Topological defect-mediated corner states and higher-order bulk topology in a two-dimensional crystalline insulator

We report the appearance of nontrivial zero-energy corner modes in the form of topological defects (trimers) in a carefully designed 2D crystalline topological insulator. The proposed scenario is developed via an unconventional stacking of 1D topological atomic chains with crystalline mirror symmetry along the diagonal $(y=x)$ line. Our analysis shows that by systematically varying the hopping parameters $t$ (intrachain), $v$ (within the unit cell) and $w$ (between unit cells), the system exhibits more than one distinct nontrivial second-order topological phase. These phases are distinguished by their zero-energy corner modes. In one of these phases, the system supports four zero modes. Two of them reside on the trimers, and the rest reside on isolated sites situated at the corners along the diagonal line. However, in the second case, the zero modes on the isolated sites persist at the corners while the zero modes on the trimers vanish. A critical look at the phase evolution of the Bloch states helps investigate the topology of these phases by using winding numbers. Our work also shows the bulk-corner correspondence that exists between the invariants and the zero modes at the corners. With four zero modes at the corners and a winding number of 2, we conclude that the system has transformed into a second-order topological insulator via tuning of the hopping amplitudes.