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

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.

19 pages, 14 figures

References in corpus (28)

• Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms
• Topological Phases for Fermionic Cold Atoms on the Lieb Lattice
• A simple method to construct Flat Band lattices
• Square-root higher-order Weyl semimetals
• Observation of pi/2 modes in an acoustic Floquet system
• Chiral Magic-Angle Twisted Bilayer Graphene in a Magnetic Field: Landau Level Correspondence, Exact Wavefunctions and Fractional Chern Insulators
• Experimental observation of Aharonov-Bohm caging using orbital angular momentum modes in optical waveguides
• Systematic construction of square-root topological insulators and superconductors
• One-dimensional topological insulators with noncentered inversion symmetry axis
• Flat bands and nontrivial topological properties in an extended Lieb lattice
• Reentrant Correlated Insulators in Twisted Bilayer Graphene at 25T ($2\pi$ Flux)
• $2^n$-root weak, Chern, and higher-order topological insulators, and $2^n$-root topological semimetals
• Interaction-induced topological properties of two bosons in flat-band systems
• Cyclotron quantization and mirror-time transition on nonreciprocal lattices
• Manipulating synthetic gauge fluxes via multicolor dressing of Rydberg-atom arrays
• Square-root Floquet topological phases and time crystals
• A Tale of Two Fractals: The Hofstadter Butterfly and The Integral Apollonian Gaskets
• Many-body Aharonov-Bohm caging in a lattice of rings
• Higher-Order Topological Insulator on a Martini Lattice and Its Square Root Descendant
• Multiplicative topological phases
• Local Hilbert space fragmentation and weak thermalization in Bose-Hubbard diamond necklaces
• Unconventional delocalization in a family of 3D Lieb lattices
• Generalized Lieb's theorem for noninteracting non-Hermitian $n$-partite tight-binding lattices
• Higher-dimensional Hofstadter butterfly on Penrose lattice
• Time evolution of localized states in Lieb lattices
• Hofstadter butterflies in magnetically modulated graphene bilayer: an algebraic approach
• Streda formula for the Hofstadter-Wilson-Dirac model in two and four dimensions
• Kitaev Chain with a Fractional Twist

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• Aharonov-Bohm caging of an electron in a quantum fractal
• Exceptionally deficient topological square-root insulators
• Impurity flat band states in the diamond chain
• High-root topological edge-state bands
• Exceptional horns in $n$-root graphene and Lieb photonic ring lattices
• Localization and Flat Bands in Bond-Inflated Lattices