Flat bands in fractal-like geometry

Abstract

We report the presence of multiple flat bands in a class of two-dimensional (2D) lattices formed by Sierpinski gasket (SPG) fractal geometries as the basic unit cells. Solving the tight-binding Hamiltonian for such lattices with different generations of a SPG network, we find multiple degenerate and non-degenerate completely flat bands, depending on the configuration of parameters of the Hamiltonian. Moreover, we find a generic formula to determine the number of such bands as a function of the generation index $\ell$ of the fractal geometry. We show that the flat bands and their neighboring dispersive bands have remarkable features, the most interesting one being the spin-1 conical-type spectrum at the band center without any staggered magnetic flux, in contrast to the Kagome lattice. We furthermore investigate the effect of the magnetic flux in these lattice settings and show that different combinations of fluxes through such fractal unit cells lead to richer spectrum with a single isolated flat band or gapless electron- or hole-like flat bands. Finally, we discuss a possible experimental setup to engineer such fractal flat band network using single-mode laser-induced photonic waveguides.

8 pages, 9 figures, accepted version

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Cited by in corpus (47)

• Artificial flat band systems: from lattice models to experiments
• Topological states on fractal lattices
• Disordered flat bands on the kagome lattice
• Topology in the Sierpiński-Hofstadter problem
• Hall conductivity of Sierpinski carpet
• Nontrivial topological flat bands in a diamond-octagon lattice geometry
• Designer Flat Bands in Quasi-One-Dimensional Atomic Lattices
• Flat bands and nontrivial topological properties in an extended Lieb lattice
• General construction of flat bands with and without band crossings based on wave function singularity
• Fractal-like photonic lattices and localized states arising from singular and nonsingular flatbands
• Flat Bands and Ferrimagnetic Order in Electronically Correlated Dice-Lattice Ribbons
• Dynamics of quasiperiodically driven spin systems
• Edge states, corner states, and flat bands in a two-dimensional $\mathcal{P}\mathcal{T}$ symmetric system
• Localized states emerging from singular and nonsingular flat bands in a frustrated fractal-like photonic lattice
• Flat bands with band crossings enforced by symmetry representation
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• Decoding flat bands from compact localized states
• Perfect localization on flat band binary one-dimensional photonic lattices
• Electronic properties and quantum transports in functionalized graphene Sierpinski carpet fractals
• Confined Electrons in Effective Plane Fractals
• Isolated flat bands in an interlocking-circles lattice
• Electronic fractal patterns in building Sierpinski-triangles molecular systems
• Designer quantum states on a fractal substrate: compact localization, flat bands and the edge modes
• Periodic and aperiodic dynamics of flat bands in diamond-octagon lattice
• Multiple flat bands and localized states in photonic super-Kagome lattices
• Flat bands and $Z_2$ topological phases in a non-Abelian kagome lattice
• Loop current states and their stability in small fractal lattices of Bose-Einstein condensates
• Anomalous quantum transport in fractal lattices
• Flux modulated flat band engineering in square-kagome ladder network
• Engineering flux-controlled flat bands and topological states in a Stagome lattice
• Emergence of Cascading Flat Bands in Breathing Superlattices
• Linearized spectral decimation in fractals
• The Fractal-Lattice Hubbard Model
• Bilayer twisting as a mean to isolate connected flat bands in a Kagome lattice through Wigner crystallization
• Homogeneous electron gas in arbitrary dimensions
• Flat Bands in Three-dimensional Lattice Models with Non-trivial Hopf Index
• Fragmentation, Zero Modes, and Collective Bound States in Constrained Models
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• Orbital magnetization in Sierpinski fractals
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• A fractal geometry immersed in a hierarchical magnetic flux distribution
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• Decorated cluster lattices: a natural framework for classical spin liquids and flat bands