Seminarium "Teoria i Modelowanie Nanostruktur"

Existing classi fication of topological phases rely on the symmetries as well as on the number of spatial dimensions being an integer. However, equipped with a notion of locality and the possibility to take a thermodynamic limit, the classi fication schemes can be extended to be suitable for quantum states on general graphs. In particular, one can consider fractal geometries characterized by (non-integer) Hausdorff dimensionand rami fication number. Here, we investigate two fractal lattices, Sierpinski carpet and gasket, exposed to an external magnetic eld. By examining spectral and localization properties, together with the Chern number calculations and level spacings analysis in the presence of disorder, we identify states with non-trivial topology that exhibit featuressimilar to the quantum Hall effect.