Topological Edge States and Fractional Quantum Hall Effect from Umklapp Scattering

We study anisotropic lattice strips in the presence of a magnetic field in the quantum Hall effect regime. At specific magnetic fields, causing resonant Umklapp scattering, the system is gapped in the bulk and supports chiral edge states in close analogy to topological insulators. These gaps result in plateaus for the Hall conductivity exactly at the known fillings n/m (both positive integers and m odd) for the integer and fractional quantum Hall effect. For double strips we find topological phase transitions with phases that support midgap edge states with flat dispersion. The topological effects predicted here could be tested directly in optical lattices

Fermionic and Majorana Bound States in Hybrid Nanowires with Non-Uniform Spin-Orbit Interaction

We study intragap bound states in the topological phase of a Rashba nanowire in the presence of a magnetic field and with non-uniform spin orbit interaction (SOI) and proximity-induced superconductivity gap. We show that fermionic bound states (FBS) can emerge inside the proximity gap. They are localized at the junction between two wire sections characterized by different directions of the SOI vectors, and they coexist with Majorana bound states (MBS) localized at the nanowire ends. The energy of the FBS is determined by the angle between the SOI vectors and the lengthscale over which the SOI changes compared to the Fermi wavelength and the localization length. We also consider double-junctions and show that the two emerging FBSs can hybridize and form a double quantum dot-like structure inside the gap. We find explicit analytical solutions of the bound states and their energies for certain parameter regimes such as weak and strong SOI. The analytical results are confirmed and complemented by an independent numerical tight-binding model approach. Such FBS can act as quasiparticle traps and thus can have implications for topological quantum computing schemes based on braiding MBSs

Giant spin orbit interaction due to rotating magnetic fields in graphene nanoribbons

We theoretically study graphene nanoribbons in the presence of spatially varying magnetic fields produced e.g. by nanomagnets. We show both analytically and numerically that an exceptionally large Rashba spin orbit interaction (SOI) of the order of 10 meV can be produced by the non-uniform magnetic field. As a consequence, helical modes exist in armchair nanoribbons that exhibit nearly perfect spin polarization and are robust against boundary defects. This paves the way to realizing spin filter devices in graphene nanoribbons in the temperature regime of a few Kelvins. If a nanoribbon in the helical regime is in proximity contact to an s-wave superconductor, the nanoribbon can be tuned into a topological phase sustaining Majorana fermions

Fractional Fermions with Non-Abelian Statistics

We introduce a novel class of low-dimensional topological tight-binding models that allow for bound states that are fractionally charged fermions and exhibit non-Abelian braiding statistics. The proposed model consists of a double (single) ladder of spinless (spinful) fermions in the presence of magnetic fields. We study the system analytically in the continuum limit as well as numerically in the tight-binding representation. We find a topological phase transition with a topological gap that closes and reopens as a function of system parameters and chemical potential. The topological phase is of the type BDI and carries two degenerate mid-gap bound states that are localized at opposite ends of the ladders. We show numerically that these bound states are robust against a wide class of perturbations

Parafermions in Interacting Nanowire Bundle

We propose a scheme to induce $\mathbb{Z}_3$ parafermion modes, exotic zero-energy bound states that possess non-Abelian statistics. We consider a minimal setup consisting of a bundle of four tunnel coupled nanowires hosting spinless electrons that interact strongly with each other. The hallmark of our setup is that it relies only on simple one-dimensional wires, uniform magnetic fields, and strong interactions, but does not require the presence of superconductivity or exotic quantum Hall phases

Integer and Fractional Quantum Hall Effect in a Strip of Stripes

We study anisotropic stripe models of interacting electrons in the presence of magnetic fields in the quantum Hall regime with integer and fractional filling factors. The model consists of an infinite strip of finite width that contains periodically arranged stripes (forming supercells) to which the electrons are confined and between which they can hop with associated magnetic phases. The interacting electron system within the one-dimensional stripes are described by Luttinger liquids and shown to give rise to charge and spin density waves that lead to periodic structures within the stripe with a reciprocal wavevector 8k_F. This wavevector gives rise to Umklapp scattering and resonant scattering that results in gaps and chiral edge states at all known integer and fractional filling factors \nu. The integer and odd denominator filling factors arise for a uniform distribution of stripes, whereas the even denominator filling factors arise for a non-uniform stripe distribution. We calculate the Hall conductance via the Streda formula and show that it is given by \sigma_H=\nu e^2/h for all filling factors. We show that the composite fermion picture follows directly from the condition of the resonant Umklapp scattering

RKKY interaction in carbon nanotubes and graphene nanoribbons

We study Rudermann-Kittel-Kasuya-Yosida (RKKY) interaction in carbon nanotubes (CNTs) and graphene nanoribbons in the presence of spin orbit interactions and magnetic fields. For this we evaluate the static spin susceptibility tensor in real space in various regimes at zero temperature. In metallic CNTs the RKKY interaction depends strongly on the sublattice and, at the Dirac point, is purely ferromagnetic (antiferromagnetic) for the localized spins on the same (different) sublattice, whereas in semiconducting CNTs the spin susceptibility depends only weakly on the sublattice and is dominantly ferromagnetic. The spin orbit interactions break the SU(2) spin symmetry of the system, leading to an anisotropic RKKY interaction of Ising and Moryia-Dzyaloshinsky form, besides the usual isotropic Heisenberg interaction. All these RKKY terms can be made of comparable magnitude by tuning the Fermi level close to the gap induced by the spin orbit interaction. We further calculate the spin susceptibility also at finite frequencies and thereby obtain the spin noise in real space via the fluctuation-dissipation theorem

Majorana Fermions in Ge/Si Hole Nanowires

We consider Ge/Si core/shell nanowires with hole states coupled to an $s$-wave superconductor in the presence of electric and magnetic fields. We employ a microscopic model that takes into account material-specific details of the band structure such as strong and electrically tunable Rashba-type spin-orbit interaction and $g$ factor anisotropy for the holes. In addition, the proximity-induced superconductivity Hamiltonian is derived starting from a microscopic model. In the topological phase, the nanowires host Majorana fermions with localization lengths that depend strongly on both the magnetic and electric fields. We identify the optimal regime in terms of the directions and magnitudes of the fields in which the Majorana fermions are the most localized at the nanowire ends. In short nanowires, the Majorana fermions hybridize and form a subgap fermion whose energy is split away from zero and oscillates as a function of the applied fields. The period of these oscillations could be used to measure the dependence of the spin-orbit interaction on the applied electric field and the $g$ factor anisotropy.Comment: 11 pages, 7 figure

Second Order Topological Superconductivity in π\pi-Junction Rashba Layers

We consider a Josephson junction bilayer consisting of two tunnel-coupled two-dimensional electron gas layers with Rashba spin-orbit interaction, proximitized by a top and bottom $s$-wave superconductor with phase difference $\phi$ close to $\pi$. We show that, in the presence of a finite weak in-plane Zeeman field, the bilayer can be driven into a second order topological superconducting phase, hosting two Majorana corner states (MCSs). If $\phi=\pi$, in a rectangular geometry, these zero-energy bound states are located at two opposite corners determined by the direction of the Zeeman field. If the phase difference $\phi$ deviates from $\pi$ by a critical value, one of the two MCSs gets relocated to an adjacent corner. As the phase difference $\phi$ increases further, the system becomes trivially gapped. The obtained MCSs are robust against static and magnetic disorder. We propose two setups that could realize such a model: one is based on controlling $\phi$ by magnetic flux, the other involves an additional layer of randomly-oriented magnetic impurities responsible for the phase shift of $\pi$ in the proximity-induced superconducting pairing

Finite-size effects in a nanowire strongly coupled to a thin superconducting shell

We study the proximity effect in a one-dimensional nanowire strongly coupled to a finite superconductor with a characteristic size which is much shorter than its coherence length. Such geometries have become increasingly relevant in recent years in the experimental search for Majorana fermions with the development of thin epitaxial Al shells which form a very strong contact with either InAs or InSb nanowires. So far, however, no theoretical treatment of the proximity effect in these systems has accounted for the finite size of the superconducting film. We show that the finite-size effects become very detrimental when the level spacing of the superconductor greatly exceeds its energy gap. Without any fine-tuning of the size of the superconductor (on the scale of the Fermi wavelength), the tunneling energy scale must be larger than the level spacing in order to reach the hard gap regime which is seen ubiquitously in the experiments. However, in this regime, the large tunneling energy scale induces a large shift in the effective chemical potential of the nanowire and pushes the topological phase transition to magnetic field strengths which exceed the critical field of Al.Comment: 14 pages, 9 figure