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Josephson junctions in a two-dimensional electron gas with spin-orbit coupling are a promising candidate to realize topological superconductivity. While it is known that the geometry of the junction strongly influences the size of the topological gap, the question of how to construct optimal geometries remains unexplored. We introduce a greedy numerical algorithm to optimize the shape of Majorana junctions. The core of the algorithm relies on perturbation theory and is embarrassingly parallel, which allows it to explore the design space efficiently. By introducing stochastic variations in the junction Hamiltonian, we avoid overfitting geometries to specific system parameters. Furthermore, we constrain the optimizer to produce smooth geometries by applying image filtering and fabrication resolution constraints. We run the algorithm in various setups and find that it reliably produces geometries with increased topological gaps over large parameter ranges. The results are robust to variations in the optimization starting point and the presence of disorder, which suggests the optimizer is capable of finding global maxima., Comment: 12 pages, 6 figures

In Weyl semimetals the application of parallel electric and magnetic fields leads to valley polarization -- an occupation disbalance of valleys of opposite chirality -- a direct consequence of the chiral anomaly. In this work, we present numerical tools to explore such nonequilibrium effects in spatially confined three-dimensional systems with a variable disorder potential, giving exact solutions to leading order in the disorder potential and the applied electric field. Application to a Weyl-metal slab shows that valley polarization also occurs without an external magnetic field as an effect of chiral anomaly "trapping": Spatial confinement produces chiral bulk states, which enable the valley polarization in a similar way as the chiral states induced by a magnetic field. Despite its finite-size origin, the valley polarization can persist up to macroscopic length scales if the disorder potential is sufficiently long ranged, so that direct inter-valley scattering is suppressed and the relaxation then goes via the Fermi-arc surface states., 14 pages, 7 figures

Platforms for creating Majorana quasiparticles rely on superconductivity and breaking of time-reversal symmetry. By studying continuous deformations to known trivial states, we find that the relationship between superconducting pairing and time reversal breaking imposes rigorous bounds on the topology of the system. Applying these bounds to $s$-wave systems with a Zeeman field, we conclude that a topological phase transition requires that the Zeeman energy at least locally exceed the superconducting pairing by the energy gap of the full Hamiltonian. Our results are independent of the geometry and dimensionality of the system., 8 pages; figure added, clarifications added to main text; submission to SciPost

Protection of topological surface states by reflection symmetry breaks down when the boundary of the sample is misaligned with one of the high symmetry planes of the crystal. We demonstrate that this limitation is removed in amorphous topological materials, where the Hamiltonian is invariant on average under reflection over any axis due to continuous rotation symmetry. While the local disorder caused by the amorphous structure weakens the topological protection, we demonstrate that the edge remains protected from localization. In order to classify such phases we perform a systematic search over all the possible symmetry classes in two dimensions and construct the example models realizing each of the proposed topological phases. Finally, we compute the topological invariant of these phases as an integral along a meridian of the spherical Brillouin zone of an amorphous Hamiltonian., Comment: 15 pages, 10 figures

With the lockdowns caused by the COVID-19 pandemic, researchers turn to online conferencing. While posing new challenges, this format also brings multiple advantages. We argue that virtual conferences will become part of our regular scientific communication and invite community members to join the movement.

We study the influence of sample termination on the electronic properties of the novel quantum spin Hall insulator monolayer $1{T}^{\ensuremath{'}}\text{\ensuremath{-}}{\mathrm{WTe}}_{2}$. For this purpose, we construct an accurate, minimal four-orbital tight-binding model with spin-orbit coupling by employing a combination of density-functional theory calculations, symmetry considerations, and fitting to experimental data. Based on this model, we compute energy bands and two-terminal conductance spectra for various ribbon geometries with different terminations, with and without a magnetic field. Because of the strong electron-hole asymmetry, we find that the edge Dirac point is buried in the bulk bands for most edge terminations. In the presence of a magnetic field, an in-gap edge Dirac point leads to exponential suppression of conductance as an edge Zeeman gap opens, whereas the conductance stays at the quantized value when the Dirac point is buried in the bulk bands. Finally, we find that disorder in the edge termination drastically changes this picture: the conductance of a sufficiently rough edge is uniformly suppressed for all energies in the bulk gap regardless of the orientation of the edge.

Majorana zero-modes in a superconductor are midgap states localized in the core of a vortex or bound to the end of a nanowire. They are anyons with non-Abelian braiding statistics, but when they are immobile one cannot demonstrate this by exchanging them in real space and indirect methods are needed. As a real-space alternative, we propose to use the chiral motion along the boundary of the superconductor to braid a mobile vortex in the edge channel with an immobile vortex in the bulk. The measurement scheme is fully electrical and deterministic: edge vortices ($\pi$-phase domain walls) are created on demand by a voltage pulse at a Josephson junction and the braiding with a Majorana zero-mode in the bulk is detected by the charge produced upon their fusion at a second Josephson junction., Comment: 9 pages, 9 figures

Recent years saw the complete classification of topological band structures, revealing an abundance of topological crystalline insulators. Here we theoretically demonstrate the existence of topological materials beyond this framework, protected by quasicrystalline symmetries. We construct a higher-order topological phase protected by a point group symmetry that is impossible in any crystalline system. Our tight-binding model describes a superconductor on a quasicrystalline Ammann-Beenker tiling which hosts localized Majorana zero modes at the corners of an octagonal sample. The Majorana modes are protected by particle-hole symmetry and by the combination of an 8-fold rotation and in-plane reflection symmetry. We find a bulk topological invariant associated with the presence of these zero modes, and show that they are robust against large symmetry preserving deformations, as long as the bulk remains gapped. The nontrivial bulk topology of this phase falls outside all currently known classification schemes., 4 pages + supplemental material

We theoretically study coherent multiple Andreev reflections in a biased three-terminal Josephson junction. We demonstrate that the direct current flowing through the junction consists of supercurrent components when the bias voltages are commensurate. This dissipationless current depends on the phase in the superconducting leads and stems from the Cooper pair transfer processes induced by nonlocal Andreev reflections of the quasiparticles originating from the superconducting leads. We identify supercurrent-enhanced lines in the current and conductance maps of the recent measurement [PNAS 115, 6991 (2018)10.1073/pnas.1800044115] on a nanowire Josephson junction and show that the magnitude of the phase-dependent current components is proportional to the junction transparency with the power corresponding to the component order.

The law of reflection states that smooth surfaces reflect waves specularly, thereby acting as a mirror. This law is insensitive to disorder as long as its length scale is smaller than the wavelength. Monolayer graphene exhibits a linear dispersion at low energies and consequently a diverging Fermi wavelength. We present proof that for a disordered graphene boundary, resonant scattering off disordered edge modes results in diffusive electron reflection even when the electron wavelength is much longer than the disorder correlation length. Using numerical quantum transport simulations, we demonstrate that this phenomenon can be observed as a nonlocal conductance dip in a magnetic focusing experiment., Comment: 18 pages, 6 figures, replaced Fig. 2, refined the corresponding discussion

The anomalous $4\pi$-periodic ac Josephson effect, a hallmark of topological Josephson junctions, was experimentally observed in a quantum spin Hall insulator. This finding is unexpected due to time-reversal symmetry preventing the backscattering of the helical edge states and therefore suppressing the $4\pi$-periodic component of the Josephson current. Here we analyze the two-particle inelastic scattering as a possible explanation for this experimental finding. We show that a sufficiently strong inelastic scattering restores the $4\pi$-periodic component of the current beyond the short Josephson junction regime. Its signature is an observable peak in the power spectrum of the junction at half the Josephson frequency. We propose to use the exponential dependence of the peak width on the applied bias and the magnitude of the dc current as means of verifying that the inelastic scattering is indeed the mechanism responsible for the $4\pi$-periodic signal., Comment: 17 pages, 16 figures

Andreev bound states in hybrid superconductor-semiconductor devices can have near-zero energy in the topologically trivial regime as long as the confinement potential is sufficiently smooth. These quasi-Majorana states show zero-bias conductance features in a topologically trivial phase, mimicking spatially separated topological Majorana states. We show that in addition to the suppressed coupling between the quasi-Majorana states, also the coupling of these states across a tunnel barrier to the outside is exponentially different for increasing magnetic field. As a consequence, quasi-Majorana states mimic most of the proposed Majorana signatures: quantized zero-bias peaks, the $4\pi$ Josephson effect, and the tunneling spectrum in presence of a normal quantum dot. We identify a quantized conductance dip instead of a peak in the open regime as a distinguishing feature of true Majorana states in addition to having a bulk topological transition. Because braiding schemes rely only on the ability to couple to individual Majorana states, the exponential control over coupling strengths allows to also use quasi-Majorana states for braiding. Therefore, while the appearance of quasi-Majorana states complicates the observation of topological Majorana states, it opens an alternative route towards braiding of non-Abelian anyons and protected quantum computation., Comment: 13 pages, 11 figures

We propose a robust and efficient algorithm for computing bound states of infinite tight-binding systems that are made up of a finite scattering region connected to semi-infinite leads. Our method uses wave matching in close analogy to the approaches used to obtain propagating states and scattering matrices. We show that our algorithm is robust in presence of slowly decaying bound states where a diagonalization of a finite system would fail. It also allows to calculate the bound states that can be present in the middle of a continuous spectrum. We apply our technique to quantum billiards and the following topological materials: Majorana states in 1D superconducting nanowires, edge states in the 2D quantum spin Hall phase, and Fermi arcs in 3D Weyl semimetals., Comment: 21 pages, 13 figures, minors changes according to the referees comments, https://scipost.org/submissions/1711.08250v1/

Symmetry is a guiding principle in physics that allows to generalize conclusions between many physical systems. In the ongoing search for new topological phases of matter, symmetry plays a crucial role because it protects topological phases. We address two converse questions relevant to the symmetry classification of systems: Is it possible to generate all possible single-body Hamiltonians compatible with a given symmetry group? Is it possible to find all the symmetries of a given family of Hamiltonians? We present numerically stable, deterministic polynomial time algorithms to solve both of these problems. Our treatment extends to all continuous or discrete symmetries of non-interacting lattice or continuum Hamiltonians. We implement the algorithms in the Qsymm Python package, and demonstrate their usefulness with examples from active research areas in condensed matter physics, including Majorana wires and Kekule graphene., Comment: 16 pages, 4 figures

Electron surface states in solids are typically confined to the outermost atomic layers and, due to surface disorder, have negligible impact on electronic transport. Here, we demonstrate a very different behavior for surface states in graphene. We probe the wavelike character of these states by Fabry-Perot (FP) interferometry and find that, in contrast to theoretical predictions, these states can propagate ballistically over micron-scale distances. This is achieved by embedding a graphene resonator formed by gate-defined p-n junctions within a graphene superconductor-normal-superconductor structure. By combining superconducting Aharanov-Bohm interferometry with Fourier methods, we visualize spatially resolved current flow and image FP resonances due to p-n-p cavity modes. The coherence of the standing-wave edge states is revealed by observing a new family of FP resonances, which coexist with the bulk resonances. The edge resonances have periodicity distinct from that of the bulk states manifest in a repeated spatial redistribution of current on and off the FP resonances. This behavior is accompanied by a modulation of the multiple Andreev reflection amplitude on-and-off resonance, indicating that electrons propagate ballistically in a fully coherent fashion. These results, which were not anticipated by theory, provide a practical route to developing electron analog of optical FP resonators at the graphene edge.

A far-reaching goal of graphene research is exploiting the unique properties of carriers to realize extreme nonclassical electronic transport. Of particular interest is harnessing wavelike carriers to guide and direct them on submicron scales, similar to light in optical fibers. Such modes, while long anticipated, have never been demonstrated experimentally. In order to explore this behavior, we employ superconducting interferometry in a graphene Josephson junction to reconstruct the real-space supercurrent density using Fourier methods. Our measurements reveal charge flow guided along crystal boundaries close to charge neutrality. We interpret the observed edge currents in terms of guided-wave states, confined to the edge by band bending and transmitted as plane waves. As a direct analog of refraction-based confinement of light in optical fibers, such nonclassical states afford new means for information transduction and processing at the nanoscale., 31 pages, including supplementary materials

The proximity effect in hybrid superconductor-semiconductor structures, crucial for realizing Majorana edge modes, is complicated to control due to its dependence on many unknown microscopic parameters. In addition, defects can spoil the induced superconductivity locally in the proximitised system which complicates measuring global properties with a local probe. We show how to use the nonlocal conductance between two spatially separated leads to probe three global properties of a proximitised system: the bulk superconducting gap, the induced gap, and the induced coherence length. Unlike local conductance spectroscopy, nonlocal conductance measurements distinguish between non-topological zero-energy modes localized around potential inhomogeneities, and true Majorana edge modes that emerge in the topological phase. In addition, we find that the nonlocal conductance is an odd function of bias at the topological phase transition, acting as a current rectifier in the low-bias limit. More generally, we identify conditions for crossed Andreev reflection to dominate the nonlocal conductance and show how to design a Cooper pair splitter in the open regime., 11 pages, 13 figures

Superconducting electronic devices have re-emerged as contenders for both classical and quantum computing due to their fast operation speeds, low dissipation and long coherence times. An ultimate demonstration of coherence is lasing. We use one of the fundamental aspects of superconductivity, the ac Josephson effect, to demonstrate a laser made from a Josephson junction strongly coupled to a multi-mode superconducting cavity. A dc voltage bias to the junction provides a source of microwave photons, while the circuit's nonlinearity allows for efficient down-conversion of higher order Josephson frequencies down to the cavity's fundamental mode. The simple fabrication and operation allows for easy integration with a range of quantum devices, allowing for efficient on-chip generation of coherent microwave photons at low temperatures., Supplementary Information available at https://qutech.nl/kouwenhoven-lab/publications/

We study the effects of strong coupling between a superconductor and a semiconductor nanowire on the creation of the Majorana bound states, when the quasiparticle dwell time in the normal part of the nanowire is much shorter than the inverse superconducting gap. This "short junction" limit is relevant for the recent experiments using the epitaxially grown aluminum characterized by a transparent interface with the semiconductor and a small superconducting gap. We find that the small superconducting gap does not have a strong detrimental effect on the Majorana properties. Specifically, both the critical magnetic field required for creating a topological phase and the size of the Majorana bound states are independent of the superconducting gap. The critical magnetic field scales with the wire cross section, while the relative importance of the orbital and Zeeman effects of the magnetic field is controlled by the material parameters only: $g$-factor, effective electron mass, and the semiconductor-superconductor interface transparency., 13 pages, 12 Figures

Majorana bound states (MBS) differ from the regular zero energy Andreev bound states in their nonlocal properties, since two MBS form a single fermion. We design strategies for detection of this nonlocality by using the phenomenon of Coulomb-mediated Majorana coupling in a simplest setting which still retains falsifiability. Focusing on the implementation of MBS based on the quantum spin Hall effect, we also design a way to probe Majoranas without the need to open a magnetic gap in the helical edge states. In the setup that we analyse, long range MBS coupling manifests in the $h/e$ magnetic flux periodicity of tunneling conductance and supercurrent. While $h/e$ is also the periodicity of Aharonov-Bohm effect and persistent current, we show how to ensure its Majorana origin by verifying that switching off the charging energy restores $h/2e$ periodicity conventional for superconducting systems., Comment: 6 pages, 3 figures

Measurement of multiple Andreev reflection (MAR) in a Josephson junction made from an InAs heterostructure with epitaxial aluminum is used to quantify the highly transparent semiconductor-superconductor interface, indicating near-unity transmission. The observed temperature dependence of MAR does not follow a conventional BCS form, but instead agrees with a model in which the density of states in the quantum well acquires an effective induced gap, in our case 180 {\mu}eV, close to that of the epitaxial superconductor. Carrier density dependence of MAR is investigated using a depletion gate, revealing the subband structure of the semiconductor quantum well, consistent with magnetotransport experiment of the bare InAs performed on the same wafer., Comment: Includes supplementary material

$ $The critical current of a Josephson junction is an oscillatory function of the enclosed magnetic flux $\Phi$, because of quantum interference modulated with periodicity $h/2e$. We calculate these Fraunhofer oscillations in a two-dimensional (2D) ballistic superconductor--normal-metal--superconductor (SNS) junction. For a Fermi circle the amplitude of the oscillations decays as $1/\Phi$ or faster. If the Fermi circle is strongly warped, as it is on a square lattice near the band center, we find that the amplitude decays slower $\propto 1/\sqrt\Phi$ when the magnetic length $l_m=\sqrt{\hbar/eB}$ drops below the separation $L$ of the NS interfaces. The crossover to the slow decay of the critical current is accompanied by the appearance of a 2D array of current vortices and antivortices in the normal region, which form a bipartite rectangular lattice with lattice constant $\simeq l_m^2/L$. The 2D lattice vanishes for a circular Fermi surface, when only the usual single row of Josephson vortices remains., Comment: 12 pages, 14 figures; V2: added three appendices, including data on the superconducting order parameter and density of states

The prospect of coupling a two-dimensional (2D) semiconductor heterostructure to a superconductor opens new research and technology opportunities, including fundamental problems in mesoscopic superconductivity, scalable superconducting electronics, and new topological states of matter. For instance, one route toward realizing topological matter is by coupling a 2D electron gas (2DEG) with strong spin-orbit interaction to an s-wave superconductor. Previous efforts along these lines have been hindered by interface disorder and unstable gating. Here, we report measurements on a gateable InGaAs/InAs 2DEG with patterned epitaxial Al, yielding multilayer devices with atomically pristine interfaces between semiconductor and superconductor. Using surface gates to form a quantum point contact (QPC), we find a hard superconducting gap in the tunneling regime, overcoming the soft-gap problem in 2D superconductor-semiconductor hybrid systems. With the QPC in the open regime, we observe a first conductance plateau at 4e^2/h, as expected theoretically for a normal-QPC-superconductor structure. The realization of a hard-gap semiconductor-superconductor system that is amenable to top-down processing provides a means of fabricating scalable multicomponent hybrid systems for applications in low-dissipation electronics and topological quantum information., includes main text, supplementary information and code for simulations. Published version

We report a new attractive critical point occurring in the Anderson localization scaling flow of symplectic models on fractals. The scaling theory of Anderson localization predicts that in disordered symplectic two-dimensional systems weak antilocalization effects lead to a metal-insulator transition. This transition is characterized by a repulsive critical point above which the system becomes metallic. Fractals possess a non-integer scaling of conductance in the classical limit which can be continuously tuned by changing the fractal structure. We demonstrate that in disordered symplectic Hamiltonians defined on fractals with classical conductance scaling $g \sim L^{-\varepsilon}$, for $0 < \varepsilon < ��_\mathrm{max} \approx 0.15$, the metallic phase is replaced by a critical phase with a scale invariant conductance dependent on the fractal dimensionality. Our results show that disordered fractals allow an explicit construction and verification of the $\varepsilon$ expansion.

We report the realization of quantum microwave circuits using hybrid superconductor-semiconductor Josephson elements comprised of InAs nanowires contacted by NbTiN. Capacitively-shunted single elements behave as transmon qubits with electrically tunable transition frequencies. Two-element circuits also exhibit transmon-like behavior near zero applied flux, but behave as flux qubits at half the flux quantum, where non-sinusoidal current-phase relations in the elements produce a double-well Josephson potential. These hybrid Josephson elements are promising for applications requiring microwave superconducting circuits operating in magnetic field., Comment: Main text: 4 pages, 4 figures; Supplement: 10 pages, 8 figures, 1 table

Studies of Majorana bound states in semiconducting nanowires frequently neglect the orbital effect of magnetic field. Systematically studying its role leads us to several conclusions for designing Majoranas in this system. Specifically, we show that for experimentally relevant parameter values orbital effect of magnetic field has a stronger impact on the dispersion relation than the Zeeman effect. While Majoranas do not require a presence of only one dispersion subband, we observe that the size of the Majoranas becomes unpractically large, and the band gap unpractically small when more than one subband is filled. Since the orbital effect of magnetic field breaks several symmetries of the Hamiltonian, it leads to the appearance of large regions in parameter space with no band gap whenever the magnetic field is not aligned with the wire axis. The reflection symmetry of the Hamiltonian with respect to the plane perpendicular to the wire axis guarantees that the wire stays gapped in the topologically nontrivial region as long as the field is aligned with the wire., Comment: 5 pages, 6 figures, data available at http://dx.doi.org/10.4121/uuid:20f1c784-1143-4c61-a03d-7a3454914abb

Hybrid graphene-superconductor devices have attracted much attention since the early days of graphene research. So far, these studies have been limited to the case of diffusive transport through graphene with poorly defined and modest quality graphene-superconductor interfaces, usually combined with small critical magnetic fields of the superconducting electrodes. Here we report graphene based Josephson junctions with one-dimensional edge contacts of Molybdenum Rhenium. The contacts exhibit a well defined, transparent interface to the graphene, have a critical magnetic field of 8 Tesla at 4 Kelvin and the graphene has a high quality due to its encapsulation in hexagonal boron nitride. This allows us to study and exploit graphene Josephson junctions in a new regime, characterized by ballistic transport. We find that the critical current oscillates with the carrier density due to phase coherent interference of the electrons and holes that carry the supercurrent caused by the formation of a Fabry-P\'{e}rot cavity. Furthermore, relatively large supercurrents are observed over unprecedented long distances of up to 1.5 $\mu$m. Finally, in the quantum Hall regime we observe broken symmetry states while the contacts remain superconducting. These achievements open up new avenues to exploit the Dirac nature of graphene in interaction with the superconducting state., Comment: Updated version after peer review. Includes supplementary material and ancillary file with source code for tight binding simulations

We show how the superconducting phase difference in a Josephson junction may be used to split the Kramers degeneracy of its energy levels and to remove all the properties associated with time reversal symmetry. The superconducting phase difference is known to be ineffective in two-terminal short Josephson junctions, where irrespective of the junction structure the induced Kramers degeneracy splitting is suppressed and the ground state fermion parity must stay even, so that a protected zero-energy Andreev level crossing may never appear. Our main result is that these limitations can be completely avoided by using multi-terminal Josephson junctions. There the Kramers degeneracy breaking becomes comparable to the superconducting gap, and applying phase differences may cause the change of the ground state fermion parity from even to odd. We prove that the necessary condition for the appearance of a fermion parity switch is the presence of a "discrete vortex" in the junction: the situation when the phases of the superconducting leads wind by $2\pi$. Our approach offers new strategies for creation of Majorana bound states as well as spin manipulation. Our proposal can be implemented using any low density, high spin-orbit material such as InAs quantum wells, and can be detected using standard tools., Comment: Source code available as ancillary files. 10 pages, 7 figures. v2: minor changes, published version

The fractionally charged quasiparticles appearing in the 5/2 fractional quantum Hall plateau are predicted to have an extra non-local degree of freedom, known as topological charge. We show how this topological charge can block the tunnelling of these particles, and how such 'topological blockade' can be used to readout their topological charge. We argue that the short time scale required for this measurement is favorable for the detection of the non-Abelian anyonic statistics of the quasiparticles. We also show how topological blockade can be used to measure braiding statistics, and to couple a topological qubit with a conventional one., Comment: Published version: one additional paragraph (on the 331 state); Figs. 1 and 4 modified; Ref. 46 added

The common approach to topological quantum computation is to implement quantum gates by adiabatically moving non-Abelian anyons around each other. Here we present an alternative perspective based on the possibility of realizing the exchange (braiding) operators of anyons by adiabatically varying pairwise interactions between them rather than their positions. We analyze a system composed by four anyons whose couplings define a T-junction and we show that the braiding operator of two of them can be obtained through a particular adiabatic cycle in the space of the coupling parameters. We also discuss how to couple this scheme with anyonic chains in order to recover the topological protection., 8 pages, 7 figures. Errors corrected, clarifications and comments added

We suggest a way to overcome the obstacles that disorder and high density of states pose to the creation of unpaired Majorana fermions in one-dimensional systems. This is achieved by splitting the system into a chain of quantum dots, which are then tuned to the conditions under which the chain can be viewed as an effective Kitaev model, so that it is in a robust topological phase with well-localized Majorana states in the outermost dots. The tuning algorithm that we develop involves controlling the gate voltages and the superconducting phases. Resonant Andreev spectroscopy allows us to make the tuning adaptive, so that each pair of dots may be tuned independently of the other. The calculated quantized zero bias conductance serves then as a natural proof of the topological nature of the tuned phase., Comment: 7 pages, 8 figures

Two-dimensional superconductors with time-reversal symmetry have a Z_2 topological invariant, that distinguishes phases with and without helical Majorana edge states. We study the topological phase transition in a class-DIII network model, and show that it is associated with a metal-insulator transition for the thermal conductance of the helical superconductor. The localization length diverges at the transition with critical exponent nu approx 2.0, about twice the known value in a chiral superconductor., Comment: 9 pages, 8 figures, 3 tables

We report electronic transport measurements of graphene contacted by NbTiN electrodes, which at low temperature remain superconducting up to at least 11 Tesla. In devices with a single superconducting contact, we find a more than twofold enhancement of the conductance at zero bias, which we interpret in terms of reflectionless tunneling. In devices with two superconducting contacts, we observe the Josephson effect, bipolar supercurrents and Fraunhofer patterns., 6 pages, 5 figures

We define a class of insulators with gapless surface states protected from localization due to the statistical properties of a disordered ensemble, namely due to the ensemble's invariance under a certain symmetry. We show that these insulators are topological, and are protected by a $\mathbb{Z}_2$ invariant. Finally, we prove that every topological insulator gives rise to an infinite number of classes of statistical topological insulators in higher dimensions. Our conclusions are confirmed by numerical simulations., Comment: 6 pages, 1 table, 5 figures, this is the final, published version

We show how to exchange (braid) Majorana fermions in a network of superconducting nanowires by control over Coulomb interactions rather than tunneling. Even though Majorana fermions are charge-neutral quasiparticles (equal to their own antiparticle), they have an effective long-range interaction through the even-odd electron number dependence of the superconducting ground state. The flux through a split Josephson junction controls this interaction via the ratio of Josephson and charging energies, with exponential sensitivity. By switching the interaction on and off in neighboring segments of a Josephson junction array, the non-Abelian braiding statistics can be realized without the need to control tunnel couplings by gate electrodes. This is a solution to the problem how to operate on topological qubits when gate voltages are screened by the superconductor.

There exists a variety of proposals to transform a conventional s-wave superconductor into a topological superconductor, supporting Majorana fermion mid-gap states. A necessary ingredient of these proposals is strong spin-orbit coupling. Here we propose an alternative system consisting of a one-dimensional chain of magnetic nanoparticles on a superconducting substrate. No spin-orbit coupling in the superconductor is needed. We calculate the topological quantum number of a chain of finite length, including the competing effects of disorder in the orientation of the magnetic moments and in the hopping energies, to identify the transition into the topologically nontrivial state (with Majorana fermions at the end points of the chain)., 7 pages, 5 figures

Recent experiments on the propagation of light over a distance L through a random packing of spheres with a power law distribution of radii (a socalled L\'evy glass) have found that the transmission probability T \propto 1/L^{\gamma} scales superdiffusively ({\gamma} < 1). The data has been interpreted in terms of a L\'evy walk. We present computer simulations to demonstrate that diffusive scaling ({\gamma} \approx 1) can coexist with a divergent second moment of the step size distribution (p(s) \propto 1/s^(1+{\alpha}) with {\alpha} < 2). This finding is in accord with analytical predictions for the effect of step size correlations, but deviates from what one would expect for a L\'evy walk of independent steps., Comment: 10 pages, 14 figures; v2: extension from 2D to 3D and comparison with experiments

The bulk microwave conductivity of a dirty d-wave superconductor is known to depend sensitively on the range of the disorder potential: long-range scattering enhances the conductivity, while short- range scattering has no effect. Here we show that the three-terminal electrical conductance of a normal-metal-d-wave superconductor-normal-metal junction has a dual behavior: short-range scattering suppresses the conductance, while long-range scattering has no effect., 5 pages

Superconducting wires without time-reversal and spin-rotation symmetries can be driven into a topological phase that supports Majorana bound states. Direct detection of these zero-energy states is complicated by the proliferation of low-lying excitations in a disordered multi-mode wire. We show that the phase transition itself is signaled by a quantized thermal conductance and electrical shot noise power, irrespective of the degree of disorder. In a ring geometry, the phase transition is signaled by a period doubling of the magnetoconductance oscillations. These signatures directly follow from the identification of the sign of the determinant of the reflection matrix as a topological quantum number., 7 pages, 4 figures; v3: added appendix with numerics for long-range disorder

A pair of counter-propagating Majorana edge modes appears in chiral p-wave superconductors and in other superconducting systems belonging to the same universality class. These modes can be described by an Ising conformal field theory. We show how a superconducting flux qubit attached to such a system couples to the two chiral edge modes via the disorder field of the Ising model. Due to this coupling, measuring the back-action of the edge states on the qubit allows to probe the properties of Majorana edge modes., 18 pages,3 figures