Higher-order topology in Fibonacci quasicrystals

In crystalline systems, higher-order topology, characterized by topological states of codimension greater than one, usually arises from the mismatch between Wannier centers and atomic sites, leading to filling anomalies. However, this phenomenon is less understood in aperiodic systems, such as quasicrystals, where bulk Wannier centers are absent. In this study, we examine a modification of Fibonacci chains and squares derived from a typical higher-order topological model, the two-dimensional Su-Schrieffer-Heeger model, to investigate their higher-order topological properties. We discover that topological interfacial states, including corner states, can emerge at the interfaces between modified Fibonacci chains and squares derived from topologically distinct parent systems. These interfacial states can be characterized by a shift in the local Wannier center spectrum, which indicates filling anomalies in finite samples. We numerically validate these interfacial states using the finite element method in phononic and photonic Fibonacci quasicrystals. Our results provide insight into the higher-order topology of quasicrystals and open avenues for exploring novel topological phases in aperiodic structures.