Higher-dimensional Hofstadter butterfly on Penrose lattice

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.

4 pages, 4 figures

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Cited by in corpus (5)

• Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries
• Physical properties of an Aperiodic monotile: Graphene-like features, chirality and zero-modes
• Kaleidoscopes of Hofstadter Butterflies and Aharonov-Bohm caging from $2^n$-root topology in decorated square lattices
• Quasiperiodic pairing in graphene quasicrystals
• Confined states and topological phases in two-dimensional quasicrystalline $\pi$-flux model