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.

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.

We calculated numerically the localization length index ν for the Chalker-Coddington model of the plateau-plateau transitions in the quantum Hall effect. By taking into account finite size effects we have obtained ν = 2.593 ± 0.0297. The calculations were carried out by two different programs that produced close results, each one within the error bars of the other. We also checked the possibility of logarithmic corrections to finite size effects and found, that they come with much larger error bars for ν. PACS numbers: 71.30.+h;71.23.An; 72.15.Rn The computation of critical indices of the plateauplateau transitions in the quantum Hall effect (QHE) (see for a review [1]) is still an open problem in modern condensed matter physics. According to the pioneering works on localization [2] the dimension two is a marginal dimension, above which delocalization can appear. Exactly at d=2 Levine, Libby and Pruisken [3][4][5] noticed, that the presence of a topological term in the nonlinear sigma model (NLSM) formulation of the problem may result in the appearance of delocalized states in strong magnetic fields. The next achievement was reached by Chalker and Coddington [6].The authors formulated and studied numerically a network model (CC model) in a random potential yielding localization-delocalization transitions. The numerical value 2.5 ± 0.5 of the Lyapunov exponent (LE) in the CC model was in good agreement with the experimentally measured localization length index ν = 2.4 in the quantum Hall effect [7]. Recently the more precise value ν = 2.38 ± 0.06 was reported in [8,9].Various aspects of the CC-model were investigated in a chain of interesting papers: In [10] the model was linked to replicated spin-chains, while in [11,12] its connection to supersymmetric spin-chains was revealed. Some links with conformal field theories of Wess-Zumino-Witten-Novikov (WZWN) type were presented in [13] and [14].In Refs. [15,16] the authors investigated the multifractal behaviour of the CC model. Both papers reported quartic deviations from the exact quadratic dependence of the multifractal indices on the parameter q, which was predicted in Refs. [13,14]. This fact points out that the validity of the simple, supersymmetric WZWN approach to plateau-plateau transitions in the quantum Hall effect is questionable and here we are still far from the application of conformal field theory.In spite of a lot of understanding that has been gained for the plateau-plateau transitions in the QHE, the final model which would allow for the calculation of the localization length index either analytically or numerically has not been formulated yet. Moreover, recently more precise numerical calculations of the localization length index of the CC-model [15,[17][18][19] show values close to t r t t t r t r t t t r t r t t t r t r t t t r t r t t t r t r t t t r r r r r r r 2j-1 2j 2j+1 2j+2 M1 M1 M1 M2 M2 Figure 1. Schematic illustration of the CC network. M1 and M2 denote the column transfer matrices as defined in (1) and (2). Multiplication with a c...

We investigate the low temperature asymptotics and the finite size spectrum of a class of Temperley-Lieb models. As reference system we use the spin-1/2 Heisenberg chain with anisotropy parameter ∆ and twisted boundary conditions. Special emphasis is placed on the study of logarithmic corrections appearing in the case of ∆ = 1/2 in the bulk susceptibility data and in the low-energy spectrum yielding the conformal dimensions. For the sl(2|1) invariant 3-state representation of the Temperley-Lieb algebra with ∆ = 1/2 we give the complete set of scaling dimensions which show huge degeneracies.

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