The electronic spectrum of a two dimensional square lattice in a perpendicular magnetic field, known as Hofstadter butterfly, was discovered in Regensburg thirty years ago [1]. We have calculated the Hofstadter butterfly for carbon nanotubes (CNTs) in the tight-binding approximation. For the case of single wall CNTs, it is straightforward to implement magnetic fields parallel to the tube axis by means of zone-folding in the graphene reciprocal lattice. We have also studied perpendicular magnetic fields which, in contrast to the parallel case, lead to a much richer, non-periodic spectrum. Moreover, we have investigated magnetic fields piercing double-wall CNTs and found strong signatures of intershell interaction in the resulting butterfly-spectrum. Ubiquitous to all perpendicular magnetic field spectra is the presence cusp-catastrophes at specific values of energy and magnetic field. At these particular points, the electronic wave function can be correspondingly visualized.
[1] D. Hofstadter, Phys. Rev. B 14, 2239 (1976)