Localization--non-ergodic transition in controllable-dimension fractal networks from diffusion-limited aggregation

TL;DR

This paper investigates spectral properties of fractal networks generated by diffusion-limited aggregation, revealing a localization--non-ergodic transition in 3D and localized states in 2D.

Contribution

It introduces a tunable model linking disordered systems and self-similar fractals, demonstrating a transition in eigenstate localization with fractal density.

Findings

All eigenstates are localized in 2D fractals.

A localization--non-ergodic transition occurs in 3D fractals as density increases.

Critical states emerge at a specific fractal density threshold.

Abstract

Our study connects the physics of disordered integer-dimensional systems and regular self-similar objects by studying spectral properties of fractal agglomerates with tunable dimension. The latter is controlled by parameter $\alpha$ of the algorithm that generates the agglomerates. We consider the nearest-neighbor tight-binding model on the agglomerates embedded in 2D and 3D, and observe that all eigenstates are localized in the 2D case, whereas in the 3D case, there is a localization--non-ergodic transition upon increasing $\alpha$,i.e., going from sparse to dense fractals: a sub-extensive number of critical states emerge in the spectrum at a certain critical value of $\alpha$. The complex geometry of the agglomerates is also responsible for a peculiar hierarchy of compact localized states and singularities in the density of states, which are typical for ordered fractals.