  The influence of bulk inversion asymmetry in [001] and [013] grown HgTe
quantum wells is investigated theoretically. The bulk inversion asymmetry leads
to an anti-crossing gap between two zero-mode Landau levels in a HgTe
topological insulator, i.e., the quantum well with inverted band structure. It
is found that this is the main contribution to the anti-crossing splitting
observed in recent experimental magneto spectroscopic measurements. The
relevant optical transitions involve different subbands, but the
electron-electron interaction induced depolarization shift is found to be
negligibly small. It is also found that the splitting of this anti-crossing
only depends weakly on the tilting angle when the magnetic field is tilted away
from the perpendicular direction to the quantum well. Thus, the strength of
bulk inversion asymmetry can be determined via a direct comparison between the
theoretical calculated one-electron energy levels and experimentally observed
anti-crossing energy gap.
