Higher-order topological insulator phase in a modified Haldane model

We explore topological properties of a modulated Haldane model (MHM) in which the strength nearest-neighbor and next-nearest-neighbor terms is made unequal three-fold rotational symmetry $\mathcal{C}_3$ broken by introducing trimerization term ($|t_{1w(2w)}|< t_{1s(2s)}$) Hamiltonian. Using parameter $\eta={t_{1w}}/{t_{1s}}= t_{2w}/t_{2s}$, we show that MHM supports transition from quantum anomalous Hall insulator (QAHI) to HOTI phase at $\eta=\pm 0.5$. The also hosts zero-energy corner mode on nano-disk can trivial without gap-closing when inversion broken. critical states are found be magnetic semimetals with single Dirac node which, unlike classic model, move along high-symmetry lines Brillouin zone. offers rich tapestry other non-topological phases.