Real-space renormalization-group approach to the integer quantum Hall effect

We review recent results based on an application of the real-space renormalization group (RG) approach to a network model for the integer quantum Hall (QH) transition. We demonstrate that this RG approach reproduces the critical distribution of the power transmission coefficients, i.e., two-terminal conductances, Pc(G), with very high accuracy. The RG flow of P (G) at energies away from the transition yields a value of the critical exponent ν that agrees with most accurate large-size lattice simulations. A description of how to obtain other relevant transport coefficients such as RL and RH is given. From the non-trivial fixed point of the RG flow we extract the critical level-spacing distribution (LSD). This distribution is close, but distinctively different from the earlier large-scale simulations. We find that the LSD obeys scaling behavior around the QH transition with ν = 2.37±0.02. Away from the transition it crosses over towards the Poisson distribution. We next investigate the plateau-to-insulator transition at strong magnetic field. For a fully quantum coherent situation, we find a quantized Hall insulator with RH ≈ h/e 2 up to RL ∼ 20h/e 2 when interpreting the results in terms of most probable value of the distribution function P (RH). Upon further increasing RL → ∞, the Hall insulator with diverging Hall resistance RH ∝ R κ L is seen. The crossover between these two regimes depends on the precise nature of the averaging procedure for the distributions P (RL) and P (RH). We also study the effect of long-ranged inhomogeneities on the critical properties of the QH transition. Inhomogeneities are modeled by a smooth random potential with a correlator which falls off with distance as a power law r−α. Similar to the classical percolation, we observe an enhancement of ν with decreasing α. These results exemplify the surprising fact that a small RG unit, containing only five nodes, accurately captures most of the correlations responsible for the localization-delocalization transition.