Quantum Hall effect in carbon nanotubes and curved graphene strips

We develop a long wavelength approximation in order to describe the low-energy states of carbon nanotubes in a transverse magnetic field. We show that, in the limit where the square of the magnetic length $l = \sqrt{\hbar c /e B}$ is much larger than the $C$-$C$ distance times the nanotube radius $R$, the low-energy theory is given by the linear coupling of a two-component Dirac spinor to the corresponding vector potential. We investigate in this regime the evolution of the band structure of zig-zag nanotubes for values of $R/l > 1$, showing that for radius $R \approx 20$ nm a clear pattern of Landau levels start to develop for magnetic field strength $B \gtrsim 10$ T. The levels tend to be four-fold degenerate, and we clarify the transition to the typical two-fold degeneracy of graphene as the nanotube is unrolled to form a curved strip. We show that the dynamics of the Dirac fermions leads to states which are localized at the flanks of the nanotube and that carry chiral currents in the longitudinal direction. We discuss the possibility to observe the quantization of the Hall conductivity in thick carbon nanotubes, which should display steps at even multiples of $2 e^2/h$, with values doubled with respect to those in the odd-integer quantization of graphene.