For an atomless frame $L$ with pointless part $E$, the cardinality of $\max L$
is bounded above by the cardinality of $\max\beta E$.

Example 8.1.7 makes the point that the pointless part of $L_{W}^{\star}$
(Theorem 8.1.4(4)) may be strictly larger than $E$.

######  Example 8.1.7.

Let $E$ be the regular open algebra of the topology $\mathscr{O}\mathbb{R}$ of
the real numbers, and let $W$ be the family of all maximal round filters on
$E$. Then $L_{W}^{\star}$ is isomorphic to $\beta E$ by Proposition 8.1.5, and
according to the proof of the proposition, we may take it to be the frame of
round ideals on $E$. Since in this case every ideal is round, $L_{W}^{\star}$
is (isomorphic to) the topology of the Stone space dual to $E$, i.e., the
topology of the Gleason cover of the real numbers.

The pointfree part of $L_{W}^{\star}$ is the sublocale consisting of those
elements which satisfy the condition of Proposition 3.2.4. The ideals which
satisfy this condition include the principal ideals, but include other ideals
as well. For instance, the reader will have no difficulty in verifying that
the ideal of regular open subsets of finite measure is nonprincipal and
satisfies the condition.

######  Question 8.1.8.

Is every pointless frame the pointless part of a compact frame?

Proposition 8.1.5, together with Example 8.1.7, motivate the consideration in
Subsection 8.3 of compact frames whose pointless parts are $C^{*}$-quotients.
This requires a brief digression in Subsection 8.2.

###  8.2. $\mathbf{sFs}$ is dually equivalent to $\mathbf{Tychs}$

Here we make the point in Proposition 8.2.1 that the restriction to skinny
frame homomorphisms, i.e., the passage from $\mathbf{F}$ to $\mathbf{Fs}$,
does not invalidate the classical dual equivalence between spatial frames and
Tychonoff spaces.

######  Definition ($\mathbf{Tychs}$,
$\max\colon\mathbf{sFs}\to\mathbf{Tychs}^{\operatorname{op}}$).

We denote the category of Tychonoff spaces with skinny continuous functions by
$\mathbf{Tychs}$. The functor

$\max\colon\mathbf{sFs}\to\mathbf{Tychs}^{\operatorname{op}}$

associates with each spatial frame $L$ the space with carrier set $\max L$
topologized by declaring

$\mathcal{O}\max L\equiv\left\\{\left\uparrow{a}\right\uparrow_{\max
L}\right\\}_{L},$

and associates with each $\mathbf{Fs}$-morphism $m\colon L\to M$ the
continuous function

$\max m\colon\max M\to\max L=\big{(}b\mapsto m_{*}(b)\big{)}.$

Proposition 8.2.1 shows that $\mathbf{sFs}$ and $\mathbf{Tychs}$ are dually
equivalent categories.

######  Proposition 8.2.1.

The functor $\max\colon\mathbf{sFs}\to\mathbf{Tychs}^{\operatorname{op}}$ is
an equivalence of categories.

###### Proof.

We have already remarked that a continuous function between Tychonoff spaces
has a frame counterpart which is skinny if and only if its fibers are
scattered, a correspondence which is evidently full and faithful. Of course,
every spatial frame is the topology of a Tychonoff space. See [1, 3.33]. ∎

###  8.3. The $qP$-coreflection in compact atomless frames

Proposition 8.3.1 points out an exotic feature of compact Hausdorff spaces
whose pointless parts are $C^{*}$-embedded.

######  Proposition 8.3.1.

Let $X$ be a compact Hausdorff space without isolated points whose pointless
part is $C^{*}$-embedded, by which we mean that its topology
$\mathcal{O}X\equiv L$ is isomorphic to $\beta\pi L$. Then the removal of any
single point does not change $\mathcal{C}X$, in the sense that for any $x\in
X$, $\mathcal{C}X=\mathcal{C}^{*}(X\setminus\\{x\\})$.

###### Proof.

Denote the open quotient corresponding to $X\setminus\\{x\\}$ by $o_{x}\colon
L\to O_{x}$. Since $\pi_{L}$ factors through $o_{x}$ by Proposition 3.1.4, the
latter is a $C^{*}$-quotient. ∎

######  Question 8.3.2.

In terms of Proposition 8.3.1, if each quotient $o_{x}\colon L\to O_{x}$,
$x\in X$, is a $C^{*}$-quotient, does it follow that $\pi_{L}$ is a
$C^{*}$-quotient?

The spaces described in Proposition 8.3.1 are reminiscent of $P$-spaces. A
point of a Tychonoff space is called a _$P$ -point_ if every real-valued
function on the space is constant on a neighborhood of the point. A $P$-space
is a space whose every point is a $P$-point. (See [8] and the references
therein for a thorough discussion of $P$-spaces.) Obviously any non-isolated
point of a $P$-space $X$ can be removed without changing $\mathcal{C}^{*}X$.
But a compact $P$-space is finite, whereas we are about to demonstrate that
the spaces of Proposition 8.3.1 abound. In fact, they are reflective in
compact Hausdorff spaces without isolated points, meaning every such space has
a canonical embedding into a space with the properties of Proposition 8.3.1.

######  Definition ($qP$-frame, $qP$-space).

A _$qP$ -frame_ is a compact frame whose pointless part is a $C^{*}$-quotient.
That is, a frame $L$ is a $qP$-frame if $\pi_{L}\colon L\to\pi L$ is
isomorphic to $\beta_{\pi L}\colon\beta\pi L\to\pi L$.333The term ‘quasi-$P$
frame’ was used in [13] with another meaning. A $qP$ space is a Tychonoff
space whose topology is a $qP$-frame.

######  Lemma 8.3.3.

A compact frame with a pointless $C^{*}$-quotient is a $qP$-frame. Otherwise
put, a $qP$-frame is a frame of the form $\beta M$ for some pointless frame
$M$.

###### Proof.

If $m\colon L\to M$ is a $C^{*}$-quotient map with $L$ compact and $M$
pointless then, because it must factor through $\pi_{L}$, it follows that
$\pi_{L}$ is a $C^{*}$-quotient isomorphic to $\beta_{\pi L}$. ∎

######  Corollary 8.3.4.

A $qP$-frame is atomless.

###### Proof.

Because a $qP$-frame $L$ is of the form $\beta M$ for a pointless frame $M$,
the coreflector $\beta_{M}\colon\beta M\to M$ is surjective and therefore
takes atoms to atoms, and because $M$ is atomless by Lemma 2.4.3(2), it
follows that $L$ is atomless. ∎

######  Definition ($\mathbf{kalFs}$, $\mathbf{qPFs}$).

We denote by $\mathbf{kalFs}$ and by $\mathbf{qPFs}$ the full subcategories of
$\mathbf{Fs}$ comprised, respectively, of the compact atomless frames and of
the $qF$-frames.

######  Theorem 8.3.5.

$\mathbf{qPFs}$ is coreflective in $\mathbf{kalFs}$; a coreflector for the
frame $L$ is the map $q_{L}\colon\beta\pi L\to L$ such that $\pi L\circ
q_{L}=\beta_{\pi_{L}}$.

###### Proof.

Given a test $\mathbf{kalFs}$-morphism $m$ with $M$ a $qP$-frame, apply the
pointless reflection to get $\pi m$, to which apply the compact coreflection
to get $\beta\pi m$, and then insert the maps $q_{L}$ and $q_{M}$ such that
$\pi_{L}\circ q_{L}=\beta_{\pi L}$ and $\pi_{M}\circ q_{M}=\beta_{\pi M}$.

${M}$${L}$${\pi M}$${\pi L}$${\beta\pi M}$${\beta\pi
L}$$\leftarrow$$\rightarrow$$\scriptstyle{m}$$\leftarrow$$\rightarrow$$\scriptstyle{\pi_{M}}$$\leftarrow$$\rightarrow$$\scriptstyle{\pi_{L}}$$\leftarrow$$\rightarrow$$\scriptstyle{\pi
m}$$\leftarrow$$\rightarrow$$\scriptstyle{\beta_{\pi
M}}$$\leftarrow$$\rightarrow$$\scriptstyle{\beta\pi
m}$$\leftarrow$$\rightarrow$$\scriptstyle{q_{M}}$$\leftarrow$$\rightarrow$$\scriptstyle{\beta_{\pi
L}}$$\leftarrow$$\rightarrow$$\scriptstyle{q_{L}}$

Then $q_{M}$ is an isomorphism because $\pi_{M}$ is a $C^{*}$-quotient map and
is therefore isomorphic to $\beta_{\pi M}$ (cf. Lemma 8.3.3), so that
$m=q_{L}\circ(\beta\pi m)\circ q_{M}^{-1}$ is the desired factorization of
$m$. ∎

######  Corollary 8.3.6.

The full subcategory of $\mathbf{Tychs}$ comprised of the $qP$-spaces is
bireflective in the full subcategory of $\mathbf{Tychs}$ comprised of the
compact spaces without isolated points.

The author would like to express his gratitude to the CECAT gang for
stimulating and encouraging discussions regarding the topics under
investigation. These Wednesday afternoon Zoom sessions were organized by
Chapman University’s Center of Excellence in Computation, Algebra, and
Topology. In particular, the author would like to thank Andrew Moshier for
suggesting the proof of Lemma 2.4.5.

## References

  * [1] J. Adámek, H. Herrlich, and G. Strecker, Abstract and Concrete Categories, The Joy of Cats, http://katmat.math.uni-bremen.de/acc, August, 2004.
  * [2] R. N. Ball, _Truncated abelian lattice-ordered groups II: the pointfree (Madden) representation_ , Topol. Appl. 178 (2014), 56–86.
  * [3] R. N. Ball and A. W. Hager, _On the localic Yosida representation of an archimedean lattice ordered group with weak order unit_ , J. Pure Appl. Algebra 70 (1990), 17–43.
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  * [6] R. N. Ball, A. W. Hager, and J. Walters Wayland, _From $\lambda$-skeletal $\kappa$-frames to $\lambda$-repletions in $\mathbf{W}$: II Continuous convergence on $\mathscr{R}L$_, in preparation.
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  * [8] R. N. Ball, J. Walters-Wayland, and E. Zenk, _The $P$-frame reflection of a completely regular frame,_ Topol. Appl. 158 No. 14 (2011), 1778–1794.
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  * [10] B. Banaschewski and C. J. Mulvey, _Stone-Čech compactification of locales, II_ , J Pure Appl Algebra 33 (1984), 107–122.
  * [11] F. Dashiell, _Shaving points,_ private communication, April 13, 2022.
  * [12] T. Dube, _Some ring theoretic properties of almost $P$-frames_, Algebra Univers. 60 (2009) 145–162.
  * [13] T. Dube and J. Nsonde Nsayi, _When certain prime ideals in rings of continuous functions are minimal or maximal_ , Topol. Apple. 192 (2015), 98–112.
  * [14] P. T. Johnstone, _Stone spaces_ , Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, 1982.
  * [15] J. Madden and H. Vermeer, _Epicomplete archimedean $\ell$-groups via a localic Wosida representation_, J. Pure and Appl. Alg. 68 (1990), 243–252.
  * [16] J. Picado and A. Pultr, Frames and Locales: Topology without Points, Frontiers in Mathematics, Birkhäuser, Springer, Basel, 2012.
  * [17] J. Picado and A. Pultr, Separation in pointfree topology, Birkhäuser, Springer Nature Switzerland AG, 2020.


# Calculations of in-gap states of ferromagnetic spin chains on s-wave wide-
band superconductors

Cristina Mier  Centro de Física de Materiales CFM/MPC (CSIC-UPV/EHU), 20018
Donostia-San Sebastián, Spain  Deung-Jang Choi  Centro de Física de Materiales
CFM/MPC (CSIC-UPV/EHU), 20018 Donostia-San Sebastián, Spain  Donostia
International Physics Center (DIPC), 20018 Donostia-San Sebastián, Spain
Ikerbasque, Basque Foundation for Science, 48013 Bilbao, Spain  Nicolás
Lorente<EMAIL_ADDRESS>Centro de Física de Materiales CFM/MPC (CSIC-
UPV/EHU), 20018 Donostia-San Sebastián, Spain  Donostia International Physics
Center (DIPC), 20018 Donostia-San Sebastián, Spain

###### Abstract

Magnetic impurities create in-gap states on superconductors. Recent
experiments explore the topological properties of one-dimensional arrays of
magnetic impurities on superconductors, because in certain regimes p-wave
pairing can be locally induced leading to new topological phases. A by-product
of the new accessible phases is the appearance of zero-energy edge states that
have non-Abelian exchange properties and can be used for topological quantum
computation. Despite the large amount of theory devoted to these systems, most
treatments use approximations that render their applicability limited when
comparing with usual experiments of 1-D impurity arrays on wide-band
superconductors. These approximations either involve tight-binding-like
approximations where the impurity energy scales match the minute energy scale
of the superconducting gap and are many times unrealistic, or they assume
strongly-bound in-gap states. Here, we present a theory for s-wave
superconductors based on a wide-band normal metal, with any possible energy
scale for the magnetic impurities. The theory is based on free-electron
Green’s functions. We include Rashba coupling and compare with recent
experimental results, permitting us to analyze the topological phases and the
experimental edge states. The infinite-chain properties can be analytically
obtained, giving us a way to compare with finite-chain calculations. We show
that it is possible to converge to the infinite limit by doing finite
numerical calculation, paving the way for numerical calculations not based on
analytical Green’s functions.

###### pacs:

74.55.+V,74.78.-w,74.90.+n

##  I Introduction

Ferromagnetic spin chains on s-wave superconductors have been shown to be able
to host Majorana bound states (MBS) at the edges of the chains by both
theoretical Choy  _et al._ (2011); Martin and Morpurgo (2012); Nadj-Perge  _et
al._ (2013); Li  _et al._ (2014); Pientka  _et al._ (2013); Braunecker and
Simon (2013); Pöyhönen  _et al._ (2014) and experimental works Nadj-Perge  _et
al._ (2014); Pawlak  _et al._ (2016); Jeon  _et al._ (2017); Kim  _et al._
(2018); Choi  _et al._ (2019). Atomic magnetic impurities produce in-gap
states Yu (1965); Shiba (1968); Rusinov (1969) because the local magnetic
interaction weakens the binding of Cooper pairs permitting the presence of
one-particle states in the superconductor gap Yazdani  _et al._ (1997); Flatté
and Byers (1997a, b). When several impurities lie on the surface such that
their induced in-gap states can overlap, in-gap bands can be formed. For two
impurities aligning ferromagnetically, the in-gap states form delocalized-
state that are analogues of molecular orbitals Flatté (2000); Morr and Yoon
(2006); Yao et al. (2014) in a one-electron picture of molecular binding. The
characterization of molecular-like in-gap states has been made possible by the
scanning tunneling microscope (STM) Kezilebieke  _et al._ (2018); Choi  _et
al._ (2018); Beck  _et al._ (2021); Ding  _et al._ (2021). As more impurities
are added, delocalized states form in ferromagnetic structures such that their
lowest eigenergies can cross the chemical potential level of the
superconductor as has been found experimentally Schneider  _et al._ (2021a);
Mier  _et al._ (2021). Thus, the spin chain can lead to closing of the
superconducting gap and to a phase transition. In the presence of spin-orbit
coupling, the pairing can change and a topological phase transition (TPT) is
induced Brydon  _et al._ (2015); Heimes  _et al._ (2015). A helical spin
structure Martin and Morpurgo (2012); Braunecker and Simon (2013, 2015);
Pientka  _et al._ (2013); Pöyhönen  _et al._ (2014); Schecter  _et al._ (2016)
has been shown to be equivalent to the effect of a Rashba interaction caused
by the superconductors spin-orbit coupling Pientka  _et al._ (2015); Schecter
_et al._ (2016); Christensen  _et al._ (2016). The possibility of emergence of
MBS in antiferromagnetically ordered chains has been also discussedKotetes
_et al._ (2016); Kobiałka  _et al._ (2021).
Recent experiments show that ferromagnetic spin chains can be assembled on
s-wave superconductors that have a substantial Rashba spin-orbit coupling
Nadj-Perge  _et al._ (2014); Pawlak  _et al._ (2016); Jeon  _et al._ (2017);
Kim  _et al._ (2018); Schneider  _et al._ (2021a); Mier  _et al._ (2021);
Schneider  _et al._ (2021b). These structures can create topological phases
that should show MBS at the edges of the structures even in the presence of
substantial disorder Awoga  _et al._ (2017). Usual s-wave superconductors are
normal metals above the superconducting transition, with large bands. Typical
parameters show that the superconducting gap is several thousands of time
smaller than the superconductor’s band. This situation is very different from
semiconductor nanowire systems proximitized by an s-wave superconductor Oreg
_et al._ (2010); Lutchyn  _et al._ (2010); Aguado (2017) where the induced
superconductor gap and the band gap can be expected to be of the same order of
magnitude. The effect of magnetic impurities is usually accounted for by an
exchange interaction acting on the superconductor’s electrons. Realistic
exchange interactions on solids range from hundreds of meV to eV, while
exchange fields in semiconducting nanowires only close the gap when they are
of the same magnitude as the superconductor gap. The large difference in
energy scales between impurities on wide-band superconductors and proximitized
semiconducting nanowires lead to very different techniques to treat each case.

The usual approach to treat wide-band superconductors is to approximate the
metallic phase by a free-electron metal Flatté and Byers (1997a, b); Pientka
_et al._ (2013); Schecter  _et al._ (2016); Li  _et al._ (2016); Choi  _et
al._ (2018); Schneider  _et al._ (2021a, b). The resulting superconductor
comes after a series of approximations to have the expected real-space
properties such as Friedel-like oscillations and coherence lengths in the
range of hundreds of nanometers. The following step is either to maintain a
Green’s function approach to characterize the electronic properties of the
superconductor, and include magnetic impurities via Dyson’s equation Flatté
and Byers (1997a, b); Choi  _et al._ (2018) (or equivalently the T-matrix
Schecter  _et al._ (2016); Sedlmayr  _et al._ (2021)), or to solve for the
wave functions using a Lippmann-Schwinger approach Pientka  _et al._ (2013);
Li  _et al._ (2016); Schneider  _et al._ (2021a, b). In the presence of
magnetic impurities and helical spin ordering, the wave-function approach is
further simplified by assuming that the induced in-gap states are strongly
bound and hence close to the Fermi energy. Despite this last approximation,
both approaches are very similar because they include the same approximations
to treat the free-electron-based superconductor.

The Green’s function approach is the one used in the present work. In section
II, we will briefly show the main approximations and features of the method.
In particular, it has the advantage of a simple and efficient numerical
implementation where the main operations are matrix inversions. Previous works
have shown that this technique permits us to rationalize the experimental
findings in wide-band superconductors with large impurity exchange coupling
Choi  _et al._ (2018); Mier  _et al._ (2021). The imaginary part of the real-
space Green’s function is the local density of states (LDOS), then using
Tersoff-Hamman’s theory Tersoff and Hamann (1983), we can evaluate the
quantities obtained in STM experiments by real-space Green’s functions. The
study of topological phases can be evaluated by determining the presence of
MBS in the LDOS. However, as in experiments, it can be difficult to prove that
a zero-energy edge mode is a MBS. There are two usual ways of identifying MBS.
One is to study the properties of the MBS. Previous works have shown that the
spin-polarization of MBS are a crucial quantity to determine Sticlet  _et al._
(2012); Jeon  _et al._ (2017); Mashkoori  _et al._ (2020); Wang  _et al._
(2021). As shown in Ref. [Mashkoori  _et al._ , 2020], the study of the
evolution of the in-gap-band structure with the changing parameter together
with the spin polarization is an excellent probe to unravel topological
phases. Comparison of the Green’s function obtained LDOS with experiments
indeed show that Cr spin chains on $\beta$-Bi2Pd wide-band superconductor can
undergo a topological phase transition when the number of atoms increase Mier
_et al._ (2021). There, it is also shown that the staggered magnetic moment
gives valuable information as predicted in Ref. [Sticlet  _et al._ , 2012]. A
second way is to use the bulk-boundary correspondence principleAsbóth  _et
al._ (2016); Kitaev (2001), where the study of topological invariants
determined by the bulk Hamiltonian implies the appearance of MBS.

In this article, we have chosen the second approach. In section III, we will
describe the impurity and Rashba Hamiltonians used here, and the derivation of
the in-gap-bands. To this end, we implement the theory developed by Tewari and
Sau Tewari and Sau (2012) using Green’s functions. We find that the usual
implementation Wang and Zhang (2012) identifying the reciprocal space
Hamiltonian with the zero-energy inverse of the Green’s function does not work
because the superconducting real-space Green’s function is not a resolvent.
Instead, we derive an effective Hamiltonian that we obtain from the
renormalization of the Green’s function and that correctly describes the
dispersion of the in-gap states. From this band structure, in section IV, we
compute the winding-number and the lower-symmetry $\mathbb{Z}_{2}$ invariant
for an infinite 1-D system. These quantities determine the topological phase
of the system, in good agreement with the emergence of MBS at the edge of
finite chains. A systematic study of ferromagnetic spin chains allows us to
define topological phase diagrams. In section V, we compare these results with
numerical calculations on finite systems. Our results show that winding
numbers can still be calculated in these systems. Indeed, our numerical
procedure seems to be robust and can be used to analyze 1-D chains in 2-D
superconductors with increasing resemblance to the experiment. As an example,
we are able to identify the non-topological or topological origin of edge
modes that oscillates around the Fermi energy as atoms are added to the chain.
Similar behavior has been recently reported on experimental resultsSchneider
_et al._ (2021a).

##  II Wide-band electronic Green’s function for superconductors

Local-basis-set approaches to treat a wide-band superconductor incurs into
numerical problems due to the large energy mismatch between the electron-band
width and the pairing energy $\Delta$. Here, we will use extended states to
describe the superconductor. The normal metal is then treated in the free-
electron approximation. We adopt the theory developed by Flatté and Byers
Flatté and Byers (1997a, b). The main features of their theory is to use a
real-space Green’s function for the superconductor based on free-electrons and
then solve for the effect of magnetic impurities using Dyson’s equation. In
this section, we are going to briefly analyze the specificities of the Green’s
functions obtained in this way.

The Bogoliubov-de Gennes approach can be succinctly expressed using Nambu’s
formalism Shiba (1968); Balatsky  _et al._ (2006); Zhu (2016); Vernier  _et
al._ (2011); Pientka  _et al._ (2013); Schecter  _et al._ (2016). Here, we
choose a plane-wave electronic basis and 4-component Nambu operators to
express the Bogoliubov-de Gennes equations in matrix form. The basis set of
our study is given by the Nambu operator Zhu (2016); Vernier  _et al._ (2011)
$\hat{\Psi}_{k}=(\hat{c}_{k,\uparrow},\hat{c}_{k,\downarrow},\hat{c}_{-k,\uparrow}^{\dagger},\hat{c}_{-k,\downarrow}^{\dagger})^{T}$,
where, $k$ is the wave-vector of the plane-wave basis function
$\phi_{k}(r)=e^{i\vec{k}\cdot\vec{r}}/\sqrt{V}$, $V$ is the normalization
volume and $\vec{r}$ are the spatial-coordinate vectors.

The Bogoliubov-de Gennes formalism is a mean-field treatment that permits us
to use one-particle equations at the expense of doubling the basis set.
Indeed, the Hamiltonian resulting from using the Nambu’s basis set is
artificially particle-hole symmetric and contains two extra redundant
solutions that are not physical. In many cases, the basis set can be reduced
to a 2-component Nambu spinor. This is not possible when spin-flip scattering
or spin-orbit interactions are present due to the mixing of spinor components
Zhu (2016).

Using free-electrons with a pairing interaction of strength $\Delta$, the
Hamiltonian matrix is expressed as Shiba (1968); Vernier  _et al._ (2011),

${H}_{BCS}=\xi_{k}\tau_{z}\sigma_{0}+\Delta\tau_{y}\sigma_{y},$ (1)

where $\xi_{k}$ is the energy from the Fermi level
($\xi_{k}=\epsilon_{k}-E_{F}$), and $\Delta$ is the superconducting pairing
potential. Here, the tensor product of Pauli matrices for the spin ($\sigma$)
and particle ($\tau$) sectors spans the $4\times 4$-matrix space if the
identity matrices ($\sigma_{0}$ and $\tau_{0}$) are included. Thanks to the
one-particle character of this Hamiltonian, we can use the resolvent to find
the superconductor’s Green’s function Vernier  _et al._ (2011):

$\displaystyle G^{0}(k,\omega)$ $\displaystyle=$ $\displaystyle\langle
k|[\omega\tau_{0}\sigma_{0}-{H}_{BCS}]^{-1}|k\rangle$ (2) $\displaystyle=$
$\displaystyle\frac{1}{\omega^{2}-\xi^{2}-\Delta^{2}}(\omega\tau_{0}\sigma_{0}+\xi_{k}\tau_{z}\sigma_{0}+\Delta\tau_{y}\sigma_{y}).$

The retarded version of Eq. (2) can be obtained replacing $\omega$ by
$\omega+i\Gamma$, where the Dynes parameter, $\Gamma$, is taken as a small and
positive real number that is phenomenologically associated with the lifetime
of quasiparticles Dynes  _et al._ (1978). The imaginary part of
$-G^{0}(k,\omega)$ becomes the one-particle density of states of the
superconductor. We do not even attempt to plot this density of states for a
realistic wide-band superconductor because it basically reduces to two tiny
gaps with value $2\Delta$ near the Fermi wave vector $k_{F}$ that disappear in
the fast dispersing bands with $k$-values.

Instead, we will Fourier transform to real space. However, the Fourier
transform does not converge. We are forced to apply the BCS-like trick of only
including states within a shell of width the Debye energy, $\hbar\omega_{D}$,
around the Fermi energy Schrieffer (1994). This can be done weighing the
integrand by Gaussian functions of width $\hbar\omega_{D}$, and at the end of
the calculation taking the limit $\hbar\omega_{D}\rightarrow 0$ because the
Debye energy is much smaller than the Fermi energy. This is done in Ref.
[Pientka  _et al._ , 2013]. This set of approximations allows us to recover
the real-space Green’s function for a free-electron-like superconductor as
presented by Flatté and Byers Flatté and Byers (1997a, b). In the above
$4\times 4$-Nambu space, the non-local Green’s function is:

$\displaystyle{G}_{BCS}$ $\displaystyle(r,\omega)=-\frac{\pi
N_{0}}{k_{F}r}e^{\frac{-\sqrt{\Delta^{2}-\omega^{2}}}{\pi\xi\Delta}r}$
$\displaystyle\times$
$\displaystyle\begin{pmatrix}\cos{k_{F}r}+\frac{\omega}{\sqrt{\Delta^{2}-\omega^{2}}}\sin{k_{F}r}&0&0&\frac{-\Delta}{\sqrt{\Delta^{2}-\omega^{2}}}\sin{k_{F}r}\\\
0&\cos{k_{F}r}+\frac{\omega}{\sqrt{\Delta^{2}-\omega^{2}}}\sin{k_{F}r}&\frac{\Delta}{\sqrt{\Delta^{2}-\omega^{2}}}\sin{k_{F}r}&0\\\
0&\frac{\Delta}{\sqrt{\Delta^{2}-\omega^{2}}}\sin{k_{F}r}&-\cos{k_{F}r}+\frac{\omega}{\sqrt{\Delta^{2}-\omega^{2}}}\sin{k_{F}r}&0\\\
\frac{-\Delta}{\sqrt{\Delta^{2}-\omega^{2}}}\sin{k_{F}r}&0&0&-\cos{k_{F}r}+\frac{\omega}{\sqrt{\Delta^{2}-\omega^{2}}}\sin{k_{F}r}\end{pmatrix}$

where $r$ is the distance between two points in the superconductor. The
prefactor includes $N_{0}$ that is the normal-metal density of states at the
Fermi energy, and the exponential behavior with distance, controlled by the
correlation length $\xi$ of the superconductor. This expression recovers known
properties of the electronic structure of an s-wave superconductor. However,
it presents a divergence at $r=0$. We can just evaluate the Fourier transform
for $r=0$ and find out the correct expression Vernier  _et al._ (2011);
Schecter  _et al._ (2016) of the Green’s function at $r=0$. In this way, we
will have the correct expression for finite $r$ and the correct $r=0$ limit.
This should not be a problem when using the above expressions on a lattice,
such that the discrete step is of the order of the underlying lattice
parameter. Since the spatial oscillations appearing in Eq. (II) are of the
order of the Fermi wavelength, $\lambda_{F}=2\pi/k_{F}$, we will be able to
use Eq. (II) in a discrete lattice Meng  _et al._ (2015) such that $\Delta
r>1/k_{F}$. In practice, we can probably use Eq. (II) even for rather small
values of $k_{F}$ and still have physically-correct results down to steps of
the order of the lattice parameter. Following Ref. [Vernier  _et al._ , 2011],
the $r=0$ limit of the Green’s function is the usual local BCS Green’s
function:

$\displaystyle G_{BCS}(r=0,\omega)$ $\displaystyle=$ $\displaystyle-\frac{\pi
N_{0}Sgn[Re(\omega)Im(\omega)]}{\sqrt{\Delta^{2}-\omega^{2}}}$ (4)
$\displaystyle\times$ $\displaystyle\begin{pmatrix}\omega&0&0&-\Delta\\\
0&\omega&\Delta&0\\\ 0&\Delta&\omega&0\\\ -\Delta&0&0&\omega\end{pmatrix}.$

In summary, the above real space Green’s function has two important
restrictions. The first one is that beyond an energy scale given by the Debye
frequency, the Green’s function is not physical. The second one is that it can
be used seamlessly from $r=0$ to finite $r$ if the discrete steps are large
enough, where the typical length scale is given by the Fermi wavelength.

In the present work, we are interested in studying the topological phases
associated with spin chains on wide-band superconductors. We will evaluate the
topological properties of the bulk superconductor. To do this, we need to
transform back our real space Green’s function to k-space. As in previous
works Pientka  _et al._ (2013); Schecter  _et al._ (2016), we are going to
assume a discrete spatial step $\Delta r=a$, the lattice parameter of our
superconductor. In this case, we have to evaluate the discrete Fourier
transform that due to the translational invariance of the underlying crystal
structure can be written as Asbóth  _et al._ (2016)

$G_{BCS}(\vec{k},\omega)=\sum_{\vec{R}}{G}_{BCS}(R,\omega)e^{i\vec{k}\cdot\vec{R}}.$
(5)

Here $\vec{R}$ are all the positions of the atoms in the crystal. We will work
on 1-D spin chains. Then it is interesting to find $G_{BCS}(k,\omega)$ in 1-D
where the other two spatial coordinates have been set to zero. This is easily
done because the sum over $\vec{R}$ can be analytically performed as explained
in Ref. [Pientka  _et al._ , 2013]. For the Green’s function this has been
done in the supplemental material of Ref. [Schecter  _et al._ , 2016].

Expression Eq. (5) lends itself to numerical implementation. This can be
interesting when trying to solve problems with spin chains that require an
all-numerical approach. We have computed Eq. (5) by considering a finite 1-D
array of sites on the superconductor and compared with the results of the
analytical calculation. The agreement is very good even for rather small sets
of the 1-D array of sites. Figure 1 shows the comparison of the density of
states $-Im\,G_{BCS}(\vec{k},\omega+i0^{+})/\pi$ computed using both schemes.
The shown case is for $k_{F}=0.15\;a_{0}^{-1}$ and $a=3.36$ Å that we have
used to describe the $\beta$-Bi2Pd superconductor Choi  _et al._ (2018); Mier
_et al._ (2021)

The calculation of the density of states
$-Im\,G_{BCS}(\vec{k},\omega+i0^{+})/\pi$ also reveals a cutoff for $k>k_{F}$
when $k_{F}$ is smaller than the Brillouin zone value $\pi/a$. This reflects
the fact that small values of $r$ are not well-taken care of by the real-space
Green’s function. As a consequence, values $k>k_{F}$ will behave
pathologically. However, the density of states becomes well-behaved as soon as
$k_{F}>\pi/a$, which corresponds to cases of substantial band folding, or
wide-band superconductors. In the case of spin chains, large folding can be
also obtained for very diluted spin chains, where the distance between
impurities is much larger than the superconductor lattice. The above procedure
then works well in the limit of diluted spin chains Pientka  _et al._ (2013);
Schecter  _et al._ (2016). The cutoff in density of states
$-Im\,G_{BCS}(\vec{k},\omega+i0^{+})/\pi$ also disappears in the case of small
correlation lengths. Showing that there are two important length scales,
$2\pi/k_{F}$ and the correlation length $\xi$. In the following calculations,
we have always used the BCS value $\xi=\hbar v_{F}/\pi\Delta$ where $v_{F}$ is
the free-electron Fermi velocity.

Figure 1:  Density of states for a free-electron-like superconductor computed
using Eq. (5), for $k_{F}=0.15\;a_{0}^{-1}$ and $\pi/a=0.5\;a_{0}^{-1}$. $(a)$
is the calculation using a finite set of 101 sites and $(b)$ is the analytical
calculation. The step for $|k|>k_{F}$ disappears as soon as $k_{F}$ becomes
larger than the first-Brillouin-zone vector $\pi/a$ or the correlation length,
$\xi$ becomes small. The normal-metal DOS at the Fermi energy is
$N_{0}=0.037$/eV, the Dynes broadening is $\Gamma=0.01$ meV, and the gap is
$\Delta=0.75$ meV.

##  III In-gap-bands with Green’s functions

In-gap or in-gap state bands are easily accounted for within the approximation
of classical spinsYu (1965); Shiba (1968); Rusinov (1969). We adopt a spin
$sd$ model, also known as Kondo Hamiltonian, that separates the impurity
action into charge and spin contributions given by the potential scattering
term $K_{j}$ and the exchange term $J_{j}$. The Kondo Hamiltonian in the
previous Nambu basis is given by

$\hat{H}_{impurity}=\sum_{j}^{N}(K_{j}\tau_{z}\sigma_{0}+J_{j}\vec{S_{j}}\cdot\vec{\alpha})\;$
(6)

where the sum over $j$ is over the impurities of the chain. The atom spin is
assumed to be classical and equal to $\vec{S_{j}}=(S_{j,x},S_{j,y},S_{j,z})$
$=S(\sin{\theta_{j}}\cos{\phi_{j}},\sin{\theta_{j}}\sin{\phi_{j}},\cos{\theta_{j}})$.
The electron spin is expressed via Pauli matrices in the Nambu basis set as:
$\vec{\alpha}=\frac{1+\tau_{z}}{2}\vec{\sigma}+\frac{1-\tau_{z}}{2}\sigma_{y}\vec{\sigma}\sigma_{y}$,
where $\vec{\sigma}$ is the spin operator Shiba (1968).

In the philosophy of the previous section, the effect of the impurity chain
can be included using Dyson’s equation. For infinite periodic chains, this is
done in reciprocal space, because the equation becomes algebraic:

$G(\vec{k},\omega)=G_{BCS}(\vec{k},\omega)+G_{BCS}(\vec{k},\omega)\Sigma(\vec{k},\omega)G(\vec{k},\omega).$
(7)

As above, the Green’s functions, $G_{BCS}$ and $G$, and the self-energy
$\Sigma$ are $4\times 4$ matrices. The arithmetic involved to solve Dyson’s
equation is just $4\times 4$-matrix algebra.

Due to the locality of the Kondo Hamiltonian, Eq. (6), and the mean-field
character of the Bogoliubov-de Gennes theory, the self-energy is easily
computed. In 1-D and assuming all impurities to be identical, it is simply

$\Sigma(\vec{k},\omega)=\sum_{\vec{R}}\langle\vec{R}|\hat{H}_{impurity}|0\rangle
e^{i\vec{k}\cdot\vec{R}}=K\tau_{z}\sigma_{0}+J\vec{S}\cdot\vec{\alpha},$ (8)

with $\vec{\alpha}$ defined above. In the above expression, we have made used
of the Bloch representation, such that the matrix element
$\langle\vec{R}|\hat{H}_{impurity}|0\rangle$ is only evaluated between unit
cell $0$ and unit cell $\vec{R}$ of the periodic system.

###  III.1 Evaluation of the effective Hamiltonian

To compute the in-gap-bands is, however, not a simple task. As we saw in the
preceding section, the superconducting Green’s function, $G_{BCS}$, is not
really a Nambu resolvent. As a consequence, the usual method of diagonalizing
a Hamiltonian extracted from the Green’s function Wang and Zhang (2012),
$\hat{H}(\vec{k})=-G^{-1}(\vec{k},\omega=0)$, does not work. This cannot work
because for $G^{0}$, this scheme gives something approximate to a flat band
for $-k_{F}<k<k_{F}$ at several hundreds of meV depending on the electron
density of the superconductor, and adding an exchange coupling in the range of
eV, just splits the band orders of magnitude away from the gap energy.

In order to solve this problem, we notice that we have to generalize the
resolvent equation. To do this, we expand $G(\vec{k},\omega)$ to first order
in $\omega$, and we identify this to the resolvent equation, Eq. (2). The
resulting Hamiltonian comes from a renormalized Green’s function and shows the
correct in-gap dependence:

$\hat{H}(\vec{k})=-\left(\frac{\partial
G^{-1}(\vec{k},\omega)}{\partial\omega}\right)^{-1}_{\omega=0}G^{-1}(\vec{k},\omega=0),$
(9)

The results are excellent, the bands perfectly match the PDOS obtained from
the imaginary party of the retarded Nambu Green’s function, and all in-gap
states properties are retrieved. This is to be expected because the condition
for finding the bands or eigenvalues of Eq. (9) is the condition of
singularity for the Green’s function for small $\omega$.

###  III.2 Calculations in real space

Here, we are going to compare with real-space calculations in order to
describe the possible topological phases as well as in-gap states of other
nature. We assume we can express the electronic states in a local basis set,
compact to the atomic sites, that do not overlap and can be taken to be a
tight-binding orthonormal basis set with a total of $N$ orbitals or sites.

In this case, Dyson’s equation is just a resolvent equation for a $4N\times
4N$ matrix:

$\hat{G}=[\hat{G}_{BCS}^{-1}-\hat{H}_{I}]^{-1}$ (10)

Where $\hat{G}_{BCS}$ is the retarded Green’s operator for the BCS Hamiltonian
from Eq.(1) and $\hat{H}_{I}=\hat{H}_{impurity}+\hat{H}_{Rashba}$. The
$\hat{H}_{Rashba}$ includes the Rashba interaction as described in the next
section.

In this case, we evaluate the real-space density of states by projecting the
density of states on the tight-binding orbitals. This projected density of
states (PDOS) on orbital $i$ or spectral function is given by

$\rho(i,\omega)=-\frac{1}{\pi}Im[{G}^{1,1}_{i,i}(\omega)+{G}^{4,4}_{i,i}(-\omega)],$
(11)

where ${G}^{\nu,\mu}_{ii}$ is the resulting Green’s function evaluated on
orbital $i$ for the Nambu components $\nu$ and $\mu$ by solving Dyson’s
equation. Thus, the calculations for finite chains are performed on a 2-D
finite mesh of the 3-D superconductor, where a few sites without impurity
interactions are left around the impurity chain. Our calculations are quite
robust against the number of free superconducting sites left around the
impurity chain, including subsurface layers, probably due to the 3-D character
of the superconducting Green’s function.

###  III.3 Rashba self-energy

In the same spirit as above, we can introduce the spin-orbit coupling for a
surface, using the self-energy for the Rashba Hamiltonian. In the tight-
binding electron basis, the non-locality of the Rashba Hamiltonian makes it
formally similar to a nearest-neighbor hopping term,

$\displaystyle\hat{H}_{Rashba}$ $\displaystyle=$ $\displaystyle
i\frac{\alpha_{R}}{2a}\sum_{i,j,\alpha,\beta}[\hat{c}^{\dagger}_{i+1,j,\alpha}(\sigma_{y})_{\alpha,\beta}\hat{c}_{i,j,\beta}$
(12) $\displaystyle-$
$\displaystyle\hat{c}^{\dagger}_{i,j+1,\alpha}(\sigma_{x})_{\alpha,\beta}\hat{c}_{i,j,\beta}+h.c.]$

where ${\alpha,\beta}$ are spin indexes. The lattice parameter of the
substrate is $a$, and the factor of $2a$ comes from a finite-difference scheme
to obtain the above discretized version of the Rashba interaction.

Transforming to a 1-D reciprocal space and using the Nambu basis set, the self
energy becomes

$\Sigma(k,\omega)=2\alpha_{R}\sin(ka)\tau_{z}\sigma_{y}$ (13)

For higher dimensions, we use the real space representation given by the above
Hamiltonian and we do the Fourier transform to reciprocal space using a
truncated unit-cell summation.

##  IV Winding number and topological phase space for spin chains on a wide-
band superconductor

The presented methodology based on Green’s function permits us to compute both
infinite and finite spin chains on superconductors. We can easily put the
bulk-boundary correspondence principleAsbóth  _et al._ (2016); Kitaev (2001)
to test as well as to characterize the topological superconducting phases
resulting from the in-gap states.

###  IV.1 Topological invariants

Tewari and Sau Tewari and Sau (2012) studied the topological properties of 1-D
spin chains in one and two dimensions. They showed that the in-gap electronic
structure induced by a spin chain on a 1-D superconductor leads to phases
compatible with the BDI classChiu  _et al._ (2016). This classification
results from the chiral symmetry characterizing the system.

Due to the presence of magnetic interactions, time-reversal symmetry is broken
in the model of a FM spin chain. However, the following antiunitary operator
may be defined, $\mathcal{T}=\tau_{0}\sigma_{0}\mathcal{K}$, where
$\mathcal{K}$ is the complex conjugate operator. The Hamiltonian satisfies
$\mathcal{T}\hat{H}(k)\mathcal{T}^{-1}=\hat{H}(-k)$, this symmetry is the so-
called generalized time reversalHeimes  _et al._ (2015) or spin-rotation time-
reversal Sato and Fujimoto (2016) symmetry. Additionally, particle-hole
symmetry is defined by the operator
$\mathcal{P}=\tau_{x}\sigma_{0}\mathcal{K}$ and it is present on every BdG
hamiltonian by construction. The combination of the two is the chiral symmetry
and the corresponding operator is the product of the two previous ones,
$\mathcal{C}=\mathcal{P}\mathcal{T}=\tau_{x}\sigma_{0}$. As a consequence, the
Hamiltonian of the system can be written, in a rotated basis, under the form:

$H(k)=\begin{pmatrix}0&A(k)\\\ A^{\dagger}(-k)&0\end{pmatrix}$ (14)

Here $A$ is a $2\times 2$ matrix in the spin sector. The representation of Eq.
(14) is easily obtained by changing the basis set from the Nambu basis
expressed in fermionic operators $\hat{c}$ and $\hat{c}^{\dagger}$ to a basis
set expressed in terms of Majorana operators
$(\hat{c}\pm\hat{c}^{\dagger})/\sqrt{2}$.

For a $2\times 2$ Hamiltonian, Eq. (14) can be written as:

$H(k)=d_{x}(k)\tau_{x}+d_{y}(k)\tau_{y},$ (15)

where the change of basis has permitted us to have a zero component of
$\tau_{z}$ because $\tau_{z}$ anticommutes with the Hamiltonian and defines
the chiral symmetry of the system Asbóth  _et al._ (2016). In the fermionic
basis set of the original Hamiltonian Heimes  _et al._ (2015); Kobiałka  _et
al._ (2021), the symmetry representation is the above
$\mathcal{C}=\tau_{x}\sigma_{0}$.

Figure 2:  In-gap states for ferromagnetic spin chains in trivial ((a) to (d))
and topological ((e) to (h)) state. The Kondo coupling is $J=2.0$ eV (left)
and $J=2.7$ eV (right), potential scattering $K=5.5$ eV and Rashba coupling
$\alpha_{R}=3.0$ eV-Å, the Fermi vector is $k_{F}=0.3\;a_{0}^{-1}$ and the
spin is 5/2 (like Cr or Mn), the metal density of states at the Fermi energy
is $N_{0}=0.037$/eV and the superconducting gap is $\Delta=0.75$ meV (like in
$\beta$-Bi2Pd [Mier  _et al._ , 2021]). (a) and (e) The infinite spin chain
has 4-bands with values close to the superconducting gap for values of $k$
between Fermi vectors, insets show the two lower bands where we can observe
the trivial (a) and topological (e) gap. (b) and (f) plot the trajectory of
the normalized $\vec{d}(k)$ as $k$ is varied. To facilitate the visualization
of the trajectory, $k$ values go from $-\pi/a$ (in cyan) to $\pi/a$ (in
magenta). In (b) all points remain in the vicinity of $(d_{x},d_{y})=(1,0)$ so
no turn is completed about zero, on the other hand, in (f) we can observe a
complete anticlockwise turn. (c) and (g) show the evolution of $d_{x}$
(orange), $d_{y}$ (green) and $w(k)$ (blue) as a function of $k$. In (c) we
observe that $w$ has a final value of $w=0$ indicating that the system is a
trivial state. In (g) $w(k)$ goes from 0 to $w=-1$, showing that here the
system is a topological state. (d) and (h) show the result of a finite 30
atomic chain calculation with same parameters as the analytical one. We plot
the PDOS in 2-D map as a function of the atomic site in the chain versus the
energy. The spectra in (d) shows in-gap states at a lowest energy of $\sim
0.1$ meV, no zero-energy edge states are obtained here. In (h), we observe one
zero energy state on each edge of the chain. The calculations for infinite
chains are 1-D calculations, however, the finite-chain calculations correspond
to finite chains on a 2-D finite mesh of the 3-D superconductor.

The BDI class has a $\mathbb{Z}$ topological invariant. This is the winding
number, $w$, related to the vector $\vec{d}=(d_{x},d_{y})$. As $k$ changes,
$\vec{d}$ describes a closed trajectory. The winding number, $w$, is an
integer that corresponds to the number of turns described by $\vec{d}$ about
the origin. In order to change topological phase, and change $w$, the
superconductor gap has to close. This happens when the determinant of the
Hamiltonian is zero. From Eq. (15), the determinant of $H(k)$ will be zero
when $\vec{d}$ is zero.

However, as Tewari and Sau Tewari and Sau (2012) emphasize, the Hamiltonian is
a $4\times 4$ matrix, and in order to keep the above description in the
particle-hole sector (the $\tau$ matrices) we identify the determinant of the
$A$ matrices with the winding vector $Det(A(k))=d_{x}(k)-id_{y}(k)$ in order
to take into account when the determinant of the full Hamiltonian becomes
zero. From the $4\times 4$-Bogoliubov-de Gennes Hamiltonian, Eq. (1) with the
addition of the impurity Hamiltonian, Eq. (6), and the Rashba term, Eq. (12),
we obtain:

$\displaystyle Det(A(k))$
$\displaystyle=(H_{1,1}(k)+H_{1,3}(k))\times(H_{2,2}(k)+H_{2,4}(k))-$ (16)
$\displaystyle(H_{1,2}(k)+H_{1,4}(k))\times(H_{2,1}(k)+H_{2,3}(k)).$

Tewari and Sau Tewari and Sau (2012) also show that a lower-symmetry class
invariant can be defined. This is the usual D-class $\mathbb{Z}_{2}$ invariant
that is given by the parity of the winding number Tewari and Sau (2012). In
2-D superconductors, the Rashba interaction leads to non-real matrix elements
and the symmetry of the 1-D system is reduced. However, we find that even for
2-D substrates, the winding number still gives results in agreement with the
appearance of MBS in finite chains. It is then interesting to classify the
topology of the spin-chain systems by their winding number, $w$.

The winding number is given by evaluating the number of turns of $\vec{d}$
about zero, given by the expression:

$w=\frac{1}{2\pi}\int^{\pi/a}_{-\pi/a}dk(d_{x}\frac{d}{dk}d_{y}-d_{y}\frac{d}{dk}d_{x}),$
(17)

where $\vec{d}$ has been previously normalized. Mathematically equivalent
expressions can be obtained by using the trajectories in the complex plane of
$z=Det(A)/|Det(A)|$ as shown in Refs. [Tewari and Sau, 2012] and [Asbóth  _et
al._ , 2016]. But they involve the evaluation of the $log(z)$ that plagues the
computation with numerical problems due to artificial discontinuities caused
by its branch cut. Expression (17) however, is numerically simple and accurate
to evaluate.

The $\mathbb{Z}_{2}$ topological invariant is calculated from the Pfaffian of
the system. For a chiral Hamiltonian written as in Eq. (14) the Pfaffian can
be easily evaluated using $Pf[H(k)]=Det[A(k)]$. And the $\mathbb{Z}_{2}$
topological invariant, ${Q}$, becomes:

$\displaystyle{Q}$ $\displaystyle=$ $\displaystyle sign\,[Pf[H(k=0)]\times
Pf[H(k=\pi/a)]]$ (18) $\displaystyle=$ $\displaystyle sign\,[d_{x}(k=0)\times
d_{x}(k=\pi/a)]$

This equation shows that the Rashba Hamiltonian at $k=0$ and $k=\pi/a$ does
not enter in the determination of the above topological invariant since it is
zero (see, e.g., Eq. (13)). For the same reason the trajectories of
$\vec{d}(k)$ wrap around zero only once, leading to winding numbers that only
take $-1$, $0$ or $+1$ valuesHeimes  _et al._ (2015); Li  _et al._ (2018).

As an example, we calculate the in-gap bands and topological invariants for an
infinite 1-D ferromagnetic chain on a superconductor. Panels (a) to (d) from
Fig. 2 correspond to a trivial state of the system. Fig. 2 (a) shows the
renormalization of the bands obtained from Eq. (9) for $-\pi/a\leq
k\leq\pi/a$. As observed, the bands reach high energies for $k$ values,
$|k|>k_{F}$, as the renormalization is not correct for $k$ in this range. The
inset shows the two lower bands in for $-k_{F}\leq k\leq k_{F}$. Figure 2 (b)
depicts the normalized trajectory described by the vector $\vec{d}$ in the
complex plane. The $k$ points are labelled by a gradient of color going from
$-\pi/a$ (in cyan) to $\pi/a$ (in magenta), in this case, $\vec{d}$ makes
small oscillations around $(d_{x},d_{y})=(1,0)$, meaning that the winding
number in this case is $w=0$. On Fig. 2 (c) we can follow this evolution:
$d_{x}$ (orange curve) stays close to 1 and $d_{y}$ (green curve) describes a
sinusoidal trajectory around 0 as we sweep $k$. The evolution takes place for
$k$ values in the range $(-k_{F},k_{F})$, however, for $|k|>k_{F}$ $d_{x}$ is
one and $d_{y}$ remains zero. Indicating that points for $|k|>k_{F}$
contribute trivially to the topology of the system. The blue curve depicts the
evolution of the cumulative value of $w$:

$w(k)=\frac{1}{2\pi}\int^{k}_{-\pi/a}dk^{\prime}(d_{x}\frac{d}{dk^{\prime}}d_{y}-d_{y}\frac{d}{dk^{\prime}}d_{x}),$
(19)

Figure 3:  Phase diagrams obtained for a ferromagnetic spin chain with normal-
metal DOS at the Fermi energy $N_{0}=0.037$/eV, $\Delta=0.75$ meV and spin
$s=5/2$. (a) and (b) Phase diagrams as a function of the Kondo coupling $J$
versus the Fermi wave vector of the system $k_{F}$, with potential scattering
$K=5.5$ eV and Rashba coupling $\alpha_{R}=3.0$ eV-Å. (a) Energy gap of the
system multiplied by the $\mathbb{Z}_{2}$ topological invariant, $Q$, allowing
for differentiation of trivial ($Q=1$) and topological ($Q=-1$) phases. (b)
Winding number, $w$ as a function of $J$ and $k_{F}$. The green areas
correspond to $w=-1$ (cases like the one shown in Fig. 2 (f)) and the magenta
areas to $w=1$, here the winding vector, $\vec{d}$ completes a turn in the
opposite direction. (c) and (d) Phase diagrams as a function of the potential
scattering, $K$ versus $k_{F}$ with Kondo coupling $J=3.0$ eV and Rashba
coupling $\alpha_{R}=3.0$ eV-Å. (e) and (f) Phase diagrams as a function of
the Rashba coupling, $\alpha_{R}$ versus $k_{F}$ with Kondo coupling $J=3.0$
eV and potential scattering $K=5.5$ eV.

The $\mathbb{Z}_{2}$ invariant, $Q$, is calculated from Eq. (18). For the
present case of $k_{F}<\pi/a$, the points $k\geq k_{F}$ contribute trivially
to the topological state of the system, such that we evaluate the Pfaffian in
Eq.(18) at $k=k_{F}$ insteaf of $k=\pi/a$. This is justified by the fact that
beyond $k=k_{F}$ the free-electron-like states disperse rapidly away from the
gap and they cannot alter the topology of the in-gap bands. This is reflected
by the absence of states beyond $k_{F}$ in the superconductor as we show on
Fig. 1. Numerically, we test that $d_{y}\approx 0$ in the k-points where we
evaluate $Q$. When $k_{F}>\pi/a$, we strictly apply Eq. (18). In the case of
Fig. 2,we obtain $Q=1$ in good agreement with $w$.

Figure 2 (d) shows a 2-D map of the PDOS calculated for a finite 30-atom chain
with the same parameters as a function of the atomic site versus the energy.
We calculate the PDOS on every site of the chain from Eq. (11). As we can
observe, the in-gap states are distributed along the chain and the lowest
energy states are found at $\sim 0.1$ meV. The absence of zero-energy edge
states is in good agreement with the trivial state of the system.

Panels (e) to (h) from Fig. 2 correspond to a topological case. Here, we have
increased the magnetic coupling to $J=2.7$ eV. The band structure has gone
through a gap closing and the bands in Fig. 2 (e) are topological. The
trajectory of $\vec{d}$ completes a turn about zero, we can better observe the
trajectory on Fig. 2 (g), where $w(k)$ evolves from zero to -1. Again, the
evolution takes place for $-k_{F}\leq k\leq k_{F}$ and the points in
$|k|>k_{F}$ only contribute trivially. The calculation in a finite chain shows
zero-energy edge states at both ends of the chain, as we expect from the bulk-
boundary correspondence principle, Fig. 2 (h).

The winding number, $w$, can be particularly difficult to evaluate because of
the large number of k-points needed. The convergence depends on the evolution
of $\vec{d}$ with $k$. At $k=k_{F}$, the band structure changes rapidly and so
does $\vec{d}$. Large values of the Rashba parameter, $\alpha$, lead to
smoother variations of $\vec{d}$, permitting a more accurate evaluation of $w$
with fewer k-points. In the same way, the evaluation of gradients depends on
the used discretization steps. It is particularly critical to use small
$\omega$ steps for the evaluation of Eq. (9) as well as a small imaginary
broadening for the Green’s functions. The behavior of $d_{y}$ with $k$ is a
stringent test to check for the convergence of the numerical calculations. Not
only should $d_{y}$ equal zero at $k=0$ and $\pm\pi/a$, but it should be odd
with $k$, as our results of Fig. 2 show.

###  IV.2 Topological phase space

By systematically evaluating the topological invariant ${Q}$ and the winding
number on a parameter space, we can create phase diagrams that we will use to
determine the topological state for any given parameters. Figure 3 shows phase
diagrams of a ferromagnetic atomic chain as a function of magnetic coupling
$J$ versus $k_{F}$ (Fig. 3 (a) and (b)), as a function of potential scattering
$K$ versus $k_{F}$ (Fig. 3 (c) and (d)) and as a function of Rashba coupling
strength, $\alpha_{R}$ versus $k_{F}$ in Fig.3 (e) and (f). The panels on the
left row of Fig. 3 depict the energy gap of the system multiplied by $Q$, like
this, the topological phases are plotted as a negative gap (in blue) and in
the trivial ones the gap is positive (in red). As expected, the topological
phases corresponding to $w=+1,-1$ perfectly match the $Q=-1$ areas.

At a TPT, the gap of the system goes to zero. On Fig. 3 (a), we can easily
observe two wide white branches corresponding to the gap closing at $k=0$
going from $k_{F}\sim 0.1\;a^{-1}_{0}$ to $k_{F}\sim 0.75\;a^{-1}_{0}$, and at
$k=\pi/a$ at low values of $J$ for $k_{F}>0.5\;a^{-1}_{0}$. In other cases,
however, the gap closing at a TPT can be difficult to observe. For example, in
Fig. 3 (a) for Fermi vector values such that $k_{F}<0.5\;a^{-1}_{0}$ and $J$
couplings going from $\sim 2.5$ eV to $\sim 4.5$ eV, the topological character
changes, but we do not see a clear zero gap in this area. Here, the gap closes
at a $k^{*}$ point close to $k_{F}$, but this transition is very abrupt
requiring a high number of k-points and a fine tuning of the parameters to
properly observe the gap closing. We have observed that the band structure
highly depends on the number of $\omega$ and k-points, this can result in
numerical artifacts in the energy gap maps. An example of this, is the
stripped structure we can observe in Fig. 3 (a) for high values of $J$ and
$k_{F}>0.6$ $a_{0}^{-1}$. When we look closely to the band structure for these
values and use a sufficiently high number of $k$ points, we conclude that
these white areas are an artifact of the non-converged bands. The phase
diagrams we show on Fig. 3 are obtained using $N_{k}=1001$ which is not
sufficient to obtain clean maps, as we have observed, calculations with at
least $N_{k}=10001$ are required to remove these artifacts. As we have
discussed in the previous section, a similar problem arises for the
convergence of the winding number.

The strong dependence of the topological character on the exchange coupling
$J$ is natural given the necessary presence of an exchange interaction to have
in-gap states. However, the potential scattering term, given by matrix-element
$K$ in Eq. (6), has an important effect on the topology of the bulk bands. In
the localized-basis set, this term appears as an on-site term, and it does the
effect of a chemical potential. It will shift the on-site energies of the
superconducting sites, and hence has an important influence on the topological
phase, Fig. 3 (c) and (d).

For $k_{F}$ values beyond the Brillouin-zone border, $\pi/a$, a stark change
of topological phase is found in Fig. 3. We have checked that this frontier is
indeed there and not some numerical artifact by testing the appearance of MBS
in finite chains. The topological regions for $k_{F}<\pi/a$ are characterized
by a negative winding number. For $k_{F}>\pi/a$ the winding number changes to
$+1$. Thus, an interface between two superconductors of very different
electron density, such that one has a $k_{F}<\pi/a$ , and the other one has
$k_{F}>\pi/a$, a spin chain straddling the interface will have a change of
winding number of 2, and hence present two MBS at the interface. Alternatively
to change the sign of the winding number, we change the sign of $\alpha_{R}$
because it changes the sign of $d_{y}$. The behavior of a ferromagnetic spin
chain with Rashba coupling can be compared with the behavior of a helical non-
collinear spin chainPientka  _et al._ (2013). Following this analogy, changing
the sign of the coupling $\alpha_{R}$, would change the chirality of the spin
helix. As a consequence, in a magnetic chain with a domain wall separating two
different chirality chains, we also find the appearance of two MBSPöyhönen
_et al._ (2014); Ojanen (2013) at the domain wall.

Figure 3 (e) and (f) show the phase diagrams as a function of the Rashba
coupling versus $k_{F}$ for $J=3.0$ eV and $K=5.5$ eV. As we can observe, the
topological phase is independent of the Rashba parameter. However, the winding
number phase diagram in Fig.3 (f) shows that the system is in the topological
state only if we have a finite, non-zero $\alpha_{R}$, showing that Eq. (18)
should not be blindly applied, as topological phases on FM chains can only be
achieved on systems with Rashba interaction, even if $\alpha_{R}$ is
infinitesimally smallHeimes  _et al._ (2015); Brydon  _et al._ (2015).
Moreover, in Fig.3 (e) we can see that the topological gap becomes bigger with
an increasing $\alpha_{R}$, giving better protection to the MBS that arise in
finite systems. Hence, the role of the Rashba interaction is to facilitate the
triplet pairing, even though the ferromagnetic ordering in the chain can
suffice to locally drive the superconductor into the topological phase.

##  V Numerical studies of topological phases

In the previous section we have shown that the topological phase can be
determined for a ferromagnetic infinite chain. We now want to study the
validity of the topological invariants in finite systems, in particular, in
tens of atom chains on 2-D superconductors, which can be compared with
experimental measurementsMier  _et al._ (2021). We create a 2-D
superconducting array, without loss of generality, the magnetic impurities are
located along the $\vec{x}$ direction in an atomic chain, all spins are
oriented perpendicular to the substrate along the $\vec{z}$ direction,
creating a ferromagnetically-ordered chain. We solve Dyson’s equation, Eq.
(7), and the PDOS is calculated on every site using Eq. (11).

Figure 4:  Numerical results of 30-atom impurity chain of a 2-D superconductor
with dimensions $N_{x}=36$ and $N_{y}=5$. (a) Spectrum obtained on the first
atom of the chain. (b) PDOS at zero energy along the chain’s axis. (c)
Renormalized bands, analytical calculation (black dashed lines) and numerical
result (in red) for a 1001-atom chain in a 1-D superconductor. (d) PDOS
spectra along the 30 atom chain. (e) Corresponding trajectory of winding
vector $\vec{d}$, the color bar shows the $k$-points where $\vec{d}(k)$ is
evaluated. (f) $d_{x}$, $d_{y}$ and cumulative winding number $w(k)$, Eq. (19)
as a fubction of $k$. Parameters: $\Delta=0.75$ meV, $N_{0}=0.037$/eV,
$k_{F}=0.7\;a_{0}^{-1}$, $\alpha=3.0$ eV-Å $J=3.5$ eV, $K=5.5$ eV.

We have verified that the in-gap states are not drastically affected by the
change in dimensionality. By performing calculations on 2-D superconductors,
we were able to observe that the extension of the in-gap states decays in
about 5 sites in the perpendicular direction to the chain. The overall PDOS
obtained along the chain and the in-gap states dispersion are largely
unaffected by the change from 1-D to 2-D. In the case of a 3-D system, 3
layers are enough for the states to decay. On the present work, the
calculations on finite chains are performed on 2-D superconducting arrays.
However, for calculations of in-gap bands and topological invariants, a big
number of atoms is required in order to attain the infinite-chain behavior,
hence we limit ourselves to 1-D systems in order to reduce the computational
time.

###  V.1 Comparison with analytical calculations

On Fig. 4 we show the results for a finite 30-atom chain located at the center
a 2-D rectangular superconducting array with dimensions $N_{x}=36$ and
$N_{y}=5$ sites. The exchange coupling is $J=3.5$ eV, the potential scattering
ampitude is $K=5.5$ eV, Rashba coupling is $\alpha_{R}=3.0$ eV-Å and the Fermi
vector is $k_{F}=0.7a_{0}^{-1}$, by looking at Fig. 3 (a) these parameters
yield a topological solution with winding number $w=1$. On Fig. 4 (a) we
depict the spectrum obtained on the first atom of the chain, here a very
pronounced peak can be observed at zero energy. On panel (b), we show the
distribution of the PDOS at zero energy along the $\vec{x}$ axis, revealing
that the zero-energy state is well localized at the ends of the chain. On Fig.
4 (d) we show a 2-D map of the spectra obtained on every atom along the
chain’s axis, we can again note the presence of zero-energy edge states,
whereas inside the chain we observe a finite energy gap. All of these features
are in good agreement with the presence of MBS. As discussed in the previous
section, the topological state of a given system can be determined from the
study of the topological invariants.

We can calculate $G(\vec{k},\omega)$ from the real-space Green’s function
$G_{i,j}(\omega)$ by using a finite Fourier transform, and using a
sufficiently high number of atoms in a 1-D finite system. We then calculate
the k-resolved Hamiltonian from the renormalized Green’s function, Eq. (9).
Figure 4 (c) depicts the numerically calculated bands (in red) for a 1001-atom
chain with the same parameters as for the 30-atom chain. We plot the infinite-
chain bands from the previous section as black dashed lines, showing good
agreement with the numerical calculations. We show the trajectory of the
vector $\vec{d}$ in Fig. 4 (e), making a complete turn about zero in the
positive sense, resulting in $w=1$ and demonstrating the topological nature of
the edge states obtained in the 30-atom chain, Fig. 4 (d). In contrast to the
infinite-chain results, the winding number determined by $d_{x}$ and $d_{y}$
show some incorrect asymmetry with $k$, Fig. 4 (f), this asymmetry can be
reduced by taking sufficiently small $\omega$ steps that improves the
numerical precision of the derivative in Eq. (9). Also, small oscillations can
appear in these curves due to the Fourier transform from the finite-chain in
real space to $k$-space, Fig. 4 (f). In order to improve the results, a
sufficiently high number of k-points and high number of atoms are required.
The Dynes parameter, $\Gamma$, needs to be adjusted for better accuracy.
Overall, these results show good agreement between finite and infinite chain
calculations that is of special interest, because it shows that the
topological state of a given system can be determined from strictly numerical
calculations in finite systems.

###  V.2 Numerical phase space

In contrast to the infinite-chain analytical calculations of previous
sections, finite-chain calculation has the advantage that the presence of MBS
can be quickly discerned in a calculation. Moreover, the phase space can be
explored by computing the in-gap electronic states projected on the first site
of the chain. In the presence of MBS, zero-energy states will appear as
parameters change.

Figure 5: Evolution of in-gap states at the edges and at the center of the
chain as a function of the couplings for exchange $J$ ((a) and (b)
respectively) and potential $K$ ((d) and (e)) interactions. For comparison,
the states at the center of an infinite chain from an analytical calculation
are also shown in (c) and (f). As expected, the agreement between the spectra
at the mid-site of the finite chain (b) and the site of the infinite chain (c)
is excellent, as well as between (e) and (f). The main differences are due to
long-range edge states that are absent from the infinite chain. Red dashed
lines indicate the TPT as obtained from phase diagrams in Fig. 3.
Topologically non-trivial phases are found between the two horizontal lines.
Parameters: $\Delta=0.75$ meV, $N_{0}=0.037$/eV, $k_{F}=0.4\;a_{0}^{-1}$,
$\alpha=3.0$ eV-Å, the finite chains ((a), (b), (d) and (e)) are 30-atom long.
The PDOS is in (1/eV) units.

To study the evolution of the edge states in the finite chains as we go
through the TPT, we calculate finite 30-atom chains as a function of the
parameters $J$ and $K$. In order to reveal the features proper to the edge of
the chain, we compare the electronic structure as a function of energy for
edge sites with the one at the center of the chain. Figure 5 depicts the
evolution of the edge states and the states at the center of the chain as the
exchange ((a) and (b) respectively) and potential ((d) and (e)) couplings are
varied. For comparison, we perform a calculation from the analytical solution
$G(k,\omega)$ of an infinite chain, and we Fourier transform to real space,
such that a site in an infinite chain can be evaluated ((c) and (f)). As
expected, the agreement between Fig. 5 (b) and (c) is excellent, as well as
between (e) and (f). There are however some differences, particularly from
states that cross the gap as the interactions change. These states are not
present in the infinite-chain calculation and can be traced back to the
projections on the edge sites, Fig. 5 (a) and (d), showing that they are edge
states extending into the center of the chain.

The red dashed lines indicate the TPT as found from the phase diagrams in Fig.
3. In good agreement, we find that MBS develop in (a) and (d) for the values
of the couplings corresponding to topological phases. Moreover, the states
that cross rapidly the Fermi energy when the couplings are changed can be
determined to have no topological origin by comparison with Fig. 3.

A closer look to Fig. 5 (a) reveals that for higher values of $J$ in the
topological state, the zero energy edge states begin to split. This is due to
the finite size of the chain, Fig. 5 corresponds to calculations with a
30-atom chain. For an increasing number of atoms, the splitting of the zero-
energy peak occurs closer to the TPT, marked by the red dashed line. The TPT
is marked by a gap closing of the bulk hamiltonian revealed by the crossing at
$J\sim 2.7$ eV of the zero-energy in-gap states, Fig. 5 (b) and (c). For the
second transition at $J\sim 4.0$ eV, we observe a narrowing of the gap, but
the gap closing is difficult to observe because a high number of k-points and
$J$ values is required to observe this gap closing. A similar situation
happens when tuning the potential scattering, $K$ in Fig. 5 (e) and (f).

###  V.3 Finite-chain spectral dependence on the number of atoms

Figure 6:  (a) to (c) ((e) to (g)) Evolution of the edge (center) states as a
function of the number of atoms in the chain and for different Kondo
couplings. For $J=2.5$ eV the lower energy in-gap states are at $\sim 0.1$ meV
and well distributed between edge (a) and center of the chain (e). For a
coupling of $J=3.0$ eV we observe a robust zero energy edge state for chains
as short as 5 atoms (b) while the bulk spectra shows an energy gap. For
$J=4.5$ eV the edge states oscillate around zero with a period of 5 atoms (c)
the oscillations are also observable in the middle of the chain (g).
Parameters: $\Delta=0.75$ meV, $N_{0}=0.037$/eV, $k_{F}=0.4\;a_{0}^{-1}$,
$\alpha=3.0$ eV-Å $K=5.5$ eV. (d) Evolution of the ingap states in a dimer
while varying the distance between the two magnetic impurities. The four in-
gap-states oscillates with the same period observed in the chain. Parameters:
Same as the chain but $J=3.2$ eV. (h) Scheme depicting the interaction between
pairs of atoms in the chain (in red) and in the dimer (blue) here the
periodicty of the interaction is set to 2 atoms for simplicity. The PDOS is in
(1/eV) units.

The study of the spin stateMashkoori  _et al._ (2020); Mier  _et al._ (2021)
of the chain while varying the magnetic coupling supports the occurrence of a
topological phase transition at $J\sim 2.7$ eV (with parameters $U=5.5$ eV,
$k_{F}=0.4$ $a_{0}^{-1}$ and $\alpha=3.0$ eV-Å) and, hence, the presence of
MBS in this case. This is also in agreement with the phase diagram from Fig. 3
(a), for $k_{F}=0.4\;a_{0}^{-1}$, the energy gap goes to zero at about $J=2.7$
eV, and the new gap changes character from trivial to topological. To further
study these finite system states, we follow the evolution of the edge states
while changing the number of atoms in the chain.

MBS are expected to be easier to detect as the chain length increases Mier
_et al._ (2021); Peng  _et al._ (2015) because the spatial overlap of their
wave functions decreases. On Fig. 6 we show the evolution of the spectra on
the first (top row) and middle atom (bottom row) of the chain as a function of
number of atoms and for different $J$ coupling values. On panels (a) and (e)
$J=2.5$ eV, the topological state has not been reached and the in-gap states
are still far from zero energy. In the middle plots (panels (b) and (f)), we
have increased the magnetic coupling to $J=3.0$ eV, this is after the system
has undergone the TPT. On panel (b) we observe a robust zero energy state for
chains as short as 5 atom-long. As the chain length increases, the edge state
stays at zero energy. If we look at the spectra on the middle of the chain
(Fig. 6 (f)), we can observe an energy gap, showing that the zero-energy state
is well localized at the chain edges. The phase diagram of Fig. 3 (a) shows
that for $J=3.0$ eV and $k_{F}=0.4\;a^{-1}_{0}$, the system is, indeed, in a
topological state.

On panels (c) and (g) from Fig. 6, the exchange coupling is $J=4.5$ eV and the
spectra on the upper panel display an edge state with an oscillatory behavior
around zero energy with a period of 5 atoms. On panel (g), we see that some of
these edge states are extended inside of the chain. Oscillations of in-gap
states has been reported by recent studiesSchneider  _et al._ (2021a),
suggesting that even for topological solutions, the MBS can interact and move
away from zero energy. To better understand the nature of the oscillations, we
look at the phase diagram on Fig. 3 (a). For these parameters the system is in
the trivial state. Despite the edge states crossing at zero energy
periodically, they are no-topological in-gap states.

Figure 6 (d) depicts the in-gap states of a dimer of magnetic atoms in a
superconductor as a function of their interatomic distance. On the $y$ axis we
vary the distance between the two atoms. Four in-gap-states results from the
hybridization of the FM dimerChoi  _et al._ (2018). As the distance changes we
observe an oscillatory behavior of the states. This points to a coupling
between atomic pairs carried by RKKY interactionRébola and Lobos (2019);
Küster  _et al._ (2021). In the case of the dimer, the amplitude of the
oscillations decays with the distance because the coupling between the two
atoms becomes smaller as the two impurities move away. For very large
interatomic distances, the dimer spectra tend to the spectra of a single
impurity. However, in the case of the atomic chain, because we keep adding
atoms, the coupling between pairs at a given distance is always present so the
oscillation amplitude does not decay, a scheme of these interactions is
depicted on Fig. 6 (h).

For different parameters, we also find oscillatory behavior about zero energy
in the topological phase when the exchange coupling is very large. In this
case, the interactions between the edge MBS are not negligible and we
reproduce the same behavior as the one reported in Ref. [Schneider  _et al._ ,
2021b]. In order to obtain topological or trivial oscillations, we find that
the exchange coupling, $J$, needs to be large enough to induce the oscillatory
behavior of the in-gap states as the number of atoms is increased.

The atomic manipulation capabilities of the STM allows us to study the
evolution of the in-gap structure as atoms are added to the chainMier  _et
al._ (2021); Schneider  _et al._ (2021a, b). Hence, the above real-space
studies permit us a direct comparison with experiments.

##  VI Summary and conclusions

In this paper we have developed a theoretical framework using a Green’s
functions approach to model wide-band superconductors that correctly describe
the band structure at the superconducting-gap energy range, despite of the
large mismatch between the normal-metal band width of the superconductor and
the pairing energy, $\Delta$.

The in-gap-bands obtained from the bulk Hamiltonian allows us to calculate the
winding number and the $\mathbb{Z}_{2}$ topological invariant that determine
the topological state of the system. We have thus computed phase diagrams that
help us to easily classify our systems. Both the exchange and the potential-
scattering interactions can drive the system in and out of the topological
phases. According to our calculations, for infinitesimally small Rashba
couplings, the topological phases can be accessed. It is worth noting the the
convergence of the band structure and the topological invariants is not
trivial, as a sufficiently high number of $\omega$ and $k$ points is generally
required.

Our numerical calculations of finite systems show good agreement with the
infinite chain, from which we could determine the topological state of edge
states. This is of special interest because it shows that the topological
state of the system can be evaluated from the finite system alone. This
further open the possibility of using this methodology using Green’s function
that are derived for free-electron metals.

We have performed calculations of the topological properties of finite spin
chains and their dependence on the number of atoms of the chain. We find
recurring in-gap state oscillations about zero energy as the number of atoms
is increased. These oscillations originate in large impurity-electron exchange
couplings, but their topological behavior actually depends on the electronic
density, fixed by the Fermi wave vector, $k_{F}$, in our studies.

In summary, the present model is a promising tool that has already
successfully described experiments on magnetic atoms manipulated with STMMier
_et al._ (2021). And will be of special interest in the search of topological
phases of spin chains on s-wave superconductors.

## Acknowledgements

Financial support from the Spanish MICINN (projects RTI2018-097895-B-C44 and
Excelencia EUR2020-112116) and Eusko Jaurlaritza (project PIBA_2020_1_0017) is
gratefully acknowledged.

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