Colored Hofstadter butterflies

Universality of Hofstadter butterflies on hyperbolic lattices

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 s…

A numerical study of the window condition for Chern numbers of Hofstadter Butterflies

The Chern numbers for Hofstadter models with rational flux 2*pi*p/q are partially determined by a Diophantine equation. A Mod q ambiguity remains. The resolution of this ambiguity is only known for the rectangular lattice with nearest neighbors hopping where it has the form of a "window condition". We study a Hofstadter butterfly on the triangular lattice for which the resolution of ambiguity is open. In the model many pairs (p,q) satisfy a window condition which is shifted relative to the window of the square model. However, we also find pairs (p,q) where the Chern numbers do not belong to an…

Butterflies and topological quantum numbers

The Hofstadter model illustrates the notion of topological quantum numbers and how they account for the quantization of the Hall conductance. It gives rise to colorful fractal diagrams of butterflies where the colors represent the topological quantum numbers.

Hofstadter蝴蝶与莫尔异质结构的金属/绝缘体转变

Hofstadter butterflies and metal/insulator transitions for moiré heterostructures

本文研究应变莫尔异质结构的紧束缚模型，通过施加层反对称剪切应变将其降为一维并运用矩阵值准周期算子理论，解释了Hofstadter型蝴蝶的出现与金属/绝缘体转变现象。

Quasiperiodic granular chains and Hofstadter butterflies

We study quasiperiodicity-induced localization of waves in strongly precompressed granular chains. We propose three different setups, inspired by the Aubry--André (AA) model, of quasiperiodic chains; and we use these models to compare the effects of on-site and off-site quasiperiodicity in nonlinear lattices. When there is purely on-site quasiperiodicity, which we implement in two different ways, we show for a chain of spherical particles that there is a localization transition (as in the original AA model). However, we observe no localization transition in a chain of cylindrical particles in …

The Colored Hofstadter Butterfly for the Honeycomb Lattice

We rely on a recent method for determining edge spectra and we use it to compute the Chern numbers for Hofstadter models on the honeycomb lattice having rational magnetic flux per unit cell. Based on the bulk-edge correspondence, the Chern number $σ_H$ is given as the winding number of an eigenvector of a $2 \times 2$ transfer matrix, as a function of the quasi-momentum $k \in (0,2 π)$. This method is computationally efficient (of order $O(n^4)$ in the resolution of the desired image). It also shows that for the honeycomb lattice the solution for $σ_H $ for flux $p/q$ in the $r$-th gap conform…