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We study the energy spectra and wavefunctions of graphene rings formed frommetallic armchair ribbons, near zero energy, to search for properties which maybe identified with "effective broken time reversal symmetry" (EBTRS).Appropriately chosen corner junctions are shown to impose phase shifts in thewavefunctions that at low energies have the same effect as effective flux tubespassing near the ribbon surface. Closing the ribbon into a ring captures thisflux and yields properties that may be understood as signatures of EBTRS. Theseinclude a gap in the spectrum around zero energy, which can be removed by theapplication of real magnetic flux through the ring. Spectra of five and sevensided rings are also examined, and it is shown these do not have particle-holesymmetry, which may also be understood as a consequence of EBTRS, and isconnected to the curvature induced in the system when such rings are formed.Effects of deviations from the ideal geometries on the spectra are alsoexamined.

Effective Time Reversal Symmetry Breaking and Energy Spectra of Graphene Armchair Rings