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

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 2n-root topology, characterizing models where n squaring operations must be applied to the Hamiltonian to arrive at the topological source of the model. In this paper, we analyze the Hofstadter regime of quasi-one-dimensional and two-dimensional 2n-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 we 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 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 2n-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.