Effect of topological defects and Coulomb charge on the low energy quantum dynamics of gapped graphene

We study the combined effect of a conical topological defect and a Coulomb charge impurity on the dynamics of Dirac fermions in gapped graphene. Beyond a certain strength of the Coulomb charge, quantum instability sets in, which demarcates the boundary between sub and supercritical values of the charge. In the subcritical regime, for certain values of the system parameters, the allowed boundary conditions in gapped graphene cone can be classified in terms of a single real parameter. We show that the observables such as local density of states, scattering phase shifts and the bound state spectra are sensitive to the value of this real parameter, which is interesting from an empirical point of view. For a supercritical Coulomb charge, we analyze the system with a regularized potential as well as with a zigzag boundary condition and find the effect of the sample topology on the observable features of the system.