Criticality without self-similarity: a 2D system with random long-range hopping

We consider a simple model of quantum disorder in two dimensions, characterized by a long-range site-to-site hopping. The system undergoes a metal–insulator transition--its eigenfunctions change from being extended to being localized. We demonstrate that at the point of the transition the nature of the eigenfunctions depends crucially on the magnitude of the hopping amplitude. At small amplitudes they are strongly multifractal. In the opposite limit of large amplitudes, the eigenfunctions do not become fractal. Their density moments do not scale as a power of the system size; instead our result suggests a power of the logarithm of the system size. In this regard, the transition differs from a similar one in the one-dimensional version of the same system, as well as from the conventional Anderson transition in more than two dimensions.