Higher-dimensional Hofstadter butterfly on the Penrose lattice

It is now possible to use quasicrystals to search for novel topological phenomena enhanced by their peculiar structure characterized by an irrational number and high-dimensional primitive vectors. Here, we extend the concept of a topological insulator with an emerging staggered local magnetic flux (i.e., without external fields), similar to Haldane's honeycomb model, to the Penrose lattice as a quasicrystal. The Penrose lattice consists of two different tiles, where the ratio of the numbers of tiles corresponds to an irrational number. Contrary to periodic lattices, the periodicity of the energy spectrum with respect to the magnetic flux no longer exists, reflecting the irrational number in the Penrose lattice. Calculating the Bott index as a topological invariant, we find topological phases appearing in a fractal energy spectrum similar to the Hofstadter butterfly. More intriguingly, by folding the one-dimensional aperiodic magnetic flux into a two-dimensional periodic flux space, the fractal structure of the energy spectrum is extended to a higher dimension, whose section corresponds to the Hofstadter butterfly.