Localization length index and subleading corrections in a Chalker-Coddington model: A numerical study

Abstract

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...

Criticality of Two-Dimensional Disordered Dirac Fermions in the Unitary Class and Universality of the Integer Quantum Hall Transition

“…They include the localization length exponent ν IQH ¼ 2.56-2.62 and the leading irrelevant exponent y ≃ 0.4 (with large error bars). At criticality, y describes the approach of the dimensionless quasi-1D Lyapunov exponent Γ to its limiting value at infinite system size Γ IQH 0 ¼ 0.77-0.82 [5][6][7]9,[11][12][13]16]. A similar exponent y was found for the average conductance ḡ of a square sample with limiting value ḡIQH ¼ 0.58-0.62 [19,20].…”

“…The integer quantum Hall transition (IQHT) at E ¼ E c is the most studied Anderson transition [2] because of its conceptual simplicity, low dimensionality, and experimental relevance. However, critical properties at the IQHT are notoriously difficult to compute analytically; they are mostly known from numerical studies which employed the Chalker-Coddington (CC) network model [3][4][5][6][7][8][9][10][11][12][13], microscopic continuous [14,15], lattice [10,[14][15][16][17], and Floquet Hamiltonians [18]. In recent works, the critical properties agree among models, indicating universality of the IQHT.…”

Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

“…40 The regular geometry of the CC model allows one to apply numerical transfer matrix techniques. 41,42 Recent implementations of this [43][44][45][46][47][48] and other methods 49,50 agree on the value ν in the range 2.56-2.62, certainly different from ν exp . The discrepancy points to the importance of the long-range electron-electron interaction, which certainly affects the scaling near the integer QH transition [51][52][53][54][55][56][57][58] and is relevant for the interpretation of experiments.…”

“…48 Since t and r appear in the denominators of the matrix elements of transfer matrices, making them zero is a singular procedure, related to the disappearance of two horizontal channels upon opening a node in the vertical direction. To overcome this difficulty, for every open node we take either t or r to be equal to ε 1.…”

Numerical evidence for marginal scaling at the integer quantum Hall transition

“…In light of the notorious difficulty with analytical approaches to the IQHT, this relation is at the heart of a long history of numerical finite-size scaling studies, mostly employing the Chalker-Coddington (CC) network model. [4][5][6][7][8][9][10][11][12][13][14] These works report ν = 2.56-2.62 but the leading irrelevant exponent y 0.4 is suprisingly small and comes with large error bars. While the above results were reproduced in a recent lattice model simulation, 15 Zhu et al reported a slightly different but incompatible value ν = 2.46-2.50 obtained from scaling the total number of conducting states in both lattice and continuum models projected to the lowest Landau level.…”