  Occurrence of topological Anderson insulator (TAI) in HgTe quantum well
suggests that when time-reversal symmetry (TRS) is maintained, the pertinent
topological phase transition, marked by re-entrant $2e^2/h$ quantized
conductance contributed by helical edge states, is driven by disorder. Here we
show that when TRS is broken, the physics of TAI becomes even richer. The
pattern of longitudinal conductance and nonequilibrium local current %Unlike
for conventional topological insulators that, in the %absence of an external
magnetic field, support only a single quantized %conductance in the quantum
anomalous Hall effect region or a single %re-entrant quantized conductance in
TAI, %our model exhibits novel TAI distribution displays novel TAI phases
characterized by nonzero Chern numbers, indicating the occurrence of multiple
chiral edge modes. Tuning either disorder or Fermi energy (in both
topologically trivial and nontrivial phases), drives transitions between these
distinct TAI phases, characterized by jumps of the quantized conductance from
$0$ to $e^2/h$ and from $e^2/h$ to $2e^2/h$. An effective medium theory based
on the Born approximation yields an accurate description of different TAI
phases in parameter space.
