Dirac states in armchair- and zigzag-edged graphene M\"obius strips
Edge structure plays an essential role in the nature of electronic states in graphene nanoribbons. By focusing on the interplay between this feature and non-trivial topology in the domain of the Dirac confinement problem, this paper proposes to examine how effects associated with edge shape manifest themselves in conjunction with the topological signature typical of M\"{o}bius strips within a low-energy regime. Aiming to provide an alternative to prevailing tight-binding approaches, zigzag and armchair M\"{o}bius strips are modeled by proposing compatible sets of boundary conditions, prescribing profiles of terminations in both transverse and longitudinal directions which are demonstrated to be coherent in describing consistently transverse edge patterns in combination with a proper M\"{o}bius periodicity. Of particular importance is the absence of constraints on the solution, in contrast with infinite mass analogues, as well as an energy spectrum with a characteristic dual structure responding exclusively to the parity associated with the transverse quantum number. Zigzag ribbons are predicted to possess an intrinsic mechanism for parity inversion, while the armchair ones carry the possibility of a coexistent gapless and gapped band structure. We also inspect the influence of the edge structure on persistent currents. In zigzag-edged configurations they are found to be sensitive to a length-dependent term which behaves as an effective flux. Armchair rings show a quite distinctive property: alternation of constant and flux-dependent currents according to the width of the ring, for a fixed transverse quantum number. In the flux-free case the effects of topology are found to be entirely suppressed, and conventional odd and even currents become undistinguishable.