Existence of robust edge currents in Sierpiński fractals

Abstract

We investigate the Hall conductivity in a Sierpiński carpet, a fractal of Hausdorff 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}=±e^{2}/h, seem to be generically present for arbitrary finite field strength, no matter 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.