Aperiodic Topological Crystalline Insulators

Topological crystalline insulators (TCIs) are usually described with topological protection imposed by the crystalline symmetry. In general, however, the existence of TCI states does not necessitate the periodicity of crystals as long as an essential lattice symmetry can be identified. Here we demonstrate the compatibility of TCIs with aperiodic systems, as exemplified by an octagonal quasicrystal. In contrast to most common topological insulators and TCIs, the proposed aperiodic TCIs are attributed to a different symmetry-invoked band inversion mechanism, which inverts states with the same parities but opposite eigenvalues of a specific symmetry (such as reflection). The nontrivial topology is characterized by a nonzero integer ``mirror Bott index''. Moreover, we demonstrate that the topological edge states and quantized conductance of the aperiodic TCI, which are robust against disorder, can be effectively manipulated by external electric fields. Our findings not only provide a better understanding of electronic topology in relation to symmetry but also extend the experimental realization of topological states to much broader material categories beyond crystals.