Dipole polarizability of relativistic Aharonov-Bohm ring: Application to graphene nanorings

The dipole polarizability of nanorings threaded by an Aharonov-Bohm (AB) magnetic flux displays an intriguing sensitivity to flux values. Thus, for certain characteristic values, neighboring angular momentum eigenstates become degenerate causing the polarizability to diverge. In nonrelativistic settings, such characteristic values correspond to half-integer multiples of the flux quantum ${\mathrm{\ensuremath{\Phi}}}_{0}=h/e$. This picture is severely modified in the Dirac model of two-dimensional materials with two distinct valleys. Thus, degeneracy occurs at valley-specific flux values and may shift as far as $\ifmmode\pm\else\textpm\fi{}0.2{\mathrm{\ensuremath{\Phi}}}_{0}$ in the massless limit representing graphene. Based on an analytic solution of the Dirac equation perturbed by both AB flux and electric fields, we derive an exact expression for the dipole polarizability. Relativistic effects on split divergencies are studied numerically for various geometries and mass terms. Finally, the Dirac equation results are verified by comparison to atomistic models of hexagonal and circular graphene nanorings.