  The inverted band structure discovered in InAs/GaSb quantum well (QW) is
found to host the topological quantum spin Hall (QSH) states. A QSH insulator
hosts counterpropagating spin-polarized edge states that are protected by the
time-reversal symmetry. The latest experiment reported a robust quantized Hall
conductance arising from these QSH states that persists in an in-plane magnetic
field as strong as $12$ Tesla. Based on the result of this experiment, we
present here a precise calculation of the effective in-plane Lande-\textit{g}
factor. We based our calculations on the tight-binding Hamiltonian projected on
a square lattice that reproduces a slightly modified Bernevig-Hughes-Zhang
(BHZ) Hamiltonian. We also study the topological phase transitions
\textit{w.r.t.} a magnetic doping. At suitable doping, one type of spin states
penetrate to the bulk of the QW and the system also enters the Quantum
Anomalous Hall (QAH) state. We further confirm this through the calculations of
quantum Hall conductance which shows a plateau at $e^2/h$ rather than $2e^2/h$
at such a doping state. The paper predicts a certain range of controllable
parameters in an inverted QW for enabling a dissipationless charge transport
needed for spintronics application.
