Anomalous quantum transport in fractal lattices

Fractal lattices are self-similar structures with repeated patterns on different scales. Quantum transport through such structures is subtle due to the possible co-existence of localized and extended states. Here, we study the dynamical properties of two fractal lattices, the Sierpiński gasket and the Sierpiński carpet. While the gasket exhibits sub-diffusive behavior, sub-ballistic transport occurs in the carpet. We show that the different dynamical behavior is in line with qualitative differences of the systems’ spectral properties. Specifically, in contrast to the Sierpiński carpet, the Sierpiński gasket exhibits an inverse power-law behavior of the level spacing distribution. As a possible technological application, we discuss a memory effect in the Sierpiński gasket which allows to read off the phase information of an initial state from the spatial distribution after long evolution times. We also show that interpolating between fractal and regular lattices allows for flexible tuning between different transport regimes. Fractal lattices have become an experimentally accessible platform to explore a rich variety of quantum transport phenomena. The authors show that, subtly depending on the specific geometry, sub-diffusive or sub-ballistic transport occurs, and relate this behavior to the spectral properties of the lattice.