# Prevalence of trivial zero-energy sub-gap states in non-uniform
helical spin chains on the surface of superconductors

Richard Hess  Henry F. Legg  Daniel Loss  Jelena Klinovaja  Department of
Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel,
Switzerland

###### Abstract

Helical spin chains, consisting of magnetic (ad-)atoms, on the surface of bulk
superconductors are predicted to host Majorana bound states (MBSs) at the ends
of the chain. Here, we investigate the prevalence of trivial zero-energy bound
states in these helical spin chain systems. The existence of trivial zero-
energy bound states can prevent the conclusive identification of MBSs and,
given the limited tunability of atomic spin chain systems, could present a
major experimental roadblock. First, we show that the Hamiltonian of a helical
spin chain on a superconductor can be mapped to an effective Hamiltonian
reminiscent of a semiconductor nanowire with strong Rashba spin-orbit
coupling. In particular, we show that a varying rotation rate between
neighbouring magnetic moments maps to smooth non-uniform potentials in the
effective nanowire Hamiltonian. Previously it has been found that trivial
zero-energy states are abundant in nanowire systems with smooth potentials.
Therefore, we perform an extensive search for zero-energy bound states in
helical spin chain systems with varying rotation rates. Although bound states
with near zero-energy do exist for certain dimensionalities and rotation
profiles, we find that zero-energy bound states are far less prevalent than in
semiconductor nanowire systems with equivalent non-uniformities. In
particular, utilising varying rotation rates, we do not find zero-energy bound
states in the most experimentally relevant setup consisting of a one-
dimensional helical spin chain on the surface of a three-dimensional
superconductor, even for profiles that produce near zero-energy states in
equivalent one- and two-dimensional systems. Although our findings do not rule
them out, the much reduced prevalence of zero-energy bound states in long non-
uniform helical spin chains compared with equivalent semiconductor nanowires,
as well as the ability to measure states locally via STM, should reduce the
experimental barrier to identifying MBSs in such systems.

††preprint: APS/123-QED

##  I Introduction

The experimental realisation of Majorana bound states (MBSs) has become one of
the most sought-after goals in modern condensed matter physics. The search for
MBSs has largely been motivated by their exotic non-Abelian braiding
statistics, which makes them a promising basis for fault tolerant quantum
computation [1, 2, 3, 4, 5, 6]. Despite this intense effort, however, there
has been no conclusive observation of MBSs so far.

Topological $p_{x}+ip_{y}$ superconductors are predicted to host MBSs at the
cores of vortices [1, 2]. However, since intrinsic $p$-wave superconductors
turn out to be rare, the main experimental focus has been on engineering
hybrid platforms based on proximity effect that can become topological
superconductors. A wide variety of engineered systems have been proposed to
host MBSs [6, 7, 8, 9, 10] such as edge or surface states of topological
insulators (TIs) [11], semiconductor nanowires [12, 13, 14, 15, 16, 17, 18],
planar Josephson junctions [19, 20, 21], TI nanowires [22, 23], graphene-based
systems [24, 25, 24, 26, 27, 28, 29, 30, 31, 32, 33], and many more.

l0.0int1pt(a)l1.62int0pt

(a)

l0.00int-1pt(b)l1.62int0.in

(b)

Figure 1: Helical spin chain system: (a) Magnetic adatoms (red) are placed on
the surface of a bulk superconductor (yellow). The adatoms form a helical spin
chain with a rotation rate that can vary along the chain. (b) Geometry of
helical spin chain used in this manuscript (number of sites shown is not to
scale). The full system has $N_{x}$ sites in the $x$ direction with magnetic
adatoms deposited on $N_{C}$ sites forming a helical spin chain. In general,
the helical spin chain is embedded in the underlying superconductor with
$n_{L}$ sites on the left side of the system and $n_{R}$ on the right. In
addition we will consider one-, two-, and three-dimensional superconductors
underlying the chain.

Although many systems have been predicted to realise MBSs, to date, the
conclusive experimental identification of MBSs has not been possible, largely
due to trivial states that can mimic the experimental signatures of MBSs [34].
Most notable is the situation in semiconductor nanowires, which are perhaps
the most mature experimental platform expected to host MBSs. Such systems
consist of a nanowire with strong Rashba spin-orbit coupling that has been
brought into proximity with a superconductor. It was predicted that a
signature of MBSs in these nanowires is a zero-bias peak in differential-
conductance measurements and that this peak is stable for a wide range of
magnetic field strengths. Although such a signature has been observed [35, 36,
37, 38, 39, 40, 41, 42, 43], it turned out that a zero-bias peak by itself is
not a unique fingerprint of MBSs. In fact, it was shown that trivial Andreev
bound states (ABSs) [44, 45] are expected to be abundant in these nanowires
[34]. These trivial states can also result in zero-bias peaks which mimic MBSs
and they occur, for instance, due non-uniformities in the nanowire parameters
[46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63].
Despite several further signatures of MBSs being proposed, due to the high
prevalence of possible zero-energy modes, it remains unclear if a conclusive
measurement of MBSs can be performed in nanowire systems.

Artificial magnetic structures, such as atomic chains, where adatoms are
placed on the surface of a superconductor have also been predicted to host
MBSs [64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77]. A single
magnetic adatom on the surface of a superconductor leads to the formation of a
Yu-Shiba-Rusinov (YSR) state, which is well studied in theory [78, 79, 80, 81,
82, 83, 84] and experiment [85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95].
Current experiments exploit an atom by atom construction technique to build
the atomic chains, allowing great control over the chain length [96, 97, 98,
99, 100, 101, 92, 102, 103, 104]. Here we will focus on long chains of
magnetic adatoms that possess a helical ordering. Helical ordering of the
magnetic moments of the adatoms can be mediated by Ruderman-Kittel-Kasuya-
Yosida (RKKY) interaction [105, 106, 107, 108, 109] which should lead to a
helix of magnetic moments with period $\pi/k_{F}$, where $k_{F}$ denotes the
Fermi momentum. Other mechanisms can also support helical ordering, for
instance the Dzyaloshinskii-Moriya interaction (DMI) [110]. Since local
measurements can be performed on such atomic chains by utilising scanning-
tunnelling microscopy techniques, the location of states can be very well
established [100, 43, 101], a significant benefit over semiconductor systems.
On the other hand, the topological transition in such chains is set by the
exchange coupling strength, $J$, between the adatoms and the superconductor,
which is not easily controlled. This lack of tunability makes exploring the
phase space of a zero-energy bound state difficult and therefore, if trivial
zero-energy states are as abundant in such chains as in nanowire systems, it
would be even more difficult to conclusively identify MBSs.

In this paper we will investigate the prevalence of trivial zero-energy modes
due to non-uniformities in long helical spin chains. In particular, we will
consider chains with varying rotation rates between adjacent magnetic moments
along the chain. We show that a mapping exists from the Hamiltonian of a
helical spin chain to an effective Hamiltonian reminiscent of a Rashba
nanowire. Under this mapping varying rotation rates of the magnetic moments
result in non-uniformities in the effective Hamiltonian that are similar to
those that result in the abundance of trivial states in Rashba nanowires.
Therefore, using this mapping as a basis, we will investigate if non-uniform
rotation rates can also lead to an abundance of zero-energy modes in helical
spin chains. Although bound states with near zero-energy do exist for certain
dimensionalities and rotation profiles, including when the spin chain remains
entirely trivial, for experimentally relevant systems we find that zero-energy
bound states are far less prevalent in spin chains than semiconductor nanowire
systems. Primarily this is due to the fact the helical chain is embedded in a
superconductor and also due to effects of the boundary.

This work is structured as follows: first, we investigate an effective one-
dimensional continuum model for a helical spin chain with a smooth spatially
varying rotation rate between adjacent magnetic moments on top of a
superconductor, showing that this maps to a Hamiltonian similar to that of a
Rashba nanowire, see Sec. II. In Sec. III we introduce one-, two- and three-
dimensional lattice models for the non-uniform helical spin chains. The
topological phase diagram of a helical periodic one-dimensional spin chain is
shown in Sec. IV and, in addition, several non-uniform profiles of the
rotation rate are suggested, which might potentially support the formation of
trivial zero-energy sub-gap states. In Secs. V, VI, and VII, we discuss the
presence and absence of trivial zero-energy sub-gap states in one-, two- and
three-dimensional systems for different types of rotation rate profiles.
Finally, we conclude in Sec. VIII. In addition, in App. A, we study the
disappearance of trivial zero-energy states in chains which are placed on the
boundary of a two-dimensional superconductor. Furthermore, we consider the
scenario of a chain deposited in the bulk of a three-dimensional
superconductor instead on the surface, see App. B.

##  II Theory of a one-dimensional spin chain with spatially varying
magnetization

In this section we show that the Hamiltonian of a helical spin chain, as shown
in Fig. 1, can be mapped to an effective Hamiltonian that is reminiscent of a
semiconductor nanowire brought into proximity with a superconductor.
Surprisingly, despite this mapping and despite the fact that spatially varying
potentials can easily be generated in the effective Hamiltonian, we will show
in the remainder of the manuscript that trivial zero-energy bound states are
far less abundant in helical spin chains than has been shown to occur in
semiconductor Rashba nanowires.

To begin, we consider a helical spin chain placed on top of a superconductor.
The effective one-dimensional (1D) model is described by the Hamiltonian
$\mathrm{H}_{\mathrm{1D}}=\int\differential x\
\Psi^{\dagger}(x)H_{\mathrm{1D}}\Psi(x)$ in the basis

$\displaystyle\Psi(x)=\begin{pmatrix}c_{x,\uparrow},&c_{x,\downarrow},&c^{\dagger}_{x,\downarrow},&-c^{\dagger}_{x,\uparrow}\\\
\end{pmatrix}^{\rm T},$ (1)

where the operator $c_{x,\nu}^{\dagger}$ ($c_{x,\nu}$) creates (annihilates)
an electron at position $x$ with spin $\nu$. The Hamiltonian density is given
by

$\displaystyle
H_{\mathrm{1D}}=\left(-\frac{\hbar^{2}}{2m}\nabla_{x}^{2}-\mu\right)\tau_{z}+J(x)\mathbf{S}(x)\cdot\bm{\sigma}+\Delta_{0}\tau_{x},$
(2)

where $m$ is the effective mass, $\mu$ the chemical potential, $\Delta_{0}$ is
the superconducting gap, and $J(x)$ is the (position dependent) exchange
coupling strength between the magnetic moments of the adatoms and the spins of
the itinerant electrons. In addition,
$\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})^{\rm T}$ is the vector of
Pauli matrices acting in spin space and $\tau_{i}$ are Pauli matrices acting
in particle-hole space. Finally, the unit-vector $\mathbf{S}(x)$ determines
the local direction of the magnetic moments of the adatoms that form the
chain. We will consider magnetic textures along the chain of the form

$\displaystyle\mathbf{S}(x)=\begin{pmatrix}\cos[2\varphi(x)],&\sin[2\varphi(x)],&0\end{pmatrix},$
(3)

such that the rotation of the magnetic moments in the $xy$-plane is described
by the total angle $\varphi(x)$ which can be described by a non-linear
function, assuming a non-uniform rotation. Applying the unitary transformation
$U=e^{-i\varphi(x)\sigma_{z}}$, one can map the Hamiltonian of the helical
chain, $H_{\mathrm{1D}}$, to that of a Rashba nanowire in a magnetic field
$\tilde{H}$ [109], such that

$\displaystyle\tilde{H}=$ $\displaystyle U^{\dagger}H_{\mathrm{1D}}U$
$\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m}\left(\nabla_{x}^{2}-\left(\\!\partialderivative{\varphi}{x}\\!\right)^{\\!2}-\left[i\partialderivative{\varphi}{x}\nabla_{x}+i\nabla_{x}\partialderivative{\varphi}{x}\right]\sigma_{z}\right)\tau_{z}$
$\displaystyle+J(x)\sigma_{x}+\Delta_{0}\tau_{x}-\mu\tau_{z}.$ (4)

When the rotation rate of the magnetic moments along the chain, which we
define as $\Phi(x)\equiv\partialderivative{\varphi}{x}$, is non-uniform then
the transformed Hamiltonian, $\tilde{H}$, contains an effective coordinate
dependent potential $\tilde{V}(x)=\hbar^{2}[\Phi(x)]^{2}/2m$ and also an
effective coordinate dependent Rashba SOI

$\displaystyle\tilde{H}_{R}(x)$
$\displaystyle=\frac{\hbar^{2}}{2m}[i\Phi(x)\nabla_{x}+i\nabla_{x}\Phi(x)]\sigma_{z}\tau_{z}$
$\displaystyle=\frac{i}{2}\left[\alpha(x)\nabla_{x}+\nabla_{x}\alpha(x)\right]\sigma_{z}\tau_{z},$
(5)

where we use the symmetrized from of the effective position-dependent Rashba
SOI [111] with the strength given by $\alpha(x)=\hbar^{2}\Phi(x)/m$. The same
unitary transformation $U$ will also yield an equivalent Hamiltonian in the
generalisation of our system to two- or three-dimensions. This is because the
kinetic terms that contain the derivative $\nabla_{i}$, acting in the
direction $i\neq x$ with $i\in\\{y,z\\}$, will commute with $U$.

The transformed Hamiltonian, $\tilde{H}$, is reminiscent of one-dimensional
semiconductor nanowires and has the same ingredients that are required for
topological superconductivity: Rashba SOI, a superconducting gap, and a Zeeman
energy [13, 14]. Using this mapping we therefore see that, in the case of a
constant rotation rate between adjacent magnetic moments along the chain,
which means $\varphi(x)=k_{r}x$, the topological phase transition for the
purely one-dimensional chain takes place at

$\displaystyle
J_{C}(k_{r})=\sqrt{\Delta_{0}^{2}+(\mu-\hbar^{2}k_{r}^{2}/[2m])^{2}}$ (6)

and depends on the spatial rotation period, which is set by $2\pi/(2k_{r})$.
If the magnetic moments of the magnetic adatoms are aligned by RKKY
interaction ($k_{r}=k_{F}$), then the period is set to $\pi/k_{r}=\pi/k_{F}$
and the system enters the topological phase at $J(k_{F})=\Delta_{0}$ [64, 65,
66]. The topological phase transition is shifted to larger values of $J$ for
$k_{r}\neq k_{F}$.

Therefore, when the rotation rate of the one-dimensional helical chain is non-
uniform, the above mapping is to an effective Hamiltonian that is reminiscent
of a one-dimensional semiconductor nanowire with spatially varying parameters.
Such nanowires have been considered extensively and it has been shown that
trivial zero-energy Andreev bound states (ABSs) are abundant in these systems
[46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 34, 59, 60]. The
existence of the mapping therefore suggests that trivial zero-energy states
could potentially be as prevalent in helical spin chains as in nanowires,
which would be a significant problem given the more limited tunability of
parameters in the atomic chain compared to a nanowire. However, although the
mapping above is rigorous, the two systems differ significantly for the
following reason: superconductivity in a semiconductor nanowire is present
only in the region that is brought into proximity with a superconductor and
the Zeeman energy is (approximately) constant throughout this region. In
contrast, the helical spin chain is embedded in a superconductor and the
exchange coupling energy – which maps to the Zeeman energy – is non-zero only
close to the position of the magnetic adatoms. As a result, as we will show in
the rest of this paper, while trivial zero-energy modes in a purely one-
dimensional chain that is aligned with the end of the superconductor are as
easily generated as this mapping suggests, the same is not true when the
atomic chain is embedded in a superconductor, especially in the experimentally
relevant scenario where the superconductor in which the chain is embedded is
three-dimensional.

##  III Lattice models of spin chain

###  III.1 One-dimensional system

In the previous section, we work in the continuum limit. In this section, we
switch to the lattice description by discretizing the Hamiltonian, which will
allow us later to solve the problem numerically. First, we construct a lattice
Hamiltonian $H_{1\mathrm{D},L}$, corresponding to $H_{1\mathrm{D}}$ defined in
Eq. (2). The lattice Hamiltonian $H_{1\mathrm{D},L}$ of the setup consisting
of an atomic chain on the superconducting surface (see Fig. 1) is given by

$\displaystyle
H_{1\mathrm{D},L}=\sum_{n=1}^{N_{x}}\Bigg{[}\sum_{\nu,\nu^{\prime}}c_{n,\nu}^{\dagger}(\\{2t-\mu\\}\delta_{\nu,\nu^{\prime}}+J_{n}\left(\mathbf{S}_{n}\cdot\bm{\sigma}\right)_{\nu\nu^{\prime}})c_{n,\nu^{\prime}}$
$\displaystyle\hskip
30.0pt-\Bigg{(}\sum_{\nu}tc_{n,\nu}^{\dagger}c_{n+1,\nu}+\Delta_{0}c_{n,\downarrow}^{\dagger}c_{n,\uparrow}^{\dagger}+\text{H.c.}\Bigg{)}\big{.}\Bigg{]},$
(7)

where $N_{x}$ denotes the total number of lattice sites and
$t=\hbar^{2}/(2ma^{2})$ denotes the matrix hopping element, which depends on
the effective lattice constant $a$. We consider an exchange coupling strength
of the form

$\displaystyle J_{n}=J\left[\Theta(n-n_{L})-\Theta(n-N_{x}+n_{R})\right],$ (8)

where $n_{L}$ ($N_{x}-n_{R}$) is the site hosting the first (last) adatom and
$\Theta(n)$ is the Heaviside step-function with $\Theta(0)=0$ and the length
of the chain of magnetic adatoms is given by $N_{C}=N_{x}-(n_{R}+n_{L})$.
Depending on the relative locations of the ends of the chain and the system
boundaries, we distinguish two different types of setups: 1) An aligned setup
is defined as the following: an end of the chain coincides with a boundary of
the system, such that $N_{C}=N_{x}$, [see Fig. 1(b)]. 2) In an embedded setup,
in contrast, magnetic adatoms are located only along a subsection of the
entire system and do not reach the system boundary. In particular, we will
consider an embedded left end, such that $N_{x}\gg N_{C}$ and $n_{R}=0$ [see
Fig. 1(b)]. We will use the same nomenclature of aligned and embedded systems
referring to setups in which the superconductor underlying the chain is two-
or three-dimensional (see below).

We define the rotation rate, $\Phi_{n}$, such that the helix has a period $\pi
a/\Phi_{n}$, as

$\displaystyle\Phi_{n}(\lambda)=\Phi_{L}+(\Phi_{R}-\Phi_{L})\
\text{sig}\left(\frac{2[n-n_{0}]}{\lambda}\right),$ (9)

where we have used the sigmoid function
$\text{sig}(x)=\frac{1}{2}\left[1+\tanh(x/2)\right]$. The profile $\Phi_{n}$
describes a helix of magnetic moments which rotates with the period $\pi
a/\Phi_{L}$ and $\pi a/\Phi_{R}$ on the left and right end of the chain,
respectively. If the sign of $\Phi_{L}$ is the opposite of $\Phi_{R}$, then
the rotation direction changes and the system contains a domain wall. The
parameter $\lambda$ controls the width and smoothness of the transition
between two sections of different rotation rate and the parameter $n_{0}$
parameterises the position of the transition. In the discretized model
$\varphi_{n}$ enters ${\bf S}(x)$ [see Eq. (3)] instead of $\varphi(x)$, where
the angle $\varphi_{n}$ relative to the magnetic moment located at $n=1$ is
given by

$\displaystyle\varphi_{n}=\sum_{m=1}^{n}\Phi_{m}(\lambda)-\Phi_{1}.$ (10)

The particular choice of a constant rotation rate, $\Phi_{R}=\Phi_{L}=\Phi$,
results in the well known case of a helical spin chain with fixed rotation
period ($\varphi_{n}=n\Phi-\Phi_{1}$). Consequently, the topological phase is
defined by the condition [73]

$\displaystyle
J^{<}_{C}(\Phi)=\sqrt{\Delta_{0}^{2}+(|\mu-2t|-2t|\cos(\Phi)|)^{2}}<J$
$\displaystyle<\sqrt{\Delta_{0}^{2}+(|\mu-2t|+2t|\cos(\Phi)|)^{2}}=J^{>}_{C}(\Phi),$
(11)

see Fig. 2a. In the limit of small angles between magnetic moments, $\Phi$,
the prediction for the lower bound of the topological phase transition,
$J_{C}^{<}$, corresponds to the analytic result, $J_{C}$, for the location of
the bulk gap closing in the continuum system as discussed in the previous
section [see Eq. (6)]. This can be shown explicitly by expanding the cosine
function and using the definition of the matrix hopping element $t$ as well as
the relation $\Phi=k_{r}a$. We note that the choice $\Phi_{k_{F}}=k_{F}a$
results in the topological phase transition criterion $J^{<}_{C}=\Delta_{0}$,
which is the lowest value of $J^{<}_{C}$ possible for any $\Phi$.

###  III.2 Two- and three-dimensional systems

Here, we extend our model from Sec. III.1 to higher dimensions, such that the
chain is deposited on top of a two- or three-dimensional (2D, 3D)
superconductor. The Hamiltonian in this case has the form

$\displaystyle
H_{\kappa\mathrm{D}}=\sum_{j}\Bigg{[}\sum_{\nu,\nu^{\prime}}c_{j,\nu}^{\dagger}\\{(2t\kappa-\mu)\delta_{\nu,\nu^{\prime}}+J_{j}(\mathbf{S}_{j}\cdot\bm{\sigma})_{\nu,\nu^{\prime}}\\}c_{j,\nu^{\prime}}$
$\displaystyle\hskip
20.0pt-\Delta_{0}c_{j,\downarrow}^{\dagger}c_{j,\uparrow}^{\dagger}-\Delta_{0}^{*}c_{j,\uparrow}c_{j,\downarrow}\Bigg{]}-\sum_{\langle
j,j^{\prime}\rangle,\nu}tc_{j,\nu}^{\dagger}c_{j^{\prime},\nu},$ (12)

where $\kappa\in\\{2,3\\}$ denotes the dimensionality of the model and the
index $j$ accounts for the $x$, $y$, and $z$ coordinates of the corresponding
lattice site. In particular, we choose $j=(n,m)$ and $j=(n,m,l)$ in two and
three dimensions, respectively, and the number of sites in $y$ ($z$) direction
is $N_{y}$ ($N_{z}$). The exchange coupling, for example in three dimensions,
is given by $J_{j}=J_{n,m,l}=J_{n}\delta_{m,m_{0}}\delta_{l,l_{0}}$, where
$m_{0}$ and $l_{0}$ denote the $y$ and $z$ position of the atomic chain,
respectively.Here, $J_{n}$ is the same as in the one-dimensional chain and the
notation $\langle j,j^{\prime}\rangle$ indicates a summation over nearest
neighbouring sites. As discussed above, we also use the terms aligned and
embedded to describe systems where $N_{C}=N_{x}$ and $N_{C}\ll N_{x}$,
respectively. Finally we note that, since the atomic chain breaks translation
symmetry in the directions perpendicular to the chain, momentum in these
directions is not a good quantum number and the location of the topological
phase transition can be expected to be different than that found in the
equivalent purely one-dimensional system. We use the Python package Kwant for
the implementation of the tight binding models [112].

##  IV Topological phase diagram and rotation rate profiles

l-0.00int-0.in(a)l3.5int-0.0in(b)l5.25int-0.05in(c)l3.5int1.12in(d)l5.25int1.12in(e)l1.67int1.76in

(a)

(b)

(c)

(d)

(e)

Figure 2: Phase diagram and non-uniform rotation profiles: (a): Phase diagram
of a one-dimensional helical spin chain with constant rotation rate
($\Phi_{n}=\Phi=\mathrm{const.}$), showing the energy difference $E_{2}-E_{1}$
between the lowest energy state and the first excited state. Orange lines
indicate the critical exchange couplings $J^{<}_{C}(\Phi)$ and
$J^{>}_{C}(\Phi)$, which separate the topological phase from the trivial phase
[see Eq. (11)], while the yellow line represents $J_{C}$ [see Eq. (6)]. In the
trivial phase, this energy difference is given by the finite-size quantization
and very small, corresponding to dark blue color. For small rates $J_{C}$
agrees well with $J^{<}_{C}$. Next, we consider spatially varying rotation
rates, with constant rates at the long right section of the chain: (b-e)
Different profiles of the rate $\Phi_{n}$ (blue) and the corresponding profile
of $J^{<}_{C}(\Phi_{n})$ (orange) along the atomic chain to be considered in
this work. The blue dashed line indicates the value of $\Phi_{k_{F}}$
corresponding to a period $\pi/k_{F}$. Numerical parameters for the phase
diagram: $t\approx 10$ meV, $N_{x}=N_{c}=322$, $\Delta_{0}=1$ meV,
$a=3\,\mbox{\AA}$, $\mu=8$ meV.

In this section, we briefly discuss the topological phase diagram of the
helical spin chain and suggest different rotation rate profiles that, based on
the mapping to semiconductor nanowires (see Sec. II), could lead to low energy
trivial sub-gap states. First, we start with an aligned one-dimensional system
and plot the energy difference $E_{2}-E_{1}$ between the first excited state
and the ground state for a chain with constant $\Phi_{n}\equiv\Phi$, such that
sub-gap states other than MBSs or ABSs close to the gap edge, emerging for
example from a shift of the chemical potential [113, 114] are not expected
(see Fig. 2a). Although in the general case this quantity does not provide
direct information about the topology of the phase, in a system with no other
sub-gap states than MBSs, this energy difference is given by the finite-size
level-spacing in the trivial phase and is very small. In contrast to that, in
the topological phases, it determines the value of the topological mini-gap,
which should be much larger than the level spacing (see, respectively, the
dark red regions of Fig. 2a). As expected, the critical value of the continuum
model $J_{C}$ (yellow line Fig. 2a), agrees well for small values of the
rotation rate $\Phi$ with the almost zero value of $E_{2}-E_{1}$ and with
$J^{<}_{C}$ (orange line Fig. 2a). The upper critical value of the lattice
model, $J^{>}_{C}$, confines the topological phase space in the regime of
strong $J$. For example, in the case of an antiferromagnetic alignment the
topological phase is absent [115, 116]. We also note that the interval spanned
by the values of $J^{<}_{C}(\Phi)$ and $J^{>}_{C}(\Phi)$ is maximal in case of
ferromagnetic ordering. However, the system with ferromagnetic configuration
stays trivial since the bulk gap does not reopen, for more details see Ref.
[73].

We now define the different non-uniform rotation profiles that we will use in
the remainder of the manuscript. In general, these non-uniform rotation rates
will involve a transition between two different rotation rates in a short and
long section of the atomic chain, mathematically described by the profile
defined in Eq. (9). We refer to a section as short if its length is comparable
or shorter than the smoothness parameter $\lambda$ (see above) and a section
as long if its length is larger than $\lambda$. Since the phase diagram shown
in Fig. 2a verifies that the topological phase transition criterion
$J^{<}_{C}$ is strongly dependent upon the rotation rate $\Phi_{n}$ between
neighbouring magnetic moments, we can expect a rich variety of states arising
due to these non-uniform profiles. Throughout we will consider only one
transition region on the left end of the chain. However, most probably, both
ends should be identical, such that there is also a trivial state on the
right. We consider only one region such that we can demonstrate the difference
between a uniform chain end and one with the transition region at the end.

The first non-uniform rotation rate profile we define is that of a smooth
decay, this is shown in Fig. 2b. In this profile, the magnetic moments rotate
in the longer section of the chain with a rate $\Phi_{R}<\Phi_{k_{F}}$, as
such this section of the chain obeys the topological phase transition
criterion for exchange couplings larger than $J^{<}_{C}(\Phi_{R})$. In
addition, the magnetic moments in the short section, e.g. on the left side of
the chain, rotate slower with a rate $\Phi_{L}<\Phi_{R}$ and, as a result, the
topological phase transition criterion in this section is shifted to
$J^{<}_{C}(\Phi_{L})$ which is larger than $J^{<}_{C}(\Phi_{R})$.
Consequently, the entire system stays in the trivial regime for exchange
couplings smaller than $J^{<}_{C}(\Phi_{R})$ and any sub-gap state in this
regime must have a trivial origin. An example of such a profile would be a
ferromagnetic ordering of the magnetic moments close to the left end of the
chain and a helical ordering on the right end. The consequences of this
profile will be discussed in detail in Sec. V.

Another rotation rate profile is one that contains a domain wall i.e. a change
of the rotation direction, for example from clockwise to anticlockwise
rotation. For instance, a rotation rate, which is negative in a short section
on the left ($\Phi_{L}<0$), but which takes the value $\Phi_{R}>0$ with

$\displaystyle\Phi_{k_{F}}>\Phi_{R}\geq|\Phi_{L}|\gg\frac{2\pi}{N_{C}}$ (13)

in the longer right section of the atomic chain. The arguments from above
apply also in this case and the topological phase transition criterion is
shifted to values of the exchange coupling larger than $J^{<}_{C}(\Phi_{R})$
in the region where the sign of $\Phi_{n}$ changes, see Fig. 2c. The entire
system therefore stays trivial below for $J<J^{<}_{C}(\Phi_{R})$ and any sub-
gap states in this limit must be of trivial origin. We refer to Sec. VI for a
detailed study of the sub-gap states resulting from such a profile.

Next, we consider a similar setup as in the case of the smooth decay, however,
for this profile $\Phi_{R}<\Phi_{L}<\Phi_{k_{F}}$ (see Fig. 2d). This choice
has a significant impact on $J^{<}_{C}$, namely, the critical value of the
exchange coupling for which the system obeys the topological phase transition
criterion in this case is smaller for the left section with the fast rotating
magnetic moments than for the long section of the chain with the slow rotating
magnetic moments. Therefore, when $J^{<}_{C}(\Phi_{L})\leq
J<J^{<}_{C}(\Phi_{R})$ the short section on the left of the atomic chain
nominally could enter the topological regime. However, these MBSs will be
strongly overlapping spatially and, thus, be hybridized. In Rashba nanowires
zero-energy bound states arising due to a section of the system entering the
topological regime have been termed as quasi-MBSs [34] and we therefore refer
to this as the quasi-MBS profile. In fact, several other quasi-MBS profiles
are possible. For example, if the rotation rate is set to
$\Phi_{L}<\Phi_{k_{F}}$ in the short section on the left side of the chain and
to $\Phi_{R}>\Phi_{k_{F}}$ in the longer section on the right side, such that
the rotation rate changes smoothly between these sections, then at some site
$n=\tilde{n}$ of the chain $\Phi_{n=\tilde{n}}=\Phi_{k_{F}}$ (see Fig. 2e). As
such, a subsystem, namely the section where $\Phi_{n}$ grows, enters the
topological regime for exchange couplings satisfying $J>\Delta$,
$J<J^{<}_{C}(\Phi_{L})$, and $J<J^{<}_{C}(\Phi_{R})$. We will investigate
quasi-MBS profiles in Sec. VII.

In summary the configurations, shown in Figs. 2b and 2c only host trivial
states since the exchange coupling is always smaller than the topological
phase transition criterion $J^{<}_{C}(\Phi_{R})$. In contrast, short sections
of the systems shown in Figs. 2d-2e could obey the topological phase
transition criterion locally. We will see that in such profiles a subsystem
hosts hybridized (quasi-)MBSs. We note that topology is defined for bulk
systems, so in general the topological phase transition criterion,
$J^{<}_{C}$, is not meaningful for a single site or small section of a chain.
Nevertheless, interpreting the behaviour of energies and wavefunctions of sub-
gap states in terms of the variation of the critical exchange coupling
$J^{<}_{C}$, such that a section of the system enters the topological regime,
agrees well with our numerical observations.

##  V Smooth Decay

This section deals with trivial zero-energy states in the one-dimensional
helical spin chain model described in Sec. III.1 using the smooth decay
rotation rate profile, as described in Sec. IV (Fig. 2b). For this profile, in
the long right section of the chain the rotation rate $\Phi_{n}$ between
adjacent magnetic moments is approximately constant such that
$\Phi_{n}\approx\Phi_{R}$, however, the rotation rate $\Phi_{n}$ decreases to
zero close to the left end of the chain (see blue line in Figs. 3e and 3f).

###  V.1 One-dimensional model

l-0.00int0.0in(a)l1.67int0.0in(b)l-0.0int0.95in(c)l1.67int0.95in(d)l0.0int1.86in(e)l1.67int1.86in(f)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3:  Energy spectrum and probability density of the lowest energy state
of a chain with a smooth decay rotation profile on a
$\underline{\text{one}}$-dimensional superconductor. (a,b) Energy spectrum as
a function of the exchange coupling $J$ for (a) the aligned and (b) embedded
systems. The black vertical solid line indicates the value
$J=J^{<}_{C}(\Phi_{R})$ at which the topological phase transition occurs.
States in the completely trivial regime [$J<J^{<}_{C}(\Phi_{R})$] are marked
yellow, while the lowest sub-gap states in the topological regime
[$J>J^{<}_{C}(\Phi_{R})$] are marked orange. Higher energy states are shown in
blue. The helical spin chain hosts trivial zero-energy states only in the
aligned system but these states are continuously pushed away from zero energy
as the number of sites without magnetic moments $n_{L}$ on the left of the
atomic chain grows. (c) [(d)] Evolution of the lowest trivial (yellow)
[topological (orange)] sub-gap state as well as of the higher states as a
function of the distance of the chain end from the system boundary at the
exchange coupling strength indicated by the yellow [orange] arrows in panels
(a) [(b)]. While the trivial sub-gap states split away from zero energy, see
panel (c), the MBSs are stable and do not substantially change in energy, see
panel (d). (e) [(f)] The probability densities of the trivial (yellow) and the
topological (orange) lowest-energy states calculated at the exchange couplings
indicated by the yellow (orange) arrows in panels (a) and (b) in case of an
aligned [embedded] system. The blue line represents the rotation rate
$\Phi_{n}$ between adjacent magnetic moments and the black dashed line
indicates the end of the chain. Moreover, the probability densities show a
clear difference between the MBSs and the trivial sub-gap states, and the
latter are approximately confined in the left section. Parameters:
$N_{C}=160$, $t\approx 10$ meV, $\mu=7$ meV, $\Delta=1$ meV, $\Phi_{L}=0$,
$2\Phi_{R}=0.1768\pi$, $\lambda=20$, $n_{0}=40$, $a=3\,\mbox{\AA}$, $n_{R}=0$.

The energy spectrum of the one-dimensional lattice model with a smooth decay
rotation profile is shown in Fig. 3a (aligned) and Fig. 3b (embedded) as a
function of exchange coupling $J$. In both cases we find that the bulk gap
closes and reopens for $J\approx J^{<}_{C}(\Phi_{R})$, indicated by a solid
black vertical line. In the regime $J<J^{<}_{C}(\Phi_{R})$ we find that sub-
gap states appear. These sub-gap states must be entirely trivial in nature,
since the topological phase transition criterion is not met for any section of
the chain.

When the chain ends coincide with the boundaries of the superconductor
(aligned system), the lowest energy sub-gap state is localized and pinned to
zero energy over a range of exchange couplings (see Fig. 3a). Such a zero-
energy bound state is reminiscent of an MBS, but here is entirely trivial.
Since the exchange coupling $J<J_{C}(\Phi)$ does not satisfy the topological
criterion in Eq. (11) for any $\Phi$, the existence of such a zero-energy
state is surprising and only arises here due to the non-uniformity of the
rotation profile. However, when the chain is embedded into the superconductor,
such that the length of the superconductor far exceeds the length of the
chain, we find that the zero-energy pinning is lifted and the sub-gap states
split away from zero energy (see Fig. 3b).

To further analyze the behavior of the low energy sub-gap states, we examine
the transition from an aligned ($n_{L}=0$) to an embedded ($n_{L}a\gg\xi$)
system by adding lattice sites to the left of the atomic chain and by
calculating the energy spectrum as a function of the distance $n_{L}$ between
the chain end and boundary of the superconductor. Here, $\xi=\hbar
v_{F}/\Delta_{0}$ denotes the superconducting coherence length with $v_{F}$
being the Fermi velocity. The resulting energy spectra for an exchange
coupling smaller than $J^{<}_{C}(\Phi_{R})$, indicated by the yellow arrow,
and for an exchange coupling larger than $J^{<}_{C}(\Phi_{R})$, indicated by
the orange arrow, are shown in Figs. 3c and 3d, respectively. This transition
from the aligned to the embedded system shows that the trivial zero-energy
state is continuously pushed away from zero energy as superconductor sites are
added to the end of the chain. In addition, the next highest state decreases
in energy such that the two lowest sub-gap states become approximately
degenerate for a sufficiently long section without magnetic adatoms on the
left of the chain (see Fig. 3c). In contrast, the zero-energy pinning of the
MBS is essentially unaffected by an increase in $n_{L}$ (see Fig. 3d).
Analyzing the probability density $|\Psi(n)|^{2}$, see Figs. 3e and 3f,
reveals that the trivial state and the left MBS tend to leak into the section
without magnetic adatoms in the case of an embedded system. More specifically,
the wavefunction $\Psi(n)$ decays exponentially into the section without
magnetic moments with $|\Psi|^{2}\sim e^{-2na/\xi}$ [117, 118].

Hence already in one dimension we observe that, in spite of the mapping of
Sec. II, the leakage of the lowest energy states of an atomic chain into the
surrounding superconductor results in an increase in energy of that state. In
general, mechanisms that suppress the leakage of the wavefunction into the
section without magnetic adatoms can restore the zero-energy pinning of this
lowest state. For instance, in one dimension, a scalar impurity at the end of
the chain reduces the leakage and a strong impurity results in a lowest energy
state that is again pinned close to zero energy (not shown).

Reducing the smoothness of the transition profile $\Phi_{n}(\lambda)$ via the
parameter $\lambda$ [see Eq. (9)], lifts the zero-energy pinning of the lowest
sub-gap state in the topologically trivial regime. In contrast, the energies
of the MBSs are unaffected by the smootheness of the profile
$\Phi_{n}(\lambda)$. The exact form of the function $\Phi_{n}(\lambda)$ is
actually not crucial for the existence of the zero-energy sub-gap states as
long as $\Phi_{n}(\lambda)$ changes sufficiently smoothly between $\Phi_{L}$
and $\Phi_{R}$. In particular, we find that $\lambda$ needs to be larger than
the rotation period of the magnetic moments on the right end of the atomic
chain to obtain a sub-gap state with energy pinned to zero. Furthermore, the
position $n_{0}$ in Eq. (9) of the smooth step should be placed at least twice
the length of $\lambda$ from the site of first magnetic adatom and the larger
the value of $n_{0}$ the more sub-gap states enter the spectrum.

Trivial sub-gap states appear at finite values of exchange coupling. By
analyzing our numerical simulations we find that their energy is only pinned
to zero when the condition

$\displaystyle\Delta_{0}<J<J^{<}_{C}(\Phi_{R})$ (14)

holds. We can understand this behaviour in the picture outlined in Ref. 46 for
semiconductor nanowires: when the exchange coupling $J$ is larger than the
superconducting gap, the superconducting correlation has an effective $p$-wave
nature and below the critical exchange coupling $J^{<}_{C}(\Phi_{R})$ two
$p$-wave channels are present. The smooth change of the rotation rate, which
maps to a smooth variation of potential and induced SOI in the effective
model, does not allow a coupling between the two channels and supports the
formation of trivial zero-energy states. In contrast, an abrupt change in the
spatial profiles couples the two channels and the sub-gap states are pushed
away from zero energy. On the other hand, for $J>J^{<}_{C}(\Phi_{R})$, the
sub-gap states evolve into MBSs [46].

If the rotation rate is set to $\Phi_{R}=\Phi_{k_{F}}$ in the longer section
of the chain, as predicted for an ordering mediated by the RKKY interaction
[105, 106, 107, 71, 72, 73, 74, 75, 76, 77, 64], and if the magnetic moments
deviate from the $k_{F}$ ordering on the very left end of the chain, as
$\Phi_{L}<\Phi_{k_{F}}$, then almost the entire system, except the section on
the left, satisfies the topological phase transition criterion when
$J_{C}(k_{r}=k_{F})\geq\Delta_{0}$, see Eq. (6). Consequently, the interval of
exchange coupling strengths for which trivial sub-gap states might appear
shrinks to zero [see Eq. (14)]. Therefore, if the magnetic moments in the long
right section of the chain form a spiral with period $\pi/k_{F}$, as RKKY
interaction suggests, then no sub-gap states are present even in the case of
an aligned system. In contrast, if the spin ordering deviates from
$k_{r}=k_{F}$ in a long section of the chain, see for example Refs. [119],
then the formation of zero-energy states of trivial nature is possible for the
aligned system.

###  V.2 Two-dimensional model

l-0.00int0.in(a)l1.67int0.in(b)l-0.0int0.88in(c)l1.67int0.88in(d)l0.0int1.8in(e)l1.75int1.8in(f)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Energy spectrum and probability density of the lowest energy state
of a chain with a smooth decay rotation profile on a
$\underline{\text{two}}$-dimensional superconductor. (a,b) Energy spectrum as
a function of the exchange coupling $J$ for (a) the aligned and (b) embedded
systems. The black vertical solid line indicates the value
$J=J^{<}_{C}(\Phi_{R})$ in the analogous one-dimensional system, this does not
match the actual bulk gap closing and reopening observed in two-dimensional
setup. (a) The lowest sub-gap state has almost zero energy when the chain is
aligned (b) but the state has finite energy when the chain is embedded. (c)
[(d)] Evolution of the lowest trivial (yellow) [topological (orange)] sub-gap
state as well as of the higher states as a function of the distance of the
chain end from the system boundary at the exchange coupling strength indicated
by the yellow [orange] arrows in panel (a). (c) The lowest trivial state is
continuously pushed away from zero energy by adding more non-magnetic sites to
the left of the end of the chain. (d) In contrast the energy of the MBSs is
essentially unaffected by the number of non-magnetic sites at the end of the
chain. (e) [f] The probability densities of the trivial (yellow) and the
topological (orange) low-energy states calculated at the exchange couplings
indicated by the yellow (orange) arrows in panels (a) and (b) in case of an
aligned [embedded] system. The wavefunction of the lowest energy trivial sub-
gap state is mostly localized at the section of the atomic chain with the
slow-rotating magnetic moments but it leaks also slightly (strongly) in the
$x$ ($y$) direction parallel (perpendicular) to the chain into the section
with no magnetic adatoms. In contrast, the probability density of the lowest
state in the topological phase reveals two peaks at the ends of the chain due
to the left and right MBS. The parameters are the same as in Fig. 3. We choose
$N_{y}=41$ and place the chain on the line $m_{0}=21$ along the $y$-direction.

We now consider a helical magnetic atomic chain placed on top of a two-
dimensional superconductor, as presented in Sec. III.2. We choose the same
smooth profile for the rotation rate between adjacent magnetic moments as in
the purely one-dimensional system discussed above. Many of our findings are
the same as for the one-dimensional system. For instance, a smooth decay
profile leads to sub-gap states with an energy close to zero over some range
of exchange coupling strengths in the trivial phase in the case of an aligned
system (see Fig. 4a). Extending the superconductor in the direction parallel
to the chain (increasing $n_{L}$) lifts the zero-energy pinning of the trivial
state, see Fig. 4c, in the same manner as in the embedded one-dimensional
system. Moreover, the two lowest-energy states become almost degenerate in the
trivial phase. Again, the energy of the MBSs is nearly independent of $n_{L}$
(see Fig. 4d) as in the one-dimensional system.

One significant difference to the analogous one-dimensional system is that the
exchange coupling strength for which the bulk gap closes and reopens is larger
than in the one-dimensional system [120], see the black vertical solid line in
Figs. 4a and 4b. This discrepancy can be understood due to the additional
leakage of the wavefunction into the direction perpendicular to the chain (see
wavefunction in Fig. 4f). Due to this leakage, low-energy states have a weaker
overlap with the sites where a finite exchange coupling is present, therefore
the effective exchange coupling strength decreases and the topological phase
transition is shifted to higher exchange coupling strengths $J$. This effect
is also reminiscent of the renormalization of the effective $g$-factor in
Rashba or TI nanowires due to metallization caused by the coupling to a thin
superconducting shell [121, 122, 123, 124]. We note that a leakage of the
wavefunction of the trivial sub-gap states in the perpendicular direction, see
Fig. 4e, in the aligned system does not significantly change the zero-energy
pinning, while the leakage in the direction parallel of the chain in the
embedded system does affect the energy of the sub-gap states.

In the most experimentally realistic setup of an embedded chain on the
superconductor, we do not find any zero-energy trivial sub-gap states. We also
note that, unlike in the one-dimensional system, even a single-site scalar
impurity at the end of the chain is now not sufficient to restore the zero-
energy pinning since, in two-dimensional systems, the confined state can
bypass the impurity.

###  V.3 Three-dimensional model

l0.int0in(a)l0.int1.25in(b)l0.2int-1.1in

(a)

(b)

l0.0int0.2in(c)

(c)

Figure 5: Energy spectrum and probability density of the lowest energy state
of a chain with a smooth decay rotation profile on a
$\underline{\text{three}}$-dimensional superconductor. (a) Energy spectrum as
a function of the exchange coupling $J$ for the aligned systems. The system
undergoes a topological phase transition indicated by the gap closing and
reopening and the appearance of zero-energy MBSs. We do not find trivial zero-
energy sub-gap states in this three-dimensional system. (b,c) Probability
density of the lowest energy state in the trivial and the topological regime,
respectively. The lowest state in the trivial regime is mostly localized in
the section of the smooth change of the rotation rate. The probability
densities are plotted for the exchange couplings strengths indicated by the
yellow and orange arrow in panel (a). The probability density in (b) shows
that the lowest state is localized in the region where the rotation rate
changes, as expected from the one- and two-dimensional case, even though the
state is energetically hardly separated from the bulk states. The MBSs are
well separated and localized at the opposite ends of the chain. Again, their
wavefuction is mostly located on the sites not covered by magnetic impurities,
allowing to effectively diminish their localization length [125, 69]. The
parameters are the same as in Fig. 3. In addition we choose $N_{y}=41$,
$m_{0}=21$, $N_{z}=25$, and $l_{0}=1$.

Finally, we consider an atomic chain placed on the surface of a three-
dimensional superconductor, utilising the model described in Sec. III.2. This
model is closest to realistic experimental set-ups, in which the adatoms are
deposited on top of a bulk three-dimensional superconductor [110, 100, 101].

As was found in the two-dimensional model, also for the three-dimensional
system there are no zero-energy states when the chain is embedded on the
surface of the superconductor. Unlike the two-dimensional system, however, in
the three-dimensional system we also do not find near zero-energy states when
the chain end is aligned with the boundary of the superconductor. In
particular, Fig. 5a shows the energy spectrum of an aligned system as a
function of the exchange coupling. The gap closes and reopens for an exchange
coupling strength $J_{C}$ that is larger than observed in both the one-
dimensional and two-dimensional systems. For coupling strengths below the
value at which the gap closes we do not observe any zero-energy sub-gap
states, even in the aligned system. We also note that the curvature of the
bulk gap closing lines differs from the curvature of the same process in the
two-dimensional superconductor (see Figs. 4a and 4b). After the gap closing,
as expected, we find MBSs appear at the end of the chain.

In the trivial regime, i.e. before the closing of the bulk gap, of the aligned
chain the probability density of the lowest state is bound to the section in
which $\Phi_{n}$ smoothly changes as we observed in one- and two dimensional
systems (see Fig. 5b), however the energy of this lowest state is essentially
equal to that of the bulk gap. Both the lowest energy state in the trivial
regime and the MBS decay exponentially into the bulk superconductor (see Figs.
5b and 5c).

We note that much of the behavior of the aligned three-dimensional system can
be understood from the fact that the chain is placed on the surface of the
three-dimensional superconductor and therefore states can scatter from the
surface into the bulk of the superconductor (along the $z$-direction). To
investigate the importance of a boundary of the system perpendicular to the
chain, we construct an analogous system in two dimensions, where the chain has
the same length as the superconductor in $x$ direction ($N_{C}=N_{x}$) and in
which the chain is placed along the boundary of the superconductor (see
Appendix A). In this scenario we also do not find trivial zero-energy sub-gap
states, similar to the three-dimensional system. In contrast, when the chain
is placed in the bulk of the superconductor rather than on the surface, then
trivial sub-gap states with energies smaller than the bulk gap do appear (see
Appendix B). Both of these results highlight the importance of scattering from
the boundary in pushing trivial zero-energy states to higher energies.

We would like to emphasize that numerical restrictions limit us to relatively
small systems in three dimensions and we cannot therefore perform extensive
numerical investigations in such systems. Nonetheless, the disappearance of
trivial zero-energy and sub-gap states in our three dimensional set-up, even
for the aligned chain, indicates that the formation of trivial zero-energy
sub-gap states is strongly suppressed when the chain is placed on top of a
three-dimensional superconductor. Especially since zero-energy states do form
in an aligned two-dimensional system of comparable width and length. As the
most realistic experimental setup is an embedded chain on the surface of a
three-dimensional superconductor we can conclude that there is a low
prevalence of zero-energy sub-gap states due to a smooth decay rotation
profile.

##  VI Domain wall

In this section, we will investigate the sub-gap states forming due to domain-
wall rotation profiles. The results from the previous section show that the
formation of trivial zero-energy states is unlikely when the length of the
superconductor ($x$ direction) exceeds the length of the chain and there is a
smooth decay in rotation rate. However, sub-gap states due to a domain wall
profile, i.e. a smooth change between a clockwise and anticlockwise rotation
of the magnetic moments within the chain, can be expected to be largely
independent of the relative position of the chain end and the superconductor
boundary. Although it should be noted that, for this profile, low energy
subgap states can be expected to form close to the domain wall rather than at
ends of the chain, as is expected for MBSs.

###  VI.1 One-dimensional model

l-0.00int-0.in(a)l1.67int-0.in(b)l-0.0int0.95in(c)l1.67int0.95in(d)l0.0int1.82in(e)l1.67int1.82in(f)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6:  Energy spectrum and probability density of the lowest energy state
of a chain with a domain wall rotation profile on a
$\underline{\text{one}}$-dimensional superconductor. The panels are arranged
in the same manner as in Fig. 3. The energies of the trivial sub-gap states
can be pinned to zero in case of (a) the aligned system and, in contrast to
the previous system, also in case of (b) the embedded system. The domain wall
allows the formation of a total of four MBSs, which hybridize and form
multiple sub-gap states in the topological regime. (c,d) The trivial state
(yellow) does not split away from zero when non-magnetic sites are added to
the left of the helical spin chain and therefore behaves similarly to the
lowest MBS (orange). (e,f) The wavefunction of the trivial state (and the left
MBS) is mainly localized in the region where the direction of the rotation
changes ($\Phi_{n}=0$). The wavefunction profiles are similar in both cases.
In particular, the wavefunction of the trivial state (yellow) is approximately
zero in the section without magnetic adatoms. Parameters: $N_{C}=164$,
$t\approx 10$ meV, $\mu=8$ meV, $\Delta=1$ meV, $2\Phi_{L}=-0.2026\pi$,
$2\Phi_{R}=0.2026\pi$, $\lambda=22$, $n_{0}=44$, $a=3\,\mbox{\AA}$, $n_{R}=0$.

First, utilising the one-dimensional model outlined in Sec. III.1 we find that
a chain with a domain wall profile does support trivial states with almost
zero energy in both the case of an aligned (see Fig. 6a) and also an embedded
chain (see Fig. 6b), which is in contrast to the system with the smooth decay
profile, for which we did not observe trivial zero-energy states in the
embedded case. In particular, within a domain wall profile, the rotation rate
of magnetic moments is set by Eqs. (9) and (13) such that the rotation rate of
the magnetic moments smoothly interpolates between
$\Phi_{L}\ll-\frac{2\pi}{N_{C}}$ on the left side and
$\Phi_{R}\gg\frac{2\pi}{N_{C}}$ on the right side of the chain. Here, we chose
for simplicity $\Phi_{L}=-\Phi_{R}$, which means that the direction of the
rotation changes along the atomic chain (see blue lines in Figs. 6e and 6f).
As such, since the critical exchange coupling at which the gap closes is
shifted to larger values for slower rotating magnetic moments, the chain is
completely trivial for an exchange coupling below $J^{<}_{C}(\Phi_{R})$ (see
also Fig. 2c). Within this trivial regime, we find two sub-gap states – as
well as their particle-hole partners – that have almost zero energy and the
transition from the aligned to the embedded system does not substantially
affect this zero-energy pinning (see Fig. 6c). Similar to the chain with a
smooth decay, the two lowest sub-gap states of the embedded chain become
almost degenerate in the case that the section without magnetic atoms to the
left of the atomic chain is sufficiently long, such that the localization
length of the states is much shorter than the length of the left section of
the superconductor without any magnetic adatoms.

Further analyzing the wavefunction of the states in this trivial regime
$J<J^{<}_{C}(\Phi)$, we find the maximum of the probability density of the
sub-gap states is localized at the position of the sign change of $\Phi_{n}$
and that the probability density is almost zero in the region where the
rotation rate reaches its maximal positive or maximal negative value (see blue
line Fig. 6e). Adding more sites to the left of the chain does not
substantially affect the energy spectrum since the weight of the wavefunction
in the section without magnetic atoms is very small (see Fig. 6f). In the
topological regime, i.e. after the closing and reopening of the bulk gap, we
find the system hosts four MBSs. In particular, one MBS appears at each end of
the chain and one MBS on each side of the domain wall. Depending on the length
of the domain wall transition the MBSs closest to the domain wall can
hybridize and form a finite-energy sub-gap state [126, 111, 127, 128].

###  VI.2 Two-dimensional model

l-0.00int-0.in(a)l1.67int-0.in(b)l-0.0int0.88in(c)l1.67int0.88in(d)l0.0int1.8in(e)l1.72int1.8in(f)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: Energy spectrum and probability density of the lowest energy state
of a chain with a domain wall rotation profile on a
$\underline{\text{two}}$-dimensional superconductor. The panels are arranged
in the same manner as in Fig. 4. The lowest sub-gap state has almost zero
energy when (a) the chain is aligned and also when (b) the chain is embedded.
The black solid vertical line indicates $J^{<}_{C}$ in an analogous one-
dimensional system, the line does not match with the actual bulk gap closing
and reopening in two dimension system. The energy of the lowest state is
neither affected in (c) the trivial nor in (d) the topological regime during
the crossover from the aligned to the embedded system if the chain is long
enough. (e) The wavefunction of the lowest energy trivial sub-gap state is
mostly localized at the domain wall but also slightly leaks into the section
with no magnetic adatoms perpendicular to the chain. In contrast in panel (f),
the probability density of the lowest state in the topological phase reveals
two additional peaks at the ends of the chain due to the left and right MBS.
Here, for our choice of parameters, the MBSs hybridize strongly. The
parameters are the same as in Fig. 6 and we chose $N_{y}=41$ and $m_{0}=21$ to
account for the two-dimensional system.

Next, we consider the helical chain with the domain wall profile that is
placed on top of a two-dimensional superconductor. The physical properties of
the sub-gap states are similar to those found in one-dimensional system,
namely trivial states are pinned to zero-energy in both cases of an aligned
and embedded chain (see Figs. 7a and 7b). The lowest energy states in the
trivial regime before the bulk gap closes and reopens (see Fig. 7c), as well
as in the topological regime after the closing and reopening of the bulk gap
(see Fig. 7d), are unaffected by the value of $n_{L}$. Furthermore, as found
in the one-dimensional system, the probability density of the near zero-energy
sub-gap states is localized to the region of the smooth change of the rotation
rate (see Fig. 7e). In the topological regime – after the reopening of the
bulk gap – strongly overlapping MBSs appear, their probability density is
maximal on the right end and in the region of the smooth transition (see Fig.
7f). We note that the wavefunction of the MBS only weakly leaks into the
$y$-direction perpendicular to the chain, such that the wavefunction is
confined very close to the position of the chain.

###  VI.3 Three-dimensional model

l0.int0in(a)l0int1.25in(b)l0.int-1.1in

(a)

(b)

l0.0int0.2in(c)

(c)

Figure 8: Energy spectrum and probability density of the lowest energy state
of a chain with a domain wall rotation profile on a
$\underline{\text{three}}$-dimensional superconductor. The panels are arranged
in the same manner as in Fig. 5. (a) No sub-gap states are present before the
topological phase transition, indicated by the gap closing and reopening. In
the topological regime, MBSs appear. The system hosts four MBSs: two on each
side of the domain wall. A zoom into the panel reveals the slightly different
energies of the two MBS pairs close to zero due to finite overlap between MBS
wavefunctions. The probability density of (b) the second lowest and (c) lowest
energy states in the topological phase. The probability densities are plotted
for the exchange couplings strength indicated by the orange arrow in panel
(a). The parameters are the same as in Fig. 6 and in addition we chose
$N_{y}=41$, $m_{0}=21$, $N_{z}=25$, and $l_{0}=1$ to account for the $y$ and
$z$ directions.

In contrast to the one- and two-dimensional systems considered above, we find
that a helical spin chain with a domain wall spin profile that is placed on
the surface of a three-dimensional superconductor does not exhibit trivial
near zero-energy states. The energy spectrum is shown in Fig. 8a. The MBSs
appear after the reopening of the bulk gap but there are no sub-gap states in
the trivial regime. This behaviour corresponds with a similar observation for
the smooth decay profile with aligned boundaries (see Sec. V.3). In the three-
dimensional system, the chain is placed on the surface of the superconductor,
thus, boundary effects such as scattering from the surface also affect the
energy of the lowest states. Finally we note, as also observed in one- and
two-dimensional systems, the probability density of the two lowest states in
the topological regime after the bulk gap has closed, reveals the presence of
a total of four MBSs, with two that are localized at the ends of the chain and
on either side of the domain wall. The MBSs close to the domain wall partially
hybridize (see Figs. 8b and 8c).

##  VII Quasi-MBS

In this section, we consider a chain in which only one section of the chain
nominally obeys the topological phase transition criterion, see the discussion
in Sec. IV and Fig. 2d. In particular, we consider a chain in which the
magnetic moments rotate with the rate $\Phi_{R}$ in the long right section of
the chain but in the left short section the rotation rate between neighbouring
magnetic moments increases smoothly up to $\Phi_{L}$, with
$\Phi_{k_{F}}>\Phi_{L}>\Phi_{R}$. The zero-energy sub-gap states that result
from a system only partially obeying the topological phase transition
criterion in a certain range of exchange couplings (see Sec. VII.1 for the
specific values of $J$ in the one-dimensional system) have been termed quasi-
MBSs [34].

###  VII.1 One-dimensional model

l-0.00int-0.in(a)l1.67int-0.in(b)l-0.0int0.95in(c)l1.67int0.95in(d)l0.0int1.86in(e)l1.67int1.86in(f)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9:  Energy spectrum and probability density of the lowest energy state
of a chain with a quasi-MBS rotation profile on a
$\underline{\text{one}}$-dimensional superconductor. The panels are arranged
in the same manner as in Fig. 3. The system hosts the so-called quasi MBSs
(yellow) for (a) the aligned and (b) the embedded system. (c) [(d)] The energy
of the quasi-MBS [the MBS] is nearly independent of the number of sites which
are placed to the left of the chain, though the probability densities shown in
panels (e) and (f) of the quasi-MBSs (yellow) and the MBSs (orange) are
substantially shifted into the section without magnetic adatoms. The
probability density reveals clearly that the quasi-MBS originates from
hybridizing MBSs which form in the section of faster rotation. Parameters:
$N_{C}=170$, $t\approx 10$ meV, $\mu=5$ meV, $\Delta=1$ meV,
$2\Phi_{L}=0.4484\pi$, $2\Phi_{R}=0.1121\pi$, $\lambda=25$, $n_{0}=50$,
$a=3\,\mbox{\AA}$, $n_{R}=0$.

First, considering a one-dimensional system, we find the bulk gap closing and
reopening appears at $J^{<}_{C}(\Phi_{R})$. The shorter left section with the
faster rotating magnetic moments, however, obeys the topological phase
transition criterion for $J^{<}_{C}(\Phi_{L})<J<J^{<}_{C}(\Phi_{R})$ (see Fig.
2d). As a result, the system contains a sub-gap state for exchange couplings
smaller than $J^{<}_{C}(\Phi_{R})$ which results from the presence of two
hybridizing MBSs. The energy of this state is well pinned to zero over a range
of exchange coupling strengths in both the aligned (see Fig. 9a) and in the
embedded setup (see Fig. 9b). This effect is shown clearly in Fig. 9d where we
observe that the energy of the lowest sub-gap states below and above
$J^{<}_{C}(\Phi_{R})$ remain close to zero as an increasing number of sites
$n_{L}$ without magnetic adatoms are added to the superconductor on the left
of the system. In contrast to the previous profiles we do not find a
degeneracy of the two lowest energy states in the embedded system.

The spatial profile of wavefunctions of these sub-gap states reveals the MBS
character of the quasi-MBSs. In particular, in the regime
$J<J^{<}_{C}(\Phi_{R})$, the probability density of the lowest state (yellow)
has two separated peaks at the ends of the section that obeys the topological
phase transition criterion (see Figs. 9e and 9f).

Increasing the exchange coupling to values of $J\approx J^{<}_{C}(\Phi_{R})$
allows one to bring the entire chain into the topological regime. We find the
quasi-MBSs transform into MBSs that are localised at the ends of the chain
(see orange line in Figs. 9e and 9f). Other rotation rate profiles, which lead
to a local reduction of the critical exchange coupling $J^{<}_{C}$, as
suggested in Fig. 2e, support quasi-MBSs as well and lead to similar results
as discussed in this section (not shown). We emphasize that the mechanism for
zero-energy states in partially topological chains crucially differs from the
one in fully trivial chains, see Secs. V and VI. However, similar to the
domain wall case, near zero-energy states appear in the partially topological
chain independent of the boundary conditions for a purely one-dimensional
system.

###  VII.2 Two-dimensional model

l-0.00int-0.in(a)l1.67int-0.in(b)l-0.0int0.88in(c)l1.67int0.88in(d)l0.0int1.8in(e)l1.72int1.8in(f)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10: Energy spectrum and probability density of the lowest energy state
of a chain with a quasi-MBS rotation profile on a
$\underline{\text{two}}$-dimensional superconductor. The panels are arranged
in the same manner as in Fig. 4. The main results are similar to ones found
for the one-dimensional model: the quasi-MBS energy is pinned close to zero in
the case of (a) the aligned and (b) the embedded system. A calculation of the
energy spectrum as a function of the length $n_{L}$ of the left section
without magnetic adatoms in (c) below and (d) above the bulk gap closing and
reopening shows that the energies are basically not affected, in contrast, to
the systems discussed in Secs. V and VI. The probability density for exchange
couplings (e) smaller and (f) larger than the coupling corresponding to the
bulk gap closing has a MBS character: The chain can partially enter the
topological phase due to the increasing rotation rate of the magnetic moments
in the left section of the chain. Consequently, MBSs appear at the ends of
this short topological section and move to the ends of the chain with
increasing exchange coupling. The parameters are the same as in Fig. 9 and in
addition we chose $N_{y}=41$ and $m_{0}=21$.

The behavior of a two-dimensional system, consisting of a one-dimensional
partially topological chain placed on top of a two-dimensional superconductor,
agrees well with the properties of the simple one-dimensional model, presented
in the previous subsection. In particular, the energy of the sub-gap states
does not depend on $n_{L}$ (see Figs. 10c and 10d) and the sub-gap states are
pinned close to zero-energy over some range of the exchange coupling strength
before the gap closes and reopens (see Figs. 10a and 10b). A subsystem of the
chain enters the topological regime for exchange couplings smaller than the
value at which the bulk gap closing and reopening appears, with quasi-MBSs
emerging at the ends of this subsystem (see Fig. 10e). The leakage of the
wavefunction in parallel or perpendicular direction to neighbouring sites of
the chain does not affect the zero-energy pinning. For sufficiently large
exchange couplings the gap closes and reopens and the right MBS is pushed to
the end of the chain (see Fig. 10f).

###  VII.3 Three-dimensional model

l0.int0in(a)l0.int1.25in(b)l0.2int-1.1in

(a)

(b)

l0.0int0.2in(c)

(c)

Figure 11: Energy spectrum and probability density of the lowest energy state
of a chain with a quasi-MBS rotation profile on a
$\underline{\text{three}}$-dimensional superconductor. The panels are arranged
in the same manner as in Fig. 5. (a) The system undergoes a topological phase
transition indicated by the gap closing and reopening and the appearance of
zero-energy MBSs. This three-dimensional system does not host any trivial
zero-energy sub-gap states. Probability density of the lowest state energy in
(b) the trivial and (c) the topological phase, respectively. Unlike the one-
and two-dimensional systems, the lowest state in the trivial state is not
localized in the region of the smooth change of the rotation rate. The
probability densities are plotted for the exchange couplings strengths
indicated by the yellow and orange arrow in panel (a). The parameters are the
same as in Fig. 9 and in addition we choose $N_{y}=41$, $m_{0}=21$,
$N_{z}=25$, $l_{0}=1$

Finally, we examine a helical chain with a quasi-MBS rotation rate profile as
in Fig. 2d on top of a three-dimensional superconductor and use the same
parameters for the chain as in the two-dimensional system. In contrast to the
one- and two-dimensional systems we do not find near zero-energy states in
this three-dimensional system, even though the parameters are the same.
Indeed, the energy spectrum shows a clear bulk gap closing and reopening as a
function of $J$ with zero-energy MBSs present only after this closing and
reopening (see Fig. 11a). The probability density of the lowest state in the
trivial and the topological regime is shown in Figs. 11b and 11c,
respectively. The lowest-energy trivial state is not bound to the region in
which $\Phi$ smoothly changes and, in contrast to the lower dimensional
systems, this lowest energy state is actually extended over the section in
which the rotation angle is almost constant, suggesting it is not of quasi-MBS
nature.

This result in combination with Secs. V.3 and VI.3 suggest that the formation
of trivial sub-gap states due to non-periodic rotations of the magnetic
moments along the chain is unlikely in three-dimensional systems. We again
note, however, that simulations in three dimensions are limited and that these
results do not exclude the presence of trivial zero-energy sub-gap states,
however, our results show that these states are less prevalent in three-
dimensional systems than in one- and two-dimensional systems and far less
abundant than in equivalent nanowire systems.

##  VIII Conclusions

In this paper, we investigated the prevalence of zero-energy bound states due
to non-periodic helical spin chains of magnetic adatoms on the surface of
superconductors. We showed that there exists a mapping of the Hamiltonian of a
one-dimensional helical spin chain to an effective Hamiltonian reminiscent of
a Rashba nanowire in the presence of a magnetic field. This mapping shows that
a spatially varying rotation rate transforms to a non-uniform Rashba SOI and a
non-uniform potential in the effective chain Hamiltonian. Since these
spatially varying potentials, when sufficiently smooth, are known to support
the formation of trivial zero-energy sub-gap states in Rashba nanowires, the
mapping therefore suggests that trivial zero-energy states in helical chains
might be as abundant as in Rashba nanowires. However, unlike any realistic
nanowire, the helical spin chain is installed on the surface of the
superconductor. As such, although it is possible to use this mapping to
generate some trivial zero-energy bound states for spin rotation profiles
which mimic known mechanisms for trivial sub-gap states in Rashba nanowires,
we found that such states are far less abundant in helical spin chains than in
nanowires. In particular, for the most experimentally relevant setup of a
helical spin chain on the surface of a three-dimensional superconductor we did
not find zero-energy bound states for any rotation profile.

Although we stress that our findings do not conclusively rule out the
appearance of trivial zero-energy sub-gap states in helical spin chain systems
due to non-uniformities, they clearly show that the same mechanisms that lead
to an abundance of zero-energy bound states in Rashba nanowires do not result
in equivalent issues in atomic chains, despite an apparent mapping between the
two systems. Further mechanisms other than the variation of the rate
$\Phi_{n}$ still can result in zero-energy sub-gap states. For instance,
another mechanism has been suggested in Ref. 129: multiple YSR states,
emerging from a magnetic chain on top of a superconductor, form a YSR band
with van Hove singularities, visible in the LDOS. The energy of these
singularities changes in the LDOS close to the chain ends and it can be tuned
to zero for sufficiently strong exchange coupling.

We also want to emphasize that our findings are only relevant for long helical
spin chains consisting of many rotation periods as, for instance, weakly
coupled YSR states in short trivial ferromagnetic chains can be tuned close to
zero energy for certain exchange coupling strengths [102, 100, 43, 101].
Nonetheless, it is a significant benefit that zero-energy bound states can be
more conclusively identified as MBSs in helical spin chains compared to in
semiconductor nanowires, especially since atomic chains have a reduced
tunability in comparison to semiconductor nanowire devices and so the phase
space of a purported MBS signal is more difficult to explore. Our findings
coupled with other benefits of helical spin chains, such as the fact that
states in atomic chains can be addressed very locally via STM measurements,
should enable one to build more confidence that a given zero-energy mode is of
topological origin rather than trivial in nature.

###### Acknowledgements.

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under Grant Agreement No 862046 and under
Grant Agreement No 757725 (the ERC Starting Grant). This work was supported by
the Georg H. Endress Foundation and the Swiss National Science Foundation.

##  Appendix A Chain on the boundary of a two-dimensional system

l0.0int0.in(a)

(a)

l0.0int0.in(b)l1.75int0.in(c)l0int1.1in(d)l1.75int1.1in(e)

(b)

(c)

(d)

(e)

Figure 12:  Sketch and energy spectra of a chain, with a quasi-MBS rotation
profile deposited at varying distances from the boundary in $y$ direction of a
two-dimensional superconductor. (a) Schematic sketch of an atomic chain (red)
deposited on a two-dimensional superconductor (orange). The length $N_{C}$ of
the atomic chain coincides with the total length $N_{x}$ of the system, the
chain is separated by a number of $N_{y}^{*}$ sites from the boundary of the
two-dimensional superconductor. (b,c) The energy spectrum of the system as a
function of the exchange coupling for $N_{y}^{*}=0$ and $N_{y}^{*}=15$,
respectively. (d,e) Energies as a function of $N_{y}^{*}$ for an exchange
coupling smaller and larger than the critical exchange coupling. The blue
[black] arrow in panels (a) and (b) indicates the value of the exchange
coupling used in panel (d) [(e)]. The trivial zero-energy state is present
only if the chain is placed away for the boundary. The parameters are the same
as in Fig. 10, except that we introduced the quantity $N_{y}^{*}$.

In this Appendix, we vary the position $m_{0}$ of the atomic chain in two-
dimensional systems. In particular, we place the chain on a boundary of a two-
dimensional superconductor. This analysis is motivated by our finding in
three-dimensional systems (see Secs. V.3, VI.3, and VII.3) that the appearance
of trivial zero-energy sub-gap states is suppressed compared to similar one-
and two-dimensional systems where such states occur due to smooth variations
of the rotation rates of neighbouring magnetic moments along the atomic chain.
This effect is likely a result of the boundary conditions, since the atomic
chain is placed on the surface of the three-dimensional superconductor. In the
two-dimensional system, see Secs. V.2, VI.2, and VII.2, however, we considered
a chain in the bulk of the superconductor ($m_{0}\approx N_{y}/2$). Placing
the chain on the boundary of a two-dimensional superconductor should,
therefore, lead to similar boundary effects as observed in three dimensions.

To further investigate the importance of boundary effects, we consider the
model from Sec. VII.2, with identical parameters except that the chain is
located at the boundary of the two-dimensional superconductor, $m_{0}=0$. The
corresponding energy spectrum as a function of the exchange coupling $J$ is
shown in Fig. 12b. The bulk gap closing and reopening at a critical exchange
coupling is accompanied by the appearance of MBSs at the ends of the chain.
The system does not, however, host trivial zero-energy sub-gap states for $J$
smaller than the critical exchange coupling.

Next, we extend the superconductor in $y$ direction, by adding
$N_{y}^{*}N_{x}$ sites, so that the atomic chain is not anymore positioned on
the boundary of the two-dimensional system. We calculate the energies as a
function of $N_{y}^{*}$ for an exchange coupling smaller (see Fig. 12d) and
larger (see Fig. 12e) than the coupling necessary for the bulk gap closing and
reopening. In the first case, there are no zero-energy states for
$N_{y}^{*}=0$, however, for finite values of $N_{y}^{*}$ a state with almost
zero energy appears. Furthermore, the size of the superconducting gap changes,
due to varying leakage of the wavefunctions to sites neighbouring the atomic
chain. For sufficiently large values of $N_{y}^{*}$ the gap and the zero-
energy state stabilize. In contrast, when the entire atomic chain enters the
topological regime, MBSs appear also for the choice $N_{y}^{*}=0$ and are
stable against changes of $N_{y}^{*}$. In both cases the size of the
superconducting gap varies as a function of $N_{y}^{*}$, but it stabilizes for
sufficient large values of $N_{y}^{*}$.

Finally, we plot the energy spectrum as a function of the exchange coupling
$J$ in the limit of large $N_{y}^{*}$, which means that the localization
length of the states in $y$ direction is shorter than the length $aN_{y}^{*}$,
see Fig. 12c. For this case, we obtain almost the same spectrum as in Fig.
10a, as expected. In addition, we calculated the energies of the systems from
Secs. V.2 and VI.2 with $m_{0}=0$. In both cases, we did not find any trivial
zero-energy sub-gap states in agreement with the results presented in Fig. 12.
In conclusion, we find that placing the chain at the boundary of the
superconductor, such that it is parallel to the chain, suppresses the
formation of trivial zero-energy sub-gap states in two dimensions.

##  Appendix B Chain in the bulk of a three-dimensional superconductor

l0.int0in(a)l0.int1.25in(b)l0.2int-1.1in

(a)

(b)

Figure 13: Energy spectrum and probability density of the lowest energy state
of a chain with a $\underline{\text{quasi-MBS}}$ rotation profile deposited in
the $\underline{\text{bulk}}$ of a $\underline{\text{three}}$-dimensional
superconductor. (a) The system hosts sub-gap states and a zoom into the figure
reveals a clear separation of a sub-gap state from the bulk states. (b)
Probability density of the sub-gap state at the exchange coupling indicated by
the yellow arrow in panel (a). The trivial sub-gap state is localized in the
region of smooth change of the rotation rate. The only changed parameters
compared to Fig. 11 are the locations of the chain sites, such that
$N_{y}=N_{z}=35$ and $m_{0}=l_{0}=17$.

In experiments, see e.g. Refs. [110, 100, 101], magnetic adatoms are deposited
on the surface of a superconductor, therefore we consider in the main text
only chains which are placed on the surface of a three-dimensional
superconductor. Here, on the contrary, we study a chain located fully inside
the bulk of a three-dimensional system. This particular situation cannot
easily be realized in experiments, however, we study this scenario to gain
further insights about the importance of boundary effects on the sub-gap state
at the surface. In particular, a sub-gap state appears in this setup for
exchange couplings smaller than the critical value at which the gap closes and
reopens, see Fig. 13a. This sub-gap state is not pinned to zero energy, but it
is separated in energy from the bulk gap, see the zoom on the right side,
which is not the case for systems in which the chain is placed on the surface
of the superconductor, cf. Secs. V.3, VI.3, and VII.3. Furthermore, we note
that the energy of the sub-gap state could potentially be pinned to zero for
sufficiently large systems. Moreover, the state is localized in the region of
smooth change of $\Phi$, see Fig. 13b. These results in combination with
Appendix A clearly show that the relative position of the chain towards the
superconductor boundary can strongly affect the energies of sub-gap states
and, in particular, trivial zero-energy states are suppressed when the chain
is placed close to a boundary, such as the surface of a three-dimensional
superconductor.

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