Real-Space Renormalization and Energy-Level Statistics at the Quantum Hall Transition

Abstract

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, 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, νG = 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 νELS = 2.37 ± 0.02.The integer quantum Hall (QH) transition is described well in terms of a delocalization-localization transition of the electronic wavefunctions. In contrast to a usual metal-insulator transition (MIT), the QH transition is characterized by a single extended state located exactly at the center ǫ = 0 of each Landau band [1]. When approaching ǫ = 0, the localization length ξ of the electron wavefunction diverges according to a power law ǫ −ν , where ǫ 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], 2.4 ± 0.2 [3], 2.35 ± 0.03 [4], and 2.39 ± 0.01 [5]. In experiments ν ≈ 2.3 has been obtained, e.g., from the frequency [6] or the sample size [7] dependence of the critical behavior of the resistance in the transition region at strong magnetic field.Recently, a semianalytical description of the integer QH transition, based on the extension of the scaling ideas for the classical percolation [8] to the Chalker-Coddington (CC) model of the quantum percolation [2], has been developed [9,10]. The key idea of this description, a real-space-renormalization group (RG) approach, is the following. Each RG step corresponds to a doubling of the system size. The RG transformation relates the conductance distribution of the sample at the next step to the conductance distribution at the previous step. The fixed point (FP) of this transformation, yields the distribution of the conductance, P c (G), of a macroscopic sample at the QH

Scaling of the quantum-Hall plateau-plateau transition in graphene

“…Our experimentally measured scaling exponent κ = 0.40 ± 0.02 is consistent with universal scaling theory. 1,2,3,7,8,9 The transition through the zeroth Landau level, however, shows no clear scaling behavior which we explain by a temperature independent intrinsic length scale governing scaling.…”