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

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, $\nu$, for which we find the value $\nu \approx 2.3-2.76$. The RG step involves only a single Hikami box, and thus can serve as a minimal RG description of the quantum Hall transition.

10 pages, 8 figures; published version

References in corpus (25)

• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• The spin quantum Hall effect in unconventional superconductors
• Exact exponents for the spin quantum Hall transition
• Random Network Models and Quantum Phase Transitions in Two Dimensions
• A Unified Model for Two Localisation Problems: Electron States in Spin-Degenerate Landau Levels, and in a Random Magnetic Field
• Quantum Hall plateau transitions in disordered superconductors
• Universal relation between longitudinal and transverse conductivities in quantum Hall effect
• Quantum and classical localisation, the spin quantum Hall effect and generalisations
• Point-Contact Conductances at the Quantum Hall Transition
• Critical Conductance and Its Fluctuations at Integer Hall Plateau Transitions
• Conductance fluctuations at the integer quantum Hall plateau transition
• Spectral Compressibility at the Metal-Insulator Transition of the Quantum Hall Effect
• Localization and conductance fluctuations in the integer quantum Hall effect: Real--space renormalization group approach
• Network Model for a 2D Disordered Electron System with Spin-Orbit Scattering
• Hall Resistivity and Dephasing in the Quantum Hall Insulator
• Real Space Renormalization of the Chalker-Coddington Model
• Landau level mixing and spin degeneracy in the quantum Hall effect
• Point-Contact Conductances from Density Correlations
• Integer quantum Hall transition in the presence of a long-range-correlated quenched disorder
• Renormalization group approach to energy level statistics at the integer quantum Hall transition
• Network Models in Class C on Arbitrary Graphs
• Quantum Breakdown of the Quantized Hall Insulator
• The Quantized Hall Insulator: a "Quantum" Signature of a "Classical" Transport Regime?
• Real-space renormalization-group approach to the integer quantum Hall effect
• Real-space renormalization at the quantum Hall transition

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• Random network models with variable disorder of geometry
• Effective Floquet model for minimally twisted bilayer graphene
• Weakly chiral networks and 2D delocalized states in a weak magnetic field
• Supersymmetry approach to delocalization transitions in a network model of the weak field quantum Hall effect and related models
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