Estados topológicos de fônons em quasicristais unidimensionais

Abstract

Theoretical models for predicting the properties of quasicrystalline materials have been of considerable interest to the scienti c community recently. However, they are mainly related to the optical and electronic characteristics of the system, and a study of the elementary oscillations, such as phonons, of one-dimensional quasicrystalline lattices is still necessary. Recently published works have shown that the localization properties of the Harper model can be modeled in a quasicrystal through the Hamiltonian of Aubry-André, considering the immeasurate potential with the lattice parameter. This model proved to present itself as a topological insulator, exhibiting border states and nontrivial phases for the electronic case. Motivated by these results, in this work, we present a study on the vibrational properties of one-dimensional quasi-crystals, highlighting the topological edge states. For this, we model a one-dimensional quasicristal through the Aubry-André model with the potential parameter de ned by the golden ratio (b = (1 + √ 5)/2). We performed the numerical calculations from the exact numerical diagonalization of the Hamiltonian. In our results, we nd the multifractal frequency spectrum known as the "Hofstadter's butter y", which emerges as a critical state of a transition from metal-insulating type states to the value of modulation of the dimensionless spring constant equal to 1.0. We also show by calculating the wavelength, that there exist certain states that cross the largest gaps of the spectrum (as a function of the phi phase) and are edge states in the system, where there are state localizations in them.