Designer quantum states on a fractal substrate: compact localization, flat bands and the edge modes

TL;DR

This paper analytically uncovers compact localized eigenstates on a fractal substrate, revealing their connection to flat bands and edge modes, with implications for photonic lattice experiments.

Contribution

It introduces an exact RSRG decimation scheme to identify localized states on a fractal substrate, showing their potential infinite number in the thermodynamic limit.

Findings

Localized states populate flat bands in fractal arrays

Number of localized states can be infinite in the thermodynamic limit

Results match experimental observations in photonic waveguide networks

Abstract

Compact localized single particle eigenstates on a deterministic fractal substrate, modelled by a triangular Sierpinski gasket of arbitrarily large size, are unravelled and examined analytically. We prescribe an exact real space renormalization group (RSRG) decimation scheme within a tight binding formalism to discern these states, and argue that the number of such states can be infinite if the fractal substrate is enlarged to its thermodynamic limit. Interestingly, these localized states turn out to populate the non-dispersive, flat bands in a periodic array of Sierpinski gasket motifs, however large they may be. Our results match and corroborate the recently observed compact localized, flat band states engineered on two dimensional photonic waveguide networks with a fractal geometry, and provide a whole subset of them, which, in principle, should be observable in fractal photonic lattice experiments.