Fractional charge and spin states in topological insulator constrictions

We theoretically investigate the properties of two-dimensional topological insulator constrictions both in the integer and fractional regimes. In the presence of a perpendicular magnetic field, the constriction functions as a spin filter with near-perfect efficiency and can be switched by electric fields only. Domain walls between different topological phases can be created in the constriction as an interface between tunneling, magnetic fields, charge density wave, or electron-electron interaction dominated regions. These domain walls host non-Abelian bound states with fractional charge and spin and result in degenerate ground states with parafermions. If a proximity gap is induced bound states give rise to an exotic Josephson current with $8\pi$ periodicity.

Figure 1

A sketch of a TIC. Spin up (down) edge states are shown in red (blue). The TIC could be either doped with magnetic impurities or subjected to a magnetic field. The pairs of helical edge states are coupled by tunneling which results in the opening of an energy gap at zero momentum. A magnetic field applied in the plane of the TI couples edge states with opposite spins and opens a gap.

Figure 2

The energy spectrum of the TIC in the presence of magnetic fields (or exchange interactions). A magnetic field in the $z$ direction, the spin quantization axis of the TIC, shifts the Dirac cone corresponding to the upper (lower) edge, $\tau =1$ ($\tau =\overline{1}$), to the left (right) such that the Dirac point is located now at wave vector $-k_Z$ ($k_Z$). The tunneling between edges opens a gap of size $2t$ at $k=0$. A magnetic field in the $x$ direction, i.e., perpendicular to the spin quantization axis, opens a gap of the size $2{\mathrm{\Delta }}_x$ at zero energy. If the chemical potential ${\mu }_1$ (${\mu }_{\overline{1}}$) is tuned inside the gap opened by tunneling at $k=0$, only spin-polarized modes with spin up (spin down) propagate through the TIC.