The proximity effect in hybrid superconductor-semiconductor 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 non-topological zero-energy 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 low-bias 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.

Symmetry is a guiding principle in physics that allows us to generalize conclusions between many physical systems. In the ongoing search for new topological phases of matter, symmetry plays a crucial role by protecting topological phases. We address two converse questions relevant to the symmetry classification of systems: is it possible to generate all possible single-body 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 non-interacting lattice or continuum Hamiltonians. We implement the algorithms in the Qsymm Python package, and demonstrate their usefulness through applications in active research areas of condensed matter physics, including Majorana wires and Kekule graphene. continuous spatial rotations, space groups, discrete onsite symmetries (such as time reversal, particle-hole and chiral symmetries), and arbitrary combinations of these. Besides static fermionic systems, our methods are also applicable to band structures in photonic crystals [25,26], magnon spectra, classical mechanical [27, 28] and electronic systems [29], and driven Floquet systems [30].The paper is structured as follows: first we review the general symmetry structure of single-particle Hamiltonians. Then we present our algorithm to find Hamiltonians with such symmetries. After that we review the symmetry finding algorithm, which, by factoring out onsite unitary symmetries, makes finding all other symmetries more efficient and guaranteed. We implement our algorithms in the Qsymm Python package [31, 32]. Finally we provide a set of examples where our method was used on problems in active areas of research. We show that Majorana wire devices may be protected against band-tilting by a magnetic symmetry, and double Dirac cones in Kekule distorted graphene are protected by point group and sublattice symmetry. We also construct model Hamiltonians for transition metal dichalcogenides and distorted spin-orbit coupled SnTe. Hamiltonian families and symmetries 2.1. Continuum and tight-binding HamiltoniansWe focus on non-interacting Hamiltonians. The quadratic Hamilton operator of a finite (zero-dimensional) system can be written as New J. Phys. 20 (2018) 093026 D Varjas et al

We calculate electronic states on a closed cylindrical surface as a model of a core-shell nanowire. The length of the cylinder can be infinite or finite. We define cardinal points on the circumference of the cylinder and consider a spatially uniform magnetic field perpendicular to the cylinder axis, in the direction South-North. The orbital motion of the electrons depends on the radial component of the field which is nonuniform around the circumference: it is equal to the total field at North and South, but vanishes at the West and East sides. For a strong field, when the magnetic length is comparable to the radius of the cylinder, the electronic states at North and South become localized cyclotron orbits, whereas at East and West the states become long and narrow snaking orbits propagating along the cylinder. The energy of the cyclotron states increases with the magnetic field whereas the energy of the snaking states is stable. Consequently, at high magnetic fields the electron density vanishes at North and South and concentrates at East and West. We include spinorbit interaction with linear Rashba and Dresselhaus models. For a cylinder of finite length the Dresselhaus interaction produces an axial twist of the charge density relative to the center of the wire, which may be amplified in the presence of the Rashba interaction.

We study the quantum mechanical states of electrons situated on a cylindrical surface of finite axial length to model a semiconductor core-shell nanowire. We calculate the conductance in the presence of a longitudinal magnetic field by weakly coupling the cylinder to semi-infinite leads. Spin effects are accounted for through Zeeman coupling and Rashba spin-orbit interaction (SOI). Emphasis is on manifestations of flux-periodic (FP) oscillations and we show how factors such as impurities, contact geometry and spin affect them. Oscillations survive and remain periodic in the presence of impurities, noncircular contacts and SOI, while Zeeman splitting results in aperiodicity, beating patterns and additional background fluctuations. Our results are in qualitative agreement with recent magnetotransport experiments performed on GaAs/InAs core-shell nanowires. Lastly, we propose methods of data analysis for detecting the presence of Rashba SOI in core-shell systems and for estimating the electron g-factor in the shell.

We model a core-shell nanowire (CSN) by a cylindrical surface of finite length. A uniform magnetic field perpendicular to the axis of the cylinder forms electron states along the lines of zero radial field projection, which can classically be described as snaking states. In a strong field, these states converge pairwise to quasidegenerate levels, which are situated at the bottom of the energy spectrum. We calculate the conductance of the CSN by coupling it to leads and predict that the snaking states govern transport at low chemical potential, forming isolated peaks, each of which may be split in two by applying a transverse electric field. If the contacts with the leads do not completely surround the CSN, as is usually the case in experiments, the amplitude of the snaking peaks changes when the magnetic field is rotated, determined by the overlap of the contacts with the snaking states.