Electric-field-generated topological states in a silicene nanotube

Applying an electric field perpendicular to the axis of a silicene armchair nanotube allows us to numerically study the formation of eight topological edge states as silicene's intrinsic spin-orbit gap is closed by the sublattice-staggered electrostatic potential created by the electric field. Following their evolution with electric field, it is revealed that, at very small fields, these eight states are very broad, spin-locked, and sublattice constrained, inheriting their properties from the K and K' states in a silicene two-dimensional honeycomb lattice. Four of those states are centered at the very top of the nanotube and the other four states are centered at the very bottom. As the field increases, each state starts to become narrower and to spread its spectral weight to the other sublattice. With further increase of the field, each state starts to spatially split, while the sublattice spreading continues. Once the spectral weight of each state is distributed evenly among both sublattices, the state has also effectively split into two spatially disconnected parts, after which, further increasing of the field will spread apart the two halves, moving them to the lateral regions of the nanotube, at the same time that the state halves become narrower. This is consistent with the formation of topological edge states, which delimit four ribbon-like topologically different regions: top and bottom topologically trivial 'ribbons' (where the electric field has induced a topological phase transition) that are adjacent to two topologically nontrivial 'ribbons' located at opposing sides of the nanotube. We also briefly access the possibility of observing these edge states by calculating the electronic properties for an electric field configuration that can be more readily produced in the laboratory.