Kaleidoscopes of Hofstadter Butterflies and Aharonov-Bohm caging from $2^n$-root topology in decorated square lattices

TL;DR

This paper explores the complex Hofstadter butterfly patterns and Aharonov-Bohm caging phenomena in $2^n$-root topological models derived from square lattices, revealing new fractal and topological features with potential experimental realizations.

Contribution

It introduces the concept of $2^n$-root topology in lattice models, analyzing their Hofstadter regimes and uncovering multiple flat bands, localized states, and fractal butterfly structures.

Findings

Multiple magnetic flux insensitive flat bands identified

Analytical eigenstates as compact localized states derived

Kaleidoscope of butterfly patterns with increasing root-degree observed

Abstract

Square-root topology describes models whose topological properties can be revealed upon squaring the Hamiltonian, which produces their respective parent topological insulators. This concept has recently been generalized to $2^n$-root topology, characterizing models where $n$ squaring operations must be applied to the Hamiltonian in order to arrive at the topological source of the model. In this paper, we analyze the Hofstadter regime of quasi-one-dimensional (quasi-1D) and two-dimensional (2D) $2^n$-root models, the latter of which has the square lattice (SL) (known for the Hofstadter Butterfly) as the source model. We show that upon increasing the root-degree of the model, there appear multiple magnetic flux insensitive flat bands, and analytically determine corresponding eigenstates. These can be recast as compact localized states (CLSs) occupying a finite region of the lattice. For a finite flux, these CLSs correspond to different harmonics contained within the same Aharonov-Bohm (AB) cage. Furthermore, as the root-degree increases, a kaleidoscope of butterflies is seen to appear in the Hofstadter diagram, with each butterfly constituting a topologically equivalent replica of the original one of the SL. As such, the index $n$, which uniquely identifies the root-degree of the model, can be seen as an additional fractal dimension of the $2^n$-root model present in its Hofstadter diagram. We discuss how these dynamics could be realized in experiments with ultracold atoms, and measured by Bragg spectroscopy or through observing dynamics of initially localized atoms in a quantum gas microscope.