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We perform tunneling measurements on indium antimonide nanowire/superconductor hybrid devices fabricated for the studies of Majorana bound states. At finite magnetic field, resonances that strongly resemble Majorana bound states, including zero-bias pinning, become common to the point of ubiquity. Since Majorana bound states are predicted in only a limited parameter range in nanowire devices, we seek an alternative explanation for the observed zero-bias peaks. With the help of a self-consistent Poission-Schr\"odinger multiband model developed in parallel, we identify several families of trivial subgap states which overlap and interact, giving rise to a crowded spectrum near zero energy and zero-bias conductance peaks in experiments. These findings advance the search for Majorana bound states through improved understanding of broader phenomena found in superconductor-semiconductor systems., Comment: In this modified version, we made small changes and added detailed device information. All changes are in red text. All original data can be found at Zenodo:https://doi.org/10.5281/zenodo.5793360

Recent tunneling conductance measurements on semiconductor-superconductor nanowires find zero-bias peaks to be ubiquitous across wide ranges of chemical potential and Zeeman energy. Motivated by this, we demonstrate that topologically-trivial Andreev abound states (ABSs) pinned near zero energy are produced rather generically in inhomogeneous systems with multi-band occupancy in the presence of inter-band coupling. We first investigate the inter-band coupling mechanism responsible for the pinning within a multi-band 1D toy model, then we confirm the findings using a 3D Schr\"{o}dinger-Poisson approach that incorporates the geometric and electrostatic details of the actual device. Our analysis shows that level-repulsion generated by inter-band coupling can lead to a rather spectacular pinning of the lowest-energy mode near zero energy in systems (or regions) characterized by very-short length scales ($\sim100~$nm).We show that level repulsion between the lowest energy levels can mimic the gap opening feature (simultaneous with the emergence of a near-zero energy mode) predicted to occur in Majorana hybrid systems. We also demonstrate that nearly-zero bias differential conductance features exhibiting particle-hole asymmetry are due to the presence of (topologically-trivial) ABSs pinned near zero-energy by level repulsion, not to Majorana zero modes, quasi-Majoranas, or any other low-energy mode that involves (partially) separated Majorana bound states. Our findings demonstrate the importance of understanding in detail multi-band physics and electrostatic effects in the context of the ongoing search for Majorana modes in semiconductor-superconductor heterostructures., Comment: 16 pages, 14 figures

We study the electronic structure of full-shell superconductor-semiconductor nanowires, which have recently been proposed for creating Majorana zero modes, using an eight-band $\vec{k} \cdot \vec{p}$ model within a fully self-consistent Schr\"{o}dinger-Poisson scheme. We find that the spin-orbit coupling induced by the intrinsic radial electric field is generically weak for sub-bands with their minimum near the Fermi energy. Furthermore, we show that the chemical potential windows consistent with the emergence of a topological phase are small and sparse and can only be reached by fine tunning the diameter of the wire. These findings suggest that the parameter space consistent with the realization of a topological phase in full-shell InAs/Al nanowires is, at best, very narrow., Comment: 5 pages, 4 figures + Supplemental Material

Higher order topological superconductors hosting Majorana-Kramers pairs (MKPs) as corner modes have recently been proposed in a two-dimensional (2D) quantum spin Hall insulator (QSHI) proximity-coupled to unconventional cuprate or iron-based superconductors. Here, we show that such MKPs can be realized using a conventional s-wave superfluid with a soliton in cold atom systems governed by the Hubbard-Hofstadter model. The MKPs emerge in the presence of interaction at the "corners" defined by the intersections of line solitons and the one-dimensional edges of the system. Our scheme is based on the recently realized cold atom Hubbard-Hofstadter lattice and will pave the way for observing Majorana corner modes and possible higher order topological superfluidity with conventional s-wave superfluids/superconductors., Comment: 5+ pages and 4 figures; Version accepted for publication in PRL

The process of learning involves interaction with the learning environment through our five senses (sight, hearing, touch, smell, and taste). Until recently, distance education focused only on the first two of those senses, sight and sound. Internet-based learning environments are predominantly visual with auditory components. With the advent of haptic technology we can now simulate/generate forces and, as a result, the sense of touch. The gaming industry has promoted the "touch" on the "wire", allowing complex multi-modal interactions online. In this article we provide a brief overview of the evolution of haptic technology, its potential for education, and existing challenges. We review recent data on 21st century students' behaviors, and share our experiences in designing interactive haptic environments for education. From the "Community of Inquiry" framework perspective, we discuss the potential impact of haptic feedback on cognitive and social presence.

In a recent high-quality experimental work on normal metal--superconducting nanowire junctions arXiv:1807.01940 (2018), strong anisotropic suppression of induced superconductivity has been observed in tunnel conductance measurements in the presence of applied magnetic field with variable orientation. Following this finding, we investigate theoretically the dependence of tunnel conductance on the direction of the Zeeman field in order to understand the operational mechanisms and to extract effective system parameters. Second, motivated by a generic discrepancy between experiment and theory, i.e., many in-gap and above-gap conductance features predicted by theory are barely observed in experiments, we study several mechanisms possibly responsible for the suppression of the theoretically predicted conductance features (e.g., length of the nanowire, self-energy effect due to the proximity effect, finite temperature, finite dissipation, and multiband effect). One essential finding in the current work is that only by a combined understanding of both suppression mechanisms can we extract effective system parameters from the experimental data (e.g., the effective nanowire-superconductor coupling, the effective Land\'e $g$ factor, and the chemical potential of the semiconducting nanowire). In addition, we consider topologically trivial Andreev bound states in hybrid nanowires in the presence of potential inhomogeneities, such as external quantum dots or potential inhomogeneities inside the nanowire. We compare the anisotropic, field-dependent features induced by these nontopological Andreev bound states with the corresponding features produced by topological Majorana zero modes in pristine nanostructures, so that we can provide guidance to differentiate between the topologically trivial and nontrivial cases., Comment: 19 pages, 15 figures

Robust topologically trivial low-energy Andreev bound states (ABSs) induced by position-dependent effective potentials have recently come under renewed focus, in light of a remarkable set of experiments observing robust quantized zero-bias conductance plateaus in semiconductor-superconductor heterostructures. We show that (1) the partial spatial separation of the wave functions of the component Majorana bound states (MBSs) is crucial for the creation and stability of topologically trivial near-zero-energy Andreev bound states, (2) the signs of the spin polarizations of the component MBSs can be either the same or opposite, depending on the profile of the inducing potential, and (3) the spin polarizations do not play a fundamental role in generating vastly different coupling strengths to local probes and/or ensuring the robustness of the near-zero-energy ABS. Consequently, in contrast to recent theoretical claims (Vuik et al., arXiv:1806.02801), we find that a robust, quantized zero-bias conductance plateau of height 2e^2/h measured in the topologically trivial regime necessarily requires partially separated ABSs (ps-ABSs), independent of the relative signs of the spin-polarizations. In addition, we show that (4) well-defined energy splitting oscillations involve MBSs characterized by exponential tails pointing toward each other, and (5) ps-ABSs generated by the tunnel barrier itself produce zero-bias conductance peaks with a characteristic width that increases strongly with the applied magnetic field. Finally, we propose (6) a quantitative scheme for analyzing the stability of Majorana modes based on probability distributions of splitting susceptibilities and show that a ps-ABS mode can be remarkably robust when judged based on its signature in a charge tunneling experiment, but, in essence, is topologically unprotected., Comment: 23 pages, 30 figures

Recent tunneling conductance measurements of Majorana nanowires show a strong variation in the magnetic field dependence of the superconducting gap among different devices. Here, we theoretically study the magnetic field dependence of the gap closing feature and establish that the degree of convexity (or concavity) of the gap closing as a function of Zeeman field can provide critical constraints on the underlying microscopic parameters of the semiconductor-superconductor hybrid system model. Specifically, we show that the gap closing feature is entirely concave only for strong spin-orbit coupling strength relative to the chemical potential. Additionally, the non-linearity (i.e. concavity or convexity) of the gap closing as a function of magnetic field complicates the simple assignment of a constant effective $ g $-factor to the states in the Majorana nanowire. We develop a procedure to estimate the effective $ g $-factor from recent experimental data that accounts for the non-linear gap closing resulting from the interplay between chemical potential and spin-orbit coupling. Thus, measurements of the magnetic field dependence of the gap closure on the trivial side of the topological quantum phase transition can provide useful information on parameters that are critical to the theoretical modeling of Majorana nanowires., Comment: 12 pages, 5 figures

Blockchain has emerged as a trusted and secure distributed ledger for transactions while also being decentralised, distributed and its legitimacy not guaranteed by a trusted authority. Since the appearance of Bitcoin, Blockchain has known many implementations based on P2P architectures. This paper presents how the blockchain and smart contracts technologies can be integrated into the SwarmESB ecosystem. SwarmESB is a framework that helps building distributed applications, which benefit from privacy and scalability features. Our proposal will present the flexibility in building not only microservices based applications, but also decen-tralised applications employing blockchain and smart-contracts by modeling a sample Dapp.

We theoretically study a spin-orbit-coupled nanowire proximitized by a superconductor in the presence of an externally applied Zeeman field ("Majorana nanowire") with zero-energy Majorana bound states localized at the two ends of the wire when the Zeeman spin splitting is large enough for the system to enter the topological phase. The specific physics of interest in the current work is the effect of having several tunnel probes attached to the wire along its length. Such tunnel probes should allow, as a matter of principle, one to observe both the predicted bulk superconducting gap closing and opening associated with the topological quantum phase transition as well as the Majorana bound states at the wire ends showing up as zero-bias conductance peaks, depending on which probes are used for the tunneling spectroscopy measurement. Because of the possible invasive nature of the tunnel probes, producing local potential fluctuations in the nanowire, we find the physical situation to be quite complex. In particular, depending on the details of the tunnel barrier operational at the probes, the Majorana nanowire could manifest additional low-energy Andreev bound states which will manifest their own almost-zero-bias peaks, complicating the interpretation of the tunneling data in multiprobe Majorana nanowires. We use two complementary microscopic models to simulate the probes, finding that the tunneling conductance spectrum depends rather sensitively on the details of the tunnel barriers at the probes. We apply our general analysis to simulate a recent multiprobe nanowire experiment commenting on the nature of the quasiparticle gaps likely controlling the experimental observations., Comment: 11 pages, 8 figures, published version

Moving towards significantly smaller nanostructures, direct structuring techniques such as electron beam lithography approach fundamental limitations in feature size and aspect ratios. Application of nanostructures like diffractive X-ray lenses requires feature sizes of below 10 nm to enter a new regime in high resolution X-raymicroscopy. As such dimensions are difficult to obtain using conventional electron beam lithography, we pursue a line-doubling approach. We demonstrate that thismethod yields structure sizes as small as 6.4 nm. X-ray lenses fabricated in this way are tested for their efficiency and microscopic resolution. In addition, the line-doubling technique is successfully extended to a six-fold scheme, where each line in a template structure written by electron beam lithography evolves into six metal lines.

For a GJR-GARCH specification with a generic innovation distribution we derive analytic expressions for the first four conditional moments of the forward and aggregated returns and variances. Moment for the most commonly used GARCH models are stated as special cases. We also the limits of these moments as the time horizon increases, establishing regularity conditions for the moments of aggregated returns to converge to normal moments. Our empirical study yields excellent approximate predictive distributions from these analytic moments, thus precluding the need for time-consuming simulations.

We solve analytically the problem of a finite length Kitaev chain coupled to a quantum dot (QD), which extends the standard Kitaev chain problem making it more closely related to the quantum dot-semiconductor-superconductor (QD-SM-SC) nanowire heterostructure that is currently under intense investigation for possible occurrence of Majorana zero modes (MZMs). Our analytical solution reveals the emergence of a robust Andreev bound state (ABSs) localized in the quantum dot region as the generic lowest energy solution in the topologically trivial phase. By contrast, in the bare Kitaev chain problem such a solution does not exist. The robustness of the ABS in the topologically trivial phase is due to a partial decoupling of the component Majorana bound states (MBSs) over the length of the dot potential. As a result, the signatures of the ABS in measurements that couple locally to the quantum dot, e.g., tunneling measurements, are identical to the signatures of topologically-protected MZMs, which arise only in the topological superconducting (TS) phase of the Kitaev chain., Comment: 11 pages, 8 figures

We show theoretically that in the generic finite chemical potential situation, the clean superconducting spin-orbit-coupled nanowire has two distinct nontopological regimes as a function of Zeeman splitting (below the topological quantum phase transition): one is characterized by finite-energy in-gap Andreev bound states, while the other has only extended bulk states. The Andreev bound state regime is characterized by strong features in the tunneling spectra creating a "gap closure" signature, but no "gap reopening" signature should be apparent above the topological quantum phase transition, in agreement with most recent experimental observations. The gap closure feature is actually the coming together of the Andreev bound states at high chemical potential rather than a simple trivial gap of extended bulk states closing at the transition. Our theoretical finding establishes the generic intrinsic Andreev bound states on the trivial side of the topological quantum phase transition as the main contributors to the tunneling conductance spectra, providing a generic interpretation of existing experiments in clean Majorana nanowires. Our work also explains why experimental tunnel conductance spectra generically have gap closing features below the topological quantum phase transition, but no gap opening features above it., Comment: 28 pages, 29 figures, accepted version

We study the low-energy physics of a one-dimensional array of superconducting quantum dots realized by proximity coupling a semiconductor nanowire to multiple superconducting islands separated by narrow uncovered regions. The effective electrostatic potential inside the quantum dots and the uncovered regions can be controlled using potential gates. By performing detailed numerical calculations based on effective tightbinding models, we find that multiple low-energy sub-gap states consisting of partially overlapping Majorana bound states emerge generically in the vicinity of the uncovered regions. Explicit differential conductance calculations show that a robust zero-bias conductance peak is not inconsistent with the presence of such states localized throughout the system, hence the observation of such a peak does not demonstrate the realization of well-separated Majorana zero modes. However, we find that creating effective potential wells in the uncovered regions traps pairs of nearby partially overlapping Majorana bound states, which become less separated and acquire a finite gap that protects the pair of Majorana zero modes localized at the ends of the system. This behavior persists over a significant parameter range, suggesting that proximitized quantum dot arrays could provide a platform for highly controllable Majorana devices., Comment: 14 pages, 14 figures

Heterogeneous ensembles built from the predictions of a wide variety and large number of diverse base predictors represent a potent approach to building predictive models for problems where the ideal base/individual predictor may not be obvious. Ensemble selection is an especially promising approach here, not only for improving prediction performance, but also because of its ability to select a collectively predictive subset, often a relatively small one, of the base predictors. In this paper, we present a set of algorithms that explicitly incorporate ensemble diversity, a known factor influencing predictive performance of ensembles, into a reinforcement learning framework for ensemble selection. We rigorously tested these approaches on several challenging problems and associated data sets, yielding that several of them produced more accurate ensembles than those that don't explicitly consider diversity. More importantly, these diversity-incorporating ensembles were much smaller in size, i.e., more parsimonious, than the latter types of ensembles. This can eventually aid the interpretation or reverse engineering of predictive models assimilated into the resultant ensemble(s)., Comment: Previously this version appeared as arXiv:2102.07344 which was submitted as a new work by accident

We consider core-shell nanowires with prismatic geometry contacted with two or more superconductors in the presence of a magnetic field applied parallel to the wire. In this geometry, the lowest energy states are localized on the outer edges of the shell, which strongly inhibits the orbital effects of the longitudinal magnetic field that are detrimental to Majorana physics. Using a tight-binding model of coupled parallel chains, we calculate the topological phase diagram of the hybrid system in the presence of non-vanishing transverse potentials and finite relative phases between the parent superconductors. We show that having finite relative phases strongly enhances the stability of the induced topological superconductivity over a significant range of chemical potentials and reduces the value of the critical field associated with the topological quantum phase transition., Comment: 24 pages, 16 figures

We show that partially separated Andreev bound states (ps-ABSs), comprised of pairs of overlapping Majorana bound states (MBSs) emerging in quantum dot-semiconductor-superconductor heterostructures, produce robust zero bias conductance plateaus in end-of-wire charge tunneling experiments. These plateaus remain quantized at $2e^2/h$ over large ranges of experimental control parameters. In light of recent experiments reporting the observation of robust $2e^2/h$-quantized conductance plateaus in local charge tunneling experiments, we perform extensive numerical calculations to explicitly show that such quantized conductance plateaus, which are obtained by varying control parameters such as the tunnel barrier height, the super gate potential, and the applied magnetic field, can arise as a result of the existence of ps-ABSs. Because ps-ABSs can form rather generically in the topologically trivial regime, even in the absence of disorder, our results suggest that the observation of a robust quantized conductance plateau does not represent sufficient evidence to demonstrate the existence of non-Abelian topologically-protected Majorana zero modes localized at the opposite ends of a wire., Comment: 8 pages, 8 figures

We propose a method for solving the Schrodinger-Poisson problem that can be efficiently implemented in realistic 3D tight-binding models of semiconductor-based Majorana devices. The method is based on two key ideas: i) For a given geometry, the Poisson problem is only solved once (for each local orbital) and the results are stored as an interaction tensor; using this Green's function scheme, the Poisson component of the iteration procedure is reduced to a few simple summations. ii) The 3D problem is mapped into an effective multi-orbital 1D problem with molecular orbitals calculated self-consistently as the transverse modes of an infinite wire with the same electrostatic potential as the local electrostatic potential of the finite 3D device. To demonstrate the capabilities of our approach, we calculate the response of the system to an external magnetic field, the dependence of the effective chemical potential on the work function difference, and the dependence of the effective semiconductor-superconductor coupling on the applied gate potential. We find that different low-energy bands are characterized by similar effective couplings, which results in induced gap features characterized by a single energy scale. We also find that electrostatic effects are responsible for a partial suppression of the Majorana energy splitting oscillations. Finally, we show that a position-dependent work function difference can produce a non-homogeneous effective potential that is not affected by the screening due to the superconductor and is only partially suppressed by the charge inside the wire. In turn, this potential can induce trivial low-energy states that mimic the phenomenology of Majorana zero modes. Thus, any position-dependent work function difference (even at the 1% level) along the nanowire must be avoided through carefully engineered semiconductor-superconductor interfaces., Comment: 28 pages, 25 figures