Higher-order topology in Fibonacci quasicrystals

In crystalline systems, higher-order topology, characterized by topological states of codimension greater than one, typically 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 Wannier centers are not well defined. In this study, we examine Fibonacci chains and squares, a quintessential type of quasicrystal, to investigate their higher-order topological properties. We discover that topological interfacial states, including corner states, can be inherited from their higher-dimensional periodic counterparts, such as the two-dimensional Su-Schrieffer-Heeger model. This finding is validated through numerical simulations of both phononic and photonic Fibonacci quasicrystals by the finite element method, revealing the emergence of topological edge and corner states at interfaces between Fibonacci quasicrystals with differing topologies inherited from their parent systems. Our results not only provide insight into the higher-order topology of quasicrystals but also open avenues for exploring novel topological phases in aperiodic structures.