  Due to nonzero intrinsic spin-orbit interaction in buckled honeycomb crystal
structures, silicene and germanene exhibit interesting topological properties,
and are therefore candidates for the realization of the quantum spin Hall
effect. We employ the Kane-Mele model to investigate the electron states in
hexagonal silicene and germanene nanorings having either zigzag or armchair
edges in the presence of a perpendicular magnetic field. We present results for
the energy spectra as function of magnetic field, the electron density of the
spin-up and spin-down states in the ring plane, and the calculation of the
probability current density. The quantum spin Hall phase is found at the edges
between the nontrivial topological phase in silicene and germanene and vacuum.
We demonstrate that the helical edge states in zigzag silicene and germanene
nanorings can be qualitatively well understood by means of classical magnetic
moments. However, this is not the case for comparable-sized armchair nanorings,
where the eigenfunctions spread throughout the ring. Finally, we note that the
energy spectra of silicene and germanene nanorings are similar and that the
differences between the two are mainly related to the difference in magnitude
of the spin-orbit coupling.
