Quasicrystals: From their discovery to their topological properties.

Abstract

Quasicrystals are a novel type of materials discovered at the end of the XX century by Dan Shechtmann. Although they are solids which are not periodic and do not have rotational symmetry, they still yield a periodic refraction pattern. It has been proved that this feature is due to a high-dimensional symmetry. This allows to treat quasiperiodic lattices as projections of high-dimensional periodic lattices. Quasicrystals are field of thorough study nowadays due to their potential applications in many different fields such as renewable energies, medicine or aerospace engineering. Moreover, their fascinating high-dimensional symmetries give a strong reason to study them theoretically.

This thesis is devoted to present and give a general overview on quasicrystals, using the last part to focus on their topological properties. In the first chapter, we make a brief summary of the history of quasicrystals, including a description of the two most important models of quasiperiodic lattices (Fibonacci and Penrose tilings) and mention their most important potential applications. In the second chapter, we introduce basic theory on crystals, in which we also introduce the tight-binding model. In the third chapter, we introduce the second quantization formalism which is a very commonly framework used nowadays in condensed matter physics. Furthermore, we include an example of the usefullness of second quantization by calculating the band structure of monolayer graphene. In the last chapter, we use the tools learned in the two previously presented chapters to discuss the topological aspects of quasicrystals. More specifically, we discuss the topological equivalence between the Harper models (also providing experimental evidence), and their topological equivalence with the 1D Fibonacci quasicrystal.