Publication list

Minimal Zeeman field requirement for a topological transition in superconductors [+]
Platforms for creating Majorana quasiparticles rely on superconductivity and breaking of timereversal 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.
Kim Pöyhönen, Daniel Varjas, Michael Wimmer, and Anton R. AkhmerovarXiv:2011.08263 [pdf], (unpublished). 
Electronic properties of InAs/EuS/Al hybrid nanowires [+]
We study the electronic properties of InAs/EuS/Al heterostructures as explored in a recent experiment [S. Vaitiekenas \emph{et al.}, Nat. Phys. (2020)], combining both spectroscopic results and microscopic device simulations. In particular, we use angleresolved photoemission spectroscopy to investigate the band bending at the InAs/EuS interface. The resulting band offset value serves as an essential input to subsequent microscopic device simulations, allowing us to map the electronic wave function distribution. We conclude that the magnetic proximity effects at the Al/EuS as well as the InAs/EuS interfaces are both essential to achieve topological superconductivity at zero applied magnetic field. Mapping the topological phase diagram as a function of gate voltages and proximityinduced exchange couplings, we show that the ferromagnetic hybrid nanowire with overlapping Al and EuS layers can become a topological superconductor within realistic parameter regimes. Our work highlights the need for a combined experimental and theoretical effort for faithful device simulation.
ChunXiao Liu, Sergej Schuwalow, Yu Liu, Kostas Vilkelis, A. L. R. Manesco, P. Krogstrup, and Michael WimmerarXiv:2011.06567 [pdf], (unpublished). 
Weyl Josephson Circuits [+]
We introduce Weyl Josephson circuits: small Josephson junction circuits that simulate Weyl band structures. We first formulate a general approach to design circuits that are analogous to Bloch Hamiltonians of a desired dimensionality and symmetry class. We then construct and analyze a sixjunction device that produces a 3D Weyl Hamiltonian with broken inversion symmetry and in which topological phase transitions can be triggered \emph{in situ}. We argue that currently available superconducting circuit technology allows experiments that probe topological properties inaccessible in condensed matter systems.
Valla Fatemi, Anton R. Akhmerov, and Landry BretheauarXiv:2008.13758 [pdf], (unpublished). 
Conductance asymmetries in mesoscopic superconducting devices due to finite bias [+]
Tunneling conductance spectroscopy in normal metalsuperconductor junctions is an important tool for probing Andreev bound states in mesoscopic superconducting devices, such as Majorana nanowires. In an ideal superconducting device, the subgap conductance obeys specific symmetry relations, due to particlehole symmetry and unitarity of the scattering matrix. However, experimental data often exhibits deviations from these symmetries or even their explicit breakdown. In this work, we identify a mechanism that leads to conductance asymmetries without quasiparticle poisoning. In particular, we investigate the effects of finite bias and include the voltage dependence in the tunnel barrier transparency, finding significant conductance asymmetries for realistic device parameters. It is important to identify the physical origin of conductance asymmetries: in contrast to other possible mechanisms such as quasiparticle poisoning, finitebias effects are not detrimental to the performance of a topological qubit. To that end we identify features that can be used to experimentally determine whether finitebias effects are the source of conductance asymmetries.
André Melo, ChunXiao Liu, Piotr Rożek, Tómas Örn Rosdahl, and Michael WimmerarXiv:2008.01734 [pdf], (unpublished). 
Josephson current via an isolated Majorana zero mode [+]
We study the equilibrium dc Josephson current in a junction between an $s$wave and a topological superconductor. Cooper pairs from the $s$wave superconducting lead can transfer to the topological side either via an unpaired Majorana zero mode localized near the junction, or via the abovegap continuum states. We find that the Majorana contribution to the supercurrent can be switched on when timereversal symmetry in the conventional lead is broken, e.g., by an externally applied magnetic field inducing a Zeeman splitting. Moreover, if the magnetic field has a component in the direction of the effective spinorbit field, there will be a Majoranainduced anomalous supercurrent at zero phase difference. This behavior may serve as a signature characteristic of Majorana zero modes, and is accessible to devices with only superconducting contacts.
ChunXiao Liu, Bernard van Heck, and Michael WimmerarXiv:2005.10657 [pdf], (unpublished). 
Twoband superconductivity with unconventional pairing symmetry in HfV$_2$Ga$_4$ [+]
In this letter, we have examined the superconducting ground state of the HfV$_2$Ga$_4$ compound using resistivity, magnetization, zerofield (ZF) and transversefield (TF) muonspin relaxation and rotation ($\mu$SR) measurements. Resistivity and magnetization unveil the onset of bulk superconductivity with $T_{\bf c}\sim$ 3.9~K, while TF$\mu$SR measurements show that the temperature dependence of the superfluid density is well described by a nodal twogap $s$+$d$wave order parameter model. In addition, ZF muon relaxation rate increases with decreasing temperature below 4.6 K, indicating the presence of weak spin fluctuations. These observations suggest an unconventional multiband nature of the superconductivity possibly arising from the distinct $d$bands of V and Hf ions with spin fluctuations playing an important role. To better understand these findings, we carry out firstprinciples electronicstructure calculations, further highlighting that the Fermi surface consists of multiple disconnected sheets with very different orbital weights and spinorbit coupling, bridging the way for a nodal multiband superconductivity scenario. In this vein, therefore, HfV$_2$Ga$_4$family stands out as an open avenue to novel unexplored unconventional superconducting compounds, such as ScV$_2$Ga$_4$ and ZrV$_2$Ga$_4$, and other many rare earths based materials.
A. Bhattacharyya, P. P. Ferreira, F. B. Santos, D. T. Adroja, J. S. Lord, L. E. Correa, A. J. S. Machado, A. L. R. Manesco, and L T. F. Eleno 
Hybrid kernel polynomial method [+]
The kernel polynomial method allows to sample overall spectral properties of a quantum system, while sparse diagonalization provides accurate information about a few important states. We present a method combining these two approaches without loss of performance or accuracy. We apply this hybrid kernel polynomial method to improve the computation of thermodynamic quantities and the construction of perturbative effective models, in a regime where neither of the methods is sufficient on its own. We demonstrate the efficiency of our approach on three examples: the calculation of supercurrent and inductance in a Josephson junction, the interaction of spin qubits defined in a two dimensional electron gas, and the calculation of the effective band structure in a realistic model of a semiconductor nanowire.
Muhammad Irfan, Sathish R. Kuppuswamy, Daniel Varjas, Pablo M. PerezPiskunow, Rafal Skolasinski, Michael Wimmer, and Anton R. AkhmerovarXiv:1909.09649 [pdf], (unpublished). 
Next steps of quantum transport in Majorana nanowire devices [+]
Majorana zero modes are localized quasiparticles that obey nonAbelian exchange statistics. Braiding Majorana zero modes forms the basis of topologically protected quantum operations which could in principle significantly reduce qubit decoherence and gate control errors in the device level. Therefore, searching for Majorana zero modes in various solid state systems is a major topic in condensed matter physics and quantum computer science. Since the first experimental signature observed in hybrid superconductorsemiconductor nanowire devices, this field has witnessed a dramatic expansion in material science, transport experiments and theory. While making the first topological qubit based on these Majorana nanowires is currently an ongoing effort, several related important transport experiments are still being pursued in the near term. These will not only serve as intermediate steps but also show Majorana physics in a more fundamental aspect. In this perspective, we summarize these key Majorana experiments and the potential challenges.
Hao Zhang, Dong E. Liu, Michael Wimmer, and Leo P. Kouwenhoven 
Supercurrentinduced Majorana bound states in a planar geometry [+]
We propose a new setup for creating Majorana bound states in a twodimensional electron gas Josephson junction. Our proposal relies exclusively on a supercurrent parallel to the junction as a mechanism of breaking timereversal symmetry. We show that combined with spinorbit coupling, supercurrents induce a Zeemanlike spin splitting. Further, we identify a new conserved quantitychargemomentum paritythat prevents the opening of the topological gap by the supercurrent in a straight Josephson junction. We propose breaking this conservation law by adding a third superconductor, introducing a periodic potential, or making the junction zigzagshaped. By comparing the topological phase diagrams and practical limitations of these systems we identify the zigzagshaped junction as the most promising option.
André Melo, Sebastian Rubbert, and Anton R. Akhmerov 
Computation of topological invariants of disordered materials using the kernel polynomial method [+]
We present an algorithm to determine topological invariants of inhomogeneous systems, such as alloys, disordered crystals, or amorphous systems. Our algorithm allows for efficient analysis of threedimensional samples with more than $10^7$ degrees of freedom, two orders of magnitude above the previous best. This performance gain is due to a localized approximation of the band projector based on the kernel polynomial method combined with the stochastic trace approximation. Our method makes it possible to study large samples and complex compounds, where disorder plays a central role, and provides a better resolution of disorderdriven phase transitions. As a case study we apply this approach to Pb$_{1x}$Sn$_{x}$Te and related alloys, and obtain the topological phase diagram of this family of threedimensional mirror Chern insulators.
Daniel Varjas, Michel Fruchart, Anton R. Akhmerov, and Pablo PerezPiskunow 
Topological phases without crystalline counterparts [+]
We construct a higherorder topological phase protected by a point group symmetry that is impossible in any crystalline system. The tightbinding model describes a superconductor on a quasicrystalline AmmannBeenker tiling which hosts localized Majorana zero modes at the corners of an octagonal sample. The Majorana modes are protected by particlehole symmetry and by the combination of an 8fold rotation and inplane 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.
Daniel Varjas, Alexander Lau, Kim Pöyhönen, Anton R. Akhmerov, Dmitry I. Pikulin, and Ion Cosma Fulga 
Enhanced proximity effect in zigzagshaped Majorana Josephson junctions [+]
High density superconductorsemiconductorsuperconductor junctions have a small induced superconducting gap due to the quasiparticle trajectories with a large momentum parallel to the junction having a very long flight time. Because a large induced gap protects Majorana modes, these long trajectories constrain Majorana devices to a low electron density. We show that a zigzagshaped geometry eliminates these trajectories, allowing the robust creation of Majorana states with both the induced gap $E_\textrm{gap}$ and the Majorana size $\xi_\textrm{M}$ improved by more than an order of magnitude for realistic parameters. In addition to the improved robustness of Majoranas, this new zigzag geometry is insensitive to the geometric details and the device tuning.
Tom Laeven, Bas Nijholt, Michael Wimmer, and Anton R. Akhmerov 
The influence of lattice termination on the edge states of the quantum spin Hall insulator monolayer $1T'$WTe$_2$ [+]
We study the influence of sample termination on the electronic properties of the novel quantum spin Hall insulator monolayer $1T'$WTe$_2$. For this purpose, we construct an accurate, minimal 4orbital tightbinding model with spinorbit coupling by employing a combination of densityfunctional theory calculations, symmetry considerations, and fitting to experimental data. Based on this model, we compute energy bands and 2terminal conductance spectra for various ribbon geometries with different terminations, with and without magnetic field. Because of the strong electronhole 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 ingap 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.
Alexander Lau, Rajyavardhan Ray, Daniel Varjas, and Anton Akhmerov 
Supercurrent carried by nonequlibrium quasiparticles in a multiterminal Josephson junction [+]
We theoretically study coherent multiple Andreev reflections in a biased threeterminal 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 form the Cooper pair transfer processes induced by nonlocal Andreev reflections of the quasiparticles originating from the superconducting leads. We identify supercurrentenhanced lines in the current and conductance maps of the recent measurement [Y. Cohen, et al., arXiv:1606.08436 (2016)] on a nanowire Josephson junction and show that the magnitude of the phasedependent current components is proportional to the junction transparency with the power corresponding to the component order.
M. P. Nowak, M. Wimmer, and A. R. Akhmerov 
Novel topological insulators from crystalline symmetries [+]
We discuss recent advances in the study of topological insulators protected by spatial symmetries by reviewing three representative, theoretical examples. In three dimensions, these states of matter are generally characterized by the presence of gapless boundary states at surfaces that respect the protecting spatial symmetry. We discuss the appearance of these topological states both in crystals with negligible spinorbit coupling and a fourfold rotational symmetry, as well as in mirrorsymmetric crystals with sizable spinorbit interaction characterized by the socalled mirror Chern number. Finally, we also discuss similar topological crystalline states in onedimensional insulators, such as nanowires or atomic chains, with mirror symmetry. There, the prime physical consequence of the nontrivial topology is the presence of quantized end charges.
Alexander Lau and Carmine Ortix 
Geometric focusing of supercurrent in hourglassshaped ballistic Josephson junctions [+]
The response of superconductornormalmetalsuperconductor junctions to magnetic field is complicated and nonuniversal because all trajectories contributing to supercurrent have a different effective area, and therefore acquire arbitrary magnetic phases. We design an hourglassshaped Josephson junction where due to the junction symmetry the magnetic phase of every trajectory is approximately equal. By doing so we are able to increase a critical field of the Josephson junction to many flux quanta per junction area. We then analyse how breaking the symmetry condition increases the sensitivity of the junction, and show that our device allows to detect supercurrent carried by ballistic trajectories of Andreev quasiparticles.
Muhammad Irfan and Anton R. AkhmerovarXiv:1810.04588 [pdf], (unpublished). 
A unified numerical approach to semiconductorsuperconductor heterostructures [+]
We develop a unified numerical approach for modeling semiconductorsuperconductor heterostructures. Our approach takes into account on equal footing important key ingredients: proximityinduced superconductivity, orbital and Zeeman effect of an applied magnetic field, spinorbit coupling as well as the electrostatic environment. As a model system, we consider indium arsenide (InAs) nanowires with epitaxial aluminum (Al) shell and demonstrate qualitative agreement of the obtained results with the existing experimental data. Finally, we characterize the topological superconducting phase emerging in a finite magnetic field and calculate the corresponding topological phase diagram.
Georg W. Winkler, Andrey E. Antipov, Bernard van Heck, Alexey A. Soluyanov, Leonid I. Glazman, Michael Wimmer, and Roman M. Lutchyn 
How to braid mobile with immobile nonAbelian anyons in a topological superconductor [+]
Majorana zeromodes 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 nonAbelian braiding statistics, but when they are immobile one cannot demonstrate this by exchanging them in real space and indirect methods are needed. As a realspace 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 zeromode in the bulk is detected by the charge produced upon their fusion at a second Josephson junction.
C. W. J. Beenakker, P. Baireuther, Y. Herasymenko, I. Adagideli, and A. R. Akhmerov 
SpinOrbit Protection of Induced Superconductivity in Majorana Nanowires [+]
Spinorbit interaction (SOI), a relativistic effect linking the motion of an electron (orbit) with its magnetic moment (spin), is an essential ingredient for various realisations of topological superconductivity, which host Majorana zeromodes, the building blocks of topological quantum computation. The prime platform for topological quantum computation is based on a semiconductor nanowire coupled to a conventional superconductor, the Majorana nanowire, in which SOI plays a key role by protecting the superconducting energy gap. Despite significant progress towards topological quantum computation, direct observation of SOI in Majorana nanowires has been challenging. Here, we observe SOI in an InSb nanowire coupled to a NbTiN superconductor. The magnetic field resilience of our superconductor allows us to track the evolution of the induced superconducting gap in a large range of magnetic field strengths and orientations, clearly revealing the presence of SOI. Numerical calculations of our devices confirm our conclusions and indicate a SOI strength of 0.15  0.35 eV\AA, sufficient to create Majorana zeromodes. We find that the direction of the spinorbit field is strongly affected by the geometry of the superconductor and can be tuned by electrostatic gating. Our study provides an important guideline to optimise Majorana circuits.
Jouri D. S. Bommer, Hao Zhang, Önder Gül, Bas Nijholt, Michael Wimmer, Filipp N. Rybakov, Julien Garaud, Donjan Rodic, Egor Babaev, Matthias Troyer, Diana Car, Sébastien R. Plissard, Erik P. A. M. Bakkers, Kenji Watanabe, Takashi Taniguchi, and Leo P. Kouwenhoven 
Qsymm: Algorithmic symmetry finding and symmetric Hamiltonian generation [+]
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 singlebody 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 noninteracting 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.
Daniel Varjas, Tomas O. Rosdahl, and Anton R. Akhmerov 
A New Platform for Nodal Topological Superconductors in Monolayer Molybdenum Dichalcogenides [+]
We propose a new platform to realize nodal topological superconductors in a superconducting monolayer of MoX$_2$ (X$=$S, Se, Te) using an inplane magnetic field. The bulk nodal points appear where the spin splitting due to spinorbit coupling vanishes near the $\pm {\bf K}$ valleys of the Brillouin zone, and are six or twelve per valley in total. In the nodal topological superconducting phase, the nodal points are connected by flat bands of zeroenergy Andreev edge states. These flat bands, which are protected by mirror symmetry in the MoX$_2$ plane, are present for all latticetermination boundaries except zigzag.
Lin Wang, Tomas Orn Rosdahl, and Doru Sticlet 
Reproducing topological properties with quasiMajorana states [+]
Andreev bound states in hybrid superconductorsemiconductor devices can have nearzero energy in the topologically trivial regime as long as the confinement potential is sufficiently smooth. These quasiMajorana states show zerobias conductance features in a topologically trivial phase, thereby mimicking spatially separated topological Majorana states. We show that in addition to the suppressed coupling between the quasiMajorana states, also the coupling of these states across a tunnel barrier to the outside is exponentially different. As a consequence, quasiMajorana states mimic most of the proposed Majorana signatures: quantized zerobias 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 quasiMajorana states for braiding. Therefore, while the appearance of quasiMajorana states complicates the observation of topological Majorana states, it opens an alternative route towards braiding of nonAbelian anyons and topological quantum computation.
A. Vuik, B. Nijholt, A. R. Akhmerov, and M. Wimmer 
Spinorbit interaction and induced superconductivity in an onedimensional hole gas [+]
Low dimensional semiconducting structures with strong spinorbit interaction (SOI) and induced superconductivity attracted much interest in the search for topological superconductors. Both the strong SOI and hard superconducting gap are directly related to the topological protection of the predicted Majorana bound states. Here we explore the onedimensional hole gas in germanium silicon (GeSi) coreshell nanowires (NWs) as a new material candidate for creating a topological superconductor. Fitting multiple Andreev reflection measurements shows that the NW has two transport channels only, underlining its onedimensionality. Furthermore, we find anisotropy of the Lande gfactor, that, combined with band structure calculations, provides us qualitative evidence for direct Rashba SOI and a strong orbital effect of the magnetic field. Finally, a hard superconducting gap is found in the tunneling regime, and the open regime, where we use the Kondo peak as a new tool to gauge the quality of the superconducting gap.
F. K. de Vries, J. Shen, R. J. Skolasinski, M. P. Nowak, D. Varjas, L. Wang, M. Wimmer, J. Ridderbos, F. A. Zwanenburg, A. Li, S. Koelling, M. A. Verheijen, E. P. A. M. Bakkers, and L. P. Kouwenhoven 
Dissipationenabled fractional Josephson effect [+]
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 timereversal 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 twoparticle 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.
Doru Sticlet, Jay D. Sau, and Anton Akhmerov 
Topological semimetals in the SnTe material class: Nodal lines and Weyl points [+]
We theoretically show that IVVI semiconducting compounds with lowtemperature rhombohedral crystal structure represent a new potential platform for topological semimetals. By means of minimal $\mathbf{k}\cdot\mathbf{p}$ models we find that the twostep structural symmetry reduction of the hightemperature rocksalt crystal structure, comprising a rhombohedral distortion along the [111] direction followed by a relative shift of the cation and anion sublattices, gives rise to topologically protected Weyl semimetal and nodal line semimetal phases. We derive general expressions for the nodal features and apply our results to SnTe showing explicitly how Weyl points and nodal lines emerge in this system. Experimentally, the topological semimetals could potentially be realized in the lowtemperature ferroelectric phase of SnTe, GeTe and related alloys.
Alexander Lau and Carmine Ortix 
Breakdown of the law of reflection at a disordered graphene edge [+]
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 a chargeneutral disordered graphene boundary results in a 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.
E. Walter, T. Ö. Rosdahl, A. R. Akhmerov, and F. Hassler 
Majoranabased fermionic quantum computation [+]
Because Majorana zero modes store quantum information nonlocally, they are protected from noise, and have been proposed as a building block for a quantum computer. We show how to use the same protection from noise to implement universal fermionic quantum computation. Our architecture requires only two Majoranas to encode a fermionic quantum degree of freedom, compared to alternative implementations which require a minimum of four Majoranas for a spin quantum degree of freedom. The fermionic degrees of freedom support both unitary coupled cluster variational quantum eigensolver and quantum phase estimation algorithms, proposed for quantum chemistry simulations. Because we avoid the JordanWigner transformation, our scheme has a lower overhead for implementing both of these algorithms, and the simulation of Trotterized Hubbard Hamiltonian in $\mathcal{O}(1)$ time per unitary step. We finally demonstrate magic state distillation in our fermionic architecture, giving a universal set of topologically protected fermionic quantum gates.
T. E. O'Brien, P. Rożek, and A. R. Akhmerov 
A general algorithm for computing bound states in infinite tightbinding systems [+]
We propose a robust and efficient algorithm for computing bound states of infinite tightbinding systems that are made up of a finite scattering region connected to semiinfinite 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.
M. Istas, C. Groth, A. R. Akhmerov, M. Wimmer, and X. waintal 
Engineering Hybrid Epitaxial InAsSb/Al Nanowire Materials for Stronger Topological Protection [+]
The combination of strong spinorbit coupling, large $g$factors, and the coupling to a superconductor can be used to create a topologically protected state in a semiconductor nanowire. Here we report on growth and characterization of hybrid epitaxial InAsSb/Al nanowires, with varying composition and crystal structure. We find the strongest spinorbit interaction at intermediate compositions in zincblende InAs$_{1x}$Sb$_{x}$ nanowires, exceeding that of both InAs and InSb materials, confirming recent theoretical studies \cite{winkler2016topological}. We show that the epitaxial InAsSb/Al interfaces allows for a hard induced superconducting gap and 2$e$ transport in Coulomb charging experiments, similar to experiments on InAs/Al and InSb/Al materials, and find measurements consistent with topological phase transitions at low magnetic fields due to large effective $g$factors. Finally we present a method to grow pure wurtzite InAsSb nanowires which are predicted to exhibit even stronger spinorbit coupling than the zincblende structure.
Joachim E. Sestoft, Thomas Kanne, Aske Nørskov Gejl, Merlin von Soosten, Jeremy S. Yodh, Daniel Sherman, Brian Tarasinski, Michael Wimmer, Erik Johnson, Mingtang Deng, Jesper Nygård, Thomas Sand Jespersen, Charles M. Marcus, and Peter Krogstrup 
Transient and Sharvin resistances of Luttinger liquids [+]
Although the intrinsic conductance of an interacting onedimensional system is renormalized by the electronelectron correlations, it has been known for some time that this renormalization is washed out by the presence of the (noninteracting) electrodes to which the wire is connected. Here, we study the transient conductance of such a wire: a finite voltage bias is suddenly applied across the wire and we measure the current before it has enough time to reach its stationary value. These calculations allow us to extract the Sharvin (contact) resistance of Luttinger and Fermi liquids. In particular, we find that a perfect junction between a Fermi liquid electrode and a Luttinger liquid electrode is characterized by a contact resistance that consists of half the quantum of conductance in series with half the intrinsic resistance of an infinite Luttinger liquid. These results were obtained using two different methods: a dynamical HartreeFock approach and a selfconsistent Boltzmann approach. Although these methods are formally approximate we find a perfect match with the exact results of Luttinger/Fermi liquid theory.
Thomas Kloss, Joseph Weston, and Xavier Waintal 
Robust Helical Edge Transport in Quantum Spin Hall Quantum Wells [+]
We show that burying of the Dirac point in semiconductorbased quantumspinHall systems can generate unexpected robustness of edge states to magnetic fields. A detailed ${\bf k\cdot p}$ bandstructure analysis reveals that InAs/GaSb and HgTe/CdTe quantum wells exhibit such buried Dirac points. By simulating transport in a disordered system described within an effective model, we further demonstrate that buried Dirac points yield nearly quantized edge conduction out to large magnetic fields, consistent with recent experiments.
Rafal Skolasinski, Dmitry I. Pikulin, Jason Alicea, and Michael Wimmer 
$h/e$ superconducting quantum interference through trivial edge states in InAs [+]
Josephson junctions defined in strong spin orbit semiconductors are highly interesting for the search for topological systems. However, next to topological edge states that emerge in a sufficient magnetic field, trivial edge states can also occur. We study the trivial edge states with superconducting quantum interference measurements on nontopological InAs Josephson junctions. We observe a SQUID pattern, an indication of superconducting edge transport. Also, a remarkable $h/e$ SQUID signal is observed that, as we find, stems from crossed Andreev states.
Folkert K. de Vries, Tom Timmerman, Viacheslav P. Ostroukh, Jasper van Veen, Arjan J. A. Beukman, Fanming Qu, Michael Wimmer, BinhMinh Nguyen, Andrey A. Kiselev, Wei Yi, Marko Sokolich, Michael J. Manfra, Charles M. Marcus, and Leo P. Kouwenhoven 
Andreev rectifier: a nonlocal conductance signature of topological phase transitions [+]
The proximity effect in hybrid superconductorsemiconductor 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 nontopological zeroenergy 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 lowbias 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.
T. Ö. Rosdahl, A. Vuik, M. Kjaergaard, and A. R. Akhmerov 
Supercurrent interference in fewmode nanowire Josephson junctions [+]
Junctions created by coupling two superconductors via a semiconductor nanowire in the presence of high magnetic fields are the basis for detection, fusion, and braiding of Majorana bound states. We study NbTiN/InSb nanowire/NbTiN Josephson junctions and find that their critical currents in the few mode regime are strongly suppressed by magnetic field. Furthermore, the dependence of the critical current on magnetic field exhibits gatetunable nodes. Based on a realistic numerical model we conclude that the Zeeman effect induced by the magnetic field and the spinorbit interaction in the nanowire are insufficient to explain the observed evolution of the Josephson effect. We find the interference between the few occupied onedimensional modes in the nanowire to be the dominant mechanism responsible for the critical current behavior. The suppression and nonmonotonic evolution of critical currents at finite magnetic field should be taken into account when designing circuits based on Majorana bound states.
Kun Zuo, Vincent Mourik, Daniel B. Szombati, Bas Nijholt, David J. van Woerkom, Attila Geresdi, Jun Chen, Viacheslav P. Ostroukh, Anton R. Akhmerov, Sebastién R. Plissard, Diana Car, Erik P. A. M. Bakkers, Dmitry I. Pikulin, Leo P. Kouwenhoven, and Sergey M. Frolov 
Spinorbit interaction in a dual gated InAs/GaSb quantum well [+]
Spinorbit interaction is investigated in a dual gated InAs/GaSb quantum well. Using an electric field the quantum well can be tuned between a single carrier regime with exclusively electrons as carriers and a twocarriers regime where electrons and holes coexist. Spinorbit interaction in both regimes manifests itself as a beating in the Shubnikovde Haas oscillations. In the single carrier regime the linear Dresselhaus strength is characterized by $\beta =$ 28.5 meV$\AA$ and the Rashba coefficient $\alpha$ is tuned from 75 to 53 meV$\AA$ by changing the electric field. In the twocarriers regime the spin splitting shows a nonmonotonic behavior with gate voltage, which is consistent with our band structure calculations.
Arjan J. A. Beukman, Folkert K. de Vries, Jasper van Veen, Rafal Skolasinski, Michael Wimmer, Fanming Qu, David T. de Vries, BinhMinh Nguyen, Wei Yi, Andrey A. Kiselev, Marko Sokolich, Michael J. Manfra, Fabrizio Nichele, Charles M. Marcus, and Leo P. Kouwenhoven 
Orbital contributions to the electron gfactor in semiconductor nanowires [+]
Recent experiments on Majorana fermions in semiconductor nanowires [Albrecht et al., Nat. 531, 206 (2016)] revealed a surprisingly large electronic Land\'e gfactor, several times larger than the bulk value  contrary to the expectation that confinement reduces the gfactor. Here we assess the role of orbital contributions to the electron gfactor in nanowires and quantum dots. We show that an LS coupling in higher subbands leads to an enhancement of the gfactor of an order of magnitude or more for small effective mass semiconductors. We validate our theoretical finding with simulations of InAs and InSb, showing that the effect persists even if cylindrical symmetry is broken. A huge anisotropy of the enhanced gfactors under magnetic field rotation allows for a straightforward experimental test of this theory.
Georg W. Winkler, Dániel Varjas, Rafal Skolasinski, Alexey A. Soluyanov, Matthias Troyer, and Michael Wimmer 
Demonstration of an ac Josephson junction laser [+]
Superconducting electronic devices have reemerged 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 multimode superconducting cavity. A dc voltage bias to the junction provides a source of microwave photons, while the circuit's nonlinearity allows for efficient downconversion 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 onchip generation of coherent microwave photons at low temperatures.
M. C. Cassidy, A. Bruno, S. Rubbert, M. Irfan, J. Kammhuber, R. N. Schouten, A. R. Akhmerov, and L. P. Kouwenhoven 
Tailoring supercurrent confinement in graphene bilayer weak links [+]
The Josephson effect is one of the most studied macroscopic quantum phenomena in condensed matter physics and has been an essential part of the quantum technologies development over the last decades. It is already used in many applications such as magnetometry, metrology, quantum computing, detectors or electronic refrigeration. However, developing devices in which the induced superconductivity can be monitored, both spatially and in its magnitude, remains a serious challenge. In this work, we have used local gates to control confinement, amplitude and density profile of the supercurrent induced in onedimensional nanoscale constrictions, defined in bilayer graphenehexagonal boron nitride van der Waals heterostructures. The combination of resistance gate maps, outofequilibrium transport, magnetic interferometry measurements, analytical and numerical modelling enables us to explore highly tunable superconducting weak links. Our study opens the path way to design more complex superconducting circuits based on this principle such as electronic interferometers or transitionedge sensors.
Rainer Kraft, Jens Mohrmann, Renjun Du, Pranauv Balaji Selvasundaram, Muhammad Irfan, Umut Nefta Kanilmaz, Fan Wu, Detlef Beckmann, Hilbert von Löhneysen, Ralph Krupke, Anton Akhmerov, Igor Gornyi, and Romain Danneau 
Valley dependent anisotropic spin splitting in silicon quantum dots [+]
Spin qubits hosted in silicon (Si) quantum dots (QD) are attractive due to their exceptionally long coherence times and compatibility with the silicon transistor platform. To achieve electrical control of spins for qubit scalability, recent experiments have utilized gradient magnetic fields from integrated micromagnets to produce an extrinsic coupling between spin and charge, thereby electrically driving electron spin resonance (ESR). However, spins in silicon QDs experience a complex interplay between spin, charge, and valley degrees of freedom, influenced by the atomic scale details of the confining interface. Here, we report experimental observation of a valley dependent anisotropic spin splitting in a Si QD with an integrated micromagnet and an external magnetic field. We show by atomistic calculations that the spinorbit interaction (SOI), which is often ignored in bulk silicon, plays a major role in the measured anisotropy. Moreover, inhomogeneities such as interface steps strongly affect the spin splittings and their valley dependence. This atomicscale understanding of the intrinsic and extrinsic factors controlling the valley dependent spin properties is a key requirement for successful manipulation of quantum information in Si QDs.
Rifat Ferdous, Erika Kawakami, Pasquale Scarlino, Michał P. Nowak, D. R. Ward, D. E. Savage, M. G. Lagally, S. N. Coppersmith, Mark Friesen, Mark A. Eriksson, Lieven M. K. Vandersypen, and Rajib Rahman 
Hard superconducting gap in InSb nanowires [+]
Topological superconductivity is a state of matter that can host Majorana modes, the building blocks of a topological quantum computer. Many experimental platforms predicted to show such a topological state rely on proximityinduced superconductivity. However, accessing the topological properties requires an induced hard superconducting gap, which is challenging to achieve for most material systems. We have systematically studied how the interface between an InSb semiconductor nanowire and a NbTiN superconductor affects the induced superconducting properties. Step by step, we improve the homogeneity of the interface while ensuring a barrierfree electrical contact to the superconductor, and obtain a hard gap in the InSb nanowire. The magnetic field stability of NbTiN allows the InSb nanowire to maintain a hard gap and a supercurrent in the presence of magnetic fields (~ 0.5 Tesla), a requirement for topological superconductivity in onedimensional systems. Our study provides a guideline to induce superconductivity in various experimental platforms such as semiconductor nanowires, two dimensional electron gases and topological insulators, and holds relevance for topological superconductivity and quantum computation.
Önder Gül, Hao Zhang, Folkert K. de Vries, Jasper van Veen, Kun Zuo, Vincent Mourik, Sonia ConesaBoj, Michał P. Nowak, David J. van Woerkom, Marina QuinteroPérez, Maja C. Cassidy, Attila Geresdi, Sebastian Koelling, Diana Car, Sébastien R. Plissard, Erik P. A. M. Bakkers, and Leo P. Kouwenhoven 
Conductance through a helical state in an InSb nanowire [+]
The motion of an electron and its spin are generally not coupled. However in a one dimensional material with strong spinorbit interaction (SOI) a helical state may emerge at finite magnetic fields, where electrons of opposite spin will have opposite momentum. The existence of this helical state has applications for spin filtering and Cooper pair splitter devices and is an essential ingredient for realizing topologically protected quantum computing using Majorana zero modes. Here we report electrical conductance measurements of a quantum point contact (QPC) formed in an indium antimonide nanowire as a function of magnetic field. At magnetic fields exceeding 3T, the $2e^2/h$ plateau shows a reentrant conductance feature towards $1e^2/h$ which increases linearly in width with magnetic field before enveloping the $1e^2/h$ plateau. Rotating the external magnetic field either parallel or perpendicular to the spinorbit field allows us to clearly attribute this experimental signature to SOI. We compare our observations with a model of a QPC incorporating SOI and extract a spinorbit energy of ~6.5meV, which is significantly stronger than the SO energy obtained by other methods.
Jakob Kammhuber, Maja C Cassidy, Fei Pei, Michal P Nowak, Adriaan Vuik, Diana Car, Sèbastien R Plissard, Erik P A M Bakkers, Michael Wimmer, and Leo P Kouwenhoven 
InSb Nanowires with BuiltIn GaxIn1xSb Tunnel Barriers for Majorana Devices [+]
Majorana zero modes (MZMs), prime candidates for topological quantum bits, are detected as zero bias conductance peaks (ZBPs) in tunneling spectroscopy measurements. Implementation of a narrow and high tunnel barrier in the next generation of Majorana devices can help to achieve the theoretically predicted quantized height of the ZBP. We propose a materialoriented approach to engineer a sharp and narrow tunnel barrier by synthesizing a thin axial segment of GaxIn1xSb within an InSb nanowire. By varying the precursor molar fraction and the growth time, we accurately control the composition and the length of the barriers. The height and the width of the GaxIn1xSb tunnel barrier are extracted from the WentzelKramersBrillouin (WKB)fits to the experimental IV traces.
Diana Car, Sonia ConesaBoj, Hao Zhang, Roy L. M. Op het Veld, Michiel W. A. de Moor, Elham M. T. Fadaly, Önder Gül, Sebastian Kölling, Sebastien R. Plissard, Vigdis Toresen, Michael T. Wimmer, Kenji Watanabe, Takashi Taniguchi, Leo P. Kouwenhoven, and Erik P. A. M. Bakkers 
Robustness of Majorana bound states in the short junction limit [+]
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 semiconductorsuperconductor interface transparency.
Doru Sticlet, Bas Nijholt, and Anton Akhmerov 
Quantized conductance and large gfactor anisotropy in InSb quantum point contacts [+]
Due to a strong spinorbit interaction and a large Land\'e gfactor, InSb plays an important role in research on Majorana fermions. To further explore novel properties of Majorana fermions, hybrid devices based on quantum wells are conceived as an alternative approach to nanowires. In this work, we report a pronounced conductance quantization of quantum point contact devices in InSb/InAlSb quantum wells. Using a rotating magnetic field, we observe a large inplane (g1=26) and outofplane (g1=52) gfactor anisotropy. Additionally, we investigate crossings of subbands with opposite spins and extract the electron effective mass from magnetic depopulation of onedimensional subbands.
Fanming Qu, Jasper van Veen, Folkert K. de Vries, Arjan J. A. Beukman, Michael Wimmer, Wei Yi, Andrey A. Kiselev, BinhMinh Nguyen, Marko Sokolich, Michael J. Manfra, Fabrizio Nichele, Charles M. Marcus, and Leo P. Kouwenhoven 
Dynamical piezoelectric and magnetopiezoelectric effects in polar metals from Berry phases and orbital moments [+]
The polarization of a material and its response to applied electric and magnetic fields are key solidstate properties with a long history in insulators, although a satisfactory theory required new concepts such as Berryphase gauge fields. In metals, quantities such as static polarization and magnetoelectric $\theta$term cease to be welldefined. In polar metals there can be analogous dynamical current responses, which we study in a common theoretical framework. We find that current responses to dynamical strain in polar metals depend on both the first and second Chern forms, related to polarization and magnetoelectricity in insulators, as well as the orbital magnetization on the Fermi surface. We provide realistic estimates that predict that the latter contribution will dominate and investigate the feasibility of experimental detection of this effect.
Daniel Varjas, Adolfo G. Grushin, Roni Ilan, and Joel E. Moore 
Transparent SemiconductorSuperconductor Interface and Induced Gap in an Epitaxial Heterostructure Josephson Junction [+]
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 semiconductorsuperconductor interface, indicating nearunity 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.
M. Kjaergaard, H. J. Suominen, M. P. Nowak, A. R. Akhmerov, J. Shabani, C. J. Palmstrøm, F. Nichele, and C. M. Marcus 
Twodimensional Josephson vortex lattice and anomalously slow decay of the Fraunhofer oscillations in a ballistic SNS junction with a warped Fermi surface [+]
$ $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 twodimensional (2D) ballistic superconductornormalmetalsuperconductor (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.
V. P. Ostroukh, B. Baxevanis, A. R. Akhmerov, and C. W. J. Beenakker 
Giant spinorbit splitting in inverted InAs/GaSb double quantum wells [+]
Transport measurements in inverted InAs/GaSb quantum wells reveal a giant spinorbit splitting of the energy bands close to the hybridization gap. The splitting results from the interplay of electronhole mixing and spinorbit coupling, and can exceed the hybridization gap. We experimentally investigate the band splitting as a function of top gate voltage for both electronlike and holelike states. Unlike conventional, noninverted twodimensional electron gases, the Fermi energy in InAs/GaSb can cross a single spinresolved band, resulting in full spinorbit polarization. In the fully polarized regime we observe exotic transport phenomena such as quantum Hall plateaus evolving in $e^2/h$ steps and a nontrivial Berry phase.
Fabrizio Nichele, Morten Kjaergaard, Henri J. Suominen, Rafal Skolasinski, Michael Wimmer, BinhMinh Nguyen, Andrey A. Kiselev, Wei Yi, Marko Sokolich, Michael J. Manfra, Fanming Qu, Arjan J. A. Beukman, Leo P. Kouwenhoven, and Charles M. Marcus 
Ballistic Majorana nanowire devices [+]
Majorana modes are zeroenergy excitations of a topological superconductor that exhibit nonAbelian statistics. Following proposals for their detection in a semiconductor nanowire coupled to an swave superconductor, several tunneling experiments reported characteristic Majorana signatures. Reducing disorder has been a prime challenge for these experiments because disorder can mimic the zeroenergy signatures of Majoranas, and renders the topological properties inaccessible. Here, we show characteristic Majorana signatures in InSb nanowire devices exhibiting clear ballistic transport properties. Application of a magnetic field and spatial control of carrier density using local gates generates a zero bias peak that is rigid over a large region in the parameter space of chemical potential, Zeeman energy, and tunnel barrier potential. The reduction of disorder allows us to resolve separate regions in the parameter space with and without a zero bias peak, indicating topologically distinct phases. These observations are consistent with the Majorana theory in a ballistic system, and exclude for the first time the known alternative explanations that invoke disorder or a nonuniform chemical potential.
Önder Gül, Hao Zhang, Jouri D. S. Bommer, Michiel W. A. de Moor, Diana Car, Sébastien R. Plissard, Erik P. A. M. Bakkers, Attila Geresdi, Kenji Watanabe, Takashi Taniguchi, and Leo P. Kouwenhoven 
Conductance Quantization at zero magnetic field in InSb nanowires [+]
Ballistic electron transport is a key requirement for existence of a topological phase transition in proximitized InSb nanowires. However, measurements of quantized conductance as direct evidence of ballistic transport have so far been obscured due to the increased chance of backscattering in one dimensional nanowires. We show that by improving the nanowiremetal interface as well as the dielectric environment we can consistently achieve conductance quantization at zero magnetic field. Additionally, studying the subband evolution in a rotating magnetic field reveals an orbital degeneracy between the second and third subbands for perpendicular fields above 1T.
Jakob Kammhuber, Maja C. Cassidy, Hao Zhang, Önder Gül, Fei Pei, Michiel W. A. de Moor, Bas Nijholt, Kenji Watanabe, Takashi Taniguchi, Diana Car, Sebastien R. Plissard, Erik P. A. M. Bakkers, and Leo P. Kouwenhoven 
Quantized conductance doubling and hard gap in a twodimensional semiconductorsuperconductor heterostructure [+]
The prospect of coupling a twodimensional (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 spinorbit interaction to an swave 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 softgap problem in 2D superconductorsemiconductor hybrid systems. With the QPC in the open regime, we observe a first conductance plateau at 4e^2/h, as expected theoretically for a normalQPCsuperconductor structure. The realization of a hardgap semiconductorsuperconductor system that is amenable to topdown processing provides a means of fabricating scalable multicomponent hybrid systems for applications in lowdissipation electronics and topological quantum information.
M. Kjaergaard, F. Nichele, H. J. Suominen, M. P. Nowak, M. Wimmer, A. R. Akhmerov, J. A. Folk, K. Flensberg, J. Shabani, C. J. Palmstrom, and C. M. Marcus 
Conductance oscillations of coreshell nanowires in transversal magnetic fields [+]
We analyze theoretically electronic transport through a coreshell nanowire in the presence of a transversal magnetic field. We calculate the conductance for a variable coupling between the nanowire and the attached leads and show how the snaking states, which are lowenergy states localized along the lines of vanishing radial component of the magnetic field, manifest their existence. In the strong coupling regime they induce AharonovBohmlike conductance oscillations, which, by decreasing the coupling to the leads, evolve into well resolved peaks. These results show that the formation of snaking states in the nanowire affects magnetoconductance measurements irrespective of the strength of the contacts with the leads. We analyze theoretically electronic transport through a coreshell nanowire in the presence of a transversal magnetic field. We calculate the conductance for a variable coupling between the nanowire and the attached leads and show how the snaking states, which are lowenergy states localized along the lines of vanishing radial component of the magnetic field, manifest their existence. In the strong coupling regime they induce AharonovBohmlike conductance oscillations, which, by decreasing the coupling to the leads, evolve into well resolved peaks. The flux periodic oscillations arise due to interference of the snaking states, which is a consequence of backscattering at either the contacts with leads or magnetic/potential barriers in the wire.
Andrei Manolescu, George Alexandru Nemnes, Anna Sitek, Tomas Orn Rosdahl, Sigurdur Ingi Erlingsson, and Vidar Gudmundsson 
Detecting Majorana nonlocality using strongly coupled Majorana bound states [+]
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 Coulombmediated 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 AharonovBohm 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.
S. Rubbert and A. R. Akhmerov 
Effects of the electrostatic environment on the Majorana nanowire devices [+]
One of the promising platforms for creating Majorana bound states is a hybrid nanostructure consisting of a semiconducting nanowire covered by a superconductor. We analyze the previously disregarded role of electrostatic interaction in these devices. Our main result is that Coulomb interaction causes the chemical potential to respond to an applied magnetic field, while spinorbit interaction and screening by the superconducting lead suppress this response. Consequently, the electrostatic environment influences two properties of Majorana devices: the shape of the topological phase boundary and the oscillations of the Majorana splitting energy. We demonstrate that both properties show a nonuniversal behavior, and depend on the details of the electrostatic environment. We show that when the wire only contains a single electron mode, the experimentally accessible inverse selfcapacitance of this mode fully captures the interplay between electrostatics and Zeeman field. This offers a way to compare theoretical predictions with experiments.
A. Vuik, D. Eeltink, A. R. Akhmerov, and M. Wimmer 
An attractive critical point from weak antilocalization on fractals [+]
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 twodimensional systems weak antilocalization effects lead to a metalinsulator transition. This transition is characterized by a repulsive critical point above which the system becomes metallic. Fractals possess a noninteger 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 $0 < \varepsilon < \beta_\mathrm{max} \approx 0.15$, 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.
Doru Sticlet and Anton Akhmerov 
Orbital effect of magnetic field on the Majorana phase diagram [+]
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.
Bas Nijholt and Anton R. Akhmerov 
Disorderinduced topological transitions in multichannel Majorana wires [+]
In this work, we investigate the effect of disorder on the topological properties of multichannel superconductor nanowires. While the standard expectation is that the spectral gap is closed and opened at transitions that change the topological index of the wire, we show that the closing and opening of a transport gap can also cause topological transitions, even in the presence of nonzero density of states across the transition. Such transport gaps induced by disorder can change the topological index, driving a topologically trivial wire into a nontrivial state or vice versa. We focus on the Rashba spinorbit coupled semiconductor nanowires in proximity to a conventional superconductor, which is an experimentally relevant system, and obtain analytical formulas for topological transitions in these wires, valid for generic realizations of disorder. Full tightbinding simulations show excellent agreement with our analytical results without any fitting parameters.
B. Pekerten, A. Teker, O. Bozat, M. Wimmer, and I. Adagideli 
Visualization of phasecoherent electron interference in a ballistic graphene Josephson junction [+]
Interference of standing waves in electromagnetic resonators forms the basis of many technologies, from telecommunications and spectroscopy to detection of gravitational waves. However, unlike the confinement of light waves in vacuum, the interference of electronic waves in solids is complicated by boundary properties of the crystal, notably leading to electron guiding by atomicscale potentials at the edges. Understanding the microscopic role of boundaries on coherent wave interference is an unresolved question due to the challenge of detecting charge flow with submicron resolution. Here we employ Fraunhofer interferometry to achieve realspace imaging of cavity modes in a graphene FabryPerot resonator, embedded between two superconductors to form a Josephson junction. By directly visualizing current flow using Fourier methods, our measurements reveal surprising redistribution of current on and off resonance. These findings provide direct evidence of separate interference conditions for edge and bulk currents and reveal the ballistic nature of guided edge states. Beyond equilibrium, our measurements show strong modulation of the multiple Andreev reflection amplitude on an off resonance, a direct measure of the gatetunable change of cavity transparency. These results demonstrate that, contrary to the common belief, electron interactions with realistic disordered edges facilitate electron wave interference and ballistic transport.
M. T. Allen, O. Shtanko, I. C. Fulga, J. I. J. Wang, D. Nurgaliev, K. Watanabe, T. Taniguchi, A. R. Akhmerov, P. JarilloHerrero, L. S. Levitov, and A. Yacoby 
Spatially resolved edge currents and guidedwave electronic states in graphene [+]
A farreaching 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 realspace 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 guidedwave states, confined to the edge by band bending and transmitted as plane waves. As a direct analog of refractionbased confinement of light in optical fibers, such nonclassical states afford new means for information transduction and processing at the nanoscale.
Monica T. Allen, Oles Shtanko, Ion Cosma Fulga, Anton Akhmerov, Kenji Watanabi, Takashi Taniguchi, Pablo JarilloHerrero, Leonid S. Levitov, and Amir Yacoby 
Realization of microwave quantum circuits using hybrid superconductingsemiconducting nanowire Josephson elements [+]
We report the realization of quantum microwave circuits using hybrid superconductorsemiconductor Josephson elements comprised of InAs nanowires contacted by NbTiN. Capacitivelyshunted single elements behave as transmon qubits with electrically tunable transition frequencies. Twoelement circuits also exhibit transmonlike behavior near zero applied flux, but behave as flux qubits at half the flux quantum, where nonsinusoidal currentphase relations in the elements produce a doublewell Josephson potential. These hybrid Josephson elements are promising for applications requiring microwave superconducting circuits operating in magnetic field.
G. de Lange, B. van Heck, A. Bruno, D. J. van Woerkom, A. Geresdi, S. R. Plissard, E. P. A. M. Bakkers, A. R. Akhmerov, and L. DiCarlo 
Electric and Magnetic Tuning Between the Trivial and Topological Phases in InAs/GaSb Double Quantum Wells [+]
Among the theoretically predicted twodimensional topological insulators, InAs/GaSb double quantum wells (DQWs) have a unique doublelayered structure with electron and hole gases separated in two layers, which enables tuning of the band alignment via electric and magnetic fields. However, the rich trivialtopological phase diagram has yet to be experimentally explored. We present an in situ and continuous tuning between the trivial and topological insulating phases in InAs/GaSb DQWs through electrical dualgating. Furthermore, we show that an inplane magnetic field shifts the electron and hole bands relatively to each other in momentum space, functioning as a powerful tool to discriminate between the topologically distinct states.
Fanming Qu, Arjan J. A. Beukman, Stevan NadjPerge, Michael Wimmer, BinhMinh Nguyen, Wei Yi, Jacob Thorp, Marko Sokolich, Andrey A. Kiselev, Michael J. Manfra, Charles M. Marcus, and Leo P. Kouwenhoven 
Ballistic Josephson junctions in edgecontacted graphene [+]
Hybrid graphenesuperconductor 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 graphenesuperconductor interfaces, usually combined with small critical magnetic fields of the superconducting electrodes. Here we report graphene based Josephson junctions with onedimensional 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 FabryP\'{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.
Victor E. Calado, Srijit Goswami, Gaurav Nanda, Mathias Diez, Anton R. Akhmerov, Kenji Watanabe, Takashi Taniguchi, Teun M. Klapwijk, and Lieven M. K. Vandersypen 
Spinorbit interaction in InSb nanowires [+]
We use magnetoconductance measurements in dualgated InSb nanowire devices together with a theoretical analysis of weak antilocalization to accurately extract spinorbit strength. In particular, we show that magnetoconductance in our threedimensional wires is very different compared to wires in twodimensional electron gases. We obtain a large Rashba spinorbit strength of $0.5 1\,\text{eV\r{A}}$ corresponding to a spinorbit energy of $0.251\,\text{meV}$. These values underline the potential of InSb nanowires in the study of Majorana fermions in hybrid semiconductorsuperconductor devices.
I. van Weperen, B. Tarasinski, D. Eeltink, V. S. Pribiag, S. R. Plissard, E. P. A. M. Bakkers, L. P. Kouwenhoven, and M. Wimmer 
Single fermion manipulation via superconducting phase differences in multiterminal Josephson junctions [+]
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 twoterminal 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 zeroenergy Andreev level crossing may never appear. Our main result is that these limitations can be completely avoided by using multiterminal 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 spinorbit material such as InAs quantum wells, and can be detected using standard tools.
B. van Heck, S. Mi, and A. R. Akhmerov 
Emergence of massless Dirac fermions in graphene's Hofstadter butterfly at switches of the quantum Hall phase connectivity [+]
The fractal spectrum of magnetic minibands (Hofstadter butterfly), induced by the moir\'e super lattice of graphene on an hexagonal crystal substrate, is known to exhibit gapped Dirac cones. We show that the gap can be closed by slightly misaligning the substrate, producing a hierarchy of conical singularities (Dirac points) in the band structure at rational values Phi = (p/q)(h/e) of the magnetic flux per supercell. Each Dirac point signals a switch of the topological quantum number in the connected component of the quantum Hall phase diagram. Model calculations reveal the scale invariant conductivity sigma = 2qe^2 / pi h and Klein tunneling associated with massless Dirac fermions at these connectivity switches.
M. Diez, J. P. Dahlhaus, M. Wimmer, and C. W. J. Beenakker 
Disorder and magneticfield induced breakdown of helical edge conduction in an inverted electronhole bilayer [+]
We calculate the conductance of a twodimensional bilayer with inverted electronhole bands, to study the sensitivity of the quantum spin Hall insulator (with helical edge conduction) to the combination of electrostatic disorder and a perpendicular magnetic field. The characteristic breakdown field for helical edge conduction splits into two fields with increasing disorder, a field $B_{c}$ for the transition into a quantum Hall insulator (supporting chiral edge conduction) and a smaller field $B'_{c}$ for the transition to bulk conduction in a quasimetallic regime. The spatial separation of the inverted bands, typical for brokengap InAs/GaSb quantum wells, is essential for the magneticfield induced bulk conduction  there is no such regime in HgTe quantum wells.
D. I. Pikulin, T. Hyart, Shuo Mi, J. Tworzydło, M. Wimmer, and C. W. J. Beenakker 
Electric control of tunneling energy in graphene double dots [+]
We theoretically investigate the spectrum of a single electron double quantum dot, defined by top gates in a graphene with a substrate induced gap. We examine the effects of electric and magnetic fields on the spectrum of localized states, focusing on the tunability of the interdot coupling. We find that the substrate induced gap allows for electrostatic control, with some limitations that for a fixed interdot distance, the interdot coupling can not be made arbitrarily small due to the Klein tunneling. On the other hand, the proximity of the valence band in graphene allows for new regimes, such as an $npn$ double dot, which have no counterparts in GaAs.
Martin Raith, Christian Ertler, Peter Stano, Michael Wimmer, and Jaroslav Fabian 
Kwant: a software package for quantum transport [+]
Kwant is a Python package for numerical quantum transport calculations. It aims to be an userfriendly, universal, and highperformance toolbox for the simulation of physical systems of any dimensionality and geometry that can be described by a tightbinding model. Kwant has been designed such that the natural concepts of the theory of quantum transport (lattices, symmetries, electrodes, orbital/spin/electronhole degrees of freedom) are exposed in a simple and transparent way: Defining a new simulation setup is very close to describing the corresponding mathematical model. Kwant offers direct support for calculations of transport properties (conductance, noise, scattering matrix), dispersion relations, modes, wave functions, various Green's functions, and outofequilibrium local quantities. Other computations involving tightbinding Hamiltonians can be implemented easily thanks to its extensible and modular nature. Kwant is free software available at http://kwantproject.org/.
Christoph W. Groth, Michael Wimmer, Anton R. Akhmerov, and Xavier Waintal