Impurities in Fibonacci Chains, Local Symmetry and Topological Order
Summary
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.