Complete escape from localization on a hierarchical lattice: A Koch fractal with all states extended

TL;DR

This paper demonstrates that a specific correlated parameter setup and magnetic flux in a Koch fractal lattice can lead to complete electron delocalization, resulting in an absolutely continuous spectrum and perfect transparency.

Contribution

It provides an exact analytical demonstration that certain correlations and magnetic flux can induce full electron delocalization on a fractal lattice, contrary to typical localization.

Findings

All states become extended under specific conditions.

The spectrum is absolutely continuous and the system is fully transparent.

Results are supported by numerical analysis of participation ratio and transmission.

Abstract

An infinitely large Koch fractal is shown to be capable of sustaining only extended, Bloch-like eigenstates, if certain parameters of the Hamiltonian describing the lattice are numerically correlated in a special way, and a magnetic flux of a special strength is trapped in every loop of the geometry. We describe the system within a tight binding formalism and prescribe the desired correlation between the numerical values of the nearest neighbor overlap integrals, along with a special value of the magnetic flux trapped in the triangular loops decorating the fractal. With such conditions, the lattice, despite the absence of translational order of any kind whatsoever, yields an absolutely continuous eigenvalue spectrum, and becomes completely transparent to an incoming electron with any energy within the allowed band. The results are analytically exact. An in-depth numerical study of the inverse participation ratio and the two-terminal transmission coefficient corroborates our findings. Our conclusions remain valid for a large set of lattice models, built with the same structural units, but beyond the specific geometry of a Koch fractal, unraveling a subtle universality in a variety of such low dimensional systems.