Carbon nanotubes band assignation, topology, Bloch states, and selection rules

Various properties of the energy band structures (electronic, phonon, etc.), including systematic band degeneracy, sticking and extremes, following from the full line group symmetry of the single-wall carbon nanotubes are established. The complete set of quantum numbers consists of the angular and linear quasimomenta and parities with respect to the U axis and, for achiral tubes, the mirror planes, The assignation of the electronic bands is performed, and the generalized Bloch symmetry adapted eigen functions are derived. The most important physical tensors are characterized by the quantum numbers. All this enables application of the presented exhaustive selection rules. The results are discussed by some examples, e.g., allowed interband transitions, conductivity, Raman tensor, etc.