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Supersymmetry approach to delocalization transitions in a network model of the weak field quantum Hall effect and related models

We consider a recently proposed network model of the integer quantum Hall (IQH) effect in a weak magnetic field. Using a supersymmetry approach, we reformulate the network model in terms of a superspin ladder. A subsequent analysis of the superspin ladder and the corresponding supersymmetric nonlinear sigma model allows to establish the phase diagram of the network model, and the form of the critical line of the weak-field IQH transition. Our results confirm the universality of the IQH transition, which is described by the same sigma model in strong and weak magnetic fields. We apply the suspersymmetry method to several related network models that were introduced in the literature to describe the quantum Hall effect in graphene, the spin-degenerate Landau levels, and localization of electrons in a random magnetic field.