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arXiv · 0810.4920

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

V. V. Mkhitaryan·M. E. Raikh
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Abstract

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 2D 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 $3\times 3$ scattering matrix at the sites. We find the fixed point of this distribution and use it to determine the critical exponent, $ν$, for which we find the value $ν\approx 2.3-2.76$. The RG step involves only a single Hikami box, and thus can serve as a minimal RG description of the quantum Hall transition.

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V. V. Mkhitaryan, M. E. Raikh. 2009-05-13. Quantum site percolation on triangular lattice and the integer quantum Hall effect. https://doi.org/10.1103/physrevb.79.125401

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