Majorana Fermions in superconducting wires: effects of long-range hopping, broken time-reversal symmetry and potential landscapes

We present a comprehensive study of two of the most experimentally relevant extensions of Kitaev's spinless model of a 1D p-wave superconductor: those involving (i) longer range hopping and superconductivity and (ii) inhomogeneous potentials. We commence with a pedagogical review of the spinless model and, as a means of characterizing topological phases exhibited by the systems studied here, we introduce bulk topological invariants as well as those derived from an explicit consideration of boundary modes. In time-reversal invariant systems, we find that the longer range hopping leads to topological phases characterized by multiple Majorana modes. In particular, we investigate a spin model, which respects a duality and maps to a fermionic model with multiple Majorana modes; we highlight the connection between these topological phases and the broken symmetry phases in the original spin model. In the presence of time-reversal symmetry breaking terms, we show that the topological phase diagram is characterized by an extended gapless regime. For the case of inhomogeneous potentials, we explore phase diagrams of periodic, quasiperiodic, and disordered systems. We present a detailed mapping between normal state localization properties of such systems and the topological phases of the corresponding superconducting systems. This powerful tool allows us to leverage the analyses of Hofstadter's butterfly and the vast literature on Anderson localization to the question of Majorana modes in superconducting quasiperiodic and disordered systems, respectively. We briefly touch upon the synergistic effects that can be expected in cases where long-range hopping and disorder are both present.Comment: 30 pages, 13 figure

Accessing nanotube bands via crossed electric and magnetic fields

We investigate the properties of conduction electrons in single-walled armchair carbon nanotubes in the presence of mutually orthogonal electric and magnetic fields transverse to the tube's axis. We find that the fields give rise to an asymmetric dispersion in the right- and left-moving electrons along the tube as well as a band-dependent interaction. We predict that such a nanotube system would exhibit spin-band-charge separation and a band-dependant tunneling density of states. We show that in the quantum dot limit, the fields serve to completely tune the quantum states of electrons added to the nanotube. For each of the predicted effects, we provide examples and estimates that are relevant to experiment.Comment: 4 pages, 2 figure