Flat bands, edge states and possible topological phases in a branching fractal

TL;DR

This paper analytically investigates flat bands, edge states, and potential topological phases in a periodic fractal array, revealing exact energy band behaviors and localized states in complex Vicsek geometries.

Contribution

It introduces an exact real space renormalization method to analyze flat bands and edge states in higher-generation Vicsek fractal arrays, highlighting potential topological phase transitions.

Findings

Existence of infinitely many flat, non-dispersive bands.

Identification of localized states associated with flat bands.

Evidence of possible topological phase transitions with edge states.

Abstract

We address the problem of analytically extracting a countable infinity of flat, non-dispersive bands in a periodic array of cells that comprise branching Vicsek geometries of higher and higher generations. Through a geometric construction, followed by an exact real space renormalization scheme we unravel clusters of compact localized states, corresponding to densely packed groups of flat bands, sometimes in close proximity with the dispersive ones, as the unit cells accommodate Vicsek fractal motifs of higher and higher generations. In such periodic arrays, energy bands close and open at energies that can be calculated exactly, and the precise correlation between the overlap integrals describing the tight binding systems can be worked out. The possibility of a topological phase transition is pointed out through an explicit construction of the edge states, weakly protected against disorder, though it is argued that the typical bulk-boundary correspondence is not holding good in such cases.