Renormalization group approach to the energy level statistics at the integer quantum Hall transition

We apply the real-space renormalization group (RG) approach to investigate the energy level statistics at the integer quantum Hall (QH) transition. Within the RG approach the macroscopic array of saddle points of the Chalker–Coddington network is replaced by a fragment consisting of only five saddle points. Previously, we have demonstrated that the RG approach reproduces the distribution of the conductance at the transition, P(G), with very high accuracy. To assess the level statistics we analyze the phases of the transmission coefficients of the saddle points. We find that, at the transition, the nearest-neighbor energy level spacing distribution (LSD) exhibits well-pronounced level repulsion. We emphasize that a metal-like LSD emerges when the fixed point distribution Pc of G is used. Studying the change of the LSD around the QH transition we observe scaling behavior. Using a one-parameter finite-size scaling analysis we are able to extract a critical exponent ν=2.38±0.04 of the localization length.