Multiple higher-order topological phases with even and odd pairs of zero-energy corner modes in a C3symmetry broken model

Abstract

Here we study emergent higher-order topological (HOTI) phases in the extended Haldane model without C 3 symmetry. For the inversion symmetric case, the QSHI and QAHI phases can embed the HOTI phases while the remaining QASHI phase does not yield any HOTI phases. Remarkably, four-fold degeneracy of zero-energy corner states can be reduced to two-fold under the application (withdrawal) of sub-lattice mass (Zeeman field) term. The sub-lattice mass and Zeeman field terms compete with each other to pin down the two mid-gap states at zero-energy. Interestingly, the bulk polarization can topologically characterize the second-order topological insulator phase with the mid-gap corner modes irrespective of their energies as long as inversion symmetry is preserved. Our study indicates that a hybrid symmetry can in principle protect the second-order topological insulator phases, however, spin-spectrum gap has to be essentially finite there.