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

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.