Flat bands in fractal-like geometry

TL;DR

This paper explores the emergence of multiple flat bands in 2D fractal-like lattices based on Sierpinski gasket geometries, revealing unique spectral features and potential for experimental realization.

Contribution

It introduces a generic formula for counting flat bands in fractal lattices and analyzes their spectral properties under magnetic flux.

Findings

Multiple flat bands depend on fractal generation and parameters.

Presence of a spin-1 conical spectrum at the band center.

Magnetic flux modifies the flat band spectrum, creating gaps or isolated bands.

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.