Quantum site percolation on triangular lattice and the integer quantum Hall effect

Generic classical electron motion in a strong perpendicular magnetic field and random potential reduces to the bond percolation on a square lattice. Here we point out that for certain smooth two-dimensional potentials with 120 degrees rotational symmetry this problem reduces to the site percolation on a triangular lattice. We use this observation to develop an approximate analytical description of the integer quantum Hall transition. For this purpose we devise a quantum generalization of the real-space renormalization group (RG) treatment of the site percolation on the triangular lattice. In quantum case, the RG transformation describes the evolution of the distribution of the 3x3 scattering matrices at the sites. We find the fixed point of this distribution and use it to determine the critical exponent, nu, for which we find the value nu approximate to 2.3/2.76. The RG step involves only a single Hikami box and thus can serve as a minimal RG description of the quantum Hall transition.