Spin connection and boundary states in a topological insulator

Abstract

We study the surface resistivity of a three-dimensional topological insulator when the boundaries exhibit a non trivial curvature. We obtain an analytical solution for a spherical topological insulator, and we show that a non trivial quantum spin connection emerges from the three dimensional band structure. We analyze the effect of the spin connection on the scattering by a bump on a flat surface. Quantum effects induced by the geometry lead to resonances when the electron wavelength is comparable to the size of the bump.Comment: 11 pages, 4 figures, submitte

Energy spectrum and broken spin-surface locking in topological insulator quantum dots

“…In fact, the explicit solution of the problem in a spherical geometry exhibits spin-surface locking. 17,18 We note in passing that for a flat TI surface, a time-dependent out-of-plane spin component can also be generated by elastic disorder. However, this component will precess around the momentum-dependent spinorbit axis (which lies in the plane) and averages to zero on time scales corresponding to the inverse Fermi energy.…”

“…For a spherical TI dot, the band structure was worked out before. [17][18][19] However, we draw attention to several features that only arise when the surface contains sharp edges, i.e., non-differentiable parts, as is the case for the cylinder. We employ three different and independent approaches to understand TI quantum dot energy levels and their spin texture: (1) For a cylindrical TI nanowire of length L and radius R closed by flat caps, we have performed detailed numerical calculations for the energy spectrum and the spin texture of the eigenstates based on the effective low-energy theory of Zhang et geometry and find qualitatively similar results.…”

“…V. Appendix A contains a derivation of the surface Dirac fermion Hamiltonian for an infinitely long nanowire using the approach of Ref. 18.…”

Random-matrix theory of Majorana fermions and topological superconductors

“…We rely on random scattering by disorder to produce a finite density of states at E = 0, but in order to preserve the chiral symmetry the disorder cannot be electrostatic (V must remain zero). Scattering by a random vector potential is one possibility (Ludwig et al, 1994;Motrunich, Damle, and Huse, 2002), or alternatively scattering by random surface deformations Parente et al, 2011). The coupling to a superconductor at Fermi energy E F → 0 introduces particle-hole symmetry without breaking the chiral symmetry 15 of the BdG Hamiltonian 15 Eq.…”

Kondo effect in a topological insulator quantum dot

“…The similar conclusion can also be applied to a three-dimensional TIQD, where its surface states can be approximated by Dirac equations with spin connection. 61,62 Therefore the low energy spectrum of a circular shaped TIQD is linear against the angular momentum quantum number m, and the low energy edge states are described by a four-band effective Hamiltonian in basis | ↑ + , | ↑ − , | ↓ + , and | ↓ − . The effective Hamiltonian then reads…”