# A proposal for realizing Majorana fermions without magnetic field in
strongly correlated nanowires

Kaushal K. Kesharpu<EMAIL_ADDRESS>Evgenii A. Kochetov  Bogoliubov
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
Dubna, Moscow Region 141980, Russia  Alvaro Ferraz  International Institute of
Physics - UFRN, Department of Experimental and Theoretical Physics - UFRN,
Natal 59078-970, Brazil

###### Abstract

We show that one dimensional (1D) topological superconductivity can be placed
in the context of phenomena associated with strongly correlated electron
systems. Here we propose a system consisting of a one-dimensional chain of
strongly correlated fermions placed on a superconducting (SC) substrate that
exhibits a spin-singlet extended $s$-wave pairing. Strong electron correlation
is shown to transform an extended $s$-wave SC into a topological SC that hosts
Majorana fermions. In contrast to the approaches based on mean-field
treatments, no Zeeman or exchange magnetic field is needed to produce such an
effect.

##  I Introduction

The topological superconductivity with Majorana zero modes (MZM) is of
fundamental scientific importance due to its proposed application in braiding-
based quantum computing [1]. The first theoretical proposal for the
realization of the one dimensional (1D) MZM was involved placing a 1D quantum
wire with spinless electrons on a $p$-wave superconductor [2]. However, from
the material science point of view this conceptually simple model is hard to
realize as a spinless electron does not exist in nature by default, and
$p$-wave superconductors are, at best, rare to find. Later several proposals
were put forward to eliminate these difficulties by ingeniously combining the
proximity induced $s$-wave superconductivity, the Rashba spin-orbit coupling
(RSOC), and the breaking of time reversal (TR) symmetry [3]. In these proposed
systems, RSOC was needed for the spin-momentum locking [4], and TR symmetry
breaking was needed to create, in effect, the spinless electrons. Using
aforementioned ideas, broadly three types of platforms have been engineered
till now, by placing [5]: (i) topological insulators on a superconductor [6,
7], (ii) semiconductors with strong spin-orbit coupling on a superconductor
[8], or (iii) chains of magnetic atoms on superconductor [9, 10].

In heterostructures, when the topological insulator is placed over the
superconductor, the 1D helical edge states provide the needed spin-momentum
locking [7]. The 1D edge states occur due to RSOC. The TR symmetry in these
heterostructures can be broken, either by applying a magnetic field [11, 12,
13], or by either doping with magnetic atoms [14], or by placing the magnetic
materials in the vicinity of the topological insulators [15, 16]. In the case
of a semiconductor on superconductor heterostructures usually a semiconductor
wire with strong RSOC and magnetic field are used to satisfy the
aforementioned requirements to generate MZM [17, 8]. For devices with a
magnetic ad-atom chain on a superconductor, the Yu-Shiba-Rusinov (YSR) states
induce spin polarized subgap in the parent superconductors [10, 9]. If the
magnetic ad-atoms are placed very close to each other the YSR states
hybridize, and the hopping between neighboring states ad-atom broadens the
particle-hole symmetric YSR pair states into bands. Once the band formed by
the positive and negative energies overlap at the center of the
superconducting gap, the $p$-wave pairing correlations will reopen the gap and
turn the ad-atom chain into a 1D topological superconductor with MZM . The
$p$-wave correlation can be induced either by RSOC in the parent
superconductor, or by the helical spin states on the magnetic ad-atom chain
[18].

From the experimental side, recent claims of observed MZM in topological
insulators (Bi/Nb [19], Bi2Te3 [20, 11], Bi2Se3 [21, 22], and Bi2Te3-xSex
[14]) have already been made. For semiconductors on superconductor platforms
usually InAs and InSb on superconductors have been extensively used in the
experiments [8]. Regarding the magnetic ad-atoms on superconductors, the chain
of Fe atoms on either Pb [10], Re [23], or Ta [24] substrates have shown signs
of the MZM. Despite these experimental successes, several concerns still
remain which have to do mainly with the requirements of the strong RSOC and
the external magnetic field. The former limits the candidate materials which
can be placed over the superconductor, while the latter limits the possible
superconducting substrates, although, moreover the crystal symmetry
consideration greatly shrink the possible superconductors supporting
topological phases [25]. The external magnetic field is the principal
hindrance, because most experiments use $s$-wave superconductors for which
might have their superconductivity destroyed in a strong enough magnetic
field. Hence, naturally the question arises, can we get rid of the constraint
imposed by the magnetic field? It is the goal of this article to suggest a
different route to realizing the MZM without making use of an external
magnetic field.

In a strong magnetic field the spins of the electrons are aligned along the
direction of the applied field. The Pauli constraint prohibits the occupation
of more than one fermion in a single quantum state. Hence no more than one
electron with the same spin can occupy a single site. Physically, it is the
same as having spinless fermions. Therefore one way of realizing the spinless
fermions is through the application of a magnetic field. Another way to
produce such an effect is to rely on the presence of a large on-site Coulomb
repulsion. In such a strong correlation regime each lattice site becomes at
most singly occupied, and the charge and spin degrees of freedom separate from
each other. The resulting fermion is naturally spinless and this is precisely
what we need. The prototype model for such a behavior is the $U\to\infty$
Hubbard model [26, 27, 28]. The main reason why such an approach was not used
before, was probably due to inadequacy of the mean-field methods to treat
strong correlation. In this paper we show that the strong electron correlation
can indeed drive the 1D system of lattice electrons into the topological phase
as the Kitaev’s toy model suggests [2]. This consideration is based on the
representation of the underlying dynamics in terms of Hubbard operators that
encode strong electron correlation effects. The basic requirement of no local
double occupancy is taken into account rigorously prior to any mean-field
treatments. To this end the $su(2|1)$ coherent-state action is employed, where
the $su(2|1)$ superalgebra incorporates the full set of Hubbard operators
[29]. We propose a device, in which a strongly correlated nanowire is placed
over an extended $s$-wave superconductor. Usually iron based superconductors
fulfill this requirement [30, 31, 32]. The details of experimental realization
is discussed in Sec. VII.

It should be mentioned that an analogous idea was presented before [33].
However, there instead of a strongly correlated nanowire, a semiconducting
nanowire with strong RSOC on an extended $s$-wave superconductor was proposed
in its place. Hence, the problem of using magnetic field still persisted
(although critical magnetic field of iron-based superconductors are quite high
$\sim 50\>T$ [34]). In another analogous work it was predicted that a
semiconducting nanowire with $C_{n}$ ($n$>4) rotational symmetry and non-zero
electronic angular momentum ($l\neq 0$) develops MZM when Coulomb repulsion is
high enough [35]. In this case magnetic field, RSOC, and even
superconductivity is not needed either.

This article is structured as follows. In Sec. II we found the effective
Hamiltonian for the physical system shown in Fig. 1. In Sec. III we derived
the expression of the Hamiltonian when the spiral spin field is present in the
wire. In Sec. IV and V we discussed the case when RSOC in the wire is both
absent and present respectively. In Sec. VI we calculated the thermodynamic
free energy of the nanowire as a function of system parameters, and discuss
its consequences. In Sec. VII we propose experiments to observe the Majorana
fermions, and the schematics for device design. In Sec. VIII we summarized our
results.

##  II Model

Figure 1: A schematic of the system under consideration. A strongly correlated
1D nanowire is placed over a superconductor with an extended $s$-wave order
parameter. Superconductivity is induced in the nanowire due to the proximity
effect. The radius of the wire ($r$) is assumed to be smaller than the Cooper
pair coherence length in the wire ($\xi_{wire}$): $r\lessapprox\xi_{wire}$.
The Majorana fermions appear at the end of the nanowire (stars) if a non-
colinear spin texture is present in the wire.

The schematics of the system under consideration is shown in Fig. 1. Here, a
strongly correlated 1D nanowire with RSOC is placed on a superconducting
substrate. Due to the proximity of the wire to the superconductor we assume
tunneling of Cooper pair into the wire (tunneling of single electron is
neglected due to the reasons discussed further). Under these assumptions, the
total Hamiltonian of the wire will be:

$\displaystyle H$ $\displaystyle=H_{1D}+H_{\alpha}+H_{\Gamma};$ (1) where,
$\displaystyle H_{1D}$ $\displaystyle=-t\sum\limits_{i,\sigma}X_{i}^{\sigma
0}X_{i+1}^{0\sigma}+\text{H.c}+\mu\sum\limits_{i}X_{i}^{00};$ $\displaystyle
H_{\alpha}$
$\displaystyle=-\imath\alpha\sum\limits_{i}X_{i}^{\sigma^{\prime}0}X_{i+1}^{0\sigma}-X_{i}^{\sigma
0}X_{i+1}^{0\sigma^{\prime}}+\text{H.c};$ $\displaystyle H_{\Gamma}$
$\displaystyle=\Gamma\sum\limits_{i,\sigma}X_{i}^{\sigma
0}d_{i\sigma}+\text{H.c}.$

Due to the strong correlation in the wire at most a single electron can occupy
a single site. Hence, the Hubbard operators $X_{i}^{\sigma\sigma^{\prime}}$
are used in Eq. (1) to construct Hamiltonian 111 The Hubbard operators
$X_{i}^{pq}\equiv\bra{p}\ ket{q}$ describes the transition at site from
$\ket{p}$ state to $\ket{q}$ state. Under strong correlation there are three
different states possible, i.e. states with up-spin $\ket{\uparrow}$, down-
spin $\ket{\downarrow}$ and empty site $\ket{0}$. Hence, for example
$X_{i}^{\uparrow 0}$ represent the transition from empty state to the up-spin
state at $i$-th site. In terms of usual electron creation
($c_{i\sigma}^{\dagger}$), annihilation ($c_{i\sigma}$), and number
($n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$) operators, the Hubbard
operators are represented as:  $\displaystyle
X_{i}^{0\sigma}=c_{i\sigma}(1-n_{i\sigma^{\prime}}),\quad X_{i}^{\sigma
0}=(1-n_{i\sigma^{\prime}})c_{i\sigma}^{\dagger}$ $\displaystyle
X_{i}^{00}=1-\sum\limits_{\sigma}\left(1-n_{i\sigma^{\prime}}\right)c_{i\sigma}^{\dagger}c_{i\sigma}^{\dagger}\left(1-n_{i\sigma^{\prime}}\right).$
Above definition of Hubbard operators shows that the no double occupancy
condition is satisfied. . In the total Hamiltonian $H$ the $H_{1D}$ part
contains the kinetic energy and chemical potential of the wire. Here, $t$ is
the electron hopping factor, and $\mu$ is the chemical potential.
$X_{i}^{\sigma 0}$ is the Hubbard operator which creates an electron at $i$-th
site with spin $\sigma$ at site. It should be remembered that the electron is
created only if the $i$-th site was empty initially. $X_{i}^{00}$ gives the
occupation number of the $i$-th site. $H_{\alpha}$ represents the RSOC in the
wire. $\alpha$ is the Rashba parameter which depends on the crystal electric
field generated due to the breaking of the inversion symmetry [37];
quantitatively, $\alpha\sim 1$ eV [4]. $H_{\Gamma}$ corresponds to the
tunneling of electrons from the superconducting substrate to the wire with
amplitude $\Gamma$. $d_{i\sigma}$ is the electron annihilation operator with
spin $\sigma$ at $i$-th site of the superconducting substrate. $X_{i}^{\sigma
0}d_{i\sigma}$ means the annihilation of the electron in the substrate and
creation of the same in the wire. It is important to note that the Hubbard
operators $X_{i}^{\sigma 0}$ transform themselves in the lowest (fundamental)
representation of the $SU(2)$ group [38], so that Eq. (1) represents a
rotationally invariant interaction.

As the dimension and physical size of the wire is smaller than the substrate,
it is natural to assume that the Fermi level of the superconductor greatly
exceeds that of the wire, $k_{\text{F}}^{\text{SC}}\gg
k_{\text{F}}^{\text{wire}}$. The large difference in Fermi levels allows the
hybridization between them to be controlled mainly by the superconducting
energy gap ($\Delta_{\text{SC}}$) and $\Gamma$. For small $\Gamma$, i.e. for
$\Gamma k_{\text{F}}^{\text{SC}}\ll\Delta_{\text{SC}}$, we can treat
$H_{\Gamma}$ peturbatively [3]. The first order perturbation terms can be
neglected, due to the suppression of the single electron tunneling 222 Due to
induced superconducting gap $\Delta$ at the Fermi level of the wire the energy
penalty for exciting a single electronic state from the top of the valence
band to the bottom of the conduction band in the wire is too high. It is due
to our assumption $k_{\text{F}}^{\text{SC}}\gg k_{\text{F}}^{\text{wire}}$,
which makes sure that $E_{F}^{\text{wire}}\sim\Delta$. . The second order
perturbation generates an effective Cooper pair hopping [40, 41]. Due to the
presence of strong correlation one can discard the single site spin-singlet
$s$-wave pairing $\left(\Delta_{i,i}^{\uparrow\downarrow}\right)$ contribution
333 A pure spin-singlet $s$-wave pairing $\Delta^{\uparrow\downarrow}_{i,i}$
does not contribute into $\delta H$ since it is multiplied by a corresponding
bilinear form of the Hubbard operators that disappear at the coinciding
lattice sites, e.g., $X^{\sigma 0}_{i}X^{\sigma^{\prime}0}_{i}=0.$ This is a
direct manifestation of strong electron correlation that prohibits on-site
double electron occupancy due to infinitely strong on-site Coulomb repulsion.
In this case, the proximity-induced pairing is only possible provided the bulk
superconductor develops either the spin-triplet $p$-wave pairing or an
extended $s$-wave one. A pure $s$-wave pairing is totally excluded. .
Therefore, on neighboring sites two types of spin pairing are possible: (i)
spin-singlet extended $s$-wave pairing and (ii) spin-triplet $p$-wave pairing.
The corresponding second order perturbed Hamiltonian is:

$\displaystyle\delta H_{\Gamma}\propto$
$\displaystyle\sum\limits_{i}\Delta_{i,i+1}^{\sigma\sigma^{\prime}}\>\left(X_{i}^{\sigma
0}X_{i+1}^{\sigma^{\prime}0}-X_{i}^{\sigma^{\prime}0}X_{i+1}^{\sigma
0}\right)$ (2)
$\displaystyle+\sum\limits_{i}D_{i,i+1}^{\sigma\sigma^{\prime}}\>\left(X_{i}^{\sigma
0}X_{i+1}^{\sigma^{\prime}0}+X_{i}^{\sigma^{\prime}0}X_{i+1}^{\sigma
0}\right)$
$\displaystyle\quad+\sum\limits_{i,\sigma}D_{i,i+1}^{\sigma\sigma}\>X_{i}^{\sigma
0}X_{i+1}^{\sigma 0}+\text{H.c.}$

Here, $D_{i,i+1}^{\sigma\sigma}$, $D_{i,i+1}^{\sigma\sigma^{\prime}}$ are the
$p$-wave order parameters, and $\Delta_{i,i+1}^{\sigma\sigma^{\prime}}$ is the
extended $s$-wave order parameter. In low energy limit we can approximate the
order parameters through ground state expectation values:

$\displaystyle D_{i,i+1}^{\sigma\sigma}$ $\displaystyle=\left\langle
d_{i,\sigma}d_{i+1,\sigma}\right\rangle$ (3) $\displaystyle
D_{i,i+1}^{\sigma\sigma^{\prime}}$ $\displaystyle=\left\langle
d_{i,\sigma}d_{i+1,\sigma^{\prime}}+d_{i,\sigma^{\prime}}d_{i+1,\sigma}\right\rangle$
$\displaystyle\Delta_{i,i+1}^{\sigma\sigma^{\prime}}$
$\displaystyle=\left\langle
d_{i,\sigma}d_{i+1,\sigma^{\prime}}-d_{i,\sigma^{\prime}}d_{i+1,\sigma}\right\rangle.$

In Eq. (3) the singlet $p$-wave order parameters are symmetric in spin
$\left(D_{i,i+1}^{\sigma\sigma^{\prime}}=D_{i,i+1}^{\sigma^{\prime}\sigma}\right)$,
but anti-symmetric in space
$\left(D_{i,i+1}^{\sigma\sigma^{\prime}}=-D_{i+1,i}^{\sigma\sigma^{\prime}}\right)$.
On the other hand the extended $s$-wave order parameters are anti-symmetric in
spin
$\left(\Delta_{i,i+1}^{\sigma\sigma^{\prime}}=-\Delta_{i,i+1}^{\sigma^{\prime}\sigma}\right)$,
but symmetric in space
$\left(\Delta_{i,i+1}^{\sigma\sigma^{\prime}}=\Delta_{i+1,i}^{\sigma\sigma^{\prime}}\right)$.
Usually $p$-wave pairing in superconductor is quite rare and unstable, on the
other hand, the extended $s$-wave pairing is predicted in high $T_{c}$ iron
based superconductors [43, 32]. Hence, from now on we assume that, the
substrate develops only extended $s$-wave pairing. Considering above
discussion we can approximate:

$\delta H_{\Gamma}=\Delta\sum\limits_{i}X_{i}^{\sigma
0}X_{i+1}^{\sigma^{\prime}0}-X_{i}^{\sigma^{\prime}0}X_{i+1}^{\sigma
0}+\text{H.c.}$ (4)

Here $\Delta$ is the induced superconducting gap in the wire. Its value can be
estimated through
$\Delta\approx\Gamma^{2}/k_{\text{F}}^{\text{SC}}\Delta_{\text{SC}}$. We have
assumed spatially independent order parameter
$\Delta_{i,i+1}^{\sigma\sigma^{\prime}}\equiv\Delta_{\text{SC}}$. Combining
Eq. (1) and Eq. (4) we write the effective Hamiltonian of the wire:

$\displaystyle H_{\text{eff}}=$
$\displaystyle-t\sum\limits_{i,\sigma}X_{i}^{\sigma 0}X_{i+1}^{0\sigma}$ (5)
$\displaystyle\quad-\imath\alpha\sum\limits_{i}X_{i}^{\sigma^{\prime}0}X_{i+1}^{0\sigma}-X_{i}^{\sigma
0}X_{i+1}^{0\sigma^{\prime}}$
$\displaystyle\qquad+\Delta\sum\limits_{i}X_{i}^{\sigma
0}X_{i+1}^{\sigma^{\prime}0}-X_{i}^{\sigma^{\prime}0}X_{i+1}^{\sigma
0}+\text{H.c.}$ $\displaystyle\qquad\quad+\mu\sum\limits_{i}X_{i}^{00}.$

To solve the Hamiltonian, Eq. (5), the $su(2|1)$ path integral representation
of the partition function will be used. This is possible due to the existence
of an injective mapping between Hubbard operators $X^{\sigma\sigma^{\prime}}$
and their $su(2|1)$ coherent-state symbols [44, 45, 29] (also known as Berezin
symbols [46]):

$X^{\sigma\sigma^{\prime}}(z,\xi)\equiv\bra{z,\xi}X^{\sigma\sigma^{\prime}}\ket{z,\xi}.$
(6)

Here $z$ and $\xi$ are the complex even and odd Grassmann parameters
resepctively 444 The state $\ket{z,\xi}$ is defined as:
$\displaystyle\ket{z,\xi}$
$\displaystyle\equiv\left(1+\bar{z}z+\bar{\xi}\xi\right)^{-1/2}\exp\left(zX^{\downarrow\uparrow}+\xi
X^{0\uparrow}\right)\ket{\uparrow}$
$\displaystyle=\left(1+\bar{z}z+\bar{\xi}\xi\right)^{-1/2}\left(\ket{\uparrow}+z\ket{\downarrow}+\xi\ket{0}\right).$
Above representation was found by acting with the lowering superspin operators
$X^{\downarrow 0}$ and $X^{\downarrow\uparrow}$ on the highest weight state
$\ket{\uparrow}$. For a complete exposition on this subject please see Ref.
[44, 49]. . Physically, $z$ represents the spinfull bosonic fields, and $\xi$
represents the spinless charged fermionic fields [48, 49]. In this new
representation and using the change of variables

$\xi\to\frac{\xi}{\sqrt{1+\left|z\right|^{2}}},\quad z\to z,$

the effective Hamiltonian Eq. (5) will be:

$\displaystyle H_{\text{eff}}(z,\xi)=$
$\displaystyle-t\sum\limits_{i}\xi_{i}\bar{\xi}_{i+1}a_{i,i+1}+\imath\alpha\sum\limits_{i}\xi_{i}\bar{\xi}_{i+1}\alpha^{\ast}_{i,i+1}$
(7)
$\displaystyle\qquad-\Delta\sum\limits_{i}\xi_{i}\xi_{i+1}\Delta^{\ast}_{i,i+1}+\text{H.c.}+\mu\sum\limits_{i}\bar{\xi}_{i}\xi_{i},$

where,

$a_{i,i+1}\equiv\frac{1+\bar{z}_{i}z_{i+1}}{\sqrt{\left(1+\left|z_{i}\right|^{2}\right)\left(1+\left|z_{i+1}\right|^{2}\right)}},$
(8a)
$\alpha^{\ast}_{i,i+1}\equiv\left[\frac{z_{i}-\bar{z}_{i+1}}{\sqrt{\left(1+\left|z_{i}\right|^{2}\right)\left(1+\left|z_{i+1}\right|^{2}\right)}}\right]^{{\ast}},$
(8b)
$\Delta^{\ast}_{i,i+1}\equiv\left[\frac{z_{i}-z_{i+1}}{\sqrt{\left(1+\left|z_{i}\right|^{2}\right)\left(1+\left|z_{i+1}\right|^{2}\right)}}\right]^{{\ast}}.$
(8c)

In the second quantization language $\xi_{i}$ ($\bar{\xi}_{i}$) creates
(annihilates) a spinless fermion at $i$-th site. Similarly, $z_{i}$
($\bar{z}_{i}$) creates (annihilates) a spinfull boson at $i$-th site. In Eq.
(7) the $\xi$ and $z$ degrees of freedom emerge as low-energy degrees of
freedom.

One can absorb the electron hopping and RSOC terms in Eq. (7) into a single
effective term. In this case the Hamiltonian, Eq. (7), transforms to:

$\displaystyle H_{\text{eff}}(z,\xi)=$
$\displaystyle-\sum\limits_{i}\bar{\xi}_{i}\xi_{i+1}\left(t\>a_{i,i+1}^{\ast}+\imath\alpha\>\alpha_{i,i+1}\right)$
(9)
$\displaystyle\quad-\Delta\sum\limits_{i}\bar{\xi}_{i}\bar{\xi}_{i+1}\Delta_{i,i+1}+\text{H.c.}+\mu\sum\limits_{i}\bar{\xi}_{i}\xi_{i}.$

In effect, electron hopping acquires a phase which is a complex function of
the spin field. One can define the modified hopping as:

$\tilde{t}_{i}=\sqrt{\left(t\>a_{i,i+1}^{\ast}\right)^{2}+\left(\alpha\>\alpha_{i,i+1}\right)^{2}}\>\exp\left[\imath\>\arctan\left(\frac{\alpha\>\alpha_{i,i+1}}{t\>a_{i,i+1}^{\ast}}\right)\right].$
(10)

##  III Hamiltonian for spiral spin fields

While treating the real systems, it is instructive physically to treat the
dynamic spin fields $z_{i}(t)$ in terms of the spin covariant symbols [44]:

$\displaystyle S_{i}^{+}=\frac{z_{i}}{1+\left|z_{i}\right|^{2}},\quad
S_{i}^{-}=\frac{\bar{z}_{i}}{1+\left|z_{i}\right|^{2}},\quad
S_{i}^{z}=\frac{1}{2}\left(\frac{1-\left|z_{i}\right|^{2}}{1+\left|z_{i}\right|^{2}}\right).$
(11)

The corresponding expressions for $S_{i}^{x}=S_{i}^{+}+S_{i}^{-}$ and
$S_{i}^{y}=-\imath\left(S_{i}^{x}-S_{i}^{y}\right)$ can be easily found. In
strongly correlated electronic systems the dynamics of the effective spin
field is much slower than that of the fermionic fields [50]. Because of this
we can apply the mean-field approximation to the spin system (from the point
of view of the fermions). In other words, the picture where spinless fermions
travel in the static background spin field is fully applicable. One of the
main difference between our approach and the previous works related to the 1D
topological superconductors [13, 51, 9, 18] is that, we first derive our
effective Hamiltonian taking full account of the strong correlation effects
and only then we apply mean-field approximation. As opposed to that in
previous works the mean-field approximation is applied to the Hamiltonian
right from the beginning.

We consider a spin field of the form:

$\vec{S}_{i}=\left(S_{i}^{x},S_{i}^{y},S_{i}^{z}\right)=\frac{1}{2}\left(\cos\theta_{i},\sin\theta_{i},0\right).$
(12)

Physically it represents the rotating spin vector on the $xy$ plane; $z$
component of the spin is zero. $\theta_{i}$ is the angle on the _xy_ plane. It
is related to the spatial coordinate through relation:
$\theta_{i}=\vec{q}\cdot\vec{r}_{i}$; with $\vec{q}$ being the spin wave
vector and $\vec{r}_{i}$ is the position vector. Substituting Eq. (12) in Eq.
(11) it is easy to see that $z_{i}=e^{\imath\theta_{i}}$. Further, using this
$z_{i}$ in Eq. (8a), (8b), and (8c) one will find:

$\displaystyle a_{i,i+1}^{\ast}$
$\displaystyle=\quad\mathrm{e}^{-\imath\left(\theta_{i+1}-\theta_{i}\right)/2}\quad\cos\frac{\left(\theta_{i+1}-\theta_{i}\right)}{2},$
(13) $\displaystyle\alpha_{i,i+1}$
$\displaystyle=\>\>\>\imath\mathrm{e}^{-\imath\left(\theta_{i+1}-\theta_{i}\right)/2}\quad\sin\frac{\left(\theta_{i+1}+\theta_{i}\right)}{2},$
$\displaystyle\Delta_{i,i+1}$
$\displaystyle=-\imath\mathrm{e}^{\imath\left(\theta_{i+1}-\theta_{i}\right)/2}\quad\>\>\sin\frac{\left(\theta_{i+1}-\theta_{i}\right)}{2}\>\mathrm{e}^{\imath\theta_{i}}.$

Substituting Eq. (13) in Eq. (9), and soon after that making the gauge
transformations
$\xi_{i}\to\xi_{i}\mathrm{e}^{\imath\left(\theta_{i}/2\>-\>\pi/4\right)}$ the
effective Hamiltonian can be written as:

$\displaystyle H_{\text{eff}}=$
$\displaystyle-t\sum\limits_{i}\bar{\xi}_{i}\xi_{i+1}\cos\frac{\left(\theta_{i+1}-\theta_{i}\right)}{2}$
(14)
$\displaystyle\quad+\alpha\sum\limits_{i}\bar{\xi}_{i}\xi_{i+1}\sin\frac{\left(\theta_{i+1}+\theta_{i}\right)}{2}$
$\displaystyle\qquad-\Delta\sum\limits_{i}\bar{\xi}_{i}\bar{\xi}_{i+1}\sin\frac{\left(\theta_{i+1}-\theta_{i}\right)}{2}+\text{H.c.}$
$\displaystyle\qquad\quad+\mu\sum\limits_{i}\bar{\xi}_{i}\xi_{i}.$

Figure 2: Spiral spin structure found from Eq. (12) by taking
$\theta_{i+1}-\theta_{i}=\theta$ as constant.

In a limiting case one can assume that the rotation of the spin through
neighboring sites $\theta_{i+1}-\theta_{i}\equiv\theta$ is constant. This spin
structure is known as the spiral spin structure. A schematic of it is shown in
Fig. 2. The Hamiltonian for the spiral spin structure then becomes:

$\displaystyle H_{\text{eff}}=$
$\displaystyle-t\sum\limits_{i}\bar{\xi}_{i}\xi_{i+1}\cos\frac{\theta}{2}+\alpha\sum\limits_{i}\bar{\xi}_{i}\xi_{i+1}\sin\left(\theta_{i}+\frac{\theta}{2}\right)$
(15)
$\displaystyle\quad-\Delta\sum\limits_{i}\bar{\xi}_{i}\bar{\xi}_{i+1}\sin\frac{\theta}{2}+\text{H.c.}+\mu\sum\limits_{i}\bar{\xi}_{i}\xi_{i}.$

The Rashba term is now real and oscillating.

Before further discussion, a explanation of why we considered the spiral spin
texture is necessary. The spiral spin texture is not new and it has been
observed in other systems, especially, in heterostructures where magnetic ad-
atoms are placed over superconducting substrates [9, 10]. In these systems
strong RSOC [52], together with RKKY interaction [53] are responsible for
generating spiral spin texture. In fact, it can be shown that the tight
binding Hamiltonian of a 1D wire with spin-orbit coupling is equivalent to the
tight binding Hamiltonian with a rotating magnetic field [54]. This result is
valid only when the _e-e_ interaction is small ($U\ll t$). When the _e-e_
interaction is strong the Hamiltonian cannot be transformed in this way. This
is due to the fractionalization of the electronic degrees of freedoms into
separate spinless fermionic field (charge) and spinful bosonic field (spin).

The mechanisms for emergence of the spiral spin texture in case of weak
interactions are not applicable to the case of strong correlations. Therefore
we suggest the following mechanisms and approximations for the emergence of
the spiral spin texture. For the strong _e-e_ interaction one can use the
Kondo lattice model, where in the limit of $J\gg t$ ($J$ being the Kondo
coupling) nearly localized effective spins emerge at individual sites behaving
as classical non interacting spins [55]. Physically we have the effective
system where at each lattice site sits a classical spin with $S=1/2$, and in
this spin background itinerant spinless fermions travel through the lattice.
The nearly localized spins create a dynamical spin field for which the
internal energy of the system is lowest.

Since the background spin field is generated dynamically, it is natural to
assume that, the system chooses the spin configuration for which the internal
energy is lowest. Keeping this in mind, for analysis of the properties of the
Hamiltonian, one can start with any non-colinear spin structure, however, for
a general complex spin textures — e.g. for skyrmions, hopfions etc — the
resulting Hamiltonian, Eq. (9), becomes too complex to be solved analytically.
Hence, some assumptions are in order. First, we assume that the _z_ component
of the spin is zero; physically, it means that the spin rotates only on the
$xy$ plane. This was motivated by the fact that the spin projection along _z_
-axis does not play any role in determining the topological properties of the
system [56]; this means that only the spin rotations on the $xy$ plane are
important. Secondly, we assume that the rotation of the spin projection on the
$xy$ plane between two neighbouring sites is constant. It is motivated by two
facts: (i) a constant rotation angle — which corresponds to the spiral spin
texture — has been observed in numerous experiments [8, 9, 10], (ii) in this
case the Hamiltonian takes a very simple form, which turns out to be solvable
analytically.

##  IV Absence of RSOC ($\alpha=0$)

(a)

(b)

(c)

(d)

Figure 3: (a,b,c) Dependence of trivial and topological phase on the angle of
the rotation of the spin through neighboring sites ($\theta$). The topological
phase appears when $\left|\mu\right|<2t\cos\left(\theta/2\right)$ is
satisfied. The range of parameters $\mu$ and $t$, for which topological phase
occurs, decreases with increase in $\theta$. (d) The condition
$\left|\mu/t\right|<2\cos\left(\theta/2\right)$ determining the topological
phase is plotted. The area for the topological phase in parameter space is
largest when spins on neighbouring atoms are almost aligned along same
direction ($\theta>0$). For ferromagnetic ordering ($\theta=0$) the
topological phase vanishes as the superconducting gap
$\left[\Delta\sin\left(\theta/2\right)\right]$ also vanishes. For anti-
ferromagnetic ordering ($\theta=\pi$) the topological phase vanishes, because
electron hopping becomes zero $\left[t\cos\left(\theta/2\right)\right]$. The
arrow marks on the boundary of the topological and trivial phase near
$\theta=0,\pi$ represent the fact that, topological phase survives as
$\theta\to 0$ and $\theta\to\pi$, however at $\theta=0,\pi$ topological phase
is absent.

First, we analyze the Eq. (15) when the RSOC is absent, i.e. we take
$\alpha=0$. The effective Hamiltonian in this case becomes:

$\displaystyle H^{\text{Kitaev}}_{\text{eff}}=$
$\displaystyle-t\sum\limits_{i}\bar{\xi}_{i}\xi_{i+1}\cos\frac{\theta}{2}-\Delta\sum\limits_{i}\bar{\xi}_{i}\bar{\xi}_{i+1}\sin\frac{\theta}{2}$
(16) $\displaystyle\qquad+\text{H.c.}+\mu\sum\limits_{i}\bar{\xi}_{i}\xi_{i}.$

This Hamiltonian does not break time reversal symmetry, as
$H^{\text{K}}_{\text{eff}}=\left(H^{\text{K}}_{\text{eff}}\right)^{\ast}$. In
this sense it is analogous to the Kitaev’s toy model of topological
superconductivity [2]555The Kitaev model is represented as:
$H^{\text{K}}=\sum\limits_{i}-tf_{i}f^{\dagger}_{i+1}+\Delta
f^{\dagger}_{i}f^{\dagger}_{i+1}+\text{H.c.}+\mu f_{i}f_{i}^{\dagger}.$ It is
also symmetric under time reversal transformation. . In fact, analogous to the
Kitaev’s toy model, we also find that, both the trivial and the topological
phases depend on the specific values of $t$, $\Delta$, $\mu$, and $\theta$.
When there is no superconductivity, the system is gapped for
$\left|\mu\right|>2t\cos(\theta/2)$; but it is gapless when
$\left|\mu\right|<2t\cos(\theta/2)$. However when superconductivity is itself
present, due to the particle hole symmetry, a topological phase appears for
$\left|\mu\right|<2t\cos(\theta/2)$. The phase diagram of the
$H_{\text{eff}}^{\text{Kitaev}}$, and its dependence on the $\theta$ —
physically, $\theta$ represents the rotation of the spin projection through
neighboring sites — is shown in Fig. 3. The mean difference between the phase
diagrams shown in Fig. 3, and the original phase diagram of the Kitaev chain
[2], is the emergence of the new parameter $\theta$ in the effective
Hamiltonian. It can be observed that with the increase in $\theta$ the area of
the topological phase in the $\mu$-$t$ phase plane decreases. This is due to
the decrease in the electron hopping parameter $t\cos\left(\theta/2\right)$.
The boundary between the topological and the trivial phase in the
$\left|\mu\right|/t$-$\theta$ phase space is given by the condition
$\left|\mu\right|/t<2\cos(\theta/2)$. The corresponding phase diagram is shown
in Fig. 3d . In this figure we observe that, when $\theta=0$, i.e. for the
ferromagnetic state, the system behaves either as a simple metal or as an
insulator depending on the relative value of $\mu$ 666If
$-2t\cos\left(\theta/2\right)\cos k<\mu<0$ then, due to unfilled band the
system behaves as metal. However, if $\mu\approx 0$ and the Fermi energy lies
inside the band gap, and the system behaves as insulator.. The reason, why the
system looses its topological property at $\theta=0$, is due to the
disappearance of the superconducting gap. In effect, the system looses the
particle-hole symmetry, which in the first place gives rise to the topological
phase. This is expected physically since the underlying extended $s$-wave
superconducting state should not survive in a ferromagnetic background. On the
other end when $\theta=\pi$, i.e. for an antiferromagnetic state, the system
behaves as an usual Mott insulator. The hopping of the electrons becomes
identically zero and this represents a state in which the fermionic particle
are frozen in their lattice sites. From the band theory point of view, the
superconducting gap never closes even if $\mu$ is modified, when $\theta=\pi$.
This means that the MZM can not be created; since in such a state Majorana
fermions only emerge during the closing and the reopening of the band gap.

(a)

(b)

(c)

(d)

Figure 4: (a,b,c) Evolution of the energy dispersion, Eq. (18), as the
superconducting gap is kept constant ($\Delta=0.7t$), and chemical potential
is decreased ($\mu$ changes from $2t$ to $0$) for $\theta=\pi/4$,
$\theta=\pi/2$, and $\theta=\pi/4$. (d) The values of $|\mu|/t$ at which the
bulk gap closes at $k=0$ for a given value of $\theta$. The values are found
by using the condition $\theta=2\arccos\left(|\mu|/2t\right)$.

Next, we discuss the effect of $\theta$ on the energy spectrum. Using Nambu
spinors in momentum space the Hamitonian Eq. (16) can be written in the usual
Bogoliubov-de Gennes form:

$\displaystyle
H_{\text{eff}}^{\text{Kitaev}}(k)=\frac{1}{2}\sum\limits_{k\in\text{BZ}}\Psi_{k}^{\dagger}\>\mathcal{H}_{k}\>\Psi_{k},$
(17) where,
$\displaystyle\Psi_{k}^{\dagger}\equiv\left[\xi_{k},\bar{\xi}_{-k}\right],$
$\displaystyle\mathcal{H}_{k}\equiv\begin{bmatrix}2t\cos\frac{\theta}{2}\cos
ka-\mu&2\Delta\sin\frac{\theta}{2}\sin ka\\\ 2\Delta\sin\frac{\theta}{2}\sin
ka&2t\cos\frac{\theta}{2}\cos ka-\mu\\\ \end{bmatrix}.$

Here, $k$ is the crystal momentum, and $a$ is the distance between two near
neighbour lattice sites. The single mode energy dispersion can now be easily
calculated:

$\displaystyle E(k)=\pm\sqrt{\left(2t\cos\frac{\theta}{2}\cos
ka-\mu\right)^{2}+\left(2\Delta\sin\frac{\theta}{2}\sin ka\right)^{2}}.$ (18)

If one compares the energy dispersion of Eq. (18), with the energy dispersion
of the Kitaev’s toy model [see Eq. (13) of Ref. 2] again, there is only single
difference; in Eq. (18) the extra parameter $\theta$ appears explicitly in our
energy dispersion. The dependence of energy dispersion for different
combination of the $\Delta$, $\mu$ and $\theta$ is shown in Fig. 4 and 5.

In Figs. 4a, 4b, and 4c the evolution of the energy dispersion are shown for
$\theta=\pi/4$, $\pi/2$, and $3\pi/4$, while the superconducting gap is kept
constant $(\Delta=0.7t)$, and the chemical potential is varied as a function
of $t$ $(\mu=2t$, $1.5t$, $0.7t$, $0)$. As is well known one of the properties
of the topological superconductors is its sensitivity to the closing and to
the reopening of the energy gap at $k=0$, $\pm\pi$ when we tune $\mu$. In the
Kitaev’s toy model this takes place precisely at $\mu=\pm 2t$. However, in our
case the gap closes for $\mu=\pm 2t\cos\left(\theta/2\right)$ — it evolves
with $\theta$. The values of $\mu/t$ where the gap closes, for a given
$\theta$ is plotted in Fig. 4d. Similarly, in Figs. 5a, 5b, and 5c the energy
dispersion are shown for $\theta=\pi/4$, $\pi/2$, and $3\pi/4$, while the
chemical potential is kept constant $(\mu=1t)$, and the superconducting gap is
varied $(\Delta=0.1t$, $0.3t$, $0.5t$, $0.6t)$. As expected for $\theta=\pi/4$
the spectrum is defined by the electron hopping term
$\left[2t\cos\left(\theta/2\right)\cos k\right]$; the superconducting term has
little effect as $\left[2t\sin\left(\theta/2\right)\sin k\right]$ is very
small. However, for $\theta=3\pi/4$ the spectrum is defined by the
superconducting term.

(a)

(b)

(c)

(d)

Figure 5: (a,b,c) Evolution of the energy dispersion, Eq. (18), as the
chemical potential is kept constant ($\mu=1t$), and the superconducting gap is
increased ($\Delta=0.1t,0.3t,0.5t,0.6t$) for $\theta=\pi/4$, $\theta=\pi/2$,
and $\theta=\pi/4$. (d) Comparison of the energy dispersion for the strongly
correlated Hamiltonian, Eq. (16), and the Kitaev’s toy model [2] at $\mu=1t$,
$\Delta=0.6t$, and $\theta=\pi/4$.

A comment is in order regarding the related work of Ref. [55]. If one compares
our Hamiltonian, Eq. (17) with Eq. (20) of the Ref. [55], they are identical.
However, there are a few fundamental differences in what the two models truly
represent. In Ref. [55] the heterostructures are produced by the magnetic ad-
atoms on a superconductor, while in our work we consider a strongly correlated
1D nanowire on the superconductor. In the magnetic ad-atom superconductor
heterostructures the whole physics is directly related to the YSR states,
while in our system the physics depends on the proximity effect induced by the
superconductivity and by the chemical potential of the nanowires. In this
sense they are completely different systems. A priori it is not obvious they
can be represented by one and the same low-energy theory. This only happens in
the limit where on-site Coulomb repulsion is infinite. If one goes away from
that limit, the relevant theories differ from each other accordingly. In our
work we have also considered the effect of the RSOC, which was not taken into
account in Ref. [55].

##  V Presence of RSOC ($\alpha\neq 0$)

Figure 6: Dependence of trivial and topological phase on $\theta$ for
different Rashba interaction $\alpha=0.1t$, $0.2t$, $0.7t$, and $0.9t$. The
plotted condition for topological phase is found from Eq. (20):
$\left|\mu\right|/t=2\cos(\theta/2)-\left|\alpha\right|/t$.

When the RSOC is present, i.e. when $(\alpha\neq 0)$, we can observe in the
$|\mu|/t$-$\theta$ phase plane the region in which the topological phase
manifest is contracted with the increase in the value of $\alpha$. To show
this more explicitly we rewrite the effective Hamiltonian, Eq. (14), as:

$\displaystyle H_{\text{eff}}(z,\xi)=$
$\displaystyle-\sum\limits_{i}\tilde{t}_{i}\>\bar{\xi}_{i}\xi_{i+1}-\Delta\sum\limits_{i}\bar{\xi}_{i}\bar{\xi}_{i+1}\sin\frac{\theta}{2}$
(19) $\displaystyle\qquad+\text{H.c.}+\mu\sum\limits_{i}\bar{\xi}_{i}\xi_{i},$
where
$\displaystyle\tilde{t}_{i}\equiv\left[t\cos\left(\frac{\theta}{2}\right)-\alpha\sin\left(\theta_{i}+\frac{\theta}{2}\right)\right]$

is the modified hopping term. Interestingly, it is spatially oscillating
through $\sim\alpha\sin\theta_{i}$. Eqs. (19) and (16) are analogous to each
other, apart from the modified hopping parameter. Therefore the boundary of
the topological phase can be established by the same process — one can
canonically represent the fermionic $\xi_{i}$ operators in terms of the
Majorana fermions, and apply the limiting conditions to find the associated
topological phase boundary (see. Ref. [2]). The resulting condition for the
topological phase is, then,
$\left|\mu\right|<\text{Min.}\left(2\tilde{t}_{i}\right)$. Explicitly we can
write:

$\left|\mu\right|<2\left(t\cos\frac{\theta}{2}-\left|\alpha\right|\right).$
(20)

The corresponding values for $\mu$, $t$, and $\theta$ which determine both the
topological and trivial phases for different values of $\alpha$ are displayed
in Fig. 6. We can note that, the parameter range for the topological phase
decreases gradually as we increase the magnitude of $\alpha$. When $\alpha\geq
t\cos\left(\theta/2\right)$ the topological phase is completely absent.
Therefore, the RSOC and non alignment of the spin vectors negatively affect
the topological phase in a strongly correlated regime. We can conclude that
for strongly correlated wire the RSOC does not add new physics to the problem.

##  VI Internal Energy

(a)

(b)

(c)

(d)

Figure 7: (a,b,c) Dependence of the internal energy ($U_{\text{Internal}}$) on
$\theta$ for constant superconducting gap ($\Delta=0.5t$) as the chemical
potential ($2t>\mu>0$) decreases. The internal energy is found by numerically
integrating Eq. (21) over Brillouin zone $k\in\left[-\pi,\pi\right]$. The
internal energies for different $\mu$, $\Delta$ are normalized. The
$\theta_{\text{Topo}}$ found from Eq. (22) is plotted as vertical bar. Values
of $\theta$ on the left of this bar has topological phase. Hence, if the
energy minima lies on the left of this vertical bar then the system
thermodynamically resides in the topological phase. (d) Dependence of the
internal energy ($U_{\text{Internal}}$) on $\theta$ for constant
superconducting gap ($\Delta=0.5t$) and for negative chemical potential.

The results found in the previous sections were applicable when the spiral
spin texture is present in the system. It was explicitly shown that the single
mode energy depends also on the spin rotation angle $\theta$. However, as
mentioned before, the background spin field is dynamic in nature. This means
that $\theta$ changes with change as we change the global system parameters
$\mu$ and $\Delta$. These two parameters are essential for the determination
of the lowest internal energy ($U_{\text{Internal}}$). The internal energy is
found by summing over the energies of all the occupied states:

$\displaystyle U_{\text{Internal}}=\sum\limits_{k\in
BZ}E(k)f\left[E(k),\mu\right],$ (21) where, $\displaystyle
f\left[E(k),\mu\right]\equiv\frac{1}{\mathrm{e}^{\left[E(k)-\mu\right]/k_{B}T}+1}.$

$f\left[E(k),\mu\right]$ is the Fermi-Dirac distribution; $k_{B}=8.6\times
10^{-5}$ eV K-1 is the Boltzman constant; $T$ is the temperature. In our
calculation we take $T\approx 0$ K. The summation in Eq. (21) is taken over
the whole Brillouin zone $k\in\left[-\pi,\pi\right]$. In Fig. 7 we show the
dependence of the internal energy on $\theta$ for constant $\Delta=0.5t$, and
for different $\mu$ values. The energies are normalized in these figures. One
can observe a clear global minimum when $\theta\in(0,\pi)$ for $\mu=1.6t$,
$1.5t$, $1.4t$, $1.3t$, $1.2t$, $1.1t$; for $\mu=0.3t$ a local minimum occurs
in $\theta\approx 5\pi/8$. For negative $\mu$ no local or global minimum
occurs at $\theta\in(0,\pi)$. It should be remembered that the system has
topological properties only when $\theta\neq 0,\pi$; this is a necessary but
not sufficient condition. The second condition is:

$\theta<\theta_{\text{Topo}}\equiv 2\>\arccos\left|\frac{\mu}{2t}\right|.$
(22)

This gives the upper boundary $\theta_{\text{Topo}}$ for a topological phase;
$\theta_{\text{Topo}}$ is a function of $\mu$. In Fig. 7 the values of
$\theta_{\text{Topo}}$ obtained from Eq. (22) are displayed as vertical lines.
Therefore if a global or local minimum lies on the left of the vertical bar
and does not occurs at $\theta=0$, then the system will thermodynamically
acquire a topological phase.

(a)

(b)

(c)

(d)

Figure 8: (a,b,c,d) Dependence of the internal energy ($U_{\text{Internal}}$)
on $\theta$ for constant chemical potential ($\mu$) as the superconducting gap
($\Delta$) increases. The internal energy is found by numerically integrating
Eq. (21) over Brillouin zone $k\in\left[-\pi,\pi\right]$. The internal
energies for different $\mu$, $\Delta$ are normalized. The
$\theta_{\text{Topo}}$ found from Eq. (22) is plotted as vertical bar. Values
of $\theta$ on the left of this bar has topological phase. Hence, if the
energy minima lies on the left of this vertical bar then the system
thermodynamically resides in the topological phase.

In Fig. 8 we show the evolution of the internal energy dependence on $\theta$
for several values of $\mu$ and $\Delta$. In Fig. 8a an interesting pattern
can be observed with the increase in $\Delta$. For $\Delta=0.1t$ the global
minimum is at $\theta=\pi$, however with increase in $\Delta$ the global
minimum is shifted towards $\theta\gtrapprox\pi/2$. Finally at $\Delta=0.7t$
the global minimum occurs at $\theta=0$ — although there is a local minimum
occurs at $\theta\lessapprox\pi/2$. The values of $\theta$ found from Eq. (22)
is shown as the vertical line. Observing both the vertical line and the minima
in Fig. 8a the topological phase appears for $\Delta=0.7t$ at local minimum.
In Fig. 8b we don’t expect any topological phase, as the minima occur either
at $\theta=0,\pi$, or do not simply satisfy the condition spelled out by Eq.
(22). In Fig. 8c for every $\Delta$ there is a local minimum at $\theta\approx
5\pi/8$, which satisfy the condition Eq. (22). Therefore we expect a
topological phase for every value of $\Delta$. In Fig. 8d we observe an
analogous behavior.

Figure 9: Phase diagram showing the values of $\mu/t$ and $\Delta/t$ for which
the topological phase occurs. Topological phase appears for the shaded
regions, however, topological phase is absent for unshaded regions. The color
coding of the shaded region represent the values of $\theta$ where the minimum
internal energy $U_{\text{Internal}}$ occurs.

As was mentioned before the following two conditions are sufficient and
necessary for the emergence of the topological phase thermodynamically: (i)
$\theta\in 0,\pi$, (ii) $\theta<\theta_{\text{Topo}}$; $\theta$ being the
angle where the minima of $U_{\text{Internal}}$ occurs. In Fig. 9 we plot the
values of $\mu$ and $\Delta$ which satisfy the aforementioned conditions. The
shaded regions represents the topological phase, while the unshaded regions
represent the trivial no topological phase. The color coding in the figure
represents the values of $\theta$ where minimum energy occurs. Interestingly,
for $0.5t\lessapprox\Delta/t\lessapprox 0.8t$ the minimum occurs at two
different value of $\mu/t$. It is due to the evolution of the minimum of the
internal energy when $\mu$ is changed accordingly as shown in Fig. 10. As
$\mu$ decreases, the minimum satisfying the aforementioned two topological
conditions occurs two times. As a result two distinct regions of topological
phase are displayed in Fig. 9.

Figure 10: Plot of the evolution of the internal energy
($U_{\text{Internal}}$) with $\theta$ when $\Delta=0.6t$ and $2t<\mu<0.2t$.
The $x$-axis represent the $\theta$, and $y$-axis represent the normalized
$U_{\text{Internal}}$. The vertical lines satisfy the condition Eq. (22). It
is plotted to explain the reason for emergence of two distinct regions where
topological phase appears in Fig. 9. With the decrease in $\mu$ the
$U_{\text{Internal}}$ develops minima satisfying the two topological
conditions ($\mu=1.4t$); with a further decrease in $\mu$ the minima do not
satisfy the topological condition ($\mu=1t$). In contrast at $\mu=0.2t$ again
a local minimum emerges satisfying the required topological condition.

The main physical conclusion from this section is that by modifying the
chemical potential we can drive the system into and out of the topological
phase thermodynamically. Although previously proposed heterostructures [8]
used the same broad idea, in this work no external magnetic field is needed to
produce such an effect.

##  VII Proposal for experimental heterostructure

(a)

(b)

Figure 11: Device schematics: (a) Tunneling experiment probing the presence of
the Majorana fermions due to dynamic spin field of the 1D nanowire. (b)
Tunneling experiment due to induced non-collinear spin texture through bottom
layer. One can use the Van der Waals magnets as the non-collinear magnetic
layer.

Usually tunneling experiments are employed to detect the Majorana fermions
[17]. The proposed experimental setups for detecting Majorana fermions in
strongly correlated wires are shown in Fig. 11. A zero bias peak in the
conductance plot is usually interpreted as the possible signature of the MZM.
However, these experiments are not at all conclusive, because Andreev bound
states also have the same zero bias peak signature [59]. Despite these
difficulties, recently a number of experiments have been performed to detect
the MZM [8, 60, 10]. We propose analogous experiments to detect MZM.

In Fig. 11a a strongly correlated wire is placed on top of an insulating
substrate. The tunneling electrons enters into the wire out through the
metallic contacts. Furthermore, they exit the wire out through the
superconducting contact, and flow directly into the current measuring device.
The chemical potential is controlled through the gate contacts. For nanowires
either heavy fermion [61, 62, 63, 64, 65, 66] or transition metal [67, 68, 69,
70, 71, 72] compounds can be used. Especially nanowires from transition metal
dichalcogenides compounds are promising candidates [72, 69, 71]. For
superconductors one can use the iron-based materials, since they have extended
_s_ -wave order parameter [43, 32, 73, 74]. Because the spiral spin field is
dynamically generated in strongly correlated 1D nanowires, we expect to see a
MZM signature in the tunneling experiments, without the need of an external
applied rotating magnetic field to generate spiral spin fields.

However one can also perform experiments by externally inducing a non-
collinear spin textures in the wire. The schematics for this corresponding
experiment is shown in Fig. 11b. Here, the only difference from the previous
arrangement (Fig. 11a) is that, we place the nanowire on a magnetic layer with
non-collinear magnetic structures. The magnetic structure of the base magnetic
material is induced into the nanowire, which is a necessary condition for the
MZM to occur. One can also use the Van der Waals magnets as magnetic layer
[75, 76]. They may host different magnetic structures, e.g. Skyrmions, spirals
[77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88].

Finally, there is yet a third way to perform the required experiment. Instead
of the nanowire one can use a single wall carbon nanotube, which has the
required strong correlation [89, 90, 91]. Interestingly, in these nanotubes
RSOC can be induced and controlled through electric field [92, 93, 94]. The
schematic of the device is analogous to the one displayed in Fig. 11a; instead
of the nanowire, a single wall carbon tube is placed over the metal contact,
the gate and the superconducting contact. One should remember, in this case
that the applied electric field also controls the RSOC.

##  VIII Conclusion

In this work, it was shown that the Majorana zero modes in nanowire-
superconductor heterostructures can emerge without application of an external
magnetic field; the only condition is that, the electrons in the nanowire
should be required to be strongly correlated. The physical system shown in
Fig. 1 was investigated for a spiral spin field, both in the presence and
absence of the Rashba spin orbit coupling (RSOC). It was found that, apart
from the chemical potential ($\mu$), an extra parameter $\theta$ — the angle
between spin projection of the neighboring sites on $xy$ plane — enters into
the Hamiltonian as a tuning parameter for allowing the system to go into and
out of the topological phase. This can be easily observed by comparing the
single mode energy as described by, Eq. (18), with the same single mode energy
displayed by the Kitaev’s toy model [3, 2]. The increase in $\theta$ decreases
the allowed values of $\mu$ and $t$ for which the system enters into the
topological phase, as shown in Fig. 3. We investigated the $\theta$ dependence
of the internal energy, and we found out that only for limited values of $\mu$
and $\Delta$ the system thermodynamically enters into the topological phase,
as shown in Fig. 7 and 8. The corresponding phase diagram is shown in Fig. 9.
We also found out that RSOC is not a necessary ingredient for the appearance
of the topological phase. This result is different from the usual
semiconductor-superconductor heterostructure used for the realisation of
Majorana fermions, where RSOC, as well as, an external magnetic field, are the
necessary ingredients. In fact, in our case with increase in intensity of RSOC
the area in the parameter phase space ($\mu/t-\theta$) for occurrence
topological phase decreases, as shown in Fig. 6. Finally, in Sec. VII we
proposed possible ways to experimentally realize the proposed system.

##  IX Acknowldgements

K.K.K wish to thank P. A. Maksimov for valuable discussion. K.K.K acknowledges
the financial support from the JINR grant for young scientists and
specialists, the Foundation for the Advancement of Theoretical Physics and
Mathematics ”Basis” for grant # 23-1-4-63-1. One of us, A.F., acknowledges
financial support from the MEC, CNPq (Brazil) and from the Simons Foundation
(USA).

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