A Higher-Order Topological Insulator Phase in a Modulated Haldane Model

Abstract

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

3 figures for main text, and 1 figure for supplementary materials