Real-Space Renormalization Group Approach to the Quantum Hall Transition

Abstract

The integer quantum Hall (QH) transition is described well in terms of a delocalization-localization transition of the electron wavefunction. In contrast to a usual metalinsulator transition (MIT), the QH transition is characterized by a single extended state located exactly at the center = 0 of each Landau band. When approaching = 0, the localization length ξ of the electron wavefunction diverges according to a power law −ν , where defines the distance to the MIT for a suitable control parameter, e.g., the electron energy. On the theoretical side, the value of ν has been extracted from various numerical simulations, e.g., ν = 2.5 ± 0.5, 2.4 ± 0.2, 2.35± 0.03, and 2.39± 0.01. In experiments ν ≈ 2.3 has been obtained, e.g., from the frequency or the sample size dependence of the critical behavior of the resistance in the transition region at strong magnetic field. We study the critical properties of the integer QH transition by employing the real-space renormalization-group (RG) approach to the Chalker-Coddington (CC) network model. We calculate the critical distribution Pc(G) of the conductance and the critical exponent ν of the QH transition for two different RG units. This allows to demonstrate that the quality of the results crucially depends on the choice of the RG unit. The CC model describes a single QH transition using a chiral network consisting of electron trajectories along equipotential lines (links) and saddle points (SP’s) of the potential (nodes). Each SP acts as a scatterer and relates the wavefunction amplitudes in two incoming and two outgoing channels. It can be characterized by a 2×2 S matrix, which depends only on the transmission and reflection coefficients ti and ri. The links correspond to random phases Φj and reflect the randomness of the potential disorder in a sample. We consider two previously studied, different RG units on a regular 2D square lattice as shown in Fig. 1. One is constructed from 4 SP’s, the other consists of 5 SP’s. The RG unit should be chosen in a way such that the essential properties of the network are taken into account. In the course of our RG approach an RG unit is then mapped onto a new single super-SP using the analytical dependence t′ = f ({ti, ri}, {Φj}) (1)