the spin
configuration of the energy bands of FM, $\Delta_{p}$ pairs electrons of the
same band, i. e. intraband pairing, realising an effectively spinless
superconducting state.

for the triplet correlations with $\bm{d}\parallel\bm{h}$. Since for $h=0$,
$n_{F}(E_{s,\pm})=n_{F}(E_{s^{\prime},\pm})$, triplet correlations vanish,
$<\Delta_{p,d\parallel\bm{h}}>=0$. Similarly, for $J=0$ we get
$<\Delta_{p,d\parallel\bm{h}}>=0$, since the expression
$k\left(n_{F}(E_{s,-})+n_{F}(E_{s^{\prime},+})\right.$
$\left.-[n_{F}(E_{s,+})+n_{F}(E_{s^{\prime},-})]\right)$ is odd in momenta
$k$. However, when both $h$ and $J$ are finite, both spin and spatial
inversion symmetries break, leading to a finite sum for triplet correlations
$<\Delta_{p,d\parallel\bm{h}}>$. Regarding correlations
$<\Delta_{p,d\perp\bm{h}}>$, matrix
$[U(k)^{{\dagger}}k\tau_{x}\bm{d}_{\perp\bm{h}}\cdot\tilde{\bm{\sigma}}i\sigma_{y}U(k)]$
has no diagonal elements, therefore they vanish. Thus, only when $h$ and $J$
are both finite in a conventional superconductor $\Delta$,
$<\Delta_{p,d\parallel\bm{h}}>$ correlations are induced. From Eq. 4 it
becomes apparent that $<\Delta_{p,d\parallel\bm{h}}>$ correlations emerge from
the imbalance among the two configurations
$\psi_{k,s}^{{\dagger}}\psi_{-k,s^{\prime}}^{{\dagger}}$ and
$\psi_{k,s^{\prime}}^{{\dagger}}\psi_{-k,s}^{{\dagger}}$ of the interband
singlet pairing (See Sup. Mat.). Moreover, since $J\rightarrow-J$ leads to
$E_{s,\pm}(k)\rightarrow E_{s,\pm}(-k)$ we get
$<\Delta_{p,d\parallel\bm{h}}>\rightarrow-<\Delta_{p,d\parallel\bm{h}}>$ or
equivalently the phase of the triplet order parameter changes by $\pi$. Most
importantly, there are no threshold values for $h$ and $J$ in order for
$<\Delta_{p,\bm{d}\parallel\bm{h}}>$ correlations to be induced. Therefore the
particular mechanism is valid even for small values of these parameters which
leave conventional superconductivity unaffected.

##  3 Topological superconductivity in superconductor-ferromagnet
heterostructures

Evidently, topological superconductivity cannot be realised simply by applying
a supercurrent in a superconducting wire under the presence of an exchange
field, since the emergent triplet pairing correlations are parallel to the
polarization of the exchange field and therefore connect different energy
bands of the wire, i.e. we get interband triplet pairing. In order to realise
MZMs we need intraband superconductivity. In our device intraband
superconductivity emerges in the FM where the triplet correlations induced in
the conventional SCs mediate by means of the proximity effect. Necessary
condition for the emergence of intraband superconductivity is that the
exchange field $h$ of the wire and the $\bm{d}$ vector of the induced triplet
correlations are misaligned. Therefore, a topological superconducting state is
realised in a FM separating two conventional SCs when supercurrents are
applied in an antiparallel configuration and the exchange fields are
perpendicular to the magnetisation of the wire and antiparallel to each other
(see Fig. 1). The antiparallel configuration guarantees that the supercurrent
and the exchange field vanish into the wire and therefore no
$<\Delta_{p,d\parallel\bm{h}_{FM}}>$ correlations are induced there. Although
this is the optimal configuration, we show that TSC is realised even for small
misalignment angles among the magnetisations of the FM and the FIs and even
for not exactly anti-parallel supercurrents and exchange fields (see Fig: 2).

In order to illustrate numerically the above arguments we employ the following
lattice Bogoliubov de Gennes equation, ${\cal
H}=\sum_{\bm{i},\bm{j}}\Psi_{\bm{i}}^{{\dagger}}H_{\bm{i},\bm{j}}^{\phantom{{\dagger}}}\Psi_{\bm{j}}^{\phantom{{\dagger}}}$,
which describes SC-FM presented in Fig. 1, with

$\displaystyle\quad
H_{\bm{i},\bm{j}}=(tf_{\bm{i},\bm{j}}+\mu_{\bm{i}})\tau_{z}+\tau_{z}\bm{h_{\bm{i}}}\cdot\tilde{\bm{\sigma}}+\Delta_{\bm{i}}\tau_{y}\sigma_{y}+\bm{J_{\bm{i}}}\cdot\bm{g}_{\bm{i},\bm{j}}$
(5)

where
$\Psi_{\bm{i}}^{{\dagger}}=(\psi_{\uparrow,\bm{i}}^{{\dagger}},\psi_{\downarrow,\bm{i}}^{{\dagger}},\psi_{\uparrow,\bm{i}}^{\phantom{{\dagger}}},\psi_{\downarrow,\bm{i}}^{\phantom{{\dagger}}})$
is the Nambu spinor, $\bm{\tau}$ and
$\tilde{\bm{\sigma}}=(\sigma_{x},\tau_{z}\sigma_{y},\sigma_{z})$ the Pauli
matrices acting on particle-hole and spin space respectively,
$f_{\bm{i},\bm{j}}=\delta_{\bm{j},\bm{i}\pm\hat{x}}+\delta_{\bm{j},\bm{i}\pm\hat{y}}$
and $\bm{g}_{\bm{i},\bm{j}}=(\pm i\delta_{\bm{j},\bm{i}\pm\hat{x}},\pm
i\delta_{\bm{j},\bm{i}\pm\hat{y}})$ the even and odd in spatial inversion,
respectively, functions connecting nearest neighbours lattice points.
Moreover, $t$ is the hopping integral, $\mu_{\bm{i}}$ the chemical potential,
$\bm{h_{\bm{i}}}$ the exchange field, $\Delta_{\bm{i}}$ the conventional
superconducting field and $J_{\bm{i}}$ the supercurrent term, on site
$\bm{i}$. Hamiltonian Eq. 5 describes the SC-FM-SC heterostructure considering
the appropriate values of $\Delta_{\bm{i}}$, $\mu_{\bm{i}}$, $\bm{h_{\bm{i}}}$
and $J_{\bm{i}}$ for each region. The proximity to the FIs is modelled by
considering a finite exchange field $\bm{h}$ over the SCs emerging from the
interfacial exchange interaction between the localized magnetic moments of the
FIs and the SCs conduction band electrons. MZMs are anticipated to emerge
localised at the edges of the wire for $|2t-\sqrt{(h_{FM})^{2}-\Delta^{\prime
2}}|<|\mu_{FM}|<|2t+\sqrt{(h_{FM})^{2}-\Delta^{\prime 2}}|$ where
$\Delta^{\prime}$ is the induced by proximity singlet superconducting field
over the FM Livanas . From Fig. 2a we conclude that the hopping term along the
wire is renormalised to $t^{\prime}\simeq 0.85t$, due to the coupling of the
wire to the SCs Peng . Figs. 2 b),c) and d) demonstrate the significant
robustness of our device against deviations from the optimal antiparallel
alignment of the magnetisation in the FIs and the antiparallel configuration
of the supercurrents in the SCs.

Figure 2: a) The low energy spectrum of the device as a function of the
chemical potential of the wire $\mu_{FM}/t$. Eigenenergies are normalised with
respect to energy gap $E_{g,0}$ for $\mu_{FM}=0$ and $h_{FM}/t=2.4$. b) The
energy gap in the wire for $h_{FM}/t=\mu_{FM}/t=2.4$ with respect to the ratio
of the magnitude of the supercurrents $|J_{l}|$,$|J_{r}|$ in the left and the
right SC, respectively. The supercurrents flow in the opposite direction. The
energy spectrum is normalised to the energy gap $E_{g,1}$ corresponding to
$J_{l}=-J_{r}$. c) Low energy excitation spectrum as a function of the ratio
$h_{x,l}/h_{x,r}$ of the exchange fields along the $x$-axis in the left and
right SC. The antiparallel configuration leads to the maximum energy gap. d)
The dependence of the excitation spectrum on the rotation angle $\theta$ of
$h_{r}$ over the right SC in the $x-z$ plane, where for $\theta=0$
$h_{r}=-h_{l}$ along the $x$-axis. Energies in c) are normalised to $E_{g,-1}$
corresponding to $h_{x,l}=-h_{x,r}$, while in d) are normalised to $E_{g,0}$
for $\theta=0$. The device appears to be more robust against deviations from
the optimal anti-parallel configuration of the exchange fields.

##  4 Controlling Majorana zero modes

To demonstrate the controllability of our setup we consider a junction between
two ferromagnetic wires FM1 and FM2 as depicted in Fig. 3. The conventional
superconductors in this configuration are separated by an insulator in order
for the supercurrents adjacent of FM1 and FM2 to be controlled independently.
First, we consider a sinusoidal variation $J=J_{0}\cos(\phi)$ of the magnitude
of the supercurrents applied in the SCs that are adjacent to the FM2. The
supercurrents always flow in opposite directions. In Fig. 3 is presented the
supercurrents configuration for $\phi=\pi$ while in Fig. 4a) is shown the low-
energy spectrum of this junction with respect to parameter $\phi$. The FM
wires are coupled through the insulating region with a hopping term
$t_{c}=0.5t$. For $-\pi/2<\phi<\pi/2$, the superconducting states in the two
FMs acquire the same phase ($0$\- Josepshon junction) and MZMs $\gamma_{1,b}$
localised in the right edge of FM1 and $\gamma_{2,a}$ localised at the left
edge of the FM2 interact with each other giving rise to an ordinary fermionic
state with finite energy. The other two MZMs $\gamma_{1,a}$ at the left edge
of FM1 and $\gamma_{2,b}$ at the right edge of FM2 remain unpaired and
therefore stick at zero energy. For $\phi=\pm\pi/2$ the energy gap in FM2
closes and MZMs $\gamma_{2,a}$ and $\gamma_{2,b}$ fuse while $\gamma_{1,a}$
and $\gamma_{1,b}$ remain uncoupled. Remarkably for $\pi/2<\phi<3\pi/2$, for
which the supercurrent flow is reversed on both sides of FM2, _the phase of
the induced p-wave superconductivity in FM2 changes by $\pi$_ leading to the
formation of a _$\pi$ -Josepshon junction_ among the FM wires. Since the
reversal of the flow of the supercurrent leads to an _interchange of the
position of $\gamma_{2,a}$ and $\gamma_{2,b}$_, and $\gamma_{2,b}$ cannot
couple with $\gamma_{1,b}$, the heterostructure hosts four MZMs reflected in
the low-energy quasi-particle spectrum. Note on the same figure that the four
MZMs are stabilised even for very small values of the supercurrent.

Figure 3:  Setup of two ferromagnets (FMs) separated by an insulator (I) in
proximity with conventional superconductors (SCs) and ferromagnetic insulators
(FIs). The junction is controlled by varying the supercurrents (blue arrows)
and/or rotating the magnetisation (black arrow) of the FIs adjacent to FM2 by
angle $\theta$.

Figure 4: Low-energy quasiparticle energy spectrums for the setup presented in
Fig. 3, a) with respect to parameter $\phi$ controlling the supercurrents
($J=J_{0}\cos(\phi)$) in the conventional SCs adjacent to FM2 and b) with
respect to $\theta$ corresponding to the angle of FIs magnetisation in the x-z
plane. Eigenenergies are normalised to the energy gap $E_{gap,\pi}$ for
$\phi=\pi$ and $\theta=\pi$ respectively. The second lowest eigenenergy (black
line) is multiplied by 10 to demonstrate clearly the splitting of the MZMs.
The spectral function $|\Psi_{z}|$ of the two lowest eigenenergies of the
heterostructure, c) for $\theta=0.45\pi$ and d) for $\theta=0.55\pi$. The
particular angles are indicated in figure b) by the dashed red line. The
function is normalised to the maximum value $|\Psi_{max}|$. For
$\theta=0.45\pi$, MZMs localised in the right edge of FM1 and left edge of FM2
form a fermionic state with finite energy $E/t=0.03$. For $\theta=0.55\pi$,
four MZMs emerge. Since in FM2 the FIs magnetisation is not in the optimal
anti-parallel configuration, MZMs appear to be less localised. In the upper c)
plot the left and the right part of the function is normalised independently
in order to compare the localisation length of MZMs in the two FMs.

Next we investigate the same junction by rotating the magnetisation and
therefore the exchange field of the FIs adjacent to FM2, in the x-z plane,
$h=h_{z}\sin(\theta)+h_{x}\cos(\theta)$ where $\theta$ is the angle from the
x-axis. Since we require that the $x$ component of the exchange fields of FIs
adjacent to FM2, point at the opposite direction for every $\theta$, the
magnetisation in the upper FI rotates clockwise while in the lower FI rotates
counter-clockwise (Fig.3). The supercurrent in the two FMs flows along the
same direction, therefore for $\theta=0$ the junction is formed by two
identical copies of FMs embedded in conventional SCs. According to our
mechanism topological SC emerges only when the magnetisation of the FIs and
that of the ferromagnetic wire is misaligned. Indeed, for $\theta=\pi/2$ the
energy gap at the wire closes (Fig. 4b)). Remarkably for $\pi/2<\theta<3\pi/2$
for which the $x$ component of the magnetisation of the FIs adjacent to FM2 is
opposite to that of the FIs adjacent to FM1, _a $\pi$-Josepshon junction is
formed_ and four Majorana zero modes emerge localised at the edges of the two
wires. In Fig. 4c) we display the spectral function $|\Psi_{z}|$ of the two
lowest eigenvalues for $\theta=0.45\pi$ and $\theta=0.55\pi$. We remark that
small rotations of the FIs magnetisation can be achieved by the application of
gate voltages.

##  5 Conclusions

We present a novel device platform for engineering MZMs based on SC-FM
heterostructures. Key elements of the underlying mechanism are the externally
applied supercurrents and the emergent exchange fields, due to proximity of
the SCs with FIs. Developments in the fabrication of SC-FI heterostructures
Liu  and ferromagnetic metallic Co Giroud ; Wang  or Fe Yazdani  nanonwires
embedded in conventional SCs, enhance the experimental feasibility of our
proposal. Several works have demonstrated both theoretically Tokuyasu ; Hijano
and experimentally Tedrow ; Hao ; Moodera  the emergence of exchange fields in
conventional SCs in proximity to FIs, e.g. in EuS/Al and EuO/Al
heterostructures. Finally, our platform does not require externally applied
magnetic fields, yet it facilitates the control of the emergent MZMs, either
through the applied supercurrents or the manipulation of FIs magnetisation by
use of currents or applied gate voltages. In this way we circumvent the
screening effects problem that undermines schemes for braiding MZMs based on
semiconductors.

This research is carried out in the context of the project ”Majorana fermions
from induced quartet states” (MIS 5049433) under the call for proposals
”Researchers’ support with an emphasis on young researchers- 2nd Cycle”. The
project is co-financed by Greece and the European Union (European Social Fund-
ESF) by the Operational Programme Human Resources Development, Education and
Lifelong Learning 2014-2020.”

### Author contribution statement

Dr. Giorgos Livanas has conducted the analysis and wrote the initial version
of the manuscript, Nikos Vanas contributed to numerical calculations, Prof.
Giorgos Varelogiannis and Prof. Manfred Sigrist have revised and wrote the
final version of the manuscript. The work was conceptualized by Dr. Giorgos
Livanas and Prof. Giorgos Varelogiannis.

## References

  * (1) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
  * (2) A.Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys.-Usp. 44, 131-136 (2001).
  * (3) D.A. Ivanov, Non-Abelian Statistics of Half-Quantum Vortices in p-Wave Superconductors, Phys. Rev. Lett. 86, 268 (2001).
  * (4) Liang Fu and C. L. Kane, Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator, Phys. Rev. Lett. 100, 096407 (2008).
  * (5) Sau, J. D., Lutchyn, R. M., Tewari, S. and Sarma, S. D., Generic new platform for topological quantum computation using semiconductor heterostructures, Phys. Rev. Lett. 104, 040502 (2010).
  * (6) J. Alicea, Majorana fermions in a tunable semiconductor device, Phys. Rev. B 81, 125318 (2010).
  * (7) J. Linder and A. Sudbo, Majorana fermions manifested as interface states in semiconductor hybrid structures, Phys. Rev. B 82, 085314 (2010).
  * (8) P. Ghosh, J. D. Sau, S. Tewari, S. Das Sarma, Non-Abelian topological order in noncentrosymmetric superconductors with broken time-reversal symmetry, Phys. Rev. B 82, 184525 (2010).
  * (9) C. Potter, P. A. Lee, Topological superconductivity and Majorana fermions in metallic surface states, Phys. Rev. B 85, 094516 (2012).
  * (10) X. L. Qi, T. L. Hughes, Zhang SC., Chiral topological superconductor from the quantum Hall state, Phys. Rev. B 82, 184516 (2010).
  * (11) M. Hell, M. Leijnse, and K. Flensberg, Two-Dimensional Platform for Networks of Majorana Bound States, Phys. Rev. Lett. 118, 107701 (2017).
  * (12) S. L. Zhu, L. B. Shao, Z. D. Wang, L. M. Duan, Probing Non-Abelian Statistics of Majorana Fermions in Ultracold Atomic Superfluid, Phys. Rev. Lett. 106, 100404 (2011).
  * (13) M. Kjaergaard, K. Wolms, K. Flensberg, Majorana fermions in superconducting nanowires without spin-orbit coupling, Phys. Rev. B 85, 020503 (2012).
  * (14) J. Klinovaja and D. Loss, Giant Spin-Orbit Interaction Due to Rotating Magnetic Fields in Graphene Nanoribbons, Phys. Rev. X 3, 011008 (2013).
  * (15) A. Matos-Abiague et al., Tunable magnetic textures From Majorana bound states to braiding, Solid State Commun. 262, 1 (2017).
  * (16) T. P. Choy, J. M. Edge, A. R. Akhmerov, C. W. J. Beenakker, Majorana fermions emerging from magnetic nanoparticles on a superconductor without spin-orbit coupling, Phys. Rev. B 84, 195442 (2011).
  * (17) F. Pientka, L. I. Glazman, F. von Oppen, Topological superconducting phase in helical Shiba chains, Phys. Rev. B 88, 155420 (2013).
  * (18) J. Klinovaja, P. Stano, A. Yazdani, D. Loss, Topological Superconductivity and Majorana Fermions in RKKY Systems, Phys. Rev. Lett. 111, 186805 (2013).
  * (19) M. M. Vazifeh, M. Franz, Self-Organized Topological State with Majorana Fermions, Phys. Rev. Lett. 111, 206802 (2013).
  * (20) M. Schecter, K. Flensberg, M. H. Christensen, B. M. Andersen, and J. Paaske Self-organized topological superconductivity in a Yu-Shiba-Rusinov chain, Phys. Rev. B 93, 140503 (2016).
  * (21) J. D. Sau, S. Tewari, Topological superconducting state and Majorana fermions in carbon nanotubes, Phys. Rev. B 88, 054503 (2013).
  * (22) T. Ojanen, Majorana states and devices in magnetic structures, Phys. Rev. B 88, 220502 (2013).
  * (23) S. B. Chung, H .J. Zhang, X. L. Qi, Zhang SC, Topological superconducting phase and Majorana fermions in half-metal/superconductor heterostructures, Phys. Rev. B 84, 060510 (2011).
  * (24) S. Takei, V. Galitski, Microscopic theory for a ferromagnetic nanowire/superconductor heterostructure: Transport, fluctuations, and topological superconductivity, Phys. Rev. B 86, 054521 (2012).
  * (25) S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, A. Yazdani, Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor, Phys. Rev. B 88, 020407 (2013).
  * (26) G. Livanas, M. Sigrist and G. Varelogiannis, Alternative paths to realize Majorana Fermions in Superconductor-Ferromagnet Heterostructures, Sci Rep 9, 6259 (2019).
  * (27) M. Sigrist and K. Ueda, Phenomenological theory of unconventional superconductivity, Rev. Mod. Phys. 63, 239 (1991)
  * (28) C. Avci, A. Quindeau, CF. Pai, et al., Current-induced switching in a magnetic insulator, Nature Mater 16, 309 (2017).
  * (29) D. Chiba, M. Sawicki, Y. Nishitani, et al., Magnetization vector manipulation by electric fields, Nature 455, 515 (2008).
  * (30) D. Chiba, Y. Nakatani, F. Matsukura, and H. Ohno, Simulation of magnetization switching by electric-field manipulation of magnetic anisotropy, Appl. Phys. Lett. 96, 192506 (2010).
  * (31) A. Romito, J. Alicea, G. Refael, and F. von Oppen, Manipulating Majorana fermions using supercurrents, Phys. Rev. B 85, 020502(R) (2012).
  * (32) A. Heimes, P. Kotetes, G. Schon, Majorana fermions from Shiba states in an antiferromagnetic chain on top of a superconductor, Phys. Rev. B 90, 060507 (2014).
  * (33) O. Dmytruk, M. Thakurathi, D. Loss and J. Klinovaja, Majorana bound states in double nanowires with reduced Zeeman thresholds due to supercurrents, Phys. Rev. B 99, 245416 (2019).
  * (34) O. Lesser, K. Flensberg, F. von Oppen, and Y. Oreg, Three-phase Majorana zero modes at tiny magnetic fields, Phys. Rev. B 103, L121116 (2021).
  * (35) Joel Röntynen and Teemu Ojanen,Tuning topological superconductivity in helical Shiba chains by supercurrent, Phys. Rev. B 90, 180503(R) (2014).
  * (36) André Melo, Sebastian Rubbert, Anton R. Akhmerov, SciPost Phys. 7, 039 (2019) Supercurrent-induced Majorana bound states in a planar geometry, SciPost Phys. 7, 039 (2019).
  * (37) G. Varelogiannis, General Rule Predicting the Hidden Induced Order Parameters and the Formation of Quartets and Patterns of Condensates, arXiv1305.2976 (2013).
  * (38) Y. Peng, F. Pientka, L.I. Glazman and F. von Oppen, Strong Localization of Majorana End States in Chains of Magnetic Adatoms, Phys. Rev. Lett. 114, 106801 (2015).
  * (39) Y. Liu, S. Vaitiekėnas, S. Martí-Sánchez, C. Koch, S. Hart, Z. Cui, T. Kanne, S. A. Khan, R. Tanta, S. Upadhyay, M. E. Cachaza, C. M. Marcus, J. Arbiol, K. A. Moler, and P. Krogstrup, Semiconductor–Ferromagnetic Insulator–Superconductor Nanowires: Stray Field and Exchange Field, Nano Lett. 2020, 20, 1, 456–462 (2019)
  * (40) M. Giroud, H. Courtois, K. Hasselbach, D. Mailly, and B. Pannetier, Superconducting proximity effect in a mesoscopic ferromagnetic wire, Phys. Rev. B 58, R11872(R) (1998).
  * (41) J. Wang, M. Singh, M. Tian, et al., Interplay between superconductivity and ferromagnetism in crystalline nanowires, Nature Phys 6, 389–394 (2010).
  * (42) B. Jäck, Y. Xie, J. Li, S. Jeon, B. A. Bernevig, A. Yazdani, Observation of a Majorana zero mode in a topologically protected edge channel, Science 364, 6447 (2019).
  * (43) T. Tokuyasu, J. A. Sauls, and D. Rainer, Proximity effect of a ferromagnetic insulator in contact with a superconductor, Phys. Rev. B 38, 8823 (1988).
  * (44) Alberto Hijano, Stefan Ilić, Mikel Rouco, Carmen González-Orellana, Maxim Ilyn, Celia Rogero, P. Virtanen, T. T. Heikkilä, S. Khorshidian, M. Spies, N. Ligato, F. Giazotto, E. Strambini, and F. Sebastián Bergeret, Coexistence of superconductivity and spin-splitting fields in superconductor/ferromagnetic insulator bilayers of arbitrary thickness, Phys. Rev. Research 3, 023131 (2021).
  * (45) P. M. Tedrow, J. E. Tkaczyk, and A. Kumar,Spin-Polarized Electron Tunneling Study of an Artificially Layered Superconductor with Internal Magnetic Field: EuO-Al, Phys. Rev. Lett. 56, 1746 1986.
  * (46) X. Hao, J. S. Moodera, and R. Meservey, Spin-filter effect of ferromagnetic europium sulfide tunnel barriers, Phys. Rev. B 42, 8235 (1990).
  * (47) E. Strambini, V. N. Golovach, G. De Simoni, J. S. Moodera, F. S. Bergeret, and F. Giazotto, Revealing the magnetic proximity effect in EuS/Al bilayers through superconducting tunneling spectroscopy, Phys. Rev. Materials 1, 054402, (2017).