Phononic topological states in 1D quasicrystals

In this work, we address the study of phonons propagating on a one-dimensional quasiperiodic lattice, where the atoms are considered bounded by springs whose strength are modulated by equivalent Aubry-Andre hoppings. As an example, from the equations of motion. we obtained the equivalent phonon spectrum of the well known Hofstadter butterfly. We have also obtained extended, critical, and localized regimes in this spectrum. By introducing the equivalent Aubry-Andre model through the variation of the initial phase phi, we have shown that border states for phonons are allowed to exist. These states can be classified as topologically protected states (topological states). By calculating the inverse participation rate, we describe the localization of phonons and verify a phase transition, characterized by the critical value of lambda = 1.0, where the states of the system change from extended to localized, precisely like in a metal-insulator phase transition.