Higher-dimensional Hofstadter butterfly on Penrose lattice

TL;DR

This paper explores topological phases in a quasicrystal Penrose lattice, revealing a higher-dimensional Hofstadter butterfly fractal spectrum driven by an irrational ratio of tiles, extending topological insulator concepts without external magnetic fields.

Contribution

It introduces a novel extension of topological insulators to quasicrystals, demonstrating fractal energy spectra and higher-dimensional Hofstadter butterfly structures in the Penrose lattice.

Findings

Topological phases appear in the Penrose lattice with a fractal energy spectrum.

The Hofstadter butterfly structure extends to higher dimensions in the quasicrystal.

Energy spectrum periodicity with respect to magnetic flux is absent due to irrational tile ratios.

Abstract

Quasicrystal is now open to search for novel topological phenomena enhanced by its 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 the 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 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 like the Hofstadter butterfly. More intriguingly, by folding the one-dimensional aperiodic magnetic flux into a two-dimensional periodic flux space, the fractal structure of energy spectrum is extended to higher dimension, whose section corresponds to the Hofstadter butterfly.