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The unusual electronic spectrum of an infinite quasiperiodic chain: extended signature of all eigenstates

We have shown in an analytical way supported by intuitive arguments that all the eigenstates of an infinite quasiperiodic chain can be of extended nature in an unusual way. The well known copper mean chain provides such an example. Earlier works, based on calculations with systems having finite sizes, showed the co-existence of localized and extended states for this particular system. Within the framework of the real space renormalization group scheme we transform a perfectly ordered chain and a copper mean chain into a period doubling sequence for which an area-preserving dynamical map is already in existence. The recursion relations for the Hamiltonian parameters of the effective period doubling sequences generated from an ordered chain and a copper mean chain are then compared to extract information regarding the true nature of the eigenstates of the latter.