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We consider a time reversal symmetry (TRS) broken Kane-Mele modelsuperimposed with Haldane model and chart out the phase diagram using spinChern number to investigate the fate of quantum anomalous Hall insulator (QAHI)and quantum spin Hall insulator (QSHI) phases. Interestingly, in addition toQSHI and QAHI phase, the phase diagram unveils quantum anomalous spin Hallinsulator (QASHI) phase where only one spin sector is topological. We also findmulticritical points where three / four topological phase boundaries coalesce.These topological phases are protected by an effective TRS and a compositeanti-unitary particle-hole symmetry leading to remarkable properties of edgemodes. We find spin-selective, spin-polarized and spin-neutral edge transportin QASHI, QSHI and QAHI phases respectively. Our study indicates that therobustness of the topological phase mainly depends on the spin gap which doesnot necessarily vanish at the Dirac points across a topological phasetransition. We believe that our proposals can be tested in near future usingrecent experimental advancements in solid state and cold atomic systems.

Eight fold quantum Hall phases in a time reversal symmetry broken tight binding model