Majorana Bound States Induced by Antiferromagnetic Skyrmion Textures
Sebastián A. Díaz, Jelena Klinovaja, Daniel Loss, Silas Hoffman
MMajorana Bound States Induced by Antiferromagnetic Skyrmion Textures
Sebasti´an A. D´ıaz, Jelena Klinovaja, Daniel Loss, and Silas Hoffman
2, 3, 4, 1 Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Department of Physics, University of Florida, Gainesville, Florida 32611, USA Quantum Theory Project, University of Florida, Gainesville, Florida 32611, USA Center for Molecular Magnetic Quantum Materials,University of Florida, Gainesville, Florida 32611, USA (Dated: February 9, 2021)Majorana bound states are zero-energy states predicted to emerge in topological superconductorsand intense efforts seeking a definitive proof of their observation are still ongoing. A standard routeto realize them involves antagonistic orders: a superconductor in proximity to a ferromagnet. Herewe show this issue can be resolved using antiferromagnetic rather than ferromagnetic order. Wepropose to use a chain of antiferromagnetic skyrmions, in an otherwise collinear antiferromagnet,coupled to a bulk conventional superconductor as a novel platform capable of supporting Majoranabound states that are robust against disorder. Crucially, the collinear antiferromagnetic regionneither suppresses superconductivity nor induces topological superconductivity, thus allowing forMajorana bound states localized at the ends of the chain. Our model introduces a new class ofsystems where topological superconductivity can be induced by editing antiferromagnetic texturesrather than locally tuning material parameters, opening avenues for the conclusive observation ofMajorana bound states.
One-dimensional topological superconductors hostzero-energy states localized at their ends called Majoranabound states (MBSs) [1–4]. An important driving forcein the research of MBSs stems from their non-Abelian ex-change statistics [5–7], a key property that makes themattractive for their potential use in topological quantumcomputing [8, 9]. Promising experimental signatures ofzero-energy states consistent with MBSs have been re-ported in nanowires [10–14] and atomic chains [15–18],which are predicted to realize one-dimensional topologi-cal superconductivity [19–24]. However, due to the inher-ent static nature of these setups, they do not easily lendthemselves to the ultimate test of measuring exchangestatistics.An alternative route toward topological superconduc-tivity is to couple a bulk superconductor to a noncollinearmagnetic texture [21–23, 25–34], such as stable topolog-ical defects known as skyrmions [35–41]. Recently, sin-gle ferromagnetic (FM) skyrmions with large topologi-cal charge [42] and FM skyrmion lattices [43–45] cou-pled to a superconductor have been predicted to supportMBSs and Majorana edge modes, respectively. Becauseof their topological stability, skyrmions are themselvesparticle-like objects which can be moved using tempera-ture gradients [46], electric currents [47–50], or magneticfield gradients from magnetic-force-microscope tips [51].Their mobility opens the door to the assembly and dy-namical manipulation of structures that can facilitatethe efforts to measure the exchange statistics of MBSs.A major hindrance, though, is that the ability to movethe skyrmions necessitates embedding them into a two-dimensional collinear FM background, which destroys theproximity-induced superconducting gap, delocalizing thezero-energy states. (b)(a)
FIG. 1. Antiferromagnetic skyrmion chains induce Majoranabound states. (a) Probability density of a Majorana boundstate (top) localized at one end of a chain of antiferromagneticskyrmions embedded in a collinear antiferromagnet (bottom).(b) Top view of the magnetic texture and Majorana boundstate probability density. The partner Majorana bound stateis localized at the other end of the chain (not shown). Textureparameters are detailed in Methods. a r X i v : . [ c ond - m a t . s t r- e l ] F e b To resolve this issue, here we consider a collinearantiferromagnetic (AFM) background into which a chainof AFM skyrmions [52, 53] is embedded, see Fig. 1.This is an ideal system because AFM skyrmions canbe stabilized without external magnetic fields and thecollinear AFM order does not have a harmful effect onsuperconductivity. We discover that in precisely theAFM system, we can modulate between topological andnontopological regions by editing between noncollinearand collinear AFM order, respectively. We also showthat its richer phase diagram exhibits more possibilitiesto tune into the topological superconducting phase.Moreover, under equivalent conditions, we find thatthe minimum length necessary to ensure well-localizedMBSs is smaller in antiferromagnetic skyrmion chains(ASCs) than in one-dimensional AFM spin helices.
Results
Model.
Using a lattice model, we describe our systemwith the Hamiltonian, H = − (cid:88) r ,σ µ c † r σ c r σ − (cid:88) (cid:104) r , r (cid:48) (cid:105) ,σ t c † r σ c r (cid:48) σ + (cid:88) r ∆( c † r ↑ c † r ↓ + c r ↓ c r ↑ ) + (cid:88) r ,µ,ν Jc † r µ n r · σ µν c r ν , (1)where c † r σ creates an electron at lattice site r with spin σ in a thin-film magnetic conductor that is coupled to asuperconductor, which induces a gap, ∆, by proximityeffect. The electronic properties of this conductor aregoverned by the chemical potential, µ , and nearestneighbor hopping amplitude, t , while the direction ofthe magnetic texture, n r , is coupled via the exchangeinteraction, with strength J , to the local spin of theconductor, s r = (1 / (cid:80) µ,ν c † r µ σ µν c r ν , where σ is thevector of Pauli matrices. For simplicity, the latticeconstant is henceforth taken as unity. Advantages of an antiferromagnet.
Although a nat-ural point of departure for topological superconductivitymay start with materials supporting a FM exchange in-teraction and, likewise, FM skyrmion chains, the criticalreader could be weary of such a system for several rea-sons. First, in order to stabilize FM skyrmions, a mag-netic field is conventionally applied perpendicular to thesurface and could destroy the superconductivity. Second,if the texture manages to survive the conditions to sup-port skyrmions, a collinear FM background will surelydestroy the superconducting correlations at least for amodest value of the exchange interaction as compared tothe AFM case. Lastly, if we are able to generate such achain, because FM skyrmions naturally repel, maintain-ing such a chain would require pinning of the individualskyrmions with a local magnetic field or impurity to lo-cally enhance the anisotropy.Because in the following analysis we are interested in localized states supported by skyrmions in a collinearmagnetic background, it is imperative that the spectrumremains gapped. However, in an infinite collinear FMlayer coupled to a superconductor, Eq. (1) with n r constant, the gap vanishes when J ≥ ∆. Consequently,any would-be localized states at the ends of a ferro-magnetic skyrmion chain are delocalized throughoutthe superconductor under the collinear ferromagneticregion. In contrast, the spectral gap of a superconductorcoupled to a collinear antiferromagnet closes only when J = (cid:112) ∆ + µ . Thus, although it is hopeless to realizeMBSs generated by ferromagnetic skyrmions residing ina larger collinear background, as we show below, chainsof antiferromagnetic skyrmions in a antiferromagneticcollinear background can host MBSs at the their ends. Antiferromagnetic skyrmion textures.
For a real-istic treatment of the skyrmionic textures, we considerthe free energy of the localized spins whose classicalminimum configuration determines n r . Including AFMexchange and interfacial Dzyaloshinskii-Moriya interac-tions, as well as easy-axis anisotropy, we use atomisticspin simulations to numerically obtain an ASC (Fig. 1) asa metastable spin configuration. We emphasize here thecomplete absence of any external magnetic field (which,for the ferromagnetic case, would be needed to stabilizea skyrmion texture).Below we also employ artificial skyrmionic tex-tures constructed using the following simple model.A single AFM skyrmion on the square latticecentered at the origin is parametrized by n r =( − i + j [sin( k S r ) cos ϕ ( r ) , sin( k S r ) sin ϕ ( r ) , cos( k S r )] T for r ≤ R S and n r = [0 , , ( − i + j ] T otherwise, where r = i ˆ x + j ˆ y and tan ϕ ( r ) = j/i with i and j beingintegers. Here k S = π/ ( R S −
1) and R S is the skyrmionradius. This model can be immediately generalized toa chain of AFM skyrmions with a spacing S betweenedges of two adjacent skyrmions. By tuning R S and S , artificial skyrmionic textures can be brought toremarkably good agreement with realistic texturesobtained by atomistic spin simulations. Induced Majorana bound states.
To confirmthat the ASC is able to induce MBSs, we constructa phase diagram with control parameters µ and J ,choosing ∆ = 0 . t for numerical simplicity, by findingthe bulk gap closing points in the periodic realisticASC. In Fig. 2(a) we plot the energy gap, E gap , scaledlogarithmically where green color indicates E gap ≈ | Ψ E ( r ) | , and their energy spectrum found (a) (b)(c) FIG. 2. Topological phases induced by an antiferromagneticskyrmion chain. (a) Topological phase diagram as a func-tion of the chemical potential µ and exchange coefficient J for ∆ = 0 . t . The logarithmically-scaled color code encodesthe energy gap, E gap , and the green curves denote E gap ≈ { µ, J } = { . t, . t } ), two MBSs (magentacircle, { µ, J } = { . t, . t } ), and four MBSs (red triangle, { µ, J } = { . t, . t } ). (c) Probability density of the lowestnonnegative energy state (left) and energy spectrum (right)of the selected phases indicated in (b) for a chain composedof 37 antiferromagnetic skyrmions. The texture parametersare the same as in Fig. 1. for a finite-size chain of AFM skyrmions. Indeed, we findtopologically distinct phases [separated by (green) gapclosing lines], in which the ASC induces zero, two, andfour MBSs, respectively. The lobular structure exhibitedby the probability density of the MBSs is reminiscentof that of the lowest-energy state induced by singleisolated AFM skyrmions (see Supplementary Note 1). Itis also similar to the clover shape of Yu-Shiba-Rusinovstates of magnetic adatoms on a superconducting surfacemeasured by scanning tunneling spectroscopy [54]. Stability of Majorana bound states.
One of the keyproperties of MBSs is that they remain at zero energy inthe presence of disorder. In the system studied here, im-purities in the sample could give rise to local variationsin the electronic properties of a material, i.e. parame-ters of our model, such as the chemical potential and theexchange interaction. Another possible source of disor-der is the underlying skyrmionic magnetic texture. Forinstance, randomly positioned impurities could locallydeform the magnetic texture. Furthermore, skyrmionicmagnetic textures at high temperatures (room tempera-ture) can be regarded as an ensemble average over small random deformations throughout the entirety of the pris-tine zero-temperature texture.We have studied the effect of disorder in the electronicproperties as well as the texture on the MBSs localizedat the ends of AFM skyrmion chains. To determine theeffect of each possible source of disorder independentlyof the others, we considered the following four disordermodels: randomly distributed chemical potential µ ,randomly distributed exchange interaction J , randomflips of the magnetic moments of the texture, and smallrandom deviations of the moments across the wholetexture. The details of these disorder studies can befound in Supplementary Note 2. Ultimately, the MBSsare robust to disorder in the electronic couplings µ and J , similar to MBSs arising in quantum wires [55],and are more sensitive to deformations in the magnetictexture. Effective Hamiltonian.
A single AFM skyrmionresiding in a collinear background hosts several localizedlow-energy states (see Supplementary Note 1). Whenskyrmions are sufficiently close together, the localizedstates hybridize and form a band. We can approximateand study the lowest energy band by projecting Eq. (1)onto the states closest to the chemical potential. Thatis, let | Ψ + L (cid:105) ( | Ψ − L (cid:105) ) be the lowest positive (negative)energy state of a single skyrmion centered at r = 0and | Ψ + R (cid:105) ( | Ψ − R (cid:105) ) be the analogous lowest positive(negative) energy state of a single skyrmion centeredat r = L ˆ x with L greater than the skyrmion radius.Upon projecting our original Hamiltonian onto thesestates, we obtain an effective Kitaev chain [1] withchemical potential µ eff = (cid:104) Ψ + L | H | Ψ + L (cid:105) , nearest-neighborhopping t eff = (cid:104) Ψ + R | H | Ψ + L (cid:105) , and superconducting pairing∆ eff = (cid:104) Ψ − R | H | Ψ + L (cid:105) . Because we are only using thelowest energy quasiparticle and its hole partner andneglecting the effect of states hosted by next-to-nearestneighbor skyrmions (and beyond), this projection givesa rough diagnosis of the presence of a pair of MBSs, i.e.if 2 | t eff | > | µ eff | , which one can confirm or disprove byexplicitly calculating the spectrum and wave functionsof the corresponding open skyrmion chain. Skyrmion density.
Although distances betweenskyrmions are fixed in the realistic texture by the spinsimulation parameters, we can study how the density ofskyrmions in the artificial texture affects the formationof MBSs by increasing the spacing between adjacentskyrmions, S = L − R S − J in a series of curves. Thefrontmost curve corresponds to a chain with S = 0 andsuccessive curves, moving back, increase the spacingby two until the penultimate curve; the last curvecorresponds to the limit S → ∞ of an isolated, singleskyrmion. The bottom face of the box is a phase diagramas a function of J interpolated along the S axis. Thetop face of the box indicates if the corresponding pointof the phase diagram below is expected to be topological(red) or nontopological (off-white) as predicted by theeffective Hamiltonian. The densest skyrmion chain, S = 0, supports MBSs near J ∈ { . t, . t, . t } , andin the range J ∈ [1 . t, . t ]. Increasing S noticeablydecreases the range of J supporting a topological phasewhich asymptotes to the case of an isolated skyrmionwhere the gap closes with the system remaining nontopo-logical. The range J ∈ [1 . t, . t ] is also topological for S = 0 with a decreasing range of J for increasing S but,in contrast, the gap closing is lifted for sufficiently large S , e.g. S = 6 near J = 1 . t , wherein the topologicalphase changes into the nontopological one. We concludethat, although a skyrmion sufficiently displaced fromits neighbors in the chain will break a single topologicalchain into two topological pieces, MBSs at the chain endscan still enjoy protection against moderate skyrmiondisplacements. Furthermore, we predict that the denserthe packing of skyrmions along the chain, the denser thetopological phase in parameter space. Experimental realization and observation.
Re-cent experiments have demonstrated the presence of FMskyrmions in Fe/Ir thin films grown on a Re substrate,the latter supporting superconductivity at sufficientlylow temperatures [41], though, the coexistence of super-conductivity with noncollinear ferromagnetism has notyet been established. Simultaneously, evidence consistentwith topological superconductivity has been measured innanowires overgrown with interlaced FM EuS and su-perconducting Al [14]. These systems are evidence ofthe experimental expertise necessary to realize magnetic-superconducting heterostructures which could guide theengineering of our proposed system.One route to experimentally realizing our setup isa heterostructure of layered transition metal dichalco-genides (TMDs). TMDs enjoy a large spin-orbit cou-pling, are amenable to being stacked, and their chemicalpotential can be shifted by the application of a back gate.Recent advances in the synthesis of TMDs have uncov-ered materials that can host superconducting or AFMorder, though not concurrently. In particular, NiSe is a promising candidate for the superconducting layeras it sustains superconductivity even as monolayer [56].On the other hand, Fe / NiS exhibits AFM [57] orderthough noncollinear AFM order remains elusive. Analo-gous to conventional FM materials [58, 59], we proposeinterfacing the antiferromagnet with a layer of heavymetals, e.g. Ir, Pa, or Pt, which could enhance thespin-orbit interaction and ultimately provide a strongDzyaloshinskii-Moriya interaction to stabilize skyrmions.A trilayer of Fe / NiS | Ir | NiSe would allow injection ordeletion of AFM skyrmions in the magnetic layer, stabi-lized by the heavy metal, which induces an effective mag- FIG. 3. Effect of spacing between adjacent skyrmions.(a) Topological phase diagram of a periodic chain of anti-ferromagnetic skyrmions, of radius R S = 4 with µ = 1 . t and∆ = 0 . t , with control parameters J and the spacing betweenadjacent skyrmions S . The curves denote the energy gap atthe specified values of S as a function of J . The bottom faceof the cube interpolates the logarithmically-scaled gap valuesof the curves along the S axis. The top face of the cube in-dicates the topological number of the effective Kitaev chain,described in the text, in which red (off-white) corresponds toone (zero). (b) Probability density of the lowest nonnegativeenergy state at selected points marked in the phase diagram(circle: S = 0; square: S = 2; triangle: S = 4; and hexagon: S = 6) for an open chain of 50 antiferromagnetic skyrmions.MBSs remain robust against a moderate spacing increase. netic exchange interaction within the NiSe supercon-ducting layer. Because the latter can be back gated [60],the topological phase space of the heterostructure, whichwe predict to be dense, can be scanned or tuned to thetopologically nontrivial regime.Two-dimensional systems hosting more complicatedgeometries of chains could provide a path for theidentification of MBSs. Rather than a simple straightchain of skyrmions, consider a quasi-one-dimensionalfigure-eight track which supports antiferromagnetismand conventional superconductivity described byEq. (1). A chain of skyrmions located initially at the FIG. 4. Exchanging MBSs on a topological racetrack.(a) MBSs, γ and γ (cid:48) residing at the ends of an antiferromag-netic skyrmion chain (red) can be shuffled to the other side ofthe structure, through the collinear antiferromagnetic back-ground (blue), utilizing a path through the cross. (b) Thesame procedure can move the MBSs back to the right via thelower leg of the structure. (c) The result is an exchange of γ and γ (cid:48) . right curve in the track supports two MBSs, γ and γ (cid:48) at each end [Fig. 4(a)]. This chain can be pinned,e.g. by a magnetic tip or a local impurity. Then thechain can be moved, by a spin current for instance,to the other side by going through the vertex of thequasi-one-dimensional structure [Fig. 4(b)]. Shufflingthe skyrmions back to the right using only the lowerleg of the structure consequently exchanges the MBSs[Fig. 4(c)] and imprints an overall phase which can bemeasured by an ancillary state. This can be generalizedto more skyrmion chains and quasi-one-dimensionalstructures with additional handles. We can effectbraiding of any two MBSs by analogous shuffling in whatmay be deemed a kind of ‘topological racetrack memory.’ Discussion
So far, AFM skyrmions have not been observed in con-ventional AFM materials. However, real-space detec-tion of skyrmions has been recently reported in ferrimag-nets [61] and synthetic antiferromagnets [62–65]. The ex-perimental assembly of chains of, albeit FM, skyrmionshas already been achieved [51, 66, 67].Our model, Eq. (1), can also describe the experimen-tally different setup of a lattice of magnetic atoms resid-ing on the surface of a superconductor. In two dimen-sions such systems are predicted to support Majoranaedge modes as in the case of so-called Shiba lattices [68]and as recently observed in van der Waals heterostruc-tures [69]. One-dimensional chains of magnetic atoms onconventional superconductors have also been experimen-tally realized and zero-energy states localized at the endsof the chains have been observed [15, 17, 18]. Owing to in-terfacial Dzyaloshinskii-Moriya interactions, the atomicchain in Ref. [18] orders in a FM spin helix, crucial forthe appearance of MBSs. Moreover, a one-dimensionalAFM spin helix can also support MBSs [70], as we ex- plain analytically in Supplementary Note 3.Even though the one-dimensional texture along thecenter of the ASC is an AFM spin helix, the former hasan important advantage over the latter. Localized MBSsare guaranteed as long as their wave functions do notoverlap, which could lead to their hybridization. For thesame induced gap, which determines MBSs localizationlength, the minimum length necessary to ensure theirlocalization is smaller in ASCs than in one-dimensionalAFM spin helices, as we show in Supplementary Note4. Intuitively, the lateral extension of ASCs provides ad-ditional spatial support for the MBSs wave function tospread out and shorten their localization length along thechain axis [71, 72].Although our analysis has been done for chains com-posed of skyrmions whose topological number is strictlyone, AFM skyrmions provide a route to connect our sys-tem to other skyrmion systems which are known to hostMBSs, e.g. skyrmions with large topological charge [42]and FM skyrmion lattices [43, 44], by adiabatically de-forming the magnetic texture. For instance, a singlechain of skyrmions can be adiabatically deformed into asingle skyrmion with large topological number. Likewise,multiple initially uncoupled chains of skyrmions can beslowly brought into proximity with each other to form alattice. In both cases, it would be interesting to see howthe spectrum and spatial distribution of the MBSs wavefunction evolve.In contrast to ferromagnets, antiferromagnets arecapable of changing the local topological phase of anantiferromagnet | superconductor bilayer by deformingbetween collinear and noncollinear magnetic textures.In particular, chains of AFM skyrmions are capable ofhosting MBSs at their ends, generating a rich phasediagram that depends on the material parameters andgeometric details of the skyrmions, further increasing thedegree of tunability into the topological superconductingphase. These end states are robust to fluctuations inthe chemical potential and magnitude of the exchangeinteraction as well as small deformations in the magnetictexture. Our system offers a new platform in whichlocal topological superconducting regions can be movedand modified by in situ manipulation of the magneticorder, and provides a potential route to measuring theexchange statistics and perform braiding of MBSs. Methods
Realistic antiferromagnetic skyrmion textures.
We use atomistic spin simulations to determine classi-cal, metastable magnetic textures governed by the energyfunction E M = (cid:88) r J n r · (cid:0) n r + ˆ x + n r + ˆ y (cid:1) − K ( n r · ˆ z ) + D ( ˆ x · n r × n r + ˆ y − ˆ y · n r × n r + ˆ x ) , (2)with n r the direction of the classical spin at site r on a square lattice, with the lattice constant taken asunity, located on the xy -plane, which includes nearestneighbor antiferromagnetic exchange J , interfacialDzyaloshinskii-Moriya interaction D , and easy-axisanisotropy K . The phase diagram of single antifer-romagnetic skyrmions modeled by a similar energyfunction has been recently discussed in Ref. [73]. Aninitial “seed texture” consisting of a chain of circularregions with downward pointing spins embedded in acollinear antiferromagnetic texture is relaxed employingthe atomistic Landau-Lifshitz-Gilbert equation. Theparameters that generate the texture used in Figs. 1-2are {D / J , K / J } = { . , . } . For a system size of360 ×
45 spins with free boundary conditions we obtaineda chain of 37 antiferromagnetic skyrmions.
Acknowledgements
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Loss, “To-wards a realistic transport modeling in a superconduct-ing nanowire with majorana fermions,” Phys. Rev. B ,024515 (2013). SUPPLEMENTARY NOTE 1: STATES INDUCEDBY A SINGLE ANTIFERROMAGNETICSKYRMION
The probability density of the MBSs presented in themain text consists of an array of lobular patterns thatfade toward the center of the antiferromagnetic skyrmionchain (ASC). We can understand how this peculiar spa-tial distribution emerges by looking at the electronicstates supported by a single, isolated antiferromagneticskyrmion. As we show in Supplementary Figure 1, theprobability density of the lowest positive energy stateinduced by the skyrmion exhibits a four-lobe structure.Comparison of Supplementary Figure 1 with the asso-ciated MBS wave functions in the main text [Fig. 2(c)]suggests that when the antiferromagnetic skyrmions arebrought close to each other to assemble the ASC, the sup-ported MBSs inherit their wave function from the lowestenergy states and their hole partners.
Supplementary Figure 1. Electronic states induced by a sin-gle antiferromagnetic skyrmion. Left panels: probability den-sity of the lowest positive energy state for the same electronicparameters as in Fig. 2 in the main text, indicated by theupper left symbol (orange square, magenta circle, and redtriangle). Right panels: the corresponding energy spectra.The antiferromagnetic skyrmion is located at the center of a45 ×
45 spin lattice generated using atomistic spin simulationsand the magnetic parameters described in Methods. For allthree cases, the lowest positive energy state shows a char-acteristic lobular structure centered at the antiferromagneticskyrmion.
SUPPLEMENTARY NOTE 2: DISORDERANALYSIS
In the absence of disorder, a linear chain of 37 AFMskyrmions with electronic model parameters { µ, ∆ , J } = { . t, . t, . t } supports four MBSs. First we studiedthe effect of disorder arising from spatial inhomogeneitiesin the chemical potential and the exchange interaction.For each disorder realization in the chemical potential,the value at each site was drawn from a Gaussian distri-bution centered at µ = 2 . t with standard deviation σ µ .The disorder strength was controlled by varying σ µ upto 5 δ , where δ is equal to half the electronic gap of theunperturbed system. Similarly, the disorder in the ex-change interaction was modeled with a Gaussian distri-bution centered at J = 1 . t and the standard deviation σ J was also varied up to 5 δ .Random flips of the magnetic moments of theskyrmionic texture were modeled using the Bernoulli dis-tribution with probability parameter p , which gives avalue of 1 with probability p or a value of 0 with prob-ability 1 − p . For a given disorder realization, a valuedrawn from the Bernoulli distribution was assigned toeach magnetic moment site. Only the moments with as-signed value of 1 were flipped. In this case the disorderstrength was controlled by p , which was swept in the in-terval [0 . , × − ].Small distortions throughout the entirety of the pris-tine magnetic texture were constructed as follows. If n r is the direction of the magnetic moment at site r , thenadding a small random vector v r , whose components aredrawn from a Gaussian distribution with zero mean andstandard deviation σ rad , results in a vector whose direc-tion is within a cone with axis n r and cone angle ≈ σ rad (in radians). Therefore, at each site the new distorteddirection n (cid:48) r is given by n (cid:48) r = n r + v r | n r + v r | . (S1)The disorder strength was in this case controlled by vary-ing σ rad . It is useful to define σ = (180 /π ) σ rad which cor-responds to an angle in degrees. The parameter σ wasincreased from 0 ◦ up to 5 ◦ .Supplementary Figure 2 shows the effect of the fourtypes of disorder on the energy states in the vicinity ofthe gap. The disorder strength parameters of each dis-order type are σ µ , σ J , p , and σ . For each value of theseparameters, the spectrum was computed for a total of100 disorder realizations. The disorder-averaged energy(black curves) and standard deviation (shades) of eachstate were then calculated. We conclude that MBSs arerobust against disorder in the electronic couplings µ and J , and they are more susceptible to deformations in themagnetic texture.0 (cid:31) E / t (cid:30) σ J /δ -3-2-101230 1 2 3 4 5 (cid:31) E / t (cid:30) σ µ /δ -3-2-101230 1 2 3 4 5 (cid:31) E / t (cid:30) p -3-2-101230 1 2 3 4 5 (cid:31) E / t (cid:30) σ -3-2-101230 1 2 3 4 5(a)(b)(c)(d) Supplementary Figure 2. Effect of disorder on energy statesin the gap vicinity. Disorder-averaged energies (black curves)with shaded standard deviations as functions of the disorderstrength parameter. Blue/Gray shaded curves correspond toin-gap/bulk states. Disorder in the chemical potential µ andexchange interaction J is shown in (a) and (b), respectively.Disorder arising from deformations of the pristine magnetictexture: (c) random flips of the magnetic moments, and (d)small random distortions about the magnetic moment direc-tions. Supplementary Figure 3. Localization of zero-energy states inone-dimensional spin helices. The probability density of thezero-energy states, | Ψ AF E ( r ) | and | Ψ FM E ( r ) | , is localized atthe interface between antiferromagnetic helical and collinearantiferromagnetic order (black solid curve) but delocalizedover the collinear ferromagnetic region (red dotted curve) forferromagnetic helices. Both helices extend over twenty-fivetimes their pitch which has a length of six lattice sites whilethe material parameters are { µ, ∆ , J } = { . t, . t, . t } . SUPPLEMENTARY NOTE 3:ONE-DIMENSIONAL SPIN HELICES
The necessary conditions to host MBSs in supercon-ductors with a noncollinear or effectively noncollinear ferro magnetic field have been extensively discussed in theliterature of one-dimensional topological materials [21–23]. Perhaps less well known are the conditions nec-essary with a noncollinear antiferro magnetic field [74].Here we use our model Hamiltonian [Eq. (1) in the maintext] to characterize a few of the important propertiesof one-dimensional spin helices. The spins reside at lat-tice sites r = i ˆ x with i an integer, and the lattice con-stant is taken as unity for simplicity. The texture ofthe helix is given by n r = ( P ) i [0 , sin( k h r ) , cos( k h r )] T ,where r = | r | , λ h = 2 π/k h is the helix pitch, and P = ± k ,the spectrum of the infinite antiferromagnetic helix ex-hibits a gap closing at k = π and supports a topo-logical phase when (cid:112) ∆ + [2 t sin( k h / − | µ | ] < | J | < (cid:112) ∆ + [2 t sin( k h /
2) + | µ | ] . This is in contrast to theferromagnetic helix in which the gap closes at k = 0and is topological when (cid:112) ∆ + [2 t cos( k h / − | µ | ] < | J | < (cid:112) ∆ + [2 t cos( k h /
2) + | µ | ] . In the antiferromag-netic (ferromagnetic) case, the region in parameter spacein which the system is topological decreases (increases)with increasing pitch. Critically, this implies that acollinear antiferromagnet, k h →
0, is always nontopolog-ical and the spectrum exhibits a trivial gap closing when J = ∆ + µ ; in contrast, the spectrum of a collinearferromagnet is gapless when (cid:112) ∆ + (2 t − | µ | ) < | J | < (cid:112) ∆ + (2 t + | µ | ) .The consequences of this are immediately realizedupon considering long helices, containing many turns,1 skyrmion chain spin helix
10 15 20 25 30 35
Supplementary Figure 4. Hybridization of MBSs. Energy dif-ference, η , between the lowest nonnegative energy state andits particle-hole partner supported by chains comprised of m antiferromagnetic skyrmions (black) and the correspondingone-dimensional antiferromagnetic spin helix (magenta) run-ning along the chain axis. The minimum length necessaryto ensure well-localized MBSs is smaller in antiferromagneticskyrmion chains than in one-dimensional antiferromagneticspin helices. flanked by a collinear region n r = ( P ) i (0 , , T . If thehelical part of the system is in the topological phase,we find zero-energy states at the ends of the helical tex-tures. In an antiferromagnet, the zero-energy states arelocalized to the interface between the helix and collineartextures [Supplementary Figure 3 (black solid line)]. Ina ferromagnet, the zero-energy states are totally delocal-ized throughout the collinear ferromagnetic region [Sup-plementary Figure 3 (red dashed line)]; paradoxically, thelarge exchange term that drives the helical portion intothe topological phase is simultaneously responsible fordestroying the gap in the collinear ferromagnetic region. SUPPLEMENTARY NOTE 4:ANTIFERROMAGNETIC SKYRMION CHAINSVS. ONE-DIMENSIONAL SPIN HELICES
There is an advantage of great importance of ASCsover one-dimensional antiferromagnetic spin helices: un- der equivalent conditions, the minimum length necessaryto host well-localized MBSs is shorter for ASCs. We com-pare the realistic ASC, obtained as explained in Methods,with the line of spins running along the chain, throughthe center of the skyrmions, which itself is an antiferro-magnetic spin helix. In order to make a meaningful com-parison devoid of finite-size effects the induced supercon-ducting gaps of the corresponding periodic textures mustbe the same. (We note that we do not determine the su-perconducting gap self-consistently in this work [75, 76].)We extract a magnetic unit cell, comprised of two antifer-romagnetic skyrmions, from within the bulk of the ASCto construct the corresponding electronic band structuresand obtain the induced gaps. For the electronic modelparameters { µ, ∆ , J } = { . t, . t, . t } the open ASCsupports two MBSs, as shown in Fig. 2 in the maintext. Fixing µ = 2 . t and ∆ = 0 . t , we find that for J = 1 . t the induced gap of the antiferromagnetic spinhelix matches that of the ASC.Naturally, to study the localization of the MBSs werequire finite textures. Starting from an open chain of37 antiferromagnetic skyrmions we sequentially removetwo skyrmions from within the bulk and compute theelectronic spectra supported by the ASC and the anti-ferromagnetic spin helix using the parameters identifiedabove. If the texture is not sufficiently long, the MBSsfrom each end may overlap and hybridize [77], hence ac-quiring a finite, nonzero energy. Therefore, as a proxyfor the MBSs localization length we use the energy dif-ference, ηη