  We propose a time reversal symmetry (TRS) broken tight binding model, being
an admixture of Kane-Mele and Haldane model where the intrinsic SOC is coupled
with a staggered magnetic flux, to investigate the fate of quantum anomalous
Hall insulator (QAHI) and quantum spin Hall insulator (QSHI) phases. We make
resort to topological invariant namely, spin Chern number
$(\rm{C}_{\uparrow},\rm{C}_{\downarrow})$ to characterize various topological
phases. The phase diagram unveils many intriguing features such as, quantum
anomalous spin Hall insulator (QASHI) phase, denoted by
$(0,\rm{C}_{\downarrow})$ and $(\rm{C}_{\uparrow},0)$, extended critical phase
where spin Chern number does not work, multicritical points where three / four
topological phase boundaries coalesce and many more in addition to the QSHI
phase [$(\rm{C}_{\uparrow},\rm{C}_{\downarrow})$ with $\rm{C}_{\uparrow} \ne
\rm{C}_{\downarrow} \ne 0 $] and QAHI phase [$(\rm{C}_{\uparrow}
\rm{C}_{\downarrow})$ with $\rm{C}_{\uparrow} = \rm{C}_{\downarrow} \ne 0$].
These topological phases are protected by an effective TRS and a composite
anti-unitary particle-hole symmetry leading to remarkable propertis of edge
modes. We find spin-selective, spin-polarized and spin-neutral edge transport
in QASHI, QSHI and QAHI phases referring to the validity of bulk-boundary
correspondence. Our analysis with low energy model gives us the correct
understanding of the various topological phases. Moreover, our study further
indicates that spin gap plays the key role in determining the robustness of the
topological invariant as it does not necessarily vanish at the Dirac points
across a topological phase transition. Based on the current experimental
advancements in solid state and cold atomic systems, we believe that our
proposals can be tested in near future.
