Defects in graphene: A topological description

Specific types of spatial defects or potentials can turn monolayer graphene into a topological material. These are classified by dimension $D$ and they systematically obtained from the Hamiltonian means its symbol $\mathcal{H}(\mathbit{k},\mathbit{r})$, an operator which generalizes Bloch contains all information. This approach, when applied to Dirac operators, allows recover tenfold classification insulators superconductors. The existence stable $\mathbb{Z}$ topology is predicted as condition on $D$, similar in thermodynamic phase transitions. Kekule distortions, vacancies, adatoms proposed examples such their equivalence discussed.