Periodic features in the dynamic structure factor of the quasiperiodic period-doubling lattice

We present an exact real-space renormalization-group method for evaluating the dynamic structure factor of an infinite one-dimensional quasiperiodic period-doubling (PB) lattice. We observe that for every normal mode frequency of the chain, the dynamic structure factor S(q,omega) always exhibits periodicity with respect to the wave vector a and the presence of such periodicity even in the absence of translational invariance in the system is quite surprising. Our analysis shows that this periodicity in S(q,omega) actually indicates the presence of delocalized phonon modes in the PD chain. The Brillouin zones of the lattice Lire found to have a hierarchical structure and the dispersion relation gives both the acoustic and optical branches. The phonon dispersion curves have a nested structure and we have shown that it is actually the superposition of the dispersion curves of an infinite set of periodic lattices.