Deterministic Aperiodic One-Dimensional Systems with All States Extended, One of Which is Periodic

A one-dimensional discrete tight-binding model with nearest-neighbour interaction is studied. We use the transfer model with variable hopping matrix elements, here assuming the two values t or - t , and constant on-site potential. Under this conditions all the eigenstates are known to be extended. It is shown that if the distribution of the off-diagonal matrix elements constitutes a deterministic aperiodic sequence, the eigenstate corresponding to the middle eigenvalue is periodic for some choices of the sequence, but not for all. The studied sequences that turn out to have a periodic middle state are the Thue-Morse sequence, the Rudin-Shapiro sequence and many of the generalised Thue-Morse sequences but not for instance the well known Fibonacci sequence.