If the variation of the primary lattice's on-site energies effectuated by the secondary lattice were truly random, that system would exhibit Anderson localization. However, with its quasiperiodic on-site modulation the Harper model actually features a metal-insulator transition at a critical modulation strength.

While this is long-standing knowledge, we have asked the question what happens in comparatively shallow cosine lattices for which next-to-nearest neighbor and even longer-range hopping matrix elements come into play, thus breaking the self-duality which underlies the Harper system's sharp transition from extended to localized energy eigenstates. Abandoning the tight-binding approximation and working with a basis of exact Wannier states of the primary lattice instead, we have discovered that in such bichromatic optical lattices the simultaneous transition of all states found in the idealized Harper system actually gives way to a sequence of mobility edges, indicating that the eigenstates disaggregate into groups each having its own transition parameters. This is exemplified in the above figure, published in Phys. Rev. A 75, 063404 (2007), which shows the average extension of the eigenstates for several depths of the primary lattice, decreasing from left to right, as functions of the amplitude of the secondary cosine potential.