Multiple higher-order topological phases with even and odd pairs of zero-energy corner modes in a $C_3$ symmetry broken model

TL;DR

This paper explores higher-order topological phases in a modified Haldane model with broken $C_3$ symmetry, revealing conditions for zero-energy corner modes and the role of various symmetry-breaking terms.

Contribution

It demonstrates that only the quantum spin Hall insulator can host second-order dipolar phases in this model, and analyzes how symmetry-breaking terms influence zero-energy states.

Findings

Four-fold degeneracy of zero-energy states can be reduced to two-fold.

Sub-lattice mass and Zeeman field compete to pin mid-gap states.

Bulk polarization characterizes the dipolar phase regardless of mid-gap state energy.

Abstract

The higher-order corner modes for quantum anomalous Hall insulators in $C_3$ symmetry broken honeycomb lattice have been engineered recently. Here we consider an extended Haldane model in presence of inversion symmetry breaking sub-lattice mass, time-reversal symmetry breaking Zeeman field and spin-orbit coupling interaction where we find that only the quantum spin Hall insulator can host the second-order dipolar phase while the remaining two first-order topological phases do not morph into the latter. Remarkably, four-fold degeneracy of zero-energy dipolar states can be reduced to two-fold under the application (withdrawn) of sub-lattice mass (Zeeman field) term when the spin-orbit coupling is already present. On the other hand, the sub-lattice mass and Zeeman field terms compete with each other to pin down the two mid-gap states at zero-energy in the absence or presence of spin-orbit coupling. Interestingly, the bulk-polarization can topologically characterize the dipolar phase irrespective of the energy of the mid-gap states as long as inversion symmetry is preserved. The effective gap criterion can qualitatively mimic the extent of SOT phase originated by the interplay between finite Zeeman exchange field, sub-lattice mass, and SOC interaction.