Spectral Properties of the Chalker-Coddington Network

Abstract

We numerically investigate the spectral statistics of pseudo-energies for the unitary network operator U of the Chalker-Coddington network. The shape of the level spacing distribution as well the scaling of its moments is compared to known results for quantum Hall systems. We also discuss the influence of multifractality on the tail of the spacing distribution.KEYWORDS: Chalker-Coddington model, level spacing distribution, scaling, quantum Hall systems, multifractality §1. IntroductionThe Chalker-Coddington network model, 1) representing systems of independent two-dimensional (2D) electrons in a strong perpendicular magnetic field and smooth disorder potential, is a convenient tool to investigate the quantum Hall universality class. 1-3) It has been used to determine various quantities characterizing the localization-delocalization (LD) transition taking place between the quantized plateaus of the Hall conductance. 4) The first approach was to use the transfer matrix method to numerically determine the characteristic exponent ν ≈ 2.35 for the divergence of the localization length ξ ∝ |E − E c | −ν when the energy of the electron E approaches the critical energy E c . 1, 5) At the critical energy itself the network model was used to calculate critical wave functions.Those were subjected to a multifractal analysis and the critical exponent α 0 ≈ 2.27 describing the scaling of typical value of their squared amplitude with the system size, exp |Ψ c | 2 ∝ L −α 0 , was determined. 2) The model can also very conveniently be used to simulate the diffusion of wave packets using a unitary time evolution operator U . By doing so it was possible to determine the scaling behavior of the local density of states. 6,7) Recently, a new approach in the investigation of network models was suggested. It was argued that network models can be used to determine the statistics of the eigenvalue spectrum of their underlying system by investigating the spectrum of the unitary network operator U . 3) So far * Permanent address: Department of Theoretical Physics, Institut of Physics, Technical University of Budapest, H-1521 Budapest, Hungary 1 the numerical investigation of the pseudo-energy statistics has been concentrating on the number variance Σ 2 (N ) = (n − n ) 2 of an energy interval containing on average N = n levels and the compressibility χ = lim N →∞ lim L→∞ dΣ 2 dN . It was found that the prediction by Chalker et al 8) that χ = (d − D 2 )/(2d), where D 2 is the correlation dimension, can be confirmed. Furthermore, it was seen that the level spacing distribution function P (s), denoting the probability to find two consecutive levels separated by an energy ǫ = s∆, ∆ being the average level spacing, deviates in its tail from the Wigner surmise. 3) This was expected at the critical energy 9-11) and was also seen in other numerical investigations. [12][13][14] In this paper we are going to take a closer look at the level spacing distribution P (s). We will show that by using the spectrum of the unitary network operator U all the kno...

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