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Creating a transmon qubit using semiconductor-superconductor hybrid materials not only provides electrostatic control of the qubit frequency, it also allows parts of the circuit to be electrically connected and disconnected in situ by operating a semiconductor region of the device as a field-effect transistor (FET). Here, we exploit this feature to compare in the same device characteristics of the qubit, such as frequency and relaxation time, with related transport properties such as critical supercurrent and normal-state resistance. Gradually opening the FET to the monitoring circuit allows the influence of weak-to-strong DC monitoring of a live qubit to be measured. A model of this influence yields excellent agreement with experiment, demonstrating a relaxation rate mediated by a gate-controlled environmental coupling.

Majorana zero modes (MZMs), fundamental building blocks for realizing topological quantum computers, can appear at the interface between a superconductor and a topological material. One of the experimental signatures that has been widely pursued to confirm the existence of MZMs is the observation of a large, quantized zero-bias conductance peak (ZBCP) in the differential conductance measurements. In this Letter, we report observation of such a large ZBCP in junction structures of normal metal (titanium/gold Ti/Au) + Dirac semimetal (cadmium arsenide Cd3As2) + conventional superconductor (aluminum Al), with a value close to four times that of the normal state conductance. Our detailed analyses suggest that this large ZBCP is most likely not caused by MZMs. We attribute the ZBCP, instead, to the existence of a supercurrent between two far-separated superconducting Al electrodes, which shows up as a zero-bias peak because of the circuitry and thermal fluctuations of the supercurrent phase, a mechanism conceived by Ivanchenko and Zil'berman more than 50 years ago [JETP 28, 1272 (1969)]. Our results thus call for extreme caution when assigning the origin of a large ZBCP to MZMs in a multiterminal semiconductor or topological insulator/semimetal setup. We thus provide criteria for identifying when the ZBCP is definitely not caused by an MZM. Furthermore, we present several remarkable experimental results of a supercurrent effect occurring over unusually long distances and clean perfect Andreev reflection features., Comment: with Supplemental Material, which includes theoretical modeling, experimental raw data sets used in Figure 3, and data fitting code

Electrons in clean macroscopic samples of graphene exhibit an astonishing variety of quantum phases when strong perpendicular magnetic field is applied. These include integer and fractional quantum Hall states as well as symmetry broken phases and quantum Hall ferromagnetism. Here we show that mesoscopic graphene flakes in the regime of strong disorder and magnetic field can exhibit another remarkable quantum phase described by holographic duality to an extremal black hole in two dimensional anti-de Sitter space. This phase of matter can be characterized as a maximally chaotic non-Fermi liquid since it is described by a complex fermion version of the Sachdev-Ye-Kitaev model known to possess these remarkable properties., Comment: 5 pages main text plus 4 pages supplement; V2 published version

System of Majorana zero modes with random infinite range interactions -- the Sachdev-Ye-Kitaev (SYK) model -- is thought to exhibit an intriguing relation to the horizons of extremal black holes in two-dimensional anti-de Sitter (AdS$_2$) space. This connection provides a rare example of holographic duality between a solvable quantum-mechanical model and dilaton gravity. Here we propose a physical realization of the SYK model in a solid state system. The proposed setup employs the Fu-Kane superconductor realized at the interface between a three dimensional topological insulator (TI) and an ordinary superconductor. The requisite $N$ Majorana zero modes are bound to a nanoscale hole fabricated in the superconductor that is threaded by $N$ quanta of magnetic flux. We show that when the system is tuned to the surface neutrality point (i.e. chemical potential coincident with the Dirac point of the TI surface state) and the hole has sufficiently irregular shape, the Majorana zero modes are described by the SYK Hamiltonian. We perform extensive numerical simulations to demonstrate that the system indeed exhibits physical properties expected of the SYK model, including thermodynamic quantities and two-point as well as four-point correlators, and discuss ways in which these can be observed experimentally., Comment: 16 pages, 9 figures; v3

A linear Josephson junction mediated by the surface states of a time-reversal-invariant Weyl or Dirac semimetal localizes Majorana flat bands protected by the time-reversal symmetry. We show that as a result, the Josephson current exhibits a discontinuous jump at $\pi$ phase difference which can serve as an experimental signature of the Majorana bands. The magnitude of the jump scales proportionally to the junction width and the momentum space distance between the Weyl nodes. It also exhibits a characteristic dependence on the junction orientation. We demonstrate that the jump is robust against the effects of non-zero temperature and weak non-magnetic disorder., Comment: 8 pages, 7 figures

When magnetic field $B$ is applied to a metal, nearly all observable quantities exhibit oscillations periodic in $1/B$. Such quantum oscillations reflect the fundamental reorganization of electron states into Landau levels as a canonical response of the metal to the applied magnetic field. We predict here that, remarkably, in the recently discovered Dirac and Weyl semimetals quantum oscillations can occur in the complete absence of magnetic field. These zero-field quantum oscillations are driven by elastic strain which, in the space of the low-energy Dirac fermions, acts as a chiral gauge potential. We propose an experimental setup in which the strain in a thin film (or nanowire) can generate pseudomagnetic field $b$ as large as 15T and demonstrate the resulting de Haas-van Alphen and Shubnikov-de Haas oscillations periodic in $1/b$., Comment: 8 pages, 7 figures

Dirac and Weyl semimetals form an ideal platform for testing ideas developed in high energy physics to describe massless relativistic particles. One such quintessentially field-theoretic idea of chiral anomaly already resulted in the prediction and subsequent observation of the pronounced negative magnetoresistance in these novel materials for parallel electric and magnetic fields. Here we predict that the chiral anomaly occurs - and has experimentally observable consequences - when real electromagnetic fields E and B are replaced by strain-induced pseudo-electromagnetic fields e and b. For example, a uniform pseudomagnetic field b is generated when a Weyl semimetal nanowire is put under torsion. In accord with the chiral anomaly equation we predict a negative contribution to the wire resistance proportional to the square of the torsion strength. Remarkably, left and right moving chiral modes are then spatially segregated to the bulk and surface of the wire forming a "topological coaxial cable". This produces hydrodynamic flow with potentially very long relaxation time. Another effect we predict is the ultrasonic attenuation and electromagnetic emission due to a time periodic mechanical deformation causing pseudoelectric field e. These novel manifestations of the chiral anomaly are most striking in the semimetals with a single pair of Weyl nodes but also occur in Dirac semimetals such as Cd3As2 and Na3Bi and Weyl semimetals with unbroken time reversal symmetry., Comment: 14 pages main text + 4 pages appendices; v2 to appear in PRX, one figure and some text added to improve clarity of presentation

Weak topological phases are usually described in terms of protection by the lattice translation symmetry. Their characterization explicitly relies on periodicity since weak invariants are expressed in terms of the momentum-space torus. We prove the compatibility of weak topological superconductors with aperiodic systems, such as quasicrystals. We go beyond usual descriptions of weak topological phases and introduce a novel, real-space formulation of the weak invariant, based on the Clifford pseudospectrum. A non-trivial value of this index implies a non-trivial bulk phase, which is robust against disorder and hosts localized zero-energy modes at the edge. Our recipe for determining the weak invariant is directly applicable to any finite-sized system, including disordered lattice models. This direct method enables a quantitative analysis of the level of disorder the topological protection can withstand., Comment: 5 pages, 2 figures + 4 pages, 4 figures appendix. This is the final, published version

Topological phases of matter that depend for their existence on interactions are fundamentally interesting and potentially useful as platforms for future quantum computers. Despite the multitude of theoretical proposals the only interaction-enabled topological phase experimentally observed is the fractional quantum Hall liquid. To help identify other systems that can give rise to such phases we present in this work a detailed study of the effect of interactions on Majorana zero modes bound to vortices in a superconducting surface of a 3D topological insulator. This system is of interest because, as was recently pointed out, it can be tuned into the regime of strong interactions. We start with a 0D system suggesting an experimental realization of the interaction-induced $\mathbb{Z}_8$ ground state periodicity previously discussed by Fidkowski and Kitaev. We argue that the periodicity is experimentally observable using a tunnel probe. We then focus on interaction-enabled crystalline topological phases that can be built with the Majoranas in a vortex lattice in higher dimensions. In 1D we identify an interesting exactly solvable model which is related to a previously discussed one that exhibits an interaction-enabled topological phase. We study these models using analytical techniques, exact numerical diagonalization (ED) and density matrix renormalization group (DMRG). Our results confirm the existence of the interaction-enabled topological phase and clarify the nature of the quantum phase transition that leads to it. We finish with a discussion of models in dimensions 2 and 3 that produce similar interaction-enabled topological phases., Comment: 14 pages, 8 figures; v2

Band-inverted electron-hole bilayers support quantum spin Hall insulator and exciton condensate phases. We investigate such a bilayer in an external magnetic field. We show that the interlayer correlations lead to formation of a helical quantum Hall exciton condensate state. In contrast to the chiral edge states of the quantum Hall exciton condensate in electron-electron bilayers, existence of the counterpropagating edge modes results in formation of a ground state spin-texture not supporting gapless single-particle excitations. This feature has deep consequences for the low energy behavior of the system. Namely, the charged edge excitations in a sufficiently narrow Hall bar are confined, i.e.~a charge on one of the edges always gives rise to an opposite charge on the other edge. Moreover, we show that magnetic field and gate voltages allow to control confinement-deconfinement transition of charged edge excitations, which can be probed with nonlocal conductance. Confinement-deconfinement transitions are of great interest, not least because of their possible significance in shedding light on the confinement problem of quarks., Comment: 19 pages, 8 figures; v3: added Landau level diagrams and discussion of quantization breakdown

Systems of strongly interacting particles, fermions or bosons, can give rise to topological phases that are not acessible to non-interacting systems. Many such interaction-enabled topological phases have been discussed theoretically but few experimental realizations exists. Here we propose a new platform for interacting topological phases of fermions with time reversal symmetry $\bar T$ (such that $\bar T^2=1$) that can be realized in vortex lattices in the surface state of a topological insulator. The constituent particles are Majorana fermions bound to vortices and antivortices of such a lattice. We explain how the $\bar T$ symmetry arises and discuss ways in which interactions can be experimentally tuned and detected. We show how these features can be exploited to realize a class of interaction-enabled crystalline topological phases that have no analog in weakly interacting systems., Comment: 5 pages with 3 figures; v2 to appear in PRB/RC. Added 2 page supplement to address robustness against disorder and other technical issues

We extend the single-particle topological classification of insulators and superconductors to include systems in which disorder preserves average reflection symmetry. We show that the topological group structure of bulk Hamiltonians and topological defects is exponentially extended when this additional condition is met, and examine some of its physical consequences. Those include localization-delocalization transitions between topological phases with the same boundary conductance, as well as gapless topological defects stabilized by average reflection symmetry., Comment: 8 pages, 5 figures, 1 table; improved section 4 "Extended topological classification" incl. example of stacked QSH layers

Interesting phases of quantum matter often arise when the constituent particles -- electrons in solids -- interact strongly. Such strongly interacting systems are however quite rare and occur only in extreme environments of low spatial dimension, low temperatures or intense magnetic fields. Here we introduce a new system in which the fundamental electrons interact only weakly but the low energy effective theory is described by strongly interacting Majorana fermions. The system consists of an Abrikosov vortex lattice in the surface of a strong topological insulator and is accessible experimentally using presently available technology. The simplest interactions between the Majorana degrees of freedom exhibit an unusual nonlocal structure that involves four distinct Majorana sites. We formulate simple lattice models with this type of interaction and find exact solutions in certain physically relevant one- and two-dimensional geometries. In other cases we show how our construction allows for the experimental realization of interesting spin models previously only theoretically contemplated., Comment: 16 pages, 5 figures. The third version has been accepted for publication in Physical Review B

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 $N\times N$ 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 $\propto\sqrt{N}$ 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)., Comment: Contribution for the JETP special issue in honor of A.F. Andreev's 75th birthday. 9 pages, 8 figures

A two-dimensional superconductor with spin-triplet p-wave pairing supports chiral or helical Majorana edge modes with a quantized (length $L$-independent) thermal conductance. Sufficiently strong anisotropy removes both chirality and helicity, doubling the conductance in the clean system and imposing a super-Ohmic $1/\sqrt{L}$ decay in the presence of disorder. We explain the absence of localization in the framework of the Kitaev Hamiltonian, contrasting the edge modes of the two-dimensional system with the one-dimensional Kitaev chain. While the disordered Kitaev chain has a log-normal conductance distribution peaked at an exponentially small value, the Kitaev edge has a bimodal distribution with a second peak near the conductance quantum. Shot noise provides an alternative, purely electrical method of detection of these charge-neutral edge modes., Comment: 11 pages, 13 figures

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 quasi-metallic 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., Comment: 4 pages, 6 figures; Corrected a factor of two error in the magnetic field scale and effective g-factor

We study the phase diagram of the inverted InAs/GaSb bilayer quantum wells. For small tunneling amplitude between the layers, we find that the system is prone to formation of an s-wave exciton condensate phase, where the spin-structure of the order parameter is uniquely determined by the small spin-orbit coupling arising from the bulk inversion asymmetry. The phase is topologically trivial and does not support edge transport. On the contrary, for large tunneling amplitude, we obtain a topologically non-trivial quantum spin Hall insulator phase with a p-wave exciton order parameter, which enhances the hybridization gap. These topologically distinct insulators are separated by an insulating phase with spontaneously broken time-reversal symmetry. Close to the phase transition between the quantum spin Hall and time-reversal broken phases, the edge transport shows quantized conductance in small samples, whereas in long samples the mean free path associated with the backscattering at the edge is temperature independent, in agreement with recent experiments., Comment: v. 2, 9 pages, 5 figures

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 2pi phase interval scales as sqrt(N) and their spacing distribution is a hybrid of the Wigner and Poisson distributions of random-matrix theory., Comment: 12 pages, 15 figures; v2 to appear in PRL, with appendix in the supplementary material

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-voltage-averaged differential conductance as a 4e^2/h zero-bias peak above a (2/3{\pi}^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., Comment: 6 pages, 4 figures

We calculate the Andreev conductance of a superconducting ring interrupted by a flux-biased Josephson junction, searching for electrical signatures of circulating edge states. Two-dimensional pair potentials of spin-singlet d-wave and spin-triplet p-wave symmetry support, respectively, (chiral) Dirac modes and (chiral or helical) Majorana modes. These produce h/e-periodic magnetoconductance oscillations of amplitude \simeq (e^{2}/h)N^{-1/2}, measured via an N-mode 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., Comment: 10 pages, 6 figures. New appendix on the gauge invariant discretization of the Bogoliubov-De Gennes equation. Accepted for publication in PRB

The helical edge state of a quantum spin-Hall insulator can carry a supercurrent in equilibrium between two superconducting electrodes (separation L, coherence length xi). We calculate the maximum (critical) current I_c that can flow without dissipation along a single edge, going beyond the short-junction restriction L<

We show that weak antilocalization by disorder competes with resonant Andreev reflection from a Majorana zero-mode to produce a zero-voltage conductance peak of order e^2/h in a superconducting nanowire. The phase conjugation needed for quantum interference to survive a disorder average is provided by particle-hole symmetry - in the absence of time-reversal symmetry and without requiring a topologically nontrivial phase. We identify methods to distinguish the Majorana resonance from the weak antilocalization effect., Comment: 13 pages, 8 figures. Addendum, February 2014: Appendix B shows results for weak antilocalization in the circular ensemble. (This appendix is not in the published version.)

We derive a generic phenomenological model of a Majorana Josephson junction that accounts for avoided crossing of Andreev states, and investigate its dynamics at constant bias voltage to reveal an unexpected pattern of any-\pi\ Josephson effect in the limit of slow decoherence.

Motivated by recent interest in the Nernst effect in cuprate superconductors, we calculate this magneto-thermo-electric effect for an arbitrary (anisotropic) quasiparticle dispersion relation and elastic scattering rate. The exact solution of the linearized Boltzmann equation is compared with the commonly used relaxation-time approximation. We find qualitative deficiencies of this approximation, to the extent that it can get the sign wrong of the Nernst coefficient. Ziman's improvement of the relaxation-time approximation, which becomes exact when the Fermi surface is isotropic, also cannot capture the combined effects of anisotropy in dispersion and scattering.

Motivated by recent developments in the field of one-dimensional topological superconductors, we investigate the topological properties of s-matrix of generic superconducting junctions where dimension should not play any role. We argue that for a finite junction the s-matrix is always topologically trivial. We resolve an apparent contradiction with the previous results by taking into account the low-energy resonant poles of s-matrix. Thus no common topological transition occur in a finite junction. We reveal a transition of a different kind that concerns the configuration of the resonant poles.

We study a penetration of an electron with high energy E<~(a/L)^2 is controlled by rare configurations of disorder, corresponding to this tail., Comment: 4 pages, 5 figures