We show how the superconducting phase difference in a Josephson junction may be used to split the Kramers degeneracy of its energy levels and to remove all the properties associated with time-reversal symmetry. The superconducting phase difference is known to be ineffective in two-terminal short Josephson junctions, where irrespective of the junction structure the induced Kramers degeneracy splitting is suppressed and the ground state fermion parity must stay even, so that a protected zero-energy Andreev level crossing may never appear. Our main result is that these limitations can be completely avoided by using 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 2π . Our approach offers strategies for creation of Majorana bound states as well as spin manipulation. Our proposal can be implemented using any low density, high spin-orbit material such as InAs quantum wells, and can be detected using standard tools.

We show how a quantum dot with a ballistic single-channel point contact to a superconductor can be created by means of a gate electrode at the edge of a quantum spin Hall insulator (such as an InAs/GaSb quantum well). A weak perpendicular magnetic field traps a Majorana zero mode, so that it can be observed in the gate-voltageaveraged differential conductance dI/dV as a 4e2 /h zero-bias peak above a ( 2 3 π 2 − 4)e 2 /h background. The one-dimensional edge does not permit the braiding of pairs of Majorana fermions, but this obstacle can be overcome by coupling opposite edges at a constriction, allowing for a demonstration of non-Abelian statistics. 4 The conducting edge of a quantum spin Hall (QSH) insulator seems like an ideal system to search for these elusive particles in a transport experiment: 5,6 Only a single mode propagates in each direction along the edge, unaffected by disorder since backscattering of these helical modes is forbidden by time-reversal symmetry. 7 The QSH edge is thus immune for the multimode and disorder effects that complicate the Majorana-fermion interpretation of transport experiments in semiconductor nanowires. 8,9 Andreev reflection at a superconducting interface has been reported in an InAs/GaSb quantum well, 10 which is a QSH insulator because of a band inversion and the appearance of edge states connecting conduction and valence bands. 11Similar experiments can be tried in HgTe/CdTe quantum wells, where the QSH effect was first discovered. 12,13 We expect a Majorana fermion to be present in these systems, delocalized along the edge connecting a normal and superconducting contact, but without a distinctive resonance in the electrical conductance. Andreev reflection of a helical edge mode doubles the current at all energies inside the band gap, so each edge contributes 2e 2 /h to the differential conductance irrespective of any midgap states.Here we present a method to restore the sensitivity of the conductance to the zero-mode resonance, by trapping the Majorana fermion near the superconducting interface. Only a minor modification of the existing experimental setup 10 is needed, essentially only a gate electrode at one of the edges, to locally push the conduction band through the Fermi level (see Fig. 1). The area under the gate then forms a two-dimensional metallic region, connected to the superconductor by the helical edge mode. Backscattering at this Andreev quantum dot in a weak magnetic field (one flux quantum or less through the dot) provides for an electrostatically tunable confinement of Majorana fermions. We discuss the detection of Majoranas as a short-term application, and braiding as a longer term perspective.There exists a variety of phase coherent backscattering mechanisms for helical edge modes, 14-20 based on different methods of time-reversal symmetry breaking to open a minigap in the edge state spectrum. A locally opened minigap forms a tunnel barrier for the edge modes and two tunnel barriers in series form a quantum dot at the QSH edge.19 For a robust Majoran...

The phase-dependent bound states (Andreev levels) of a Josephson junction can cross at the Fermi level if the superconducting ground state switches between even and odd fermion parity. The level crossing is topologically protected, in the absence of time-reversal and spin-rotation symmetry, irrespective of whether the superconductor itself is topologically trivial or not. We develop a statistical theory of these topological transitions in an N-mode quantum-dot Josephson junction by associating the Andreev level crossings with the real eigenvalues of a random non-Hermitian matrix. The number of topological transitions in a 2π phase interval scales as √[N], and their spacing distribution is a hybrid of the Wigner and Poisson distributions of random-matrix theory.

The quasi-bound states of a superconducting quantum dot that is weakly coupled to a normal metal appear as resonances in the Andreev reflection probability, measured via the differential conductance. We study the evolution of these Andreev resonances when an external parameter (such as magnetic field or gate voltage) is varied, using a random-matrix model for the N × N scattering matrix. We contrast the two ensembles with broken time-reversal symmetry, in the presence or absence of spin-rotation symmetry (class C or D). The poles of the scattering matrix in the complex plane, encoding the center and width of the resonance, are repelled from the imaginary axis in class C. In class D, in contrast, a number ∝ √ N of the poles has zero real part. The corresponding Andreev resonances are pinned to the middle of the gap and produce a zero-bias conductance peak that does not split over a range of parameter values (Y-shaped profile), unlike the usual conductance peaks that merge and then immediately split (X-shaped profile). Contribution for the JETP special issue in honor of A.F. Andreev's 75th birthday.

We calculate the conductance of a two-dimensional bilayer with inverted electron-hole bands to study the sensitivity of the quantum spin Hall insulator (with helical edge conduction) to the combination of electrostatic disorder and a perpendicular magnetic field. The characteristic breakdown field for helical edge conduction splits into two fields with increasing disorder, a field B c for the transition into a quantum Hall insulator (supporting chiral edge conduction) and a smaller field B c for the transition to bulk conduction in a quasimetallic regime. The spatial separation of the inverted bands, typical for broken-gap InAs/GaSb quantum wells, is essential for the magnetic-field-induced bulk conduction-there is no such regime in HgTe quantum wells. A two-dimensional band insulator can support two types of conducting edge states: counterpropagating (helical) edge states in zero magnetic field and unidirectional (chiral) edge states in a sufficiently strong perpendicular field. These two topologically distinct phases are referred to as a quantum spin Hall (QSH) and quantum Hall (QH) insulator, respectively [1,2]. The physics of the QSH-to-QH transition is governed by band inversion [3][4][5]: The electronlike and holelike subbands near the Fermi level are interchanged in a QSH insulator, so that the band gap in the bulk becomes smaller rather than larger with increasing perpendicular magnetic field [6,7]. The gap closing at a characteristic field B c signals the transition from an inverted QSH gap with helical edge states to a noninverted QH gap supporting chiral edge states.The early experiments on the QSH effect were performed in HgTe layers with CdTe barriers (type-I quantum wells) [8,9]. Recently the effect has also been observed in InAs/GaSb bilayers with AlSb barriers (type-II quantum wells) [10][11][12][13]. Both types of quantum wells can have electron-hole subbands in inverted order, but while these are strongly coupled in type-I quantum wells, they are spatially separated and weakly coupled in the broken-gap quantum wells of type II (see Fig. 1). Although the difference has no consequences in zero magnetic field, we will show here that the breakdown of helical edge conduction in a magnetic field becomes qualitatively different.In both type-I and type-II quantum wells we find an increase with disorder of the characteristic field B c for the QSH-to-QH transition, as a consequence of the same mechanism that is operative in topological Anderson insulators [14]: a disorderinduced renormalization of the band gap [15]. Basically, in a narrow-gap semiconductor the effect of disorder on the bulk band gap is opposite in the inverted and noninverted cases. While a noninverted band gap is reduced by disorder, the inverted band gap is increased. Since B c is proportional to the zero-field band gap, it is pushed to larger fields by impurity scattering.As a consequence, disorder increases the robustness of helical edge conduction in type-I quantum wells, such as HgTe. In contrast, we find that in broken-gap quantum...