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

Recently, a two-channel random network model was proposed for quantum Hall effect in a weak (non-quantizing) magnetic field. For the anisotropic variant of this network, we analytically average over the randomness using the supersymmetry method, and map the network onto a model of interacting $u(1,1|2)$ superspins. We analyze the resulting superspin model by relating it to a nonlinear sigma-model. We argue that the phase diagram of the superspin model does not change qualitatively if we identify $u(1,1|2)$ superspins with ordinary $su(2)$ spin-1/2's. This allows us to develop a deeper insight and extend the phase diagram into the regions inaccessible to a sigma-model approach. We further extend our approach to a related two-channel network models and discuss their physical implications.