21 May 2026 Thu 4 pm

The phenomenon of localization of elementary excitations in disordered quantum systems is well-known for a long time. Disorder can be realized not only by variation of Hamiltonian parameters such as on-site energies or hoppings, but also in geometry of the respective structures. The famous example of this type of problems is so-called quantum percolation, where vacancies are randomly created in a lattice. Moreover, models on complex networks, including self-similar fractals, were considered. 

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 α 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 α, 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 α. 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

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