Controlling edge states of zigzag carbon nanotubes by the Aharonov-Bohm flux

It has been known theoretically that localized states exist around the zigzag edges of a graphite ribbon and of a carbon nanotube, whose energy eigenvalues are located between conduction and valence hands, We find that in metallic, zigzag single-walled, carbon nanotubes two of the localized states become critical, and that their localization length is sensitive to the mean curvature of a tube, and it can be controlled by the Aharonov-Bohm flux. The curvature-induced mini gap closes by the relatively weak magnetic field. A conductance measurement in the presence of the Aharonov-Bohm flux can give information about the curvature effect and the critical states.