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Zero-mass lines result in appearance of linear dispersion modes for Dirac fermions. These modes play an important role in various physical systems. However, a Dirac fermion may not precisely follow a single zero-mass line, due to either tunneling between different lines or centrifugal forces. Being shifted from a zero-mass line the Dirac fermion acquires mass which can substantially influence its expected "massless" behavior. In the paper we calculate the energy gap caused by the tunneling between two zero-mass lines and show that its opening leads to the delocalization of linear dispersion modes. The adiabatic bending of a zero-mass line gives rise to geometric phases. These are the Berry phase, locally associated with a curvature, and a new phase resulting from the mass square asymmetry in the vicinity of a zero-mass line.
Comment: 12 pages, 7 figures. Discussion about reduction to pure graphene and several references added. Methodological part expanded. Accepted to J. Am. Chem. Soc
Comment: 12 pages, 4 figures, accepted for publication by J. Phys. Soc. Jpn
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