Anomalous quantum transport in fractal lattices

Fractal lattices are self-similar structures with repeated patterns on different scales. As in other aperiodic lattices, the absence of translational symmetry can give rise to quantum localization effects. In contrast to low-dimensional disordered systems, co-existence of localized and extended states is possible in fractal structures, and can lead to subtle transport behavior. Here, we study the dynamical properties of two fractal lattices, the Sierpiński gasket and the Sierpiński carpet. Despite their geometric similarity, the transport turns out to behave very differently: In the Sierpiński gasket, we find a sub-diffusive behavior, whereas the Sierpiński carpet exhibits sub-ballistic transport properties. 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. From the point of view of technological applications, we demonstrate that the sub-diffusive behavior in the Sierpiński gasket can be used as a quantum memory. By interpolating between fractal and regular lattices, a flexible tuning between different transport regimes becomes possible.