Publication list

Pymablock: an algorithm and a package for quasidegenerate perturbation theory [+]
A common technique in the study of complex quantummechanical systems is to reduce the number of degrees of freedom in the Hamiltonian by using quasidegenerate perturbation theory. While the SchriefferWolff transformation achieves this and constructs an effective Hamiltonian, its scaling is suboptimal, and implementing it efficiently is both challenging and errorprone. We introduce an algorithm for constructing an equivalent effective Hamiltonian as well as a Python package, Pymablock, that implements it. Our algorithm combines an optimal asymptotic scaling with a range of other improvements. The package supports numerical and analytical calculations of any order and it is designed to be interoperable with any other packages for specifying the Hamiltonian. We demonstrate how the package handles constructing a k.p model, analyses a superconducting qubit, and computes the lowenergy spectrum of a large tightbinding model. We also compare its performance with reference calculations and demonstrate its efficiency.
Isidora Araya Day, Sebastian Miles, Hugo K. Kerstens, Daniel Varjas, and Anton R. AkhmerovarXiv:2404.03728 [pdf], (unpublished). 
Chiral adiabatic transmission protected by Fermi surface topology [+]
We demonstrate that Andreev modes that propagate along a transparent Josephson junction have a perfect transmission at the point where three junctions meet. The chirality and the number of quantized transmission channels is determined by the topology of the Fermi surface and the vorticity of the superconducting phase differences at the trijunction. We explain this chiral adiabatic transmission (CAT) as a consequence of the adiabatic evolution of the scattering modes both in momentum and real space. We identify an effective energy barrier that guarantees quantized transmission. We expect that CAT is observable in nonlocal conductance and thermal transport measurements. Furthermore, because it does not rely on particlehole symmetry, CAT is also possible to observe directly in metamaterials.
Isidora Araya Day, Kostas Vilkelis, Antonio L. R. Manesco, A. Mert Bozkurt, Valla Fatemi, and Anton R. AkhmerovarXiv:2311.17160 [pdf], (unpublished). 
Isotropic 3D topological phases with broken time reversal symmetry [+]
Axial vectors, such as current or magnetization, are commonly used order parameters in timereversal symmetry breaking systems. These vectors also break isotropy in three dimensional systems, lowering the spatial symmetry. We demonstrate that it is possible to construct a fully isotropic and inversionsymmetric threedimensional medium where timereversal symmetry is systematically broken. We devise a cubic crystal with scalar timereversal symmetry breaking, implemented by hopping through chiral magnetic clusters along the crystal bonds. The presence of only the spatial symmetries of the crystal  finite rotation and inversion symmetry  is sufficient to protect a topological phase. The realization of this phase in amorphous systems with average continuous rotation symmetry yields a statistical topological insulator phase. We demonstrate the topological nature of our model by constructing a bulk integer topological invariant, which guarantees gapless surface spectrum on any surface with several overlapping Dirac nodes, analogous to crystalline mirror Chern insulators. We also show the expected transport properties of a threedimensional statistical topological insulator, which remains critical on the surface for odd values of the invariant.
Helene Spring, Anton R. Akhmerov, and Daniel VarjasarXiv:2310.18400 [pdf], (unpublished). 
Fermionic quantum computation with Cooper pair splitters [+]
We propose a practical implementation of a universal quantum computer that uses local fermionic modes (LFM) rather than qubits. Our design consists of quantum dots tunnel coupled by a hybrid superconducting island together with a tunable capacitive coupling between the dots. We show that coherent control of Cooper pair splitting, elastic cotunneling, and Coulomb interactions allows us to implement the universal set of quantum gates defined by Bravyi and Kitaev. Finally, we discuss possible limitations of the device and list necessary experimental efforts to overcome them.
Kostas Vilkelis, Antonio Manesco, Juan Daniel Torres Luna, Sebastian Miles, Michael Wimmer, and Anton AkhmerovarXiv:2309.00447 [pdf], (unpublished). 
Lack of nearsightedness principle in nonHermitian systems [+]
The nonHermitian skin effect is a phenomenon in which an extensive number of states accumulates at the boundaries of a system. It has been associated to nontrivial topology, with nonzero bulk invariants predicting its appearance and its position in real space. Here we demonstrate that the nonHermitian skin effect is not a topological phenomenon in general: when translation symmetry is broken by a single nonHermitian impurity, skin modes are depleted at the boundary and accumulate at the impurity site, without changing any bulk invariant. This may occur even for a fully Hermitian bulk.
Helene Spring, Viktor Könye, Anton R. Akhmerov, and Ion Cosma FulgaarXiv:2308.00776 [pdf], (unpublished). 
DoubleFourier engineering of Josephson energyphase relationships applied to diodes [+]
We present a systematic method to design arbitrary energyphase relations using parallel arms of two series Josephson tunnel junctions each. Our approach employs Fourier engineering in the energyphase relation of each arm and the position of the arms in real space. We demonstrate our method by engineering the energyphase relation of a nearideal superconducting diode, which we find to be robust against the imperfections in the design parameters. Finally, we show the versatility of our approach by designing various other energyphase relations.
A. Mert Bozkurt, Jasper Brookman, Valla Fatemi, and Anton R. Akhmerov 
Design of a Majorana trijunction [+]
Braiding of Majorana states demonstrates their nonAbelian exchange statistics. One implementation of braiding requires control of the pairwise couplings between all Majorana states in a trijunction device. In order to have adiabaticity, a trijunction device requires the desired pair coupling to be sufficently large and the undesired couplings to vanish. In this work, we design and simulate of a trijunction device in a twodimensional electron gas with a focus on the normal region that connects three Majorana states. We use an optimisation approach to find the operational regime of the device in a multidimensional voltage space. Using the optimization results, we simulate a braiding experiment by adiabatically coupling different pairs of Majorana states without closing the topological gap. We then evaluate the feasibility of braiding in a trijunction device for different shapes and disorder strengths.
Juan Daniel Torres Luna, Sathish R. Kuppuswamy, and Anton R. Akhmerov 
Landau quantization near generalized van Hove singularities: magnetic breakdown and orbit networks [+]
We develop a theory of magnetic breakdown (MB) near highorder saddle points in the dispersions of twodimensional materials, where two or more semiclassical cyclotron orbits approach each other. MB occurs due to quantum tunneling between several trajectories, which leads to nontrivial scattering amplitudes and phases. We show that for any saddle point this problem can be solved by mapping it to a scattering problem in a 1D tightbinding chain. Moreover, the occurrence of magnetic breakdown on the edges of the Brillouin zone facilitates the delocalization of the bulk Landau level states and the formation of 2D orbit networks. These extended network states compose dispersive minibands with finite energy broadening. This effect can be observed in transport experiments as a strong enhancement of the longitudinal bulk conductance in a quantum Hall bar. In addition, it may be probed in STM experiments by visualizing bulk current patterns.
V. A. Zakharov, A. Mert Bozkurt, A. R. Akhmerov, and D. O. Oriekhov 
AharonovBohm magnetism in open Fermi surfaces [+]
Orbital diamagnetism requires closed orbits according to the LiftshiftzKosevich theory. Therefore, one might expect that open Fermi surfaces do not have a diamagnetic response. Contrary to this expectation, we show that open orbits in finite systems do contribute a magnetic response which oscillates between diamagnetism and paramagnetism. The oscillations are similar to the AharonovBohm effect, because the oscillation phase is set by the number of flux quanta through the area defined by the width of the sample and the distance between adjacent atomic layers. The magnetic response originates from the closed trajectories formed by counterpropagating open orbits coupled via specular boundary reflections. The phenomenon acts as a probe of the phase coherence of open electron trajectories.
Kostas Vilkelis, Ady Stern, and Anton AkhmerovarXiv:2303.04310 [pdf], (unpublished). 
Phase transitions of wave packet dynamics in disordered nonHermitian systems [+]
Disorder can localize the eigenstates of onedimensional nonHermitian systems, leading to an Anderson transition with a critical exponent of 1. We show that, due to the lack of energy conservation, the dynamics of individual, realspace wave packets follows a different behavior. Both transitions between localization and unidirectional amplification, as well as transitions between distinct propagating phases become possible. The critical exponent of the transition equals $1/2$ in propagatingpropagating transitions.
Helene Spring, Viktor Könye, Fabian A. Gerritsma, Ion Cosma Fulga, and Anton R. AkhmerovarXiv:2301.07370 [pdf], (unpublished). 
Pfaffian invariant identifies magnetic obstructed atomic insulators [+]
We derive a $\mathbb{Z}_4$ topological invariant that extends beyond symmetry eigenvalues and Wilson loops and classifies twodimensional insulators with a $C_4 \mathcal{T}$ symmetry. To formulate this invariant, we consider an irreducible Brillouin zone and constrain the spectrum of the open Wilson lines that compose its boundary. We fix the gauge ambiguity of the Wilson lines by using the Pfaffian at high symmetry momenta. As a result, we distinguish the four $C_4 \mathcal{T}$protected atomic insulators, each of which is adiabatically connected to a different atomic limit. We establish the correspondence between the invariant and the obstructed phases by constructing both the atomic limit Hamiltonians and a $C_4 \mathcal{T}$symmetric model that interpolates between them. The phase diagram shows that $C_4 \mathcal{T}$ insulators allow $\pm 1$ and $2$ changes of the invariant, where the latter is overlooked by symmetry indicators.
Isidora Araya Day, Anastasiia Varentcova, Daniel Varjas, and Anton R. Akhmerov 
Impact of disorder on the distribution of gate coupling strengths in a spin qubit device [+]
A scalable spinbased quantum processor requires a suitable semiconductor heterostructure and a gate design, with multiple alternatives being investigated. Characterizing such devices experimentally is a demanding task, with the full development cycle taking at least months. While numerical simulations are more timeefficient, their predictive power is limited due to unavoidable disorder and devicetodevice variation. We develop a spinqubit device simulation for determining the distribution of the coupling strengths between the electrostatic gate potentials and the effective device Hamiltonian in presence of disorder. By comparing our simulation results with the experimental data, we demonstrate that the coupling of the gate voltages to the dot chemical potential and the interdot tunnel coupling match up to disorderinduced variance. To demonstrate the flexibility of our approach, we also analyze an alternative nonplanar geometry inspired by FinFET devices.
Sathish R. Kuppuswamy, Hugo Kerstens, ChunXiao Liu, Lin Wang, and Anton AkhmerovarXiv:2208.02190 [pdf], (unpublished). 
Breathing mode in openorbit magnetotransport: a magnetic lens with a quantum mechanical focal length [+]
We consider the propagation of electrons in a lattice with an anisotropic dispersion in the $x$$y$ plane (lattice constant $a$), such that it supports open orbits along the $x$axis in an outofplane magnetic field $B$. We show that a point source excites a "breathing mode", a state that periodically spreads out and refocuses after having propagated over a distance $\ell =(eaB/h)^{1}$ in the $x$direction. Unlike known magnetic focusing effects, governed by the classical cyclotron radius, this is an intrinsically quantum mechanical effect with a focal length $\propto\hbar$.
D. O. Oriekhov, T. T. Osterholt, T. Vakhtel, A. R. Akhmerov, and C. W. J. Beenakker 
Greedy optimization of the geometry of Majorana Josephson junctions [+]
Josephson junctions in a twodimensional electron gas with spinorbit coupling are a promising candidate to realize topological superconductivity. While it is known that the geometry of the junction strongly influences the size of the topological gap, the question of how to construct optimal geometries remains unexplored. We introduce a greedy numerical algorithm to optimize the shape of Majorana junctions. The core of the algorithm relies on perturbation theory and is embarrassingly parallel, which allows it to explore the design space efficiently. By introducing stochastic variations in the junction Hamiltonian, we avoid overfitting geometries to specific system parameters. Furthermore, we constrain the optimizer to produce smooth geometries by applying image filtering and fabrication resolution constraints. We run the algorithm in various setups and find that it reliably produces geometries with increased topological gaps over large parameter ranges. The results are robust to variations in the optimization starting point and the presence of disorder, which suggests the optimizer is capable of finding global maxima.
André Melo, Tanko Tanev, and Anton R. Akhmerov 
Topological defects in a doublemirror quadrupole insulator displace diverging charge [+]
We show that topological defects in quadrupole insulators do not host quantized fractional charges, contrary to what their Wannier representation indicates. In particular, we test the charge quantization hypothesis based on the Wannier representation of a parametric defect and a disclination. Against the expectations, we find that the local charge density decays as $\sim 1/r^2$ with distance, leading to a diverging defect charge. We identify sublattice symmetry and not higher order topology as the origin of the previously reported charge quantization.
Isidora Araya Day, Anton R. Akhmerov, and Daniel Varjas 
Multiplet supercurrent in Josephson tunneling circuits [+]
The multiterminal Josephson effect allows DC supercurrent to flow at finite commensurate voltages. Existing proposals to realize this effect rely on nonlocal Andreev processes in superconductornormalsuperconductor junctions. However, this approach requires precise control over microscopic states and is obscured by dissipative current. We show that standard tunnel Josephson circuits also support multiplet supercurrent mediated only by local tunneling processes. Furtheremore, we observe that the supercurrents persist even in the high charging energy regime in which only sequential Cooper transfers are allowed. Finally, we demonstrate that the multiplet supercurrent in these circuits has a quantum geometric component that is distinguinshable from the wellknown adiabatic contribution.
André Melo, Valla Fatemi, and Anton R. Akhmerov 
Chiral Anomaly Trapped in Weyl Metals: Nonequilibrium Valley Polarization at Zero Magnetic Field [+]
In Weyl semimetals the application of parallel electric and magnetic fields leads to valley polarization  an occupation disbalance of valleys of opposite chirality  a direct consequence of the chiral anomaly. In this work, we present numerical tools to explore such nonequilibrium effects in spatially confined threedimensional systems with a variable disorder potential, giving exact solutions to leading order in the disorder potential and the applied electric field. Application to a Weylmetal slab shows that valley polarization also occurs without an external magnetic field as an effect of chiral anomaly "trapping": Spatial confinement produces chiral bulk states, which enable the valley polarization in a similar way as the chiral states induced by a magnetic field. Despite its finitesize origin, the valley polarization can persist up to macroscopic length scales if the disorder potential is sufficiently long ranged, so that direct intervalley scattering is suppressed and the relaxation then goes via the Fermiarc surface states.
Pablo M. PerezPiskunow, Nicandro Bovenzi, Anton R. Akhmerov, and Maxim Breitkreiz 
Mechanisms of Andreev reflection in quantum Hall graphene [+]
We perform realistic simulations of a hybrid superconductorgraphene device in the quantum Hall regime to identify the origin of downstream resistance oscillations in a recent experiment [Zhao et. al. Nature Physics 16, (2020)]. A comparison between the simulations and the experimental data suggests that disorderinduced intervalley scattering at the normalsuperconductor (NS) interface can be the dominant cause of oscillations. We also show conductance oscillations due to additional edge states on clean interfaces with Fermi level mismatch. However, the regular pattern as a function of external parameters is not visible in the presence of disorder. Our work provides a way to qualitatively probe the quality of NS interfaces on multiterminal quantum Hall devices.
Antonio L. R. Manesco, Ian Matthias Flór, ChunXiao Liu, and Anton R. Akhmerov 
Amorphous topological phases protected by continuous rotation symmetry [+]
Protection of topological surface states by reflection symmetry breaks down when the boundary of the sample is misaligned with one of the high symmetry planes of the crystal. We demonstrate that this limitation is removed in amorphous topological materials, where the Hamiltonian is invariant on average under reflection over any axis due to continuous rotation symmetry. While the local disorder caused by the amorphous structure weakens the topological protection, we demonstrate that the edge remains protected from localization. In order to classify such phases we perform a systematic search over all the possible symmetry classes in two dimensions and construct the example models realizing each of the proposed topological phases. Finally, we compute the topological invariant of these phases as an integral along a meridian of the spherical Brillouin zone of an amorphous Hamiltonian.
Helene Spring, Anton R. Akhmerov, and Daniel Varjas 
BlochLorentz magnetoresistance oscillations in delafossites [+]
Recent measurements of the outofplane magnetoresistance of delafossites (PdCoO$_2$ and PtCoO$_2$) observed oscillations which closely resemble the AharanovBohm effect. We develop a semiclassical theory of these oscillations and show that they are a consequence of the quasi2D dispersion of delafossites. We observe that the Lorentz force created by an inplane magnetic field makes the outofplane motion of electrons oscillatory, similarly to Bloch oscillations. Analysis of the visibility of these BlochLorentz oscillations reveals the meanfree path to be $l \approx 4.4 \mu m$ in comparison to the literature inplane mean free path of $20 \mu m$. The meanfree path is reduced as a consequence of the outofplane relaxation and sample wall scattering. Our theory offers a way to design an experimental geometry that is better suited for probing the phenomenon and to investigate the outofplane dynamics of ballistic quasitwodimensional materials.
Kostas Vilkelis, Lin Wang, and Anton Akhmerov 
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. Akhmerov 
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 Bretheau 
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). 
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., PNAS 115, 6991 (2018)] 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 
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). 
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 
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 
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 
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 
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 it allows to simulate a Trotterized Hubbard Hamiltonian in O(1) time. 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 
The 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.
K. Zuo, V. Mourik, D. B. Szombati, B. Nijholt, D. J. van Woerkom, A. Geresdi, J. Chen, V. P. Ostroukh, A. R. Akhmerov, S. R. Plissard, D. Car, E. P. A. M. Bakkers, D. I. Pikulin, L. P. Kouwenhoven, and S. M. Frolov 
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.
R. Kraft, J. Mohrmann, R. Du, P. B. Selvasundaram, M. Irfan, U. N. Kanilmaz, F. Wu, D. Beckmann, H. von Löhneysen, R. Krupke, A. R. Akhmerov, I. Gornyi, and R. Danneau 
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.
D. Sticlet, B. Nijholt, and A. R. Akhmerov 
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 $Ф$, 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 $1/Ф$ 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 $\propto 1/\sqrt Ф$ 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 $\simeq l_m²/L$. 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 
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 4e²/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 
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.
D. Sticlet and A. R. 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.
B. Nijholt and A. R. Akhmerov 
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.
M. T. Allen, O. Shtanko, I. C. Fulga, A. R. Akhmerov, K. Watanabi, T. Taniguchi, P. JarilloHerrero, L. S. Levitov, and A. 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 
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.
V. E. Calado, S. Goswami, G. Nanda, M. Diez, A. R. Akhmerov, K. Watanabe, T. Taniguchi, T. M. Klapwijk, and L. M. K. Vandersypen 
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 
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/.
C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal 
Fluxcontrolled quantum computation with Majorana fermions [+]
Majorana fermions hold promise for quantum computation, because their nonAbelian braiding statistics allows for topologically protected operations on quantum information. Topological qubits can be constructed from pairs of wellseparated Majoranas in networks of nanowires. The coupling to a superconducting charge qubit in a transmission line resonator (transmon) permits braiding of Majoranas by external variation of magnetic fluxes. We show that readout operations can also be fully fluxcontrolled, without requiring microscopic control over tunnel couplings. We identify the minimal circuit that can perform the initializationbraidingmeasurement steps required to demonstrate nonAbelian statistics. We introduce the Random Access Majorana Memory, a scalable circuit that can perform a joint parity measurement on Majoranas belonging to a selection of topological qubits. Such multiqubit measurements allow for the efficient creation of highly entangled states and simplify quantum error correction protocols by avoiding the need for ancilla qubits.
T. Hyart, B. van Heck, I. C. Fulga, M. Burrello, A. R. Akhmerov, and C. W. J. Beenakker 
Phaselocked magnetoconductance oscillations as a probe of Majorana edge states [+]
We calculate the Andreev conductance of a superconducting ring interrupted by a fluxbiased Josephson junction, searching for electrical signatures of circulating edge states. Twodimensional pair potentials of spinsinglet dwave and spintriplet pwave symmetry support, respectively, (chiral) Dirac modes and (chiral or helical) Majorana modes. These produce h/eperiodic magnetoconductance oscillations of amplitude \simeq (e²}/h)N^{1/2}, measured via an Nmode point contact at the inner or outer perimeter of the grounded ring. For Dirac modes the oscillations in the two contacts are independent, while for an unpaired Majorana mode they are phase locked by a topological phase transition at the Josephson junction.
M. Diez, I. C. Fulga, D. I. Pikulin, M. Wimmer, A. R. Akhmerov, and C. W. J. Beenakker 
Statistical topological insulators [+]
We define a class of insulators with gapless surface states protected from localization due to the statistical properties of a disordered ensemble, namely due to the ensemble's invariance under a certain symmetry. We show that these insulators are topological, and are protected by a $\mathbb{Z}_2$ invariant. Finally, we prove that every topological insulator gives rise to an infinite number of classes of statistical topological insulators in higher dimensions. Our conclusions are confirmed by numerical simulations.
I. C. Fulga, B. van Heck, J. M. Edge, and A. R. Akhmerov 
Adaptive tuning of Majorana fermions in a quantum dot chain [+]
We suggest a way to overcome the obstacles that disorder and high density of states pose to the creation of unpaired Majorana fermions in onedimensional systems. This is achieved by splitting the system into a chain of quantum dots, which are then tuned to the conditions under which the chain can be viewed as an effective Kitaev model, so that it is in a robust topological phase with welllocalized Majorana states in the outermost dots. The tuning algorithm that we develop involves controlling the gate voltages and the superconducting phases. Resonant Andreev spectroscopy allows us to make the tuning adaptive, so that each pair of dots may be tuned independently of the other. The calculated quantized zero bias conductance serves then as a natural proof of the topological nature of the tuned phase.
I. C. Fulga, A. Haim, A. R. Akhmerov, and Y. Oreg 
Braiding of nonAbelian anyons using pairwise interactions [+]
The common approach to topological quantum computation is to implement quantum gates by adiabatically moving nonAbelian anyons around each other. Here we present an alternative perspective based on the possibility of realizing the exchange (braiding) operators of anyons by adiabatically varying pairwise interactions between them rather than their positions. We analyze a system composed by four anyons whose couplings define a Tjunction and we show that the braiding operator of two of them can be obtained through a particular adiabatic cycle in the space of the coupling parameters. We also discuss how to couple this scheme with anyonic chains in order to recover the topological protection.
M. Burrello, B. van Heck, and A. R. Akhmerov 
Topological blockade and measurement of topological charge [+]
The fractionally charged quasiparticles appearing in the 5/2 fractional quantum Hall plateau are predicted to have an extra nonlocal degree of freedom, known as topological charge. We show how this topological charge can block the tunnelling of these particles, and how such 'topological blockade' can be used to readout their topological charge. We argue that the short time scale required for this measurement is favorable for the detection of the nonAbelian anyonic statistics of the quasiparticles. We also show how topological blockade can be used to measure braiding statistics, and to couple a topological qubit with a conventional one.
B. van Heck, M. Burrello, A. Yacoby, and A. R. Akhmerov 
Thermal metalinsulator transition in a helical topological superconductor [+]
Twodimensional superconductors with timereversal symmetry have a Z_2 topological invariant, that distinguishes phases with and without helical Majorana edge states. We study the topological phase transition in a classDIII network model, and show that it is associated with a metalinsulator transition for the thermal conductance of the helical superconductor. The localization length diverges at the transition with critical exponent nu approx 2.0, about twice the known value in a chiral superconductor.
I. C. Fulga, A. R. Akhmerov, J. Tworzydło, B. Béri, and C. W. J. Beenakker 
Zerobias conductance peak and Josephson effect in grapheneNbTiN junctions [+]
We report electronic transport measurements of graphene contacted by NbTiN electrodes, which at low temperature remain superconducting up to at least 11 Tesla. In devices with a single superconducting contact, we find a more than twofold enhancement of the conductance at zero bias, which we interpret in terms of reflectionless tunneling. In devices with two superconducting contacts, we observe the Josephson effect, bipolar supercurrents and Fraunhofer patterns.
M. Popinciuc, V. E. Calado, X. L. Liu, A. R. Akhmerov, T. M. Klapwijk, and L. M. K. Vandersypen 
Coulombassisted braiding of Majorana fermions in a Josephson junction array [+]
We show how to exchange (braid) Majorana fermions in a network of superconducting nanowires by control over Coulomb interactions rather than tunneling. Even though Majorana fermions are chargeneutral quasiparticles (equal to their own antiparticle), they have an effective longrange interaction through the evenodd electron number dependence of the superconducting ground state. The flux through a split Josephson junction controls this interaction via the ratio of Josephson and charging energies, with exponential sensitivity. By switching the interaction on and off in neighboring segments of a Josephson junction array, the nonAbelian braiding statistics can be realized without the need to control tunnel couplings by gate electrodes. This is a solution to the problem how to operate on topological qubits when gate voltages are screened by the superconductor.
B. van Heck, A. R. Akhmerov, F. Hassler, M. Burrello, and C. W. J. Beenakker 
Topological quantum number and critical exponent from conductance fluctuations at the quantum Hall plateau transition [+]
The conductance of a twodimensional electron gas at the transition from one quantum Hall plateau to the next has samplespecific fluctuations as a function of magnetic field and Fermi energy. Here we identify a universal feature of these mesoscopic fluctuations in a Corbino geometry: The amplitude of the magnetoconductance oscillations has an e²/h resonance in the transition region, signaling a change in the topological quantum number of the insulating bulk. This resonance provides a signed scaling variable for the critical exponent of the phase transition (distinct from existing positive definite scaling variables).
I. C. Fulga, F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker 
Dirac boundary condition at the reconstructed zigzag edge of graphene [+]
Edge reconstruction modifies the electronic properties of finite graphene samples. We formulate a lowenergy theory of the reconstructed zigzag edge by deriving the modified boundary condition to the Dirac equation. If the unit cell size of the reconstructed edge is not a multiple of three with respect to the zigzag unit cell, valleys remain uncoupled and the edge reconstruction is accounted for by a single angular parameter $\vartheta$. Dispersive edge states exist generically, unless $\vartheta = \pi/2$. We compute $\vartheta$ from a microscopic model for the "reczag" reconstruction (conversion of two hexagons into a pentagonheptagon pair) and show that it can be measured via the local density of states. In a magnetic field there appear three distinct edge modes in the lowest Landau level, two of which are counterpropagating.
J. A. M. van Ostaay, A. R. Akhmerov, C. W. J. Beenakker, and M. Wimmer 
Coulomb stability of the 4πperiodic Josephson effect of Majorana fermions [+]
The Josephson energy of two superconducting islands containing Majorana fermions is a 4\piperiodic function of the superconducting phase difference. If the islands have a small capacitance, their ground state energy is governed by the competition of Josephson and charging energies. We calculate this ground state energy in a ring geometry, as a function of the flux \Ф enclosed by the ring, and show that the dependence on the AharonovBohm phase 2e\Ф/\hbar remains 4\piperiodic regardless of the ratio of charging and Josephson energies  provided that the entire ring is in a topologically nontrivial state. If part of the ring is topologically trivial, then the charging energy induces quantum phase slips that restore the usual 2\piperiodicity.
B. van Heck, F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker 
Majorana fermions emerging from magnetic nanoparticles on a superconductor without spinorbit coupling [+]
There exists a variety of proposals to transform a conventional swave superconductor into a topological superconductor, supporting Majorana fermion midgap states. A necessary ingredient of these proposals is strong spinorbit coupling. Here we propose an alternative system consisting of a onedimensional chain of magnetic nanoparticles on a superconducting substrate. No spinorbit coupling in the superconductor is needed. We calculate the topological quantum number of a chain of finite length, including the competing effects of disorder in the orientation of the magnetic moments and in the hopping energies, to identify the transition into the topologically nontrivial state (with Majorana fermions at the end points of the chain).
T.P. Choy, J. M. Edge, A. R. Akhmerov, and C. W. J. Beenakker 
Scattering theory of topological insulators and superconductors [+]
The topological invariant of a topological insulator (or superconductor) is given by the number of symmetryprotected edge states present at the Fermi level. Despite this fact, established expressions for the topological invariant require knowledge of all states below the Fermi energy. Here, we propose a way to calculate the topological invariant employing solely its scattering matrix at the Fermi level without knowledge of the full spectrum. Since the approach based on scattering matrices requires much less information than the Hamiltonianbased approaches (surface versus bulk), it is numerically more efficient. In particular, is bettersuited for studying disordered systems. Moreover, it directly connects the topological invariant to transport properties potentially providing a new way to probe topological phases.
I. C. Fulga, F. Hassler, and A. R. Akhmerov 
Transmission probability through a Lévy glass and comparison with a Lévy walk [+]
Recent experiments on the propagation of light over a distance L through a random packing of spheres with a power law distribution of radii (a socalled L\'evy glass) have found that the transmission probability T \propto 1/L^{\gamma} scales superdiffusively ({\gamma} < 1). The data has been interpreted in terms of a L\'evy walk. We present computer simulations to demonstrate that diffusive scaling ({\gamma} \approx 1) can coexist with a divergent second moment of the step size distribution (p(s) \propto 1/s^(1+{\alpha}) with {\alpha} < 2). This finding is in accord with analytical predictions for the effect of step size correlations, but deviates from what one would expect for a L\'evy walk of independent steps.
C. W. Groth, A. R. Akhmerov, and C. W. J. Beenakker 
Toptransmon: hybrid superconducting qubit for parityprotected quantum computation [+]
Qubits constructed from uncoupled Majorana fermions are protected from decoherence, but to perform a quantum computation this topological protection needs to be broken. Parityprotected quantum computation breaks the protection in a minimally invasive way, by coupling directly to the fermion parity of the system  irrespective of any quasiparticle excitations. Here we propose to use a superconducting charge qubit in a transmission line resonator (a socalled transmon) to perform parityprotected rotations and readout of a topological (top) qubit. The advantage over an earlier proposal using a flux qubit is that the coupling can be switched on and off with exponential accuracy, promising a reduced sensitivity to charge noise.
F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker 
Spintriplet supercurrent carried by quantum Hall edge states through a Josephson junction [+]
We show that a spinpolarized Landau level in a twodimensional electron gas can carry a spintriplet supercurrent between two spinsinglet superconductors. The supercurrent results from the interplay of Andreev reflection and Rashba spinorbit coupling at the normalsuperconductor (NS) interface. We contrast the currentphase relationship and the Fraunhofer oscillations of the spintriplet and spinsinglet Josephson effect in the lowest Landau level, and find qualitative differences.
J. A. M. van Ostaay, A. R. Akhmerov, and C. W. J. Beenakker 
Majorana fermions in equilibrium and driven cold atom quantum wires [+]
We introduce a new approach to create and detect Majorana fermions using optically trapped 1D fermionic atoms. In our proposed setup, two internal states of the atoms couple via an optical Raman transitionsimultaneously inducing an effective spinorbit interaction and magnetic fieldwhile a background molecular BEC cloud generates swave pairing for the atoms. The resulting cold atom quantum wire supports Majorana fermions at phase boundaries between topologically trivial and nontrivial regions, as well as `Floquet Majorana fermions' when the system is periodically driven. We analyze experimental parameters, detection schemes, and various imperfections.
L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. Ignacio Cirac, E. Demler, M. D. Lukin, and P. Zoller 
Quantum point contact as a probe of a topological superconductor [+]
We calculate the conductance of a ballistic point contact to a superconducting wire, produced by the swave proximity effect in a semiconductor with spinorbit coupling in a parallel magnetic field. The conductance G as a function of contact width or Fermi energy shows plateaus at halfinteger multiples of 4e²/h if the superconductor is in a topologically nontrivial phase. In contrast, the plateaus are at the usual integer multiples in the topologically trivial phase. Disorder destroys all plateaus except the first, which remains precisely quantized, consistent with previous results for a tunnel contact. The advantage of a ballistic contact over a tunnel contact as a probe of the topological phase is the strongly reduced sensitivity to finite voltage or temperature.
M. Wimmer, A. R. Akhmerov, J. P. Dahlhaus, and C. W. J. Beenakker 
Scattering formula for the topological quantum number of a disordered multimode wire [+]
The topological quantum number Q of a superconducting or chiral insulating wire counts the number of stable bound states at the end points. We determine Q from the matrix r of reflection amplitudes from one of the ends, generalizing the known result in the absence of timereversal and chiral symmetry to all five topologically nontrivial symmetry classes. The formula takes the form of the determinant, Pfaffian, or matrix signature of r, depending on whether r is a real matrix, a real antisymmetric matrix, or a Hermitian matrix. We apply this formula to calculate the topological quantum number of N coupled dimerized polymer chains, including the effects of disorder in the hopping constants. The scattering theory relates a topological phase transition to a conductance peak, of quantized height and with a universal (symmetry class independent) line shape. Two peaks which merge are annihilated in the superconducting symmetry classes, while they reinforce each other in the chiral symmetry classes.
I. C. Fulga, F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker 
Probing Majorana edge states with a flux qubit [+]
A pair of counterpropagating Majorana edge modes appears in chiral pwave superconductors and in other superconducting systems belonging to the same universality class. These modes can be described by an Ising conformal field theory. We show how a superconducting flux qubit attached to such a system couples to the two chiral edge modes via the disorder field of the Ising model. Due to this coupling, measuring the backaction of the edge states on the qubit allows to probe the properties of Majorana edge modes.
C.Y. Hou, F. Hassler, A. R. Akhmerov, and J. Nilsson 
Effects of disorder on the transmission of nodal fermions through a dwave superconductor [+]
The bulk microwave conductivity of a dirty dwave superconductor is known to depend sensitively on the range of the disorder potential: longrange scattering enhances the conductivity, while short range scattering has no effect. Here we show that the threeterminal electrical conductance of a normalmetaldwave superconductornormalmetal junction has a dual behavior: shortrange scattering suppresses the conductance, while longrange scattering has no effect.
J. K. Asboth, A. R. Akhmerov, M. V. Medvedyeva, and C. W. J. Beenakker 
Randommatrix theory of Andreev reflection from a topological superconductor [+]
We calculate the probability distribution of the Andreev reflection eigenvalues R_n at the Fermi level in the circular ensemble of randommatrix theory. Without spinrotation symmetry, the statistics of the electrical conductance G depends on the topological quantum number Q of the superconductor. We show that this dependence is nonperturbative in the number N of scattering channels, by proving that the pth cumulant of G is independent of Q for p
C. W. J. Beenakker, J. P. Dahlhaus, M. Wimmer, and A. R. Akhmerov 
Quantized conductance at the Majorana phase transition in a disordered superconducting wire [+]
Superconducting wires without timereversal and spinrotation symmetries can be driven into a topological phase that supports Majorana bound states. Direct detection of these zeroenergy states is complicated by the proliferation of lowlying excitations in a disordered multimode wire. We show that the phase transition itself is signaled by a quantized thermal conductance and electrical shot noise power, irrespective of the degree of disorder. In a ring geometry, the phase transition is signaled by a period doubling of the magnetoconductance oscillations. These signatures directly follow from the identification of the sign of the determinant of the reflection matrix as a topological quantum number.
A. R. Akhmerov, J. P. Dahlhaus, F. Hassler, M. Wimmer, and C. W. J. Beenakker 
Geodesic scattering by surface deformations of a topological insulator [+]
We consider the classical ballistic dynamics of massless electrons on the conducting surface of a threedimensional topological insulator, influenced by random variations of the surface height. By solving the geodesic equation and the Boltzmann equation in the limit of shallow deformations, we obtain the scattering cross section and the conductivity {\sigma}, for arbitrary anisotropic dispersion relation. At large surface electron densities n this geodesic scattering mechanism (with {\sigma} propto sqrt{n}) is more effective at limiting the surface conductivity than electrostatic potential scattering.
J. P. Dahlhaus, C.Y. Hou, A. R. Akhmerov, and C. W. J. Beenakker 
Flatlens focusing of electrons on the surface of a topological insulator [+]
We propose the implementation of an electronic Veselago lens on the conducting surface of a threedimensional topological insulator (such as Bi2Te3). The negative refraction needed for such a flat lens results from the sign change of the curvature of the Fermi surface, changing from a circular to a snowflakelike shape across a sufficiently large electrostatic potential step. No interband transition (as in graphene) is needed. For this reason, and because the topological insulator provides protection against backscattering, the potential step is able to focus a broad range of incident angles. We calculate the quantum interference pattern produced by a point source, generalizing the analogous optical calculation to include the effect of a noncircular Fermi surface (having a nonzero conic constant).
F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker 
Anyonic interferometry without anyons: How a flux qubit can read out a topological qubit [+]
Proposals to measure nonAbelian anyons in a superconductor by quantum interference of vortices suffer from the predominantly classical dynamics of the normal core of an Abrikosov vortex. We show how to avoid this obstruction using coreless Josephson vortices, for which the quantum dynamics has been demonstrated experimentally. The interferometer is a flux qubit in a Josephson junction circuit, which can nondestructively read out a topological qubit stored in a pair of anyons  even though the Josephson vortices themselves are not anyons. The flux qubit does not couple to intravortex excitations, thereby removing the dominant restriction on the operating temperature of anyonic interferometry in superconductors.
F. Hassler, A. R. Akhmerov, C.Y. Hou, and C. W. J. Beenakker 
Topological quantum computation away from the ground state with Majorana fermions [+]
We relax one of the requirements for topological quantum computation with Majorana fermions. Topological quantum computation was discussed so far as manipulation of the wave function within degenerate many body ground state. The simplest particles providing degenerate ground state, Majorana fermions, often coexist with extremely low energy excitations, so keeping the system in the ground state may be hard. We show that the topological protection extends to the excited states, as long as the Majorana fermions do not interact neither directly, nor via the excited states. This protection relies on the fermion parity conservation, and so it is generic to any implementation of Majorana fermions.
A. R. Akhmerov 
Robustness of edge states in graphene quantum dots [+]
We analyze the single particle states at the edges of disordered graphene quantum dots. We show that generic graphene quantum dots support a number of edge states proportional to circumference of the dot over the lattice constant. Our analytical theory agrees well with numerical simulations. Perturbations breaking electronhole symmetry like nextnearest neighbor hopping or edge impurities shift the edge states away from zero energy but do not change their total amount. We discuss the possibility of detecting the edge states in an antidot array and provide an upper bound on the magnetic moment of a graphene dot.
M. Wimmer, A. R. Akhmerov, and F. Guinea 
Majorana bound states without vortices in topological superconductors with electrostatic defects [+]
Vortices in twodimensional superconductors with broken timereversal and spinrotation symmetry can bind states at zero excitation energy. These socalled Majorana bound states transform a thermal insulator into a thermal metal and may be used to encode topologically protected qubits. We identify an alternative mechanism for the formation of Majorana bound states, akin to the way in which Shockley states are formed on metal surfaces: An atomicscale electrostatic line defect can have a pair of Majorana bound states at the end points. The Shockley mechanism explains the appearance of a thermal metal in vortexfree lattice models of chiral pwave superconductors and (unlike the vortex mechanism) is also operative in the topologically trivial phase.
M. Wimmer, A. R. Akhmerov, M. V. Medvedyeva, J. Tworzydło, and C. W. J. Beenakker 
Absence of a metallic phase in chargeneutral graphene with a random gap [+]
It is known that fluctuations in the electrostatic potential allow for metallic conduction (nonzero conductivity in the limit of an infinite system) if the carriers form a single species of massless twodimensional Dirac fermions. A nonzero uniform mass $\bar{M}$ opens up an excitation gap, localizing all states at the Dirac point of charge neutrality. Here we investigate numerically whether fluctuations $\delta M \gg \bar{M} \neq 0$ in the mass can have a similar effect as potential fluctuations, allowing for metallic conduction at the Dirac point. Our negative conclusion confirms earlier expectations, but does not support the recently predicted metallic phase in a randomgap model of graphene.
J. H. Bardarson, M. V. Medvedyeva, J. Tworzydło, A. R. Akhmerov, and C. W. J. Beenakker 
Theory of nonAbelian FabryPerot interferometry in topological insulators [+]
Interferometry of nonAbelian edge excitations is a useful tool in topological quantum computing. In this paper we present a theory of a nonAbelian edge state interferometer in a 3D topological insulator brought in proximity to an swave superconductor. The nonAbelian edge excitations in this system have the same statistics as in the previously studied 5/2 fractional quantum Hall (FQH) effect and chiral pwave superconductors. There are however crucial differences between the setup we consider and these systems, like the need for a converter between charged and neutral excitations and the neutrality of the nonAbelian excitations. These differences manifest themselves in a temperature scaling exponent of 7/4 for the conductance instead of 3/2 as in the 5/2 FQH effect.
J. Nilsson and A. R. Akhmerov 
Domain wall in a chiral pwave superconductor: a pathway for electrical current [+]
Superconductors with p+ip pairing symmetry are characterized by chiral edge states, but these are difficult to detect in equilibrium since the resulting magnetic field is screened by the Meissner effect. Nonequilibrium detection is hindered by the fact that the edge excitations are unpaired Majorana fermions, which cannot transport charge near the Fermi level. Here we show that the boundary between p_x+ip_y and p_xip_y domains forms a oneway channel for electrical charge. We derive a product rule for the domain wall conductance, which allows to cancel the effect of a tunnel barrier between metal electrodes and superconductor and provides a unique signature of topological superconductors in the chiral pwave symmetry class.
I. Serban, B. Béri, A. R. Akhmerov, and C. W. J. Beenakker 
Pseudodiffusive transmission of nodal Dirac fermions through a clean dwave superconductor [+]
We calculate the transmission of electrons and holes between two normalmetal electrodes (N), separated over a distance L by an impurityfree superconductor (S) with dwave symmetry of the order parameter. Nodal lines of vanishing excitation gap form ballistic conduction channels for coupled electronhole excitations, described by an anisotropic twodimensional Dirac equation. We find that the transmitted electrical and thermal currents, at zero energy, both have the pseudodiffusive 1/L scaling characteristic of massless Dirac fermions  regardless of the presence of tunnel barriers at the NS interfaces. Tunnel barriers reduce the slope of the 1/L scaling in the case of the electrical current, while leaving the thermal current unaffected.
J. K. Asboth, A. R. Akhmerov, A. C. Berceanu, and C. W. J. Beenakker 
Theory of the topological Anderson insulator [+]
We present an effective medium theory that explains the disorderinduced transition into a phase of quantized conductance, discovered in computer simulations of HgTe quantum wells. It is the combination of a random potential and quadratic corrections proportional to p² sigma_z to the Dirac Hamiltonian that can drive an ordinary band insulator into a topological insulator (having an inverted band gap). We calculate the location of the phase boundary at weak disorder and show that it corresponds to the crossing of a band edge rather than a mobility edge. Our mechanism for the formation of a topological Anderson insulator is generic, and would apply as well to threedimensional semiconductors with strong spinorbit coupling.
C. W. Groth, M. Wimmer, A. R. Akhmerov, J. Tworzydło, and C. W. J. Beenakker 
Switching of electrical current by spin precession in the first Landau level of an invertedgap semiconductor [+]
We show how the quantum Hall effect in an invertedgap semiconductor (with electron and holelike states at the conduction and valenceband edges interchanged) can be used to inject, precess, and detect the electron spin along a onedimensional pathway. The restriction of the electron motion to a single spatial dimension ensures that all electrons experience the same amount of precession in a parallel magnetic field, so that the full electrical current can be switched on and off. As an example, we calculate the magnetoconductance of a pn interface in a HgTe quantum well and show how it can be used to measure the spin precession due to bulk inversion asymmetry.
A. R. Akhmerov, C. W. Groth, J. Tworzydło, and C. W. J. Beenakker 
Electrically detected interferometry of Majorana fermions in a topological insulator [+]
We show how a chiral Dirac fermion (a massless electron or hole) can be converted into a pair of neutral chiral Majorana fermions (a particle equal to its own antiparticle). These two types of fermions exist on the metallic surface of a topological insulator, respectively, at a magnetic domain wall and at a magnetsuperconductor interface. Interferometry of Majorana fermions is a key operation in topological quantum computation, but the detection is problematic since these particles have no charge. The DiracMajorana converter enables electrical detection of the interferometric signal.
A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker 
Quantum GoosHänchen effect in graphene [+]
The GoosHänchen (GH) effect is an interference effect on total internal reflection at an interface, resulting in a shift sigma of the reflected beam along the interface. We show that the GH effect at a pn interface in graphene depends on the pseudospin (sublattice) degree of freedom of the massless Dirac fermions, and find a sign change of sigma at angle of incidence alpha*=arcsin[sin alpha_c]^1/2 determined by the critical angle alpha_c for total reflection. In an ndoped channel with pdoped boundaries the GH effect doubles the degeneracy of the lowest propagating mode, introducing a twofold degeneracy on top of the usual spin and valley degeneracies. This can be observed as a stepwise increase by 8e²/h of the conductance with increasing channel width.
C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydło 
Nonalgebraic length dependence of transmission through a chain of barriers with a Lévy spacing distribution [+]
The recent realization of a "Lévy glass" (a threedimensional optical material with a Lévy distribution of scattering lengths) has motivated us to analyze its onedimensional analogue: A linear chain of barriers with independent spacings s that are Lévy distributed: p(s)~1/s^(1+alpha) for s to infinity. The average spacing diverges for 0
C. W. J. Beenakker, C. W. Groth, and A. R. Akhmerov 
Splitting of a Cooper pair by a pair of Majorana bound states [+]
Majorana bound states are spatially localized superpositions of electron and hole excitations in the middle of a superconducting energy gap. A single qubit can be encoded nonlocally in a pair of spatially separated Majorana bound states. Such Majorana qubits are in demand as building blocks of a topological quantum computer, but direct experimental tests of the nonlocality remain elusive. Here we propose a method to probe the nonlocality by means of crossed Andreev reflection, which is the injection of an electron into one bound state followed by the emission of a hole by the other bound state (equivalent to the splitting of a Cooper pair over the two states). We have found that, at sufficiently low excitation energies, this nonlocal scattering process dominates over local Andreev reflection involving a single bound state. As a consequence, the lowtemperature and lowfrequency fluctuations $\delta I_{i}$ of currents into the two bound states $i=1,2$ are maximally correlated: $\overline{\delta I_{1}\delta I_{2}}=\overline{\delta I_{i²}}$.
J. Nilsson, A. R. Akhmerov, and C. W. J. Beenakker 
Theory of the valleyvalve effect in graphene nanoribbons [+]
A potential step in a graphene nanoribbon with zigzag edges is shown to be an intrinsic source of intervalley scattering  no matter how smooth the step is on the scale of the lattice constant a. The valleys are coupled by a pair of localized states at the opposite edges, which act as an attractor/repellor for edge states propagating in valley K/K'. The relative displacement Delta along the ribbon of the localized states determines the conductance G. Our result G=(e²}/h)[1\cos(N\pi+2\pi\Delta/3a)] explains why the ``valleyvalve'' effect (the blocking of the current by a pn junction) depends on the parity of the number N of carbon atoms across the ribbon.
A. R. Akhmerov, J. H. Bardarson, A. Rycerz, and C. W. J. Beenakker 
Boundary conditions for Dirac fermions on a terminated honeycomb lattice [+]
We derive the boundary condition for the Dirac equation corresponding to a tightbinding model on a twodimensional honeycomb lattice terminated along an arbitary direction. Zigzag boundary conditions result generically once the boundary is not parallel to the bonds. Since a honeycomb strip with zigzag edges is gapless, this implies that confinement by lattice termination does not in general produce an insulating nanoribbon. We consider the opening of a gap in a graphene nanoribbon by a staggered potential at the edge and derive the corresponding boundary condition for the Dirac equation. We analyze the edge states in a nanoribbon for arbitrary boundary conditions and identify a class of propagating edge states that complement the known localized edge states at a zigzag boundary.
A. R. Akhmerov and C. W. J. Beenakker 
Correspondence between Andreev reflection and Klein tunneling in bipolar graphene [+]
Andreev reflection at a superconductor and Klein tunneling through an np junction in graphene are two processes that couple electrons to holes  the former through the superconducting pair potential Delta and the latter through the electrostatic potential U. We derive that the energy spectra in the two systems are identical, at low energies E<
C. W. J. Beenakker, A. R. Akhmerov, P. Recher, and J. Tworzydło 
Valleyisospin dependence of the quantum Hall effect in a graphene pn junction [+]
We calculate the conductance G of a bipolar junction in a graphene nanoribbon, in the highmagnetic field regime where the Hall conductance in the pdoped and ndoped regions is 2e²/h. In the absence of intervalley scattering, the result G=(e²/h)(1cos Ф) depends only on the angle Ф between the valley isospins (= Bloch vectors representing the spinor of the valley polarization) at the two opposite edges. This plateau in the conductance versus Fermi energy is insensitive to electrostatic disorder, while it is destabilized by the dispersionless edge state which may exist at a zigzag boundary. A straininduced vector potential shifts the conductance plateau up or down by rotating the valley isospin.
J. Tworzydło, I. Snyman, A. R. Akhmerov, and C. W. J. Beenakker 
Detection of valley polarization in graphene by a superconducting contact [+]
Because the valleys in the band structure of graphene are related by timereversal symmetry, electrons from one valley are reflected as holes from the other valley at the junction with a superconductor. We show how this Andreev reflection can be used to detect the valley polarization of edge states produced by a magnetic field. In the absence of intervalley relaxation, the conductance G_NS=2(e²/h)(1cos(Theta)) of the junction on the lowest quantum Hall plateau is entirely determined by the angle Theta between the valley isospins of the edge states approaching and leaving the superconductor. If the superconductor covers a single edge, Theta=0 and no current can enter the superconductor. A measurement of G_NS then determines the intervalley relaxation time.
A. R. Akhmerov and C. W. J. Beenakker 
Pseudodiffusive conduction at the Dirac point of a normalsuperconductor junction in graphene [+]
A ballistic strip of graphene (width W>> length L) connecting two normal metal contacts is known to have a minimum conductivity of 4e²}/pi h at the Dirac point of charge neutrality. We calculate what happens if one of the two contacts becomes superconducting. While the ballistic conductance away from the Dirac point is increased by Andreev reflection at the normalsuperconductor (NS) interface, we find that the minimum conductivity stays the same. This is explained as a manifestation of pseudodiffusive conduction at the Dirac point. As a generalization of our results for a ballistic system, we provide a relation between the conductance G_NS of an arbitrarily disordered normalsuperconductor junction in graphene and its value G_N when both contacts are in the normal state.
A. R. Akhmerov and C. W. J. Beenakker 
Universal temperature dependence of the conductivity of a strongly disordered granular metal [+]
A disordered array of metal grains with large and random intergrain conductances is studied within the oneloop accuracy renormalization group approach. While at low level of disorder the dependence of conductivity on log T is nonuniversal (it depends on details of the array's geometry), for strong disorder this dependence is described by a universal nonlinear function, which depends only on the array's dimensionality. In two dimensions this function is found numerically. The dimensional crossover in granular films is discussed.
A. R. Akhmerov and A. S. Ioselevich