Scaling and Crossover Functions for the Conductance in the Directed Network Model of Edge States

Abstract

We consider the directed network (DN) of edge states on the surface of a cylinder of length L and circumference C. By mapping it to a ferromagnetic superspin chain, and using a scaling analysis, we show its equivalence to a one-dimensional supersymmetric nonlinear sigma model in the scaling limit, for any value of the ratio L/C, except for short systems where L is less than of order C^{1/2}. For the sigma model, the universal crossover functions for the conductance and its variance have been determined previously. We also show that the DN model can be mapped directly onto the random matrix (Fokker-Planck) approach to disordered quasi-one-dimensional wires, which implies that the entire distribution of the conductance is the same as in the latter system, for any value of L/C in the same scaling limit. The results of Chalker and Dohmen are explained quantitatively.

10 pages, REVTeX, 2 eps figures

References in corpus (11)

• Finite Temperature Properties of Quantum Antiferromagnets in a Uniform Magnetic Field in One and Two Dimensions
• Three-dimensional disordered conductors in a strong magnetic field: surface states and quantum Hall plateaus
• Chiral Surface States in the Bulk Quantum Hall Effect
• Exact Solution for the Distribution of Transmission Eigenvalues in a Disordered Wire and Comparison with Random-Matrix Theory
• Continuum quantum ferromagnets at finite temperature and the Quantum Hall effect
• Towards a theory of the integer quantum Hall transition: from the nonlinear sigma model to superspin chains
• Quantum Transport in Disordered Wires: Equivalence of One-Dimensional Sigma Model and Dorokhov-Mello-Pereyra-Kumar Equation
• Chiral Metal as a Ferromagnetic Super Spin Chain
• Chiral Metal as a Heisenberg Ferromagnet
• Universal low-temperature properties of quantum and classical ferromagnetic chains
• Nonuniversal Critical Conductance Fluctuations of Chiral Surface States in the Bulk Integral Quantum Hall Effect -- An Exact Calculation

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