Proximity superconductivity in atom-by-atom crafted quantum dots.

Gapless materials in electronic contact with superconductors acquire proximity-induced superconductivity in a region near the interface1,2. Numerous proposals build on this addition of electron pairing to originally non-superconducting systems and predict intriguing phases of matter, including topological3-7, odd-frequency8, nodal-point9 or Fulde-Ferrell-Larkin-Ovchinnikov10 superconductivity. Here we investigate the most miniature example of the proximity effect on only a single spin-degenerate quantum level of a surface state confined in a quantum corral11 on a superconducting substrate, built atom by atom by a scanning tunnelling microscope. Whenever an eigenmode of the corral is pitched close to the Fermi energy by adjusting the size of the corral, a pair of particle-hole symmetric states enters the gap of the superconductor. We identify these as spin-degenerate Andreev bound states theoretically predicted 50 years ago by Machida and Shibata12, which had-so far-eluded detection by tunnel spectroscopy but were recently shown to be relevant for transmon qubit devices13,14. We further find that the observed anticrossings of the in-gap states are a measure of proximity-induced pairing in the eigenmodes of the quantum corral. Our results have direct consequences on the interpretation of impurity-induced in-gap states in superconductors, corroborate concepts to induce superconductivity into surface states and further pave the way towards superconducting artificial lattices.

Unique Signatures of Topological Phases in Two-Dimensional THz Spectroscopy.

We develop a microscopic theory for the two-dimensional (2D) spectroscopy of one-dimensional topological superconductors. We consider a ring geometry of an archetypal topological superconductor with periodic boundary conditions, bypassing energy-specific differences caused by topologically protected or trivial boundary modes that are hard to distinguish. We show numerically and analytically that the cross-peak structure of the 2D spectra carries unique signatures of the topological phases of the chain. Our work reveals how 2D spectroscopy can identify topological phases in bulk properties.

Bosonic Continuum Theory of One-Dimensional Lattice Anyons.

Anyons with arbitrary exchange phases exist on 1D lattices in ultracold gases. Yet, known continuum theories in 1D do not match. We derive the continuum limit of 1D lattice anyons via interacting bosons. The theory maintains the exchange phase periodicity fully analogous to 2D anyons. This provides a mapping between experiments, lattice anyons, and continuum theories, including Kundu anyons with a natural regularization as a special case. We numerically estimate the Luttinger parameter as a function of the exchange angle to characterize long-range signatures of the theory and predict different velocities for left- and right-moving collective excitations.

Winding Up Quantum Spin Helices: How Avoided Level Crossings Exile Classical Topological Protection.

A magnetic helix can be wound into a classical Heisenberg chain by fixing one end while rotating the other one. We show that in quantum Heisenberg chains of finite length, the magnetization slips back to the trivial state beyond a finite turning angle. Avoided level crossings thus undermine classical topological protection. Yet, for special values of the axial Heisenberg anisotropy, stable spin helices form again, which are nonlocally entangled. Away from these sweet spots, spin helices can be stabilized dynamically or by dissipation. For half-integer spin chains of odd length, a spin slippage state and its Kramers partner define a qubit with a nontrivial Berry connection.

Skyrmion-Anti-Skyrmion Pair Creation by in-Plane Currents.

Magnetic Skyrmions can be considered as localized vortexlike spin textures which are topologically protected in continuous systems. Because of their stability, their small size, and the possibility to move them by low electric currents, they are promising candidates for spintronic devices. Without changing the topological charge, it is possible to create Skyrmion-anti-Skyrmion pairs. We derive a Skyrmion equation of motion which reveals how spin-polarized charge currents create Skyrmion-anti-Skyrmion pairs. It allows us to identify general prerequisites for the pair creation process. We corroborate these general principles by numerical simulations. On a lattice, where the concept of topological protection has to be replaced by that of a finite energy barrier, the anti-Skyrmion partner of the pairs is annihilated and only the Skyrmion survives. This eventually changes the total Skyrmion number and yields a new way of creating and controlling Skyrmions.

Coexisting edge states and gapless bulk in topological states of matter.

We consider two-dimensional systems in which edge states coexist with a gapless bulk. Such systems may be constructed, for example, by coupling a gapped two-dimensional state of matter that carries edge states to a gapless two-dimensional system in which the spectrum is composed of a number of Dirac cones. We find that, in the absence of disorder, the edge states could be protected even when the two systems are coupled, due to momentum and energy conservation. We distinguish between weak and strong edge states by the level of their mixing with the bulk. In the presence of disorder, the edge states may be stabilized when the bulk is localized or destabilized when the bulk is metallic. We analyze the conditions under which these two cases occur. Finally, we propose a concrete physical realization for one of our models based on bilayer Hg(Cd)Te quantum wells.

Exact results for the Kondo screening cloud of two helical liquids.

We analyze the screening of a magnetic quantum dot with spin 1/2 coupled to two helical liquids. Interestingly, we find two qualitatively different sets of Toulouse points, i.e., nontrivial parameters for which we can solve the two channel Kondo model exactly. This enables us to calculate the temperature and voltage dependent Kondo screening cloud, which develops oscillations for an applied spin voltage µ(s). Such a spin voltage can be conveniently applied by a charge bias in a four-terminal helical liquid setup.

Probing the topologically trivial nature of end states in antiferromagnetic atomic chains on superconductors.

Spin chains proximitized by s-wave superconductors are predicted to enter a mini-gapped phase with topologically protected Majorana modes (MMs) localized at their ends. However, the presence of non-topological end states mimicking MM properties can hinder their unambiguous observation. Here, we report on a direct method to exclude the non-local nature of end states via scanning tunneling spectroscopy by introducing a locally perturbing defect on one of the chain's ends. We apply this method to particular end states observed in antiferromagnetic spin chains within a large minigap, thereby proving their topologically trivial character. A minimal model shows that, while wide trivial minigaps hosting end states are easily achieved in antiferromagnetic spin chains, unrealistically large spin-orbit coupling is required to drive the system into a topologically gapped phase with MMs. The methodology of perturbing candidate topological edge modes in future experiments is a powerful tool to probe their stability against local disorder.

Precursors of Majorana modes and their length-dependent energy oscillations probed at both ends of atomic Shiba chains.

Isolated Majorana modes (MMs) are highly non-local quantum states with non-Abelian exchange statistics, which localize at the two ends of finite-size 1D topological superconductors of sufficient length. Experimental evidence for MMs is so far based on the detection of several key signatures: for example, a conductance peak pinned to the Fermi energy or an oscillatory peak splitting in short 1D systems when the MMs overlap. However, most of these key signatures were probed only on one of the ends of the 1D system, and firm evidence for an MM requires the simultaneous detection of all the key signatures on both ends. Here we construct short atomic spin chains on a superconductor-also known as Shiba chains-up to a chain length of 45 atoms using tip-assisted atom manipulation in scanning tunnelling microscopy experiments. We observe zero-energy conductance peaks localized at both ends of the chain that simultaneously split off from the Fermi energy in an oscillatory fashion after altering the chain length. By fitting the parameters of a low-energy model to the data, we find that the peaks are consistent with precursors of MMs that evolve into isolated MMs protected by an estimated topological gap of 50 µeV in chains of at least 35 nm length, corresponding to 70 atoms.

Rotating edge-field driven processing of chiral spin textures in racetrack devices.

Topologically distinct magnetic structures like skyrmions, domain walls, and the uniformly magnetized state have multiple applications in logic devices, sensors, and as bits of information. One of the most promising concepts for applying these bits is the racetrack architecture controlled by electric currents or magnetic driving fields. In state-of-the-art racetracks, these fields or currents are applied to the whole circuit. Here, we employ micromagnetic and atomistic simulations to establish a concept for racetrack memories free of global driving forces. Surprisingly, we realize that mixed sequences of topologically distinct objects can be created and propagated over far distances exclusively by local rotation of magnetization at the sample boundaries. We reveal the dependence between chirality of the rotation and the direction of propagation and define the phase space where the proposed procedure can be realized. The advantages of this approach are the exclusion of high current and field densities as well as its compatibility with an energy-efficient three-dimensional design.

Controlling in-gap end states by linking nonmagnetic atoms and artificially-constructed spin chains on superconductors.

Chains of magnetic atoms with either strong spin-orbit coupling or spiral magnetic order which are proximity-coupled to superconducting substrates can host topologically non-trivial Majorana bound states. The experimental signature of these states consists of spectral weight at the Fermi energy which is spatially localized near the ends of the chain. However, topologically trivial Yu-Shiba-Rusinov in-gap states localized near the ends of the chain can lead to similar spectra. Here, we explore a protocol to disentangle these contributions by artificially augmenting a candidate Majorana spin chain with orbitally-compatible nonmagnetic atoms. Combining scanning tunneling spectroscopy with ab-initio and tight-binding calculations, we realize a sharp spatial transition between the proximity-coupled spiral magnetic order and the non-magnetic superconducting wire termination, with persistent zero-energy spectral weight localized at either end of the magnetic spiral. Our findings open a new path towards the control of the spatial position of in-gap end states, trivial or Majorana, via different chain terminations, and the realization of designer Majorana chain networks for demonstrating topological quantum computation.

Toward tailoring Majorana bound states in artificially constructed magnetic atom chains on elemental superconductors.

Realizing Majorana bound states (MBS) in condensed matter systems is a key challenge on the way toward topological quantum computing. As a promising platform, one-dimensional magnetic chains on conventional superconductors were theoretically predicted to host MBS at the chain ends. We demonstrate a novel approach to design of model-type atomic-scale systems for studying MBS using single-atom manipulation techniques. Our artificially constructed atomic Fe chains on a Re surface exhibit spin spiral states and a remarkable enhancement of the local density of states at zero energy being strongly localized at the chain ends. Moreover, the zero-energy modes at the chain ends are shown to emerge and become stabilized with increasing chain length. Tight-binding model calculations based on parameters obtained from ab initio calculations corroborate that the system resides in the topological phase. Our work opens new pathways to design MBS in atomic-scale hybrid structures as a basis for fault-tolerant topological quantum computing.