Abstract
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–André 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–André model through the variation of the initial phase , 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 , where the states of the system change from extended to localized, precisely like in a metal-insulator phase transition.
Highlights
The discovery of a new class of materials by Shechtman et al in 1984 [1], when they were studying the diffraction figures for an alloy of Aluminum and Manganese, had started up a new and very rich research area
In the icosahedral and decagonal quasicrystal√s, the selfsimilarity is related to the Golden Ratio ((1 + 5)/2), so that the atoms are separated by distances that represent the Fibonacci sequence
We study the phase transition through the Inverse Participation Rate (IPR), where the system changes from extended to localized
Summary
The discovery of a new class of materials by Shechtman et al in 1984 [1], when they were studying the diffraction figures for an alloy of Aluminum and Manganese ( that give him the Nobel Prize in Chemistry of 2011 [2]), had started up a new and very rich research area. Many works on one-dimensional quasicrystals showed that the localization property of the Harper model [36] could be found in a quasicrystal through the Hamiltonian of Aubry-Andr, considering the potential incommensurable with the lattice parameter [37,38,39] This model proved to present itself as a topological insulator which exhibits border states and non-trivial phases, experimentally verified in the works of Kraus et al [35], which used waveguides to obtain the frequency spectrum in a quasicrystal, indicating the existence of a photonic gap [40, 41].
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