Band inversion at critical magnetic fields in a silicene quantum dot

We have found out that the band inversion in a silicene quantum dot (QD), in perpendicular magnetic $B$ and electric ${\mathrm{\Delta }}_z$ fields, drastically depends on the strength of the magnetic field. We study the energy spectrum of the silicene QD where the electric field provides a tunable band gap $\mathrm{\Delta }$. Boundary conditions introduce chirality, so that negative and positive angular momentum $m$ zero Landau level (ZLL) edge states show a quite different behavior regarding the band-inversion mechanism underlying the topological insulator transition. We show that, whereas some ZLLs suffer band inversion at $\mathrm{\Delta }=0$ for any $B>0$, other ZLLs only suffer band inversion above critical values of the magnetic field at nonzero values of the gap.