Universality of Hofstadter butterflies on hyperbolic lattices

TL;DR

This paper investigates the Hofstadter butterfly spectrum on hyperbolic lattices, revealing universal features determined by the fundamental tile's edges and discussing experimental verification possibilities.

Contribution

It provides the first large-scale computation of the Hofstadter butterfly on hyperbolic lattices, showing universal spectral features and explaining their relation to hyperbolic Landau levels.

Findings

Butterfly spectrum persists with large gapped regions.

Shape determined by the number of edges of the fundamental tile.

Fractal structure is absent in hyperbolic case.

Abstract

Motivated by recent experimental breakthroughs in realizing hyperbolic lattices in superconducting waveguides and electric circuits, we compute the Hofstadter butterfly on regular hyperbolic tilings. By utilizing large hyperbolic lattices with periodic boundary conditions, we obtain the true hyperbolic bulk spectrum that is unaffected by contributions from boundary states. Our results reveal that the butterfly spectrum with large extended gapped regions prevails and that its shape is universally determined by the number of edges of the fundamental tile, while the fractal structure is lost in such a non-Euclidean case. We explain how these intriguing features are related to the nature of Landau levels in hyperbolic space, and how they could be verified experimentally.