Topological properties of electronic states in quasiperiodic chains

In recent years, topological aspects of quasiperiodic systems have attracted much attention. However, due to the lack of translational symmetry, the conventional formulae for topological invariants are inapplicable to such systems and many of their topological properties remain unrevealed. In this study, we analyze topological properties of a tight-binding model on the Thue-Morse chain, which is one of the quasiperiodic chains. To diagnose the properties, we use local Berry phases, which are defined for individual bonds and quantized to 0 or π in the presence of symmetries such as spatial inversion. Physically, the π Berry phase corresponds to a local covalent bond and implies that topologically nontrivial edge states appear if the bond is broken. We verify that this is indeed the case for the bonds with inversion points. More remarkably, we find that some edge states appear to be nearly topological even if the broken bond is a bond without an inversion center, and the pattern of such bonds is quasiperiodic. These behaviors can be explained by the presence of partial inversion regions, where a part of atomic arrangements are inversion symmetric like palindromes.