Effective time-reversal symmetry breaking and energy spectra of graphene armchair rings

We study the energy spectra and wave functions of graphene rings formed from metallic armchair ribbons, near zero energy, to search for properties which may be identified with "effective broken time-reversal symmetry" (EBTRS). Appropriately chosen corner junctions are shown to impose phase shifts in the wave functions that at low energies have the same effect as effective flux tubes passing near the ribbon surface. Closing the ribbon into a ring captures this flux and yields properties that may be understood as signatures of EBTRS. These include a gap in the spectrum around zero energy, which can be removed by the application of real magnetic flux through the ring. Spectra of five- and seven-sided rings are also examined, and it is shown these do not have particle-hole symmetry, which may also be understood as a consequence of EBTRS, and is connected to the curvature induced in the system when such rings are formed. Effects of deviations from the ideal geometries on the spectra are also examined.