Impurities in Fibonacci Chains, Local Symmetry and Topological Order

Abstract

Topological order in condensed matter physics has grown to become an important concept in studying novel materials. It shows resilience of quantum properties (e.g. quantized resistivity in quantum-hall systems) against perturbations and therefore has many potential applications. Recently, some claims have been made that this topological order also exists in one-dimensional quasicrystals, and that its effects have been observed [1]. Due to the lack of periodicity in the lattice structure of quasicrystals, it is hard to study topological properties in conventional ways. One question that arises is how well protected topological phases are in this class of materials. In order to start addressing this question, we chose to investigate the effect of placing impurities in an otherwise quasiperiodic lattice. To do so, we consider the tight-binding Fibonacci chain from the perspective of a real-space renormalization procedure. Two implementations of the quasiperiodic modulations are possible: the on-site potential or the hopping parameter. It is however shown that they are equivalent under renormalization. The effect of impurities in the hopping model is then analyzed and it is found that the amount by which quasiperiodic order is disturbed is highly dependent on what is called the renormalization path of the site at which the impurity is placed. We point to a link in the theory of local symmetry resonators in aperiodic binary chains and finally, the topological properties of the chain are overviewed. These include the appearance of a topological invariant, which is characterized by the presence of edge states.