Marta Brzezinska (1, 2)
(1) Department of Physics, University of Zurich,
Winterthurerstrasse 190, 8057 Zurich, Switzerland
(2) Department of Theoretical Physics, Wroc law University of Science and Technology,
Wybrze_ze Wyspianskiego 27, 50-370 Wroc law, Poland
Existing classications of topological phases are based on the presence (or absence) of
symmetries and the number of spatial dimensions being an integer. However, equipped with
a notion of locality and the possibility to take a thermodynamic limit, the classication
schemes can be extended in order to include quantum states on general graphs. In particular,
one can consider a class of self-similar geometries characterized by a fractional dimension.
In this talk, I will focus on two fractal lattices, Sierpinski carpet and gasket, exposed to an
external magnetic eld and described within tight-binding approximation. By investigating
spectral and localization properties, together with the real-space Chern number calculations
and level spacings analysis in the presence of disorder, I will show that these systems exhibit
features similar to quantum Hall states in almost two dimensions.