Interplay of topological phases in magnetic adatom-chains on top of a Rashba superconducting surface

We investigate the topological properties and the accessible Majorana fermion (MF) phases arising in a hybrid device consisting of a chain of magnetic adatoms placed on the surface of a conventional superconductor with Rashba spin-orbit coupling (SOC). By identifying the favored classical magnetic ground state of the adatom chain, we extract the corresponding phase diagram which exhibits an interplay of ferromagnetic (FM), antiferromagnetic (AFM) and spiral orders. We determine the parameter regime for which the FM or AFM phases dominate over the spiral and additionally become stable against thermal and quantum fluctuations. For the topological analysis we focus on the FM and AFM cases and employ a low-energy effective model relying on Shiba bound states. We find that for both magnetic patterns the hybrid system behaves as a topological superconductor which can harbor one or even two MFs per edge, due to chiral symmetry. As we show, the two magnetic orderings lead to qualitatively and quantitatively distinct topological features that are reflected in the spatial profile of the MF wavefunctions. Finally, we propose directions on how to experimentally access the diverse MF phases by varying the adatom spacing, the SOC strength, or the magnetic moment of the adatoms in consideration.

This new perspective opened the door for new MF devices based on magnetic adatoms on the surface of conventional superconductors. One finds implementations with magnetic adatoms where the ordering is random [19], spiral [20][21][22][23][24][25][26][27][28][29][30], antiferromagnetic (AFM) with SOC induced by the combination of Zeeman fields and supercurrents [31], and ferromagnetic (FM) on top of a superconducting surface with Rashba SOC [32,33]. According to very recent experimental findings [34], MFs seem to indeed emerge in magnetic adatom hybrid devices, where the ordering of the chain appears to be ferromagnetic. This type of ordering can lead to MFs only if Rashba SOC is present, arising from the broken inversion asso- * Electronic address: andreas.heimes@kit.edu ciated with the Pb superconducting substrate. In fact, this is a plausible scenario for Pb which owes already a non-negligible intrinsic SOC [35]. Evenmore, it has been shown that the Rashba SOC arising in Pb quantum well structures can be considerably large and tunable [36][37][38][39]. The related SOC strength can even reach a corresponding momentum splitting of the order of δk ∼ 0.05 k F , where k F is the Fermi-momentum ( = 1).
In this work we focus on a platform directly related to the recent experiment of Ref. [34]. Specifically we consider a single chain consisting of classical magnetic adatoms deposited on top of the surface of a SC with Rashba SOC. We first infer the energetically favored classical magnetic order of the chain, out of the possible FM, AFM and spiral profiles. Secondly, we investigate the topological properties of the arising engineered TSCs, particularly focusing on the topological FM and AFM chains.
In the first part of the manuscript, we explore the competition of the three aforementioned magnetic profiles by assuming identical adatoms owing a fixed spin S. The magnetic atoms interact via a Ruderman-Kittel-Kasuya-Yosida (RKKY)-type superexchange [40], which is mediated by the electrons of the SC. Due to the presence of SOC, the resulting superechange interaction is anisotropic and includes a Dzyaloshinsky-Moriya (DM) contribution [41] which favors spiral ordering. On the other hand, FM and AFM orders are stabilized by Isingtype anisotropy terms, induced by the crystal field of the substrate, which favor an easy axis for the magnetic ordering (see Fig. 1). By taking into account the various interactions, we extract the resulting magnetic phase diagram by additionally varying the distance of the adatoms. In this manner, our results address implementations with alternative substrates, either due to a different supercon- ducting material or orientation of the surface involved.
In the second part, we focus on the topological properties of these platforms, and concentrate on the FM and AFM cases. This is justified, as the findings of Ref. [34] indicate a strong Ising anisotropy, which as we show here, can additionally render the FM and AFM phases inert to quantum and thermal fluctuations in spite of the onedimensional character of the chain. For extracting the topological phase diagram, we first retrieve an effective low-energy model based on Shiba states [42], which constitute midgap electronic states of the SC localized at the sites of the adatoms. The symmetry properties of the system gives rise to a rich phase diagram of MFphases with 0, 1, or 2 MFs per chain edge. One can access the three phases via varying the adatom distance, the strength of the SOC and the value of the magnetic moment. The phases with 2 MFs per chain edge are topologically protected by chiral [33,[43][44][45][46] symmetry, and they indeed become accessible here for the parameters adopted. For illustrating the relevant mechanism driving the diverse topological phases, we identify the relevant gap closings in the Shiba bandstructure, which provide insight for manipulating the MFs and tailoring the topological properties of these platforms.
Our paper is structured as follows: In Sec. I we obtain the magnetic phase diagram for a magnetic chain on top of a metallic surface with Rashba SOC. We consider that the magnetic adatoms interact via an RKKY interaction, while at the same time they experience a crystal field induced Ising anisotropy. In Sec. II we extend the previous analysis for the case of a superconducting substrate and discuss the modifications on the interplay of the spiral, FM and AFM phases. In Sec. III, we retrieve an effec-tive one-dimensional low-energy Hamiltonian of the hybrid device for the FM and AFM implementations, relying on Shiba bound states. In Secs. IV and V, we extract the topological phase diagrams and study the arising MF wavefunction characteristics, for the FM and AFM Shiba chains, respectively. Finally, we present our conclusions in Sec. VI.

I. MAGNETIC PHASES OF AN ADATOM CHAIN ON A RASHBA METALLIC SURFACE
In this section we discuss the favored ordering of a chain of magnetic atoms placed on top of a metallic surface with Rashba SOC. We first retrieve the RKKY superexchange interaction between the magnetic atoms, which is mediated by the substrate electrons. By additionally taking into account an Ising-like anisotropy term due to the crystal field, we retrieve the classical magnetic phase diagram, which consists of FM, AFM and spiral phases. Finally, we investigate the impact of quantum and thermal fluctuations on the FM and AFM magnetic orders, and show that they are stable.

A. RKKY interaction
We start with the Hamiltonian of a two dimensional metallic substrate with Rashba SOC: where h k = ξ k + α(k ×ẑ) · σ is a 2 × 2 matrix in spinspace and ψ † k = (c † k↑ , c † k↓ ) is the corresponding spinor. Furthermore, c † kσ creates an electron with momentum k and spin projection σ. The quadratic electronic dispersion, ξ k = k 2 /2m − µ, can be linearized around the Fermi-momentum (k F = √ 2mµ), i.e. ξ k = v F (k − k F ), where k = |k| and v F is the Fermi-velocity. The Hamiltonian h k can be readily diagonalized via a π/2-rotation about thek-axis: wherek = k/k. The respective eigenenergies are given by corresponding to the two helicity bands λ = ±1. Thus, the effective momentum splitting δk corresponds to a SOC strength α = v F δk/k F . In order to proceed, we define the Matsubara Green's function in the helicity subspace: g λ (k, iω) = (iω−ξ kλ ) −1 and with that we obtain According to the result above, the electronic Green's function is given by At this point, we assume a certain arrangement for the magnetic adatoms on the metallic substrate. Here we consider classical spins S i with magnitude |S i | = S, placed at positions R i = iax, with i = 1, ..., N . In addition, we consider that the interaction between the adatoms is driven by an exchange interaction mediated by the conduction electrons of the substrate. The coupling between adatoms and conduction electrons can be parametrized by an exchange energy J, i.e.
Given that J is a small coupling constant and that the local modifications of the electronic spectrum in the substrate are negligible, we can follow a standard one-loop expansion and obtain an effective spin-spin interaction. The so called RKKY interaction reads [40] where the spin susceptibility can be derived using the Green's function given in Eq. (4): where G(R, iω) = dk (2π) 2 e ik·R G(k, iω). In the following we will consider a chain of adatoms with magnetic moments oriented along the x direction. In Appendix A we present in detail the steps which yield the well known result [47] for the RKKY interaction: where ν F = m/2π is the density of states at the Fermilevel for each spin-band and r ij ≡ (i−j)a. Eq. (8) holds in the limit k F a 1. For vanishing SOC, we recover the usual spin rotationally invariant Heisenberg interaction, proportional to S i · S j . On the other hand, a finite SOC produces both an additional Ising interaction S y i S y j and a DM interaction (S i × S j ) y . Note, that the particular form for the RKKY interaction could have been readily retrieved by considering all the bilinear spin-spin interaction terms, which are allowed by the C 2v point group symmetry of the system in the non-magnetic phase.
We may write the RKKY interaction in a compact fashion, by taking into account that the rotation of a classical spin S j by an angle θ ij = 2mαr ij with respect to S i , is given by Thus we may rewrite Eq. (8) as follows [47]: This implies that the SOC would generally establish a spiral configuration, with a tilting angle θ i+1,i = 2maα between successive spins. However, the substrate crystal field effects (CFEs), violate spin rotational invariance so that the magnetic moment of adatoms tends to point along the axis perpendicular to the surface (z axis here). This anisotropy gives rise to an additional term appearing in the total adatom Hamiltonian, which depends on the microscopic details of the substrate and can generally assume a rather complicated form. However, here we will consider the simplest allowed term with the form which accounts for the broken spin-rotational invariance.
The parameter D has been already estimated experimentally for some cases, by means of spin-polarized STM [48,49].

B. Classical magnetic ground state
In this paragraph, we discuss the competition of the possible magnetic phases of the adatom chain (see Fig. 2), arising from the interplay of the SOC and the CFEs. The former favors a spiral ordering while the latter, if large enough, can stabilize a FM or AFM ordering depending on the adatom spacing. For the rest, we treat the spins classically, thus assuming that |S i | = S with a fixed magnitude S. In the classical limit: S → ∞ whereas J → 0, so that JS remains finite. Later we will discuss the stability of the classical ground state against quantum and thermal fluctuations.
There are various ways to determine the classical ground state of the Hamiltonian H classical = H CFE + H RKKY . In this section we pursue a rather qualitative discussion and we prefer to apply a trial configuration S i (ϑ) = S sin(ϑi)x + S cos(ϑi)ẑ, with the orientation of the spins confined in the xz plane. The latter form is fixed due to i. the CFEs which energetically favor the appearance of finite magnetization along the easy z axis and ii. the mixing of the x and z magnetization components induced by the DM interaction as an indirect result of the Rashba SOC. Therefore, the particular form of the Hamiltonian implies that in the magnetic ground state, the spins are lying in the xz plane. Under these conditions, the ground state is defined by the optimal value of the angle ϑ, which minimizes the classical Hamiltonian: We minimize this Hamiltonian with respect to ϑ for an  infinite chain. In Fig. 3 we see that depending on the relation between: i. exchange energy JS, ii. CF anisotropy D, iii. SOC strength αk F , and iv. adatom spacing a, the classical ground state can assume a FM (ϑ = 0), AFM (ϑ = π) or spiral configuration (ϑ = 0, π). The stronger the SOC, the stronger the CF anisotropy that has to be present, in order to compensate the tendency of the system to form a spiral. Moreover, we observe that by tuning the SOC strength, as for instance by applying an electric field along the z axis, we can realize a FM ↔ AFM quantum phase transition. This can be clearly seen in Fig. 4 where the different phases are shown for two different values of the atomic spacing. The particular characteristic of the phase diagram implies that electric fields can be also employed for altering the topological properties of the Shiba chains yielding a rich landscape of MF phases. Finally, note that the arising phase transitions are first order.

C. Thermal and quantum fluctuations
In this paragraph we will investigate the robustness of the FM and AFM phases against thermal and quantum fluctuations. To this end we assume that the system resides either in the FM or the AFM phase, with ϑ ± = 0, π respectively. We will retrieve the dispersion of the magnetic fluctuations for each magnetic phase, by employing a Holstein-Primakoff (HP) transformation [50]. In the limit of large S, the HP transformation reads The linear term H quantum vanishes, whereas the bilinear term is given by In momentum space the Hamiltonian reads with the combinations q , immediately provides the eigenenergies of the spin wave modes, In order to investigate the stability of the FM and AFM phases, we calculate the sublattice magnetization, i.e.
Using the Bogoliubov operators and by introducing the Bose-Einstein distribution n q = β † q β q , we obtain the deviation of the sublattice magnetization from its groundstate value where we have assumed that sgn( S z 1 ) = 1. In Fig. 5 we show ∆S for different temperatures and SOC strength.
For the atomic spin we use S = 15/2 that has been realized in clusters of few magnetic atoms [51]. We find that for T = 0 and α = 0 only the AFM configuration exhibits quantum fluctuations, which are absent in the FM case (∆S = 0). In both cases we find that quantum as well as thermal fluctuations are suppressed with increasing anisotropy D. For both AFM and FM configurations, the sum in Eq. (18) diverges for finite temperatures when taking the limit D → 0, and thus thermal fluctuations destroy the magnetic order. In STM experiments the nearest neighbor exchange energy as well as the crystal field anisotropy can be measured. The next neighbor RKKY interaction of various metals is of the order [48,49] The crystal field anisotropy D has been determined in Ref. [49] to be approximately given by ∼ 1 meV or even larger [48]. In terms of the parametersJ and D our calculation covers the parameter regime D/J 10, which is consistent with the aforementioned experimental realization. Furthermore, the so far explored temperatures are within the range T ∼ 0 − 1 K, which are typical for the MF experiments. As a conclusion, FM or AFM magnetic chains may be established, even in the presence of strong SOC without being destroyed by fluctuations.

II. MAGNETIC PHASES OF AN ADATOM CHAIN ON A RASHBA SUPERCONDUCTOR
Here we extend the previous analysis in order to investigate the effect of superconductivity on the magnetic phase diagram. Once again, the magnetic adatoms interact via an RKKY interaction which is mediated by the electrons of the substrate superconductor, while they also feel a spin anisotropy due to the crystal field.

A. Gor'kov-Nambu Green's function
In this section we derive the Green's function for the superconducting substrate degrees of freedom in the presence of SOC. As before, we exclusively discuss Rashba SOC, although other couplings between spin and momentum may be intrinsically present. This type of SOC can be engineered and can be considerably large for instance in quantum wells [36][37][38][39]. In fact, superconducting thin films of Pb feature both intrinsic and Rashba types of SOC. Starting from the Hamiltonian of Eq. (1) for a twodimensional metallic surface with SOC, we consider here an additional spin singlet s-wave pairing term ∆ (here real and positive) where the Pauli matrices τ are defined in particle-hole space and is the corresponding spinor. Following the procedure of Sec. I A we perform a rotation, i.e.
Mind that the representation of the spin operator in the extended space is given byσ/2 = (τ z σ x , σ y , τ z σ z )/2. By we obtain and with the above, the electronic Gor'kov-Nambu Green's function becomeŝ Thus the presence of the Rashba SOC induces triplet pairing correlations [52][53][54] where tan ϕ k = k y /k x . The emergence of triplet correlations can be also understood within the theory of induced orders and patterns of coexisting phases [53,54,56,57]. In this work, we assume only a local pairing interaction leading to a spin singlet superconducting order parameter ∆, which is accompanied by the triplet correlations above. However, in the presence of suitable non-local interactions which contribute to the above superconducting triplet channel, the s-wave singlet and p-wave triplet order parameters necessarily coexist at a microscopic level due to the SOC [52][53][54]. In the latter case, a p-wave spin triplet order parameter has to be taken into account and determined self-consistently, as it can lead to modifications of the topological phase diagram [5].
In Eq. (22) one can identify the electronic Gor'kov-Nambu Green's function kλ and the anomalous one, By focusing on positions along the x axis, i.e. r = rx, we find These expressions are valid if ω D v F /r ∼ E F /k F r, where ω D is the Debye frequency. Mind that for r = 0, we basically recover the electronic bulk Green's function

B. RKKY interaction
In order to discuss the effective RKKY interaction mediated by the quasiparticles of a superconducting substrate with SOC, one can simply replace the Green's function appearing in Eq. (7) by the one of Eq. (23) and the spin Pauli matrices σ α and σ β by the corresponding components of the spin Pauli vector in the new representatioñ For a superconducting substrate, the RKKY interaction owes an additional term, which does not arise in the case of metallic substrates. This distinct RKKY term is associated with Shiba states [42], appearing due to the presence of the magnetic adatoms on the superconducting surface. The latter constitute localized states at the sites of the adatoms, with energies ε 0 which are smaller than the superconducting gap. In spite of the fact that the number of Shiba states is relatively small compared to the number of the bulk accessible states, it has been recently shown that their contribution to the RKKY interaction can become important, favoring an AFM ordering [58]. Nonetheless, in order for the Shiba term to dominate over the bulk RKKY contribution the adatom spacing has to be rather long, since the former decays as (k F r) −1 whereas the latter decays as (k F r) −2 . The authors of Ref. [58] showed that the Shiba contribution dominates if the condition k F r > ξ 0 /r is fulfilled, holding for the material parameters and the atomic spacing (r ∼ 100nm), which they focused on.
In stark contrast, here we assume an adatom spacing of the order of 1 nm and a coherence length of ξ 0 ∼ 80nm. Therefore we find that k F r < ξ 0 /r and conclude that the Shiba bound state contribution is negligible in our case. Its inclusion would only move the phase boundaries slightly deeper into the AFM region. Moreover, since the main contribution to the RKKY interaction arises for energies quite above the gap ∆, there is also no quantitative modification of the results found previously in Sec. I for a normal metallic substrate. Therefore, the phase diagrams presented in Figs. 3 and 4, also hold for the case of a superconducting substrate.

III. EFFECTIVE MODEL FOR FM AND AFM SHIBA CHAINS
After having discussed the contribution of Shiba states to the effective spin-spin interaction, we proceed with investigating in more detail their characteristics for the FM and AFM magnetic orderings of the spins S i , with |S i | = S. The magnetic exchange Hamiltonian now becomes We introduced M j = JS(±1) j , corresponding to FM (+) and AFM (−) ordering, respectively. In order to find the electronic spectrum we solve the Bogoliubov -de Gennes (BdG) equation [27,31,32,[58][59][60] where the wavefunction φ k = (u k↑ , u k↓ , v k↑ , v k↓ ) T contains the spin-dependent particle-and hole-components u and v. We assume that the continuum states are only slightly affected by the presence of the magnetic atoms and thus assume the usual spectrum, E k = ∆ 2 + ξ 2 k , for the superconductor. By defining φ j = (1/2π) dk e ik·Rj φ k we trace out the continuum states and end up with the equation where V j = M j τ z σ z . By performing an expansion both in the normalized energy, ε/∆, as well as the couplings to higher order neighbors, we obtain from Eq. (23) The integrals in Eq. (23) together with the approximate form for the Bessel functions given in Eq. (A2), yield the coefficients where ξ 0 is the coherence length of the superconductor. The indices s and a denote functions which are symmetric or anti-symmetric under inversion r → −r. With this we rewrite Eq. (27) in the form of a Schrödinger equation with the Hamiltonian where we have compactly denoted f i−j ≡ f (r ij ) and set M = JS. The solution of Eq. (34) determines the energies and wavefunctions of the Shiba midgap states.

IV. TOPOLOGICAL FM SHIBA CHAIN
It has been well established, already from earlier proposals involving topological insulators [6] and semiconductors [8][9][10][11], that the combined presence of SOC, swave superconductivity and magnetism, can induce topological superconductivity. Note also that a recent symmetry classification [5] has presented further directions of how to combine these ingredients for engineering TSCs. Both cases of topological FM and AFM Shiba chains fall into this classification scheme. In fact, the effective model for a topological Shiba chain (see also [32]) resembles previous continuum models describing TSCs using nanowires with Rashba SOC [10,11]. In the present case, the effective Zeeman field is provided by the magnetic adatoms (classical here) and the SOC occurs due to the involvement of a superconducting surface. However, there are also important differences. First, the perpendicular local magnetic field felt by the Shiba states is generally less harmful for superconductivity compared to a perpendicular magnetic field, due to the additional contribution of the orbital effects in the latter situation. In the case of Shiba states, in spite of the fact that the superconducting gap becomes locally suppressed, it generally survives even when the magnetic exchange energy becomes comparable to it [60]. In addition, note that the FM ordering is more likely to suppress superconductivity locally compared to the AFM ordering.
Moreover, another distinctive feature for the effective model of topological Shiba chains is that they incorporate triplet pairing correlations, which can generally lead to a significant quantitative modification of the phase diagram (see Ref. [5]). In addition, the topological Shiba chain models are lattice models involving higher order neighbor couplings, thus strongly depending on the adatom spacing. Consequently, one can not always restrict to a nearest neighbor model but instead, depending on the ratio ξ 0 /a, a large number of neighbors can become relevant. Evenmore, the inherent presence of additional chiral symmetries, leads to a rich variety of topologically phases even with 2 MFs per edge.
In the following paragraph, we first discuss the symmetries of a topological FM Shiba chain which are crucial for performing a topological classification of the accessible MF phases. Furthermore, we extract the topological phase diagram and study numerically the MF wavefunc-tions for a finite chain, in order to discuss aspects related to the experimental realization of this scenario.

A. Symmetry classification
As already mentioned, in the absence of magnetism, the point group symmetry of the hybrid structure (as in Fig. 1), consisting of the chain on top of an infinite substrate surface, is C 2v . Since the effective model describing the Shiba chains is embedded in the two-dimensional geometry, rather than being a stictly one dimensional system, it inherits the same point group properties. This is reflected in the C 2v point group symmetry of the nonmagnetic part of the Hamiltonian in Eq. (35). This point group consists of the: 1. identity element E : (x, y, z) → (x, y, z) , 2. reflection operation σ yz : (x, y, z) → (−x, y, z) , 3. reflection operation σ xz : (x, y, z) → (x, −y, z) , 4. z axis π-rotation C 2 : (x, y, z) → (−x, −y, z).
Note that for the effective Shiba state model of Eq. (35) only the inversion operation I : x → −x is accessible, and corresponds to Ii = −i and Ij = −j, with i, j denoting adatom sites. Therefore, within our spinor formalism the aforementioned symmetries are generated by the unitary operators:Ê = I,σ yz = iτ z σ x I,σ xz = iσ y andĈ 2 = iτ z σ z I. The term associated with the presence of FM ordering, M j τ z σ z = M τ z σ z , transforms under the C 2v elements in the following manner:σ † yz τ z σ zσyz = −τ z σ z , σ † xz τ z σ zσxz = −τ z σ z ,Ĉ † 2 τ z σ zĈ2 = τ z σ z . Moreover, the FM chain is invariant under the action of the discrete translation operator,t a , which leads to shift i → i + 1, i.e. equal to the adatom spacing a.
In contrast, the usual time-reversal operation T with generatorT = iσ yK , is broken as the FM term satis-fiesT † τ z σ zT = −τ z σ z . HereK denotes the anti-unitary complex-conjugation operator. As it becomes evident from the above relations, the FM term is invariant under the action of the following combined symmetry operations: T σ xz and T σ yz , i.e. consisting of operations under which the rest of the Hamiltonian is invariant. Usually, this type of symmetries are called hidden symmetries [5,61], as they are a combination of symmetry operations which, separately, do not leave the Hamiltonian invariant. In the particular case only the action of the operator Θ ≡σ xzT =K, which coincides with the complex conjugation, leaves the total BdG Hamiltonian invariant. Together with the built-in charge-conjugation symmetry of the BdG Hamiltonian, effected by the operatorΞ ≡ τ xK , they give rise to the chiral symmetry operatorΠ ≡ τ x . The presence of this set of symmetries implies that the system resides in the BDI symmetry class, which in one dimension can support topologically non-trivial phases, characterized by a Z invariant [62][63][64]. The latter allows an integer number of MFs per chain edge that, as we show in the next paragraph, is indeed an accessible topological phase for the particular system.

B. FM Shiba chain Hamiltonian
In order to study the topological properties of a FM Shiba chain, we will transfer to momentum space, defined in the FM Brillouin zone (BZ) k ∈ (−π/a, π/a]. At this point we introduce the corresponding BdG momentum space Hamiltonian, H k = H 0 k + H m k , consisting of the i. non-magnetic H 0 k and ii. magnetic H m k parts: where we have introduced B = ∆/(πν F JS) and The above Hamiltonian acts on the wavefunction φ k = (u k↑ , u k↓ , v k↑ , v k↓ ) T . In addition, t δ corresponds to the δ-order nearest neighbor hopping, v δ corresponds to the δ-order nearest neighbor SOC, D δ to the δ-order nearest neighbor extended s-wave spin-singlet superconducting gap and d δ to the δ-order nearest neighbor spin-triplet superconducting gap oriented along the y axis.

C. Topological invariant
For exploring the topological phase diagram, we reside on the presence of chiral symmetryΠ = τ x and block offdiagonalize the BdG Hamiltonian [43,64], via a rotation about the τ y axis effected by the unitary transformation (τ z + τ x )/ √ 2. We obtain The upper block off-diagonal block is given by The determinant of A k is a complex number and reads With the vectors g k = (Re Det[A k ], Im Det[A k ], 0) and g k = g k /|g k |, the related Z topological invariant is defined by the winding number [31,65] Note that by considering a Z classification which also takes into account phases with 2 MFs, we manage to go beyond the study of a FM Shiba chain performed in Ref. [32], which assumed a Z 2 classification and thus restricted to the cases with 1 MF per edge.

D. Topological phase diagram -Results
In Fig. 6 we show the winding number as a function of the adatom spacing a, magnetic exchange energy JS and SOC strength α. Phases with zero, one or two MFs per edge are accessible. When the ground state of the system resides in a phase near a boundary of the topological phase diagram, one can employ a weak perpendicular Zeeman (electric) field to tune the magnetic exchange energy (SOC strength) in order to achieve transitions between phases with different number of MFs. We additionally observe in Fig. 6(b) that the phase diagram exhibits MF bound states even for very small values of α. This is similar to the nanowire case [10,11], where α mainly determines the spatial profile and localization of the MFs at the edges of a finite system. Interestingly, this also holds for the case of 2 MFs per edge.
In Fig. 7 we compare the winding number calculation shown in panel (a), with the evolution of the two low- In both cases, we find topological phases harboring 1 or 2 MFs per chain edge. Note in (a) that tuning the magnetic exchange energy can be used to switch between 1 and 2 MF phases. When close to the phase boundary, the latter could be for instance achieved by applying a weak perpendicular Zeeman field. In (b) we observe that for an infinitessimally small SOC strength, both 1 and 2 MF phases are accessible. This is anticipated for the single MF phase where α does not enter the topological criterion, but quite remarkably, it also takes place for the 2 MF situation. Here electrical tuning of α can be used for realizing topological quantum phase transitions. Note that in order to obtain well localized MF bound states and validate the bulk-boundary correspondence predictions, quite long chains are required. This is particularly the case for the transition to the 2 MF phase, which is protected by chiral symmetry. In panel (c) we show the corresponding gap closings of the energy dispersions ε k , occuring exactly at the phase transition points I and II. The transition (I) N = 0 → 2 arises from gap closings at the non-inversion-symmetric points ±k * , connected to each other by inversion. Instead, the transition (II) N = 2 → 1 arises due to a gap closing at the inversion symmetric momentum k = 0. est positive eigenenergies, shown in panel (b) that was obtained from the open chain Hamiltonian for different lengths. As follows from bulk-boundary correspondence, the number of MF bound states agrees with the value of N , although long chains are required here in order to obtain quantitative accordance with the predicted phase boundaries. As a matter of fact, this is the case for the gap closing that occurs at the transition from trivial to N = 2. Here, two truly-zero energy bound states appear only for very long chains. This has to be contrasted with the region where N = 1. There, the zero energy bound state become stabilized already for shorter lengths of the chain, which can be seen through the different decays of the wavefunctions in Fig. 8(a) and (b).
To shed more light on the above findings, we complementary demonstrate in Fig. 7(c) the gap closings of the bulk band structure, for the parameters where the topological quantum phase transitions occur. One observes that the phase transition involving a single MF corresponds to gap closings at the inversion symmetric wavevector k = 0, whereas in the case involving 2MFs, the dispersion shows gap closings at two non-inversionsymmetric points ±k * .
For even better understanding, let us investigate in more detail the behavior of the topological invariant. The gap closing conditions and therefore the phase boundaries, are given by setting Det[A k ] = 0, which requires the following two equations to be simultaneously satisfied To obtain some analytical results, we will focus on a simplified situation. For instance, by considering a short superconducting coherence length, ξ 0 , we can restrict ourselves only up to nearest neighbor terms in the Hamiltonian of Eq. (36). Under these conditions we have t k = t 1 cos(ka), v k = v 1 sin(ka), D k = D 1 cos(ka) and d k = d 1 sin(ka). Eq. (46) is satisfied for the inversion symmetric points k = 0, π and the pair of noninversion-symmetric points ±k * , given by cos(k * a) = v 1 ∆/(d 1 t 1 −v 1 D 1 ). By setting these k-values in Eq. (47), we obtain the gap closing conditions, or equivalently the phase boundaries. For k = 0, π we obtain the condition t 2 1 +(∆±D 1 ) 2 = B 2 , akin to the criteria found in nanowire models [10,11]. A similar procedure can provide the gap closing conditions for ±k * points, which however is quite lengthy and will not be presented here.
It is important to comment on the form of the wavefunctions in the case of 2 MFs. In this case, the noninversion-symmetric points ±k * , will give rise to zeroenergy wavefunctions, which however are complex and proportional to e ±ik * aj (j index of chain site). Nonetheless, MF wavefunctions should be real, and this can only achieved by making linear combinations of the wavefunctions, so that they finally obtain a dependence cos(k * aj) and sin(k * aj). This explains i. the oscillating behavior shown in Fig. 8 with a period determined by k * and ii. the fact that when one MF wavefunction shows a maximum, the other shows a minimum. As expected, the MF wavefunction for a single MF phase does not show this type of feature.

V. TOPOLOGICAL AFM SHIBA CHAIN
The presence of Rashba SOC due to the superconducting substrate, is an ingredient capable of engineering MFs also for other magnetic phases of the chain, apart from the FM one. As we already discussed in previous paragraphs, the same conditions which favor the FM ordering, also provide fertile ground for the establishment of AFM ordering. A crucial requirement is the presence of strong Ising anisotropy in order to overcome the DM interaction. If this is the case, other details such as the adatom spacing, will decide on the FM or AFM type of ordering.
The possibility of topological AFM Shiba chains was recently discussed in Ref. [31] for superconducting substrates in the absence of SOC. In that case, a new mechanism for engineering topological superconductivity was proposed, were SOC was induced by a supercurrent flow along the chain together with an in plane Zeeman field. Note that AFM and FM Shiba chains can only host MFs in the presence of some kind of SOC [5]. In stark contrast, spiral Shiba chains do not require SOC, but exhibit MF in a self-tuned manner. Nonetheless, any realistic manipulation of MFs in spiral Shiba chains will unavoidably require the application of external Zeeman fields [28] or supercurrents [29], counterbalancing the advantage of self-tunability.
In the rest of the manuscript we will focus on the topological phases supported in AFM Shiba chains. As we will present in the next paragraphs, some of the generic results which we reported earlier for the FM case are also relevant for the AFM order. For instance, AFM Shiba chains also support MF phases with one or two MFs per chain edge. However, as we explain in the AFM analysis, the underlying mechanism and the topologically relevant k-space points, differ in each case. The reason can be traced back to the i. different magnetic wavevector Q = 0 or π and ii. the presence of additional hidden symmetries which appear in the AFM case.
As in the FM case, the AFM chain is invariant under the hidden symmetry operatorΘ ≡σ xzT =K. More importantly, the distinct propertyt † a (−1) j τ z σ zta = −(−1) j τ z σ z can yield additional hidden symmetries, when t a is combined with σ yz , σ xz or T . Indeed we find three additional symmetries: i. the anti-unitary symmetry Θ = T t a ii. the unitary symmety O = σ xz t a and iii. the unitary symmety O = σ yz t a . On the other hand, unitary symmetries allow to block-diagonalize the Hamiltonian and label it with the eigenvalues of the respective operators. Here we may use only one of the two unitary symmetry operators for block diagonalizing the Hamiltonian. Note that the presence of two anti-unitary symmetries Θ and Θ , does not allow the classification of the Hamiltonian according to the ten existing symmetry classes [62][63][64]. The latter classification methods can be only applied on Hamiltonians with no additional unitary symmetries present. However, after the block diagonalization of the Hamiltonian relying on the unitary symmetry, a symmetry classification is possible [5]. This is exactly the tactic which we will follow in the next paragraph, by first transferring to the RBZ.

B. AFM Shiba chain Hamiltonian
By transferring to momentum space, we obtain the following Schrödinger equation, which provides the singleparticle spectrum in the AFM case: with Q = π/a, k ∈ BZ, H 0 k and H m k given in Eq. (36). By passing to the RBZ we obtain where by additionally introducing the ρ Pauli matrices in the AFM space we end up with the Hamiltonian defined in the RBZ, while we introduced The explicit form reads where the parameters appearing can be directly retrieved by the definitions of t k , v k , D k and d k . For completeness, we present their expression below At this point, we move on with the symmetry classification. In the particular basis, the translation operatort a has the representation For simplicity, we will drop the U (1) phase factor, since it is irrelevant for the present discussion. On the other hand, complex conjugation has the following representation in this basisK = ρ xK , withK not acting on the wavevector Q. Under these conditions we obtain the representation for the following operators:Θ = ρ xK , Θ = ρ y σ yK andÔ = ρ z σ y . We directly confirm that the Hamiltonian is invariant under the action of these operators, as discussed in the previous paragraph. However, there are additional symmetries. We find two chiral symmetries:Π ≡ τ x andΠ ≡ τ x ρ z σ y , as also two chargeconjugation symmetries:Ξ ≡ τ x ρ xK andΞ ≡ τ x ρ y σ yK . In this representation both time-reversal symmetry operators satisfyΘ 2 = (Θ ) 2 = I, yielding the symmetry class BDI⊕BDI.
The particular symmetry class of the Hamiltonian can alternatively retrieved by block diagonalizing the Hamiltonian via the transformation which yields U H k U † = 1 2 σ (1 + σσ z ) ⊗ H k,σ , with the blocks H k,σ = (t k,+ − σv k,− )τ z + (t k,− − σv k,+ )τ z ρ y + Bτ z ρ z + [σ(∆ + D k,+ ) − d k,− ]τ y ρ y + (σD k,− − d k,+ )τ y . (63) Interestingly we find that for each subspace, spin up and down, the Hamiltonian possesses the form of two decoupled topological FM Shiba chain models (see Eq. (36)), but with the AFM Pauli matrices playing the role of the spin Pauli matrices. Note, that with the particular choice of the spinor, the functions in front of the matrices have a similar behavior under inversion (k → −k), as in the FM Shiba case studied earlier or related nanowire models [10,11,33,43]. Therefore, we anticipate at least an equally rich phase diagram, exhibiting an interplay of topological phases with one or two MFs per edge of the chain.

C. Topological invariant
Each of the σ subblocks reside in the BDI symmetry class and can be off-block diagonalized, similar to the procedure followed in the FM case. Therefore, we effect the transformation (τ z + τ x )/( √ 2) which yields with the upper off-diagonal block given by By introducing the determinants Det[A k,σ ], as also the related vectors g k,σ = (Re Det[A k,σ ], Im Det[A k,σ ], 0), we can define the quantities with the unit vectorsĝ k,σ = g k,σ /|g k,σ |. However, the quantities above do not constitute topological invariants because the g k,σ vectors are not compactified in the RBZ, i.e. do not have the same value for the RBZ edges k = ±π/2a. The latter occurs because we chose to work in the AFM space {k−Q/2, k+Q/2}, instead of the bandindex space. Only Hamiltonians defined in the band index space satisfy the compactification condition. In the present situation, the folding of k-space has been performed in a convenient manner, which however does not meet the above criterion. Therefore, a topological invariant can be only defined by combining the two σ sectors. Essentially we have to start from the total Hamiltonian H k , block off-diagonalize it, introduce the upper off-diagonal blockÃ k and define a corresponding vector g k = (Re Det[Ã k ], Im Det[Ã k ], 0). This procedure yields the topologically invariant quantity with N σ being R, instead of Z. Note this procedure was circumvented in Ref. [31] by extending the integration to the BZ. However, the method presented in this paragragh is the most general and we conclude that only N is capable of providing the related Z number of MFs per edge which are protected by chiral symmetry. Nonetheless, there can be situations where additional terms in the Hamiltonian can violate chiral symmetry while at the same time preserving the unitary symmetry O. In this case, each Hamiltonian block H k,σ belongs to symmetry class D, which is characterized by a strong Z 2 invariant in one momentum space dimension. However, due to interdependence of the two blocks, only phases with 0 or 1 MFs are accessible. The phase diagram is retrieved by introducing a total Z 2 invariant obtained by muliplying the Z 2 invariants of each block.

D. Topological phase diagram -Results
In Fig. 9 we present the calculated winding number N of Eq. (67), with varying adatom spacing a and (a) magnetic exchange energy πν F JS or (b) normalized SOC strength α. As in the FM case, we also encounter phases with zero, one or two MFs per edge. The modification of the magnetic exchange energy, effected for instance by applying of a Zeeman field perpendicular to the ordered spins (x axis), can tune the phase diagram. Similar functionality appears with the variation of the SOC strength, where its increase can extend the window for phases with 2 MFs.
However, in contrast to the FM case, we observe that generally a critical SOC strength is required for realizing a transition to the topological phases. The latter feature will be explained below, by considering a nearest neighbor model for the AFM Shiba chain. In short, the apparent difference relies on the fact that for the 1MF phase of the FM and AFM chains, different k points are topologically involved. For the FM case, the inversionsymmetric points k = 0, π become relevant. In contrast, the 1MF phase in the AFM case arise from gap closings of each σ =↑, ↓ block Hamiltonian at the k = 0 point of the RBZ, which coincides with the π/2a point of the original BZ. Therefore, the topological criteria are retrieved from different points, with the SOC strength not involved in the FM case but crucially appearing in the AFM case.
We now proceed with examining in more detail the topological properties of the system for two values of the magnetic exchange energy. First we consider a cut of Fig. 9 for α = 0.03 v F and πν F JS = 0.75. In Fig. 10 we present: (a) the topological invariant N and (b) the relevant gap closings in RBZ associated with the changes of N . We observe in Fig. 10(b) that the transition N = 0 → 2 occurs due to the gap closings at the points In both cases, we find topological phases harboring 1 or 2 MFs per chain edge. Note in (a), that tuning the magnetic exchange energy can be used to switch between 1 and 2 MF phases. When close to the phase boundary, this could achieved with a weak perpendicular Zeeman field (x axis). Observe also that a threshold SOC strength is generally required for both 1 and 2 MF phases to become accessible. This is in contrast to the FM case and arises because the strength for the SOC α appears now in the topological criterion for the 1MF phase. Therefore, also tuning of α can be exploited for realizing topological quantum phase transitions, but via a different mechanism.
±k * for σ =↓. The particular phase with 2MFs is protected by chiral symmetry. Upon increasing the adatom distance in phase II, the ±k * points converge to k = 0 and merge, exactly when another topological phase transition occurs N = 2 → 1. The latter transition and change in N is possible due to the recombination of the two ±k * points at the inversion symmetric point k = 0  of the RBZ. The last transition to the trivial superconducting phase occurs via a gap closing at k = 0 of the σ =↓ subblock. Note generally that the dependence of N on the adatom spacing a is quite complicated, as all the coefficients are functions of the latter. In Fig. 10(c) we depict the two-lowest positive eigenenergies of the AFM Hamiltonian for an open chain. Note that, the appearance of a single zero eigenenergy agrees very well with the bulk predictions for the 1MF phase. In contrast, the bulk results for the 2MF phase are retrieved for quite long chains.
In Fig. 11 we present the arising MF wavefunctions in the 1 MF and 2 MF cases. In the case with 2 MFs, we retrieve once again the oscillatory behavior of the wavefunctions associated with the cos(k * aj) and sin(k * aj), related to chiral symmetry. However, the MF wavefunction for the 1MF phase, shows also a particular oscillatory behaviour due to different reasons. Since the latter topological phase is arising from the k = 0 point of the RBZ, which coincides with the k = π/2a point of the BZ, the wavefunctions show close to this transition point a characteristic ocillatory behavior given by the lattice constant, i.e. it assumes the form cos(jπ/2). This oscillatory behavior, with a wavelength given by the adatom spacing, still persists even deep inside the 1MF phase (see Fig. 11(a)).
To obtain further insight, we will retrieve some analytical results by restricting to the nearest and next nearest neighbor versions of the Hamiltonian in Eq. (36). For the nearest neighbor model we have t k = t 1 cos(ka), v k = v 1 sin(ka), D k = D 1 cos(ka) and d k = d 1 sin(ka).
Each Hamiltonian block now obtains the form H k,σ = −σv 1 cos(ka)τ z − t 1 sin(ka)τ z ρ y + Bτ z ρ z +[σ∆ − d 1 cos(ka)]τ y ρ y − σD 1 sin(ka)τ y . (68) The apparent exchange of roles between t 1 ↔ v 1 and D 1 ↔ d 1 happens because the k = 0 point of the RBZ corresponds to the k = π/2a of the BZ. This is exactly the reason for the distinctly different dependence on the SOC, that we obtain in the AFM topological phase diagrams. Therefore, gap closings at k = 0, connected to a 1MF phase, will occur when v 2 1 + (σ∆ − d 1 ) 2 = B 2 depending on each σ block. Obviously the topological phase boundaries for the 1MF phase depends on the SOC strength, in contrast to the FM case and nanowires proposals.
On the other hand, the chiral symmetry protected points are given by cos(k σ, * a) = σt 1 ∆/(t 1 d 1 − v 1 D 1 ). Since k * ∈ (−π/2a, π/2a], we obtain sgn[cos(k σ, * a)] = sgn [σ]. This implies that for each σ block we obtain a single k σ, * satisfying the gap closing criterion. Even more, chiral symmetry here implies that for a point k σ, * , there exists another in the −σ spin block for k −σ, * = −k σ, * . Thus the ±k * pair of chiral symmetry protected points found in the FM case, translates now into the (k σ, * , k −σ, * ) pair of points, i.e. inversion connects the two subblocks. This also explains why we can not generally consider the quantities N σ as independent chiral symmetry related topologically invariant quantities.
Nonetheless, a direct comparison with the results presented in Fig. 10, shows that a nearest neighbor model is inadequate for capturing the physics of the exact model, since the chiral symmetry protected 2MF phase originates from ±k * points of the same subblock. This can only occur if we take into account the next nearest neighbor contributions. In fact, for a model with only next nearest neighbor terms, the Hamiltonian coincides with that of Eq. (68) but with k → 2k or a → 2a. Thus we may equivalently make use of Eq. (68) but now with k in the original BZ. As a result, the equation cos(k σ, * a) = σt 1 ∆/(t 1 d 1 − v 1 D 1 ) can now provide a set of ±k * points, for each σ subblock, explaining our findings.
We now proceed with a cut of the phase diagram in Fig. 9, for α = 0.03 v F and πν F JS = 0.85. In Fig. 12 we present: (a) the topological invariant N and (b) the relevant gap closings in RBZ associated with the changes of N . For the particular value of the magnetic exchange energy, the possibility of 2MF phases still appears, but has a different origin. This is clearly reflected in the fact that the topological invariant changes always by 1. This implies that only inversion-symmetric point k = 0 can yield gap closings. This is indeed the case, as shown in Fig. 9(b). We find that the different transitions occur due to the gap closings at the k = 0 for the two different σ =↑, ↓ sub-blocks. Interestingly we observe that only after two successive gap closings at k = 0 for σ =↑, the system becomes topologically trivial. This counter intuitive result can be naturally explained when next nearest neighbors are present, leading to a quadratic gap closing at k = 0 [66]. Essentially, the 2MF phase also in the present case, constitutes a manifestation of chiral symmetry.
In Fig. 13 we show representative MF wavefunctions for the N = 1, 2 regions of Fig. 12. Indeed, we find the  appearance of two MF when k F a/π = 5.59. Each wavefunction is oscillating in magnitude and becomes exponentially suppressed in the bulk. As in previous sections, we denote the left and right Majorana wavefunction by γ L/R,i (i = 1, 2), respectively. Both γ L and γ R appear to be shifted spatially, with one becoming maximum at the points where the other is minimized. Furthermore, as we show in Fig. 14, the oscillatory behavior of the wavefunctions can be attributed to the band mimina of the gapped 2MF phase. Close to transition II depicted Fig. 12(a), the wavefunctions exhibit a periodicity cos(jπ/2) which originates from the k = 0 point of RBZ, that coincides to the k = π/2a point of the original BZ. In contrast, deep inside the 2MF phase and inbetween transitions I and II, the dispersion shows minima away from the k = 0 wavevector of the RBZ, leading to oscillations with smaller frequency (see Fig. 14).

VI. CONCLUSIONS
In summary, we investigated the magnetic phase diagram and the emergence of Majorana fermions in chains of magnetic adatoms deposited on a superconducting substrate with Rashba spin-orbit coupling (SOC). By considering classical magnetic adatoms, that interact via a superexchange interaction in the additional presence of magnetic anisotropy, we identified the parameter space for which ferromagnetic (FM), antiferromagnetic (AFM) or spiral order is stabilized. The presence of magnetic anisotropy, which arise from the crystal field of the substrate, promotes the FM and AFM phases and renders them robust against thermal as also quantum fluctuations.
Motivated by recent experiments which confirmed the possibility of strong magnetic anisotropy in such type of devices, we explored the occurence of topological superconductivity for the most prominent FM and AFM configurations. Our findings reveal a rich topological phase diagram for both cases, which can support phases of 1 or 2 MFs per edge, and can open perspectives for novel quantum computing applications. The phases with two MFs per edge are protected by chiral symmetries, which differ for each magnetic pattern. Remarkably, the topological phase diagrams for the two cases exhibit a different dependence on the strength of SOC which is directly related to the magnetic wavevector, Q = 0 or π. In fact, depending on the value of Q, different points of the Shiba bandstructure become topologically relevant. As a consequence, the MF wavefunctions demonstrate a variety of oscillatory characteristics which reflect the type of the underlying magnetic order.
The thorough parameter exploration performed in this work, addressing i. the competition of magnetic phases for the adatom chain and ii. the detailed topological phase diagram of the hybrid device, can motivate new experiments by employing alternative superconducting substrates or types of adatoms. In particular, the topological phase diagram can be tailored via tuning the SOC strength, the adatom spacing and the magnitude of the atomic spin. Thus the emergent interplay of magnetic and therefore topological phases in Shiba chains predicted in this work, can open the door for novel versatile and functional MF platforms.