Numerical Study of the Localization Length Critical Index in a Network Model of Plateau-Plateau Transitions in the Quantum Hall Effect

Abstract

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...

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.…”

“…[45]. In the standard transfer matrix method one multiplies many transfer matrixs for a single realization of disorder and relies on the self-averaging property of Lyapunov exponents.…”

“…Following Ref. [45] we have produced large ensembles of the Lyapunov exponent γ by simulating many disorder realizations for many combinations of x and M . We calculated 624 disorder realizations for any combination of M = 20, 40, 60, 80, 100, 120, 140, 160, 180, 200 and x = 0.08/12 · [0, 1, 2, 3,4,5,6,7,8,9,10,11,12] for fixed L = 5 000 000.…”

Percolation transitions in bilayer graphene encapsulated by hexagonal boron nitride

“…[5] and references therein). Recently, more accurate calculations yielded γ ≈ 2.6 for the noninteracting critical point [5][6][7]. The experimentally measured values for the PPT exponents in a two-dimensional electron gas (2DEG) read κ = 0.42, p = 2, and γ = 2.38 [8,9].…”

Quantum Hall transitions: An exact theory based on conformal restriction

“…The nature of the critical state at and the critical phenomena near the IQH transition are at the focus of intense experimental [8][9][10][11][12][13] and theoretical research. [14][15][16][17][18][19][20][21][22][23] In spite of much effort over several decades, an analytical treatment of most of the critical conducting states in disordered electronic systems, including in particular that of the mentioned IQH transition, has been elusive (although some proposals [14][15][16] have been put forward, but see Refs. 18 and 19).…”