The existence of robust edge currents in Sierpinsky Fractals*

We investigate the Hall conductivity in a Sierpinski carpet, a fractal of Hausdor dimension d_f = ln(8)/ ln(3) ≈ 1.893, subject to a perpendicular magnetic field. We compute the Hall conductivity using linear response and the recursive Green function method. Our main finding is that edge modes, corresponding to a maximum Hall conductivity of at least σ_xy = ± e²/h, seems to be generically present for arbitrary finite field strength, no mater how one approaches the thermodynamic limit of
the fractal. We discuss a simple counting rule to determine the maximal number of edge modes in terms of paths through the system with a fixed width. This quantized edge conductance, as in the case of the conventional Hofstadter problem, is stable with respect to disorder and thus a robust feature of the system.