Self-dual model for one-dimensional incommensurate crystals including next-nearest-neighbor hopping, and its relation to the Hofstadter model

A one-dimensional tight-binding model for incommensurate crystals including next-nearest-neighbor interactions is studied. The model is self-dual, and includes as special cases the self-dual Aubry model as well as the earlier studied non-self-dual extensions of the Aubry model of Soukoulis and Economou, and Riklund, Liu, Wahlström, and Zheng. As for the Aubry model, we find no mobility edges in the energy spectrum for our model. Instead there is an energy-independent metal-insulator transition at the fixed point of the duality transformation, where all wave functions change from extended to localized. It is shown by generalizing the Thouless formula to include next-nearest-neighbor hopping that, in contrast to the Aubry model, the localization length of the states in the localized regime is dependent on the eigenenergy. The energy-independent localization length in the Aubry model is therefore not solely due to self-duality. We also relate our model to a model for two-dimensional Bloch electrons in a constant magnetic field, and show how the extension of the hopping range beyond nearest neighbors breaks the symmetry of the Hofstadter butterfly.