For nontopological quantum materials, introducing defects can significantly alter their properties by modifying symmetry and generating a nonzero analytical index, thus transforming the material into a topological one. We present a method to construct the potential matrix configuration with the purpose of obtaining a nonzero analytical index, akin to a topological invariant like a winding or Chern number. We establish systematic connections between these potentials, expressed in the continuum limit, and their initial tight-binding model description. We apply our method to graphene with an adatom, a vacancy, and both as key examples illustrating our comprehensive description. This method enables analytical differentiation between topological and nontopological zero-energy modes and allows for the construction of defects that induce topology.

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• Point defects
• Topological phases of matter
• Valley degrees of freedom
• Graphene
• Honeycomb lattice
• Green's function methods