 that hopping terms are only allowed between atomic YSR states with the same spin, i.e. between NNNs in the present model (see Methods and Fig. 4e). In contrast, the effective superconducting pairing ∆1 can easily be induced between adjacent atoms, since their spins are anti-aligned, and will be suppressed for NNNs (∆2), where the spins have a parallel alignment. For non-zero SOC these restrictions are lifted (see Methods), and the ratio of the coefficients induced by the SOC and present without the SOC may be estimated by the dimensionless parameter R /ℏ F, where R is the Rashba parameter and F is the Fermi velocity. This parameter was estimated to be h / F,0 = 0.094 for Mn/Nb(110) in Ref. 26. We consequently pay particular attention to the case t1<< t2, and ∆1>>∆2 of our minimal model and further assume 0 ≈ 0.0 meV, which is motivated by the experimental YSR state energies for N = 1,2,3 (Fig. 1c).
Experimentally, information about the sub-gap bands' dispersion can be extracted from one-dimensional quasiparticle interference (QPI) measurements 26,31 (see Supplementary Note 4 and the extracted dispersion of the scattering vectors in Fig. 4c). We adjust the parameters of the minimal model such that the dispersions of the possible scattering vectors q1, q2 in Fig. 4b extracted from the calculated dispersion in Fig. 4a reasonably fit the experimental dispersions of Fig. 4c. Using these parameters, we calculate the spatially resolved LDOS for various chain lengths N (Fig. 4d, Supplementary Fig. 4, and Methods) which, given the few free parameters, agree surprisingly well with the experimental data in Fig. 1c,d. In particular, the minimal model nicely reproduces the appearance of non-zero energy end states with an oscillatory decay towards the chain center at energies +/− in a large minigap b, the fast convergence of these end states towards +/− already for short chains ( Supplementary Fig. 4), as well as the dispersive nature of the bulk states. Having substantiated the minimal model by comparison to the experimental data, we investigate its parameterdependent phases and the topological nature of the edge states in the following. Figs. 4f-h show the energy spectrum of a 40-site chain vs. the on-site energy 0 for different values of the hopping terms t1 and t2. It can be seen that the topologically non-trivial phase of the Kitaev chain is entirely quenched for 1 → 0 which corresponds to zero SOC (Fig. 4f). This is a consequence of the ETRS, which results in a Kramers degeneracy of all states, hence to a topologically trivial Majorana number. This finding agrees with previous studies, showing that topologically non-trivial phases cannot be found in antiferromagnetic YSR chains without either strong SOC or additional supercurrents 9,10 breaking the ETRS. However, it is found that the end states at finite energy are formed even in this topologically trivial regime for t2 > t1. Each of these end states is twofold degenerate for even-length chains and shows a small splitting for odd-length chains where the ETRS is broken (cf. the observed experimental splittings in the regime 8 < N < 14 for odd N in Fig. 1c); the degeneracy is restored for semi-infinite chains ( → ∞). The fact that they merge with the continuum of states in Figs. 4f-h for large 0 without a gap closure further proves their topologically trivial origin.
For non-zero values of t1, corresponding to finite SOC, the non-zero-energy end states in some part of the topologically non-trivial phase are preserved (Fig. 4g). But now, also a topologically non-trivial phase with zeroenergy MMs is recovered, which only takes up a large proportion of the phase space for unrealistically large values of SOC (Fig. 4h).

Conclusions & Outlook
In conclusion, we have shown that the in-gap quasiparticle structure of dense, antiferromagnetic YSR chains can be qualitatively described by a minimal model. In contrast to ferromagnetic chains, the antiferromagnetic structure facilitates the formation of a large minigap in the YSR band even without SOC. This observation may be qualitatively explained by the fact that all YSR states in a ferromagnetic chain possess the same spin polarization, meaning that the s-wave superconducting pairing in the substrate may only open a gap in the spectrum for nonzero SOC. For the antiferromagnetic chain the localized YSR states at the sites have an alternating spin polarization, which enables pairing between these states without SOC and can lead to a considerably larger minigap. This large size of the minigap naturally leads to a strong localization of potential end states, since their extension is inversely proportional to the minigap width 26,49. Therefore, antiferromagnetic chains appear to be more suitable for the realization of well-localized end states than their ferromagnetic counterparts. However, as visible from our simulations in Fig. 4f and as was shown in Refs. 9 and 10, the formation of the topologically nontrivial phase now is a threshold effect: the SOC has to compete with the pairing potential, meaning that unrealistically large values of SOC have to be assumed in order to enable the formation of MMs. This is in contrast to the ferromagnetic chain, which is gapless in the absence of SOC and consequently an arbitrarily weak SOC may drive it into the topologically non-trivial regime. The most promising path towards combining the large minigap of antiferromagnetic chains with the easily achievable topologically non-trivial phases of ferromagnetic chains lies in the realization of YSR bands formed by non-collinear spin configurations 23,50.
Our minimal model furthermore reproduces the presence of intriguing finite-energy end states in the antiferromagnetic chains which are not topologically protected and whose energy can be tuned by local potentials. A corresponding topologically trivial phase has been characterized by Pientka et al. as a two-channel p-wave superconducting wire where the interaction between two pairs of MMs lifts their energy to finite values and destroys their topological protection 3,51. However, we want to emphasize that this separation into two channels is not straightforward if SOC is present, and thus, this argument must not be taken at face value. Coincidentally, the trivial end states may appear at near-zero energy by local potentials where they could be misinterpreted as MMs. We have shown here that a local perturbation of the end states with defects is a distinct way to prove their topologically trivial or non-trivial nature. This methodology can be used on other sample systems to probe the stability of candidate topological edge modes against local disorder.

Experimental procedures
The experiments were performed in a home-built STM setup under ultra-high-vacuum at a base temperature of T = 320 mK 52. Nb(110) and Ta(110) single crystals were used as a substrate and cleaned by high-temperature flashes to T > 2700 K with an e-beam heater. In this way, atomically clean surfaces with only few residual oxygen impurities on the surface can be obtained for both materials, as shown previously 37,53. Subsequently, single Mn atoms were deposited onto the surface of the clean substrates held at low temperatures (T < 7 K), resulting in a statistical distribution of adatoms. We use superconducting Nb tips made from mechanically cut and sharpened high-purity Nb wire. The tips were flashed in situ to about 1500 K to remove residual contaminants. STM images were obtained by regulating the tunneling current Istab to a constant value with a feedback loop while applying a constant bias voltage Vstab across the tunneling junction. For measurements of differential tunneling conductance (dI/dV) spectra, the tip was stabilized at bias voltage Vstab and current Istab as individually noted in the figure captions. In a next step, the feedback loop was switched off and the bias voltage was swept from -Vstab to +Vstab. The dI/dV signal was measured using standard lock-in techniques with a small modulation voltage Vmod (RMS) of frequency f = 4.142 kHz added to Vstab. dI/dV line-profiles and maps were acquired recording multiple dI/dV spectra along a one-dimensional line or a two-dimensional grid of lateral positions on the sample, respectively. Note that we chose stabilization parameters at which the contribution of Andreev reflections and direct Cooper pair tunneling can be neglected (see Supplementary Note 3). The use of superconducting Nb tips increases the effective energy resolution of the experiment beyond the Fermi-Dirac limit 54. However, the differential tunneling conductance dI/dV measured with superconducting tips is proportional to the convolution of the sample's local density of states (LDOS) and the superconducting tip density of states (DOS). Consequently, STS data need to be numerically deconvoluted in order to resemble the sample's LDOS, as it is typically known for the interpretation of STS experiments. After careful deconvolution of the spectra, the superconducting gaps of the Nb and Ta surfaces are found to be ∆Nb = 1.51 meV and ∆Ta = 0.64 meV, respectively (see Refs. 26,31 for Nb and Supplementary  Fig. 3 for Ta). We show only deconvoluted data throughout the manuscript (see Supplementary Note 3 for details). Mn chains were constructed using lateral atom manipulation 36,37 techniques at low tunneling resistances of R ≈ 30 -60 kΩ.

Minimal model for YSR bands in antiferromagnetic chains
Pientka et al. showed in Ref. 3 that the low-energy electronic structure of a single orbital chain of classical magnetic moments with