Vibrational properties of a general aperiodic Thue-Morse lattice: Role of the pseudoinvariant of the trace map

Abstract

Using the real-space renormalization group (RSRG) scheme of Ghosh and Karmakar [Phys. Rev. B 58, 2586 (1998)], we analytically determine the trace-map relation for a general spring-mass model of the aperiodic Thue-Morse (TM) lattice, and, interestingly observe that this map has a pseudoinvariant. This pseudoinvariant has a crucial role on the nature of the eigenmodes of this lattice. When the pseudoinvariant vanishes identically, as in the case of the on-site, transfer, or mixed model of the TM lattice, all normal modes are found to be delocalized, whereas the eigenmodes are critical for more general models with nonzero pseudoinvariant. Our RSRG scheme also gives the average phonon density of states $\ensuremath{\rho}(\ensuremath{\omega})$ and Lyapunov exponent $\ensuremath{\gamma}(\ensuremath{\omega})$ (= inverse localization length) of the eigenmodes.