We calculate numerically the localization length critical index within the Chalker-Coddington model of the plateau-plateau transitions in the quantum Hall effect. We report a finite-size scaling analysis using both the traditional power-law corrections to the scaling function and the inverse logarithmic ones, which provided a more stable fit resulting in the localization length critical index ¼ 2:616 AE 0:014. We observe an increase of the critical exponent with the system size, which is possibly the origin of discrepancies with early results obtained for smaller systems. DOI: 10.1103/PhysRevLett.107.066402 PACS numbers: 71.30.+h, 71.23.An, 72.15.Rn Plateau-plateau transitions in the quantum Hall effect have been one of the most challenging problems in condensed matter physics during the past two decades. It is an interesting example of the localization-delocalization transition in two-dimensional disordered systems, where a quantum critical point appears due to the breaking of time reversal symmetry. One of the important problems in this area of research is the formulation of a quantum field theory describing the transition. The first suggestion in this respect appeared in Ref. [1], where the authors noticed that the presence of the topological term in the nonlinear sigma model formulation of the problem can result in the occurrence of delocalized states under strong magnetic fields.Later, Chalker and Coddington [2] formulated a phenomenological model of quantum percolation based on a transfer-matrix approach (referred to as the CC model hereafter). The numerical value 2:5 AE 0:5 of the critical index of the Lyapunov exponent (LE) calculated within the CC model (see Ref.[3] for a review) was in good agreement with the experimentally measured localization length index ¼ 2:4 in the quantum Hall effect [4]. This success motivated considerable interest in the CC model and stimulated its further investigation until the present day [5][6][7][8][9][10][11][12][13][14]. In early studies the continuum limit of the CC model was related to replicas of ordinary spin chains [6], a Hubbard-like model [7], and supersymmetric spin chains [8,9]. In Refs. [10,11], the continuum limit was also related to the conformal field theory of the Wess-Zumino-Witten-Novikov (WZWN) type. Analyzing the representations of the PSLð2j2Þ conformal field theory, they found one which gives a reasonable value of 16=7 ' 2:286 for the localization length index. Moreover, multifractal scaling indices of the CC model were predicted to depend quadratically on the parameter q of the multifractal analysis within the WZWN model.Most intriguing developments in the plateau-plateau transition problem were reported later in Refs. [15,16], where the multifractal behavior of the CC model was investigated. In both papers, approximately quartic deviation from the exact quadratic dependence of the multifractal indices on the parameter q, predicted in Refs. [10,11], was observed. The latter suggested that the validity of the supersymmetric WZWN approach to plate...

Recent results for the critical exponent of the localization length at the integer quantum Hall transition differ considerably between experimental (νexp ≈ 2.38) and numerical (νCC ≈ 2.6) values obtained in simulations of the Chalker-Coddington (CC) network model. The difference is at least partially due to effects of the electron-electron interaction present in experiments. Here we propose a mechanism that changes the value of ν even within the single-particle picture. We revisit the arguments leading to the CC model and consider more general networks with structural disorder. Numerical simulations of the new model lead to the value ν ≈ 2.37. We argue that in a continuum limit the structurally disordered model maps to free Dirac fermions coupled to various random potentials (similar to the CC model) but also to quenched two-dimensional quantum gravity. This explains the possible reason for the considerable difference between critical exponents for the CC model and the structurally disordered model. We extend our results to network models in other symmetry classes.

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Recently it was shown (I. A. Gruzberg, A. Klümper, W. Nuding and A. Sedrakyan, Phys. Rev. B 95, 125414 (2017)) that taking into account random positions of scattering nodes in the network model with U (1) phase disorder yields a localization length exponent 2.37 ± 0.011 for plateau transitions in the integer quantum Hall effect. This is in striking agreement with the experimental value of 2.38 ± 0.06. Randomness of the network was modeled by replacing standard scattering nodes of a regular network by pure tunneling resp. reflection with probability p where the particular value p = 1/3 was chosen. Here we investigate the role played by the strength of the geometric disorder, i.e. the value of p. We consider random networks with arbitrary probability 0 < p < 1/2 for extreme cases and show the presence of a line of critical points with varying localization length indices having a minimum located at p = 1/3.