Trace-map invariant and zero-energy states of the tight-binding Rudin-Shapiro model

The localization of electrons in a one-dimensional tight-binding model with diagonal aperiodicity given by the Rudin-Shapiro sequence is studied in the frame of the transfer-matrix formalism. It is proved that the trace map of the model possesses a polynomial invariant. It is also proved that an infinite sequence of values of the on-site potential exists such that for each of these the energy E=0 lies in the spectrum of the periodic approximants corresponding to the even generations of the chain. Accurate numerical computations show that the states associated to the center of the spectrum are weaker than exponentially localized even for rather small amplitudes of the on-site potential. Scaling laws that govern the spatial decay and self-similarity of these states are derived for various values of the potential strength. It is also shown that the decrease of the potential amplitude qualitatively changes the self-similarity of the wave functions on increasing length scales.