Real-space renormalization at the quantum Hall transition

TL;DR

This paper reviews the application of real-space renormalization group methods to the integer quantum Hall transition, accurately reproducing critical conductance distributions and extracting critical exponents consistent with other simulations.

Contribution

It demonstrates the effectiveness of the RG approach applied to the Chalker-Coddington model in analyzing quantum Hall critical phenomena and energy-level statistics.

Findings

Accurately reproduces critical conductance distribution P_c(G).

Determines critical exponents nu_G=2.39 and nu_ELS=2.37 with high precision.

Provides insights into energy-level statistics at the quantum Hall transition.

Abstract

We review recent applications of the real-space renormalization group (RG) approach to the integer quantum Hall (QH) transition. The RG approach, applied to the Chalker-Coddington network model, reproduces the critical distribution of the power transmission coefficients, i.e., two-terminal conductances, P_c(G), with very high accuracy. The RG flow of P(G) at energies away from the transition yields a value of the critical exponent, nu_G=2.39 +/- 0.01, that agrees with most accurate large-size lattice simulations. Analyzing the evolution of the distribution of phases of the transmission coefficients upon a step of the RG transformation, we obtain information about the energy-level statistics (ELS). From the fixed point of the RG transformation we extract a critical ELS. Away from the transition the ELS crosses over towards a Poisson distribution. Studying the scaling behavior of the ELS around the QH transition, we extract the critical exponent nu_ELS=2.37 +/- 0.02.