  We consider a time reversal symmetry (TRS) broken Kane-Mele model
superimposed with Haldane model and chart out the phase diagram using spin
Chern number to investigate the fate of quantum anomalous Hall insulator (QAHI)
and quantum spin Hall insulator (QSHI) phases. Interestingly, in addition to
QSHI and QAHI phase, the phase diagram unveils quantum anomalous spin Hall
insulator (QASHI) phase where only one spin sector is topological. We also find
multicritical points where three / four topological phase boundaries coalesce.
These topological phases are protected by an effective TRS and a composite
anti-unitary particle-hole symmetry leading to remarkable properties of edge
modes. We find spin-selective, spin-polarized and spin-neutral edge transport
in QASHI, QSHI and QAHI phases respectively. Our study indicates that the
robustness of the topological phase mainly depends on the spin gap which does
not necessarily vanish at the Dirac points across a topological phase
transition. We believe that our proposals can be tested in near future using
recent experimental advancements in solid state and cold atomic systems.
