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

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