Recently, it has been suggested that topological superconductivity and Majorana end states can be realized in a chain of magnetic impurities on the surface of an s-wave superconductor when the magnetic moments form a spin helix as a result of the RKKY interaction mediated by the superconducting substrate. Here, we investigate this scenario theoretically by developing a tight-binding Bogoliubov-de Gennes description starting from the Shiba bound states induced by the individual magnetic impurities. While the resulting model Hamiltonian has similarities with the Kitaev model for one-dimensional spinless p-wave superconductors, there are also important differences, most notably the long-range nature of hopping and pairing as well as the complex hopping amplitudes. We use both analytical and numerical approaches to explore the consequences of these differences for the phase diagram and the localization properties of the Majorana end states when the Shiba chain is in a topological superconducting phase.

We combine scanning-tunneling-spectroscopy experiments probing magnetic impurities on a superconducting surface with a theoretical analysis of the tunneling processes between (superconducting) tip and substrate. We show that the current through impurity-induced Shiba bound states is carried by single-electron tunneling at large tip-substrate distances and Andreev reflections at smaller distances. The single-electron current requires relaxation processes, allowing us to extract information on quasiparticle transitions and lifetimes.

We consider a two-dimensional electron gas with strong spin-orbit coupling contacted by two superconducting leads, forming a Josephson junction. We show that in the presence of an in-plane Zeeman field, the quasi-one-dimensional region between the two superconductors can support a topological superconducting phase hosting Majorana bound states at its ends. We study the phase diagram of the system as a function of the Zeeman field and the phase difference between the two superconductors (treated as an externally controlled parameter). Remarkably, at a phase difference of π, the topological phase is obtained for almost any value of the Zeeman field and chemical potential. In a setup where the phase is not controlled externally, we find that the system undergoes a first-order topological phase transition when the Zeeman field is varied. At the transition, the phase difference in the ground state changes abruptly from a value close to zero, at which the system is trivial, to a value close to π, at which the system is topological. The critical current through the junction exhibits a sharp minimum at the critical Zeeman field and is therefore a natural diagnostic of the transition. We point out that in the presence of a symmetry under a mirror reflection followed by time reversal, the system belongs to a higher symmetry class, and the phase diagram as a function of the phase difference and the Zeeman field becomes richer.

A recent experiment [Nadj-Perge et al., Science 346, 602 (2014)] provides evidence for Majorana zero modes in iron (Fe) chains on the superconducting Pb(110) surface. Here, we study this system by scanning tunneling microscopy using superconducting tips. This high-resolution technique resolves a rich subgap structure, including zero-energy excitations in some chains. We compare the symmetry properties of the data under voltage reversal against theoretical expectations and provide evidence that the putative Majorana signature overlaps with a previously unresolved low-energy resonance. Interpreting the data within a Majorana framework suggests that the topological gap is significantly smaller than previously believed. Aided by model calculations, we also analyze higherenergy features of the subgap spectrum and their relation to high-bias peaks which we associate with the Fe d-bands.Building on advances in nanofabrication [1], engineering topological phases by proximity in superconducting hybrid structures has come within reach of current experiments. A major motivation for realizing such phases are their non-abelian Majorana quasiparticles [2][3][4], and their subsequent applications. The underlying topological superconducting phases can be realized in onedimensional (1d) helical liquids contacted by conventional s-wave superconductors [5][6][7][8][9]. Among the most promising platforms studied in experiment are semiconductor nanowires [10][11][12][13][14][15], edges of two-dimensional topological insulators [16,17], and chains of magnetic adatoms [18,19]. While the proximity coupling to a superconductor is needed to induce a gap protecting the topological phase, it also has more subtle consequences. Magnetic interactions mediated by the superconductor can stabilize magnetic order in the 1d system [20][21][22][23]. Conversely, the spin structure may affect the superconductor. This is particularly apparent for adatom chains, where a band of subgap Shiba states [24][25][26][27] may strongly modify the low-energy properties of the system [8,[28][29][30][31][32] and possibly induce trivial zero-energy features at the chain end [33]. At strong coupling, the 1d states bleed substantially into the superconductor, reducing the effective coherence length at low energies [34]. Nadj-Perge et al.[18] recently provided intriguing evidence for Majorana states in Fe chains on Pb(110). Here, we present data on the same system employing scanning tunneling microscopy/spectroscopy (STM/STS) with superconducting tips (see also [18]). We show that the use of superconducting tips not only provides enhanced resolution of the subgap structure, but also allows for additional consistency checks on the interpretation of the data in terms of Majorana quasiparticles. Our observations indicate a subgap spectrum comprising a flat Shiba band and strongly dispersing Fe states. An interpretation in terms of Majorana states suggests that the induced gap is considerably smaller than previously believed.We carried out the experiments in a Specs JT-ST...

Topological superconductors can support localized Majorana states at their boundaries. These quasi-particle excitations have non-Abelian statistics that can be used to encode and manipulate quantum information in a topologically protected manner. While signatures of Majorana bound states have been observed in one-dimensional systems, there is an ongoing effort to find alternative platforms that do not require fine-tuning of parameters and can be easily scalable to large numbers of states. Here we present a novel experimental approach towards a two-dimensional architecture. Using a Josephson junction made of HgTe quantum well coupled to thin-film aluminum, we are able to tune between a trivial and a topological superconducting state by controlling the phase difference φ across the junction and applying an in-plane magnetic field. We determine the topological state of the induced superconductor *