We show that the combination of spin-orbit coupling with a Zeeman field or strong interactions may lead to the formation of a helical electron liquid in single-channel quantum wires, with spin and velocity perfectly correlated. We argue that zero-energy Majorana bound states are formed in various situations when such wires are situated in proximity to a conventional s-wave superconductor. This occurs when the external magnetic field, the superconducting gap, or, most simply, the chemical potential vary along the wire. These Majorana states do not require the presence of a vortex in the system. Experimental consequences of the helical liquid and the Majorana states are also discussed.

Topological quantum computation provides an elegant way around decoherence, as one encodes quantum information in a non-local fashion that the environment finds difficult to corrupt. Here we establish that one of the key operations-braiding of non-Abelian anyons-can be implemented in one-dimensional semiconductor wire networks. Previous work 1,2 provided a recipe for driving semiconducting wires into a topological phase supporting long-sought particles known as Majorana fermions that can store topologically protected quantum information. Majorana fermions in this setting can be transported, created, and fused by applying locally tunable gates to the wire. More importantly, we show that networks of such wires allow braiding of Majorana fermions and that they exhibit non-Abelian statistics like vortices in a p + ip superconductor. We propose experimental setups that enable the Majorana fusion rules to be probed, along with networks that allow for efficient exchange of arbitrary numbers of Majorana fermions. This work paves a new path forward in topological quantum computation that benefits from physical transparency and experimental realism. PACS numbers:arXiv:1006.4395v2 [cond-mat.mes-hall]

We show that Franck-Condon physics leads to a significant current suppression at low bias voltages (termed Franck-Condon blockade) in transport through single molecules with strong coupling between electronic and vibrational degrees of freedom. Transport in this regime is characterized by remarkably large Fano factors (10(2)-10(3) for realistic parameters), which arise due to avalanchelike transport of electrons. Avalanches occur in a self-similar manner over a wide range of time scales, leading to power-law dependences of the current noise on frequency and vibrational relaxation rate.

Second-order topological insulators are crystalline insulators with a gapped bulk and gapped crystalline boundaries, but topologically protected gapless states at the intersection of two boundaries. Without further spatial symmetries, five of the ten Altland-Zirnbauer symmetry classes allow for the existence of such second-order topological insulators in two and three dimensions. We show that reflection symmetry can be employed to systematically generate examples of second-order topological insulators and superconductors, although the topologically protected states at corners (in two dimensions) or at crystal edges (in three dimensions) continue to exist if reflection symmetry is broken. A three-dimensional second-order topological insulator with broken time-reversal symmetry shows a Hall conductance quantized in units of e 2 /h.Introduction.-After the discovery of topological insulators and superconductors and their classification for the ten Altland-Zirnbauer symmetry classes [1][2][3], the concept of nontrivial topological band structures has been extended to materials in which the crystal structure is essential for the protection of topological phases [4]. This includes weak topological insulators [5], which rely on the discrete translation symmetry of the crystal lattice, and topological crystalline insulators [6], for which other crystal symmetries are invoked to protect a topological phase. Whereas the original strong topological insulators always have topologically protected boundary states, weak topological insulators or topological crystalline insulators have protected boundary states for selected surfaces/edges only.In a recent publication, Schindler et al. [7] proposed another extension of the topological insulator (TI) family: a higher-order topological insulator. Being crystalline insulators, these have well-defined faces and well-defined edges or corners at the intersections between the faces. An nth order topological insulator has topologically protected gapless states at the intersection of n crystal faces, but is gapped otherwise [7]. For example, a second-order topological insulator in two dimensions (d = 2) has zeroenergy states at corners, but a gapped bulk and no gapless edge states. Earlier examples of higher-order topological insulators and superconductors avant la lettre appeared in works by (see also [11,12]), who considered insulators and superconductors with protected corner states in d = 2 and d = 3 [13]. Sitte et al. showed that a threedimensional topological insulator in a magnetic field of generic direction also acquires the characteristics of a second-order topological Chern insulator, with chiral states moving along the sample edges [14].Since a second-order TI has a topologically trivial d-dimensional bulk, from a topological point of view its boundaries are essentially stand-alone (d − 1)-dimensional insulators, so that topologically protected states at corners (for d = 2) or edges (for d = 3) arise naturally as "domain walls" at the intersection of two boundaries if these ar...

Recently, it has been suggested that topological superconductivity and Majorana end states can be realized in a chain of magnetic impurities on the surface of an s-wave superconductor when the magnetic moments form a spin helix as a result of the RKKY interaction mediated by the superconducting substrate. Here, we investigate this scenario theoretically by developing a tight-binding Bogoliubov-de Gennes description starting from the Shiba bound states induced by the individual magnetic impurities. While the resulting model Hamiltonian has similarities with the Kitaev model for one-dimensional spinless p-wave superconductors, there are also important differences, most notably the long-range nature of hopping and pairing as well as the complex hopping amplitudes. We use both analytical and numerical approaches to explore the consequences of these differences for the phase diagram and the localization properties of the Majorana end states when the Shiba chain is in a topological superconducting phase.

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Quantum mechanics predicts that the equilibrium state of a resistive electrical circuit contains a dissipationless current. This persistent current has been the focus of considerable theoretical and experimental work, but its basic properties remain a topic of controversy. The main experimental challenges in studying persistent currents have been the small signals they produce and their exceptional sensitivity to their environment. To address these issues we have developed a new technique for detecting persistent currents which offers greatly improved sensitivity and reduced measurement back action. This allows us to measure the persistent current in metal rings over a wider range of temperature, ring size, and magnetic field than has been possible previously. We find that measurements of both a single ring and arrays of rings agree well with calculations based on a model of non-interacting electrons.An electrical current induced in a resistive circuit will rapidly decay in the absence of an applied voltage. This decay reflects the tendency of the circuit's electrons to dissipate energy and relax to their ground state. However quantum mechanics predicts that the electrons' many-body ground state (and, at finite temperature, their thermal equilibrium state) may itself contain a "persistent" current which flows through the resistive circuit without dissipating energy or decaying. A dissipationless equilibrium current flowing through a resistive circuit is highly counterintuitive, but it has a familiar analog in atomic physics: some atomic species' electronic ground states possess non-zero orbital angular momentum, equivalent to a current circulating around the atom.Theoretical treatments of persistent currents (PC) in resistive metal rings have been developed over a number of decades (see [1,2] and references therein). Calculations which take 1

Strong coupling of electronic and vibrational degrees of freedom entails a low-bias suppression of the current through single-molecule devices, termed Franck-Condon blockade. In the limit of slow vibrational relaxation, transport in the Franck-Condon-blockade regime proceeds via avalanches of large numbers of electrons, which are interrupted by long waiting times without electron transfer. The avalanches consist of smaller avalanches, leading to a self-similar hierarchy which terminates once the number of transferred electrons per avalanche becomes of the order of unity. Experimental signatures of self-similar avalanche transport are strongly enhanced current (shot) noise, as expressed by giant Fano factors, and a power-law noise spectrum. We develop a theory of the Franck-Condonblockade regime with particular emphasis on effects of electron cotunneling through highly excited vibrational states. As opposed to the exponential suppression of sequential tunneling rates for low-lying vibrational states, cotunneling rates suffer only a power-law suppression. This leads to a regime where cotunneling dominates the current for any gate voltage. Including cotunneling within a rate-equation approach to transport, we find that both the Franck-Condon blockade and self-similar avalanche transport remain intact in this regime. We predict that cotunneling leads to absorptioninduced vibrational sidebands in the Coulomb-blockaded regime as well as intrinsic telegraph noise near the charge degeneracy point.