  We investigate theoretically the electron spin states in disk-shaped HgTe
topological insulator quantum dots (TIQDs) containing a single magnetic
$Mn^{2+}$ ion. We show that the energy spectrum and the electron density
distribution of the topological edge states in HgTe TIQD can be modulated
significantly by the position of the magnetic $Mn^{2+}$ ion. The numerical
results further demonstrate that the electric fields not only tune the spin
splittings of edge states via the $\emph{sp-d}$ exchange interaction between
the electron (hole) and the magnetic $Mn^{2+}$ ion but also give rise to the
bright-to-dark transitions and anti-crossing behaviors in the photoluminescence
(PL) spectra. Such spin properties of HgTe TIQDs with single $Mn^{2+}$ ion as
illustrated in this work could offer a new platform for topological
electro-optical devices.
