Anderson transition driven by running fractal dimensions in a fractal-shaped structure

Anderson transition in a fractal-shaped structure is driven by running fractal dimension from two to three. We analyzed the spectral statistics of a quantum particle confined in a fractal-shaped structure generated using the dielectric breakdown model in three dimensions. When the fractal dimension is almost three, the spectral statistics of low-energy states are similar to those of a Gaussian orthogonal ensemble. As the fractal dimension decreases to two, the spectral statistics become Poissonian. This indicates the onset of quantum localization in the fractal-shaped structure, so the Anderson transition takes place as the fractal dimension changes. The effects of introducing random potentials are also analyzed. When a random magnetic field affects quantum states extended over our fractal-shaped structure, the spectral statistics are similar to those of a Gaussian unitary ensemble. On the other hand, a random on-site potential encourages quantum localization in our fractal-shaped structure.