Topological Phase Transition in 2D Brickwall Lattices Containing Defects

We examine a 2D brickwall lattice with a point vacancy defect generating a topological zero-energy state. This model, classified under BDI topological insulators, is described by means of a low-energy Dirac approach near the Fermi energy. We demonstrate that a gap emerges at a critical parameter, causing a topological phase transition within the symmetry class. At the transition, the spatial wavefunction of the zero mode undergoes a qualitative change, characterized by the inverse participation ratio. We propose a method to detect the transition in analog systems and two-dimensional materials like graphene, by observing the change in the dislocation pattern in the electronic charge density.