Observing Topology in Bipartite Lattices Using Wavefunction Dislocations

We introduce an innovative approach for detecting topological states in bipartite lattices that include point defects or vacancies. By employing the tight-binding model to simulate wavefunctions within bipartite lattices, dislocations manifest in the wavefunction around the point defect. Employing Atiyah-Singer index theorems, we relate the analytical index, which pertains to observed dislocations in zero modes, to the topological index that represents the system's inherent topology.

We demonstrate that dislocations in wavefunctions are linked to a topological edge state, with the count of modes exhibiting a dislocation equating to zero modes tied to an analytical index. However, zero modes originating from boundary conditions, like the graphene zigzag edge, neither influence the topological invariant nor manifest dislocations.