Quasiperiodic quantum systems are realized in several ways in condensed matter physics, such as layered materials with lattice mismatch and ultracold gases trapped in bichromatic optical lattices. Here first I briefly review the status of theoretical and experimental studies of topological phenomena involving quasiperiodic quantum systems.

A quasiperiodic modulation on a low-dimensional lattice can work both as a controlled inhomogeneity and as a window to a higher dimension. In this talk I illustrate this viewpoint by presenting some of the results of our theoretical studies: The topological phase diagram of a quasiperiodically modulated Bose-Hubbard model [1], and the effect of quasiperiodic modulation on a model of the topological superconductor [2], focusing on the effect of interaction between particles. Finally, I would like to comment on our more recent projects.