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 quasione-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. 73.20.Dx, 73.40.Hm, 72.15.Rn

Localization in disordered superconducting wires with broken spin-rotation symmetry

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Localization and conductance fluctuations in the integer quantum Hall effect: Real-space renormalization-group approach

“…Recent works [21][22][23][24][25][26][27][28] explored the correspondence between the network model and certain limits of the models that were already studied (spin chains [21][22][23][24][25] , Hubbard chains 26 and Dirac fermions 27,28 ). The network model 12 is illustrated in Fig.…”

Feynman’s propagator applied to network models of localization

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