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arXiv:0905.1163 · Physical Review B

Critical exponent for the quantum Hall transition

Keith Slevin, Tomi Ohtsuki

Published Jul 9, 2009• 153 citations
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Abstract

We report an estimate $\ensuremath{\nu}=2.593$ [2.587,2.598] of the critical exponent of the Chalker-Coddington model of the integer quantum Hall effect that is significantly larger than previous numerical estimates and in disagreement with experiment. This suggests that models of noninteracting electrons cannot explain the critical phenomena of the integer quantum Hall effect.

DOI
10.1103/physrevb.80.041304

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We report an estimate \ensuremathν=2.593\ensuremath{\nu}=2.593\ensuremathν=2.593 [2.587,2.598] of the critical exponent of the Chalker-Coddington model of the integer quantum Hall effect that is significantly larger than previous numerical estimates and in disagreement with experiment. This suggests that models of noninteracting electrons cannot explain the critical phenomena of the integer quantum Hall effect.
The quantum Hall effect is one of the most extensively studied topological effects in solid‐state physics. The transitions between different quantum Hall states exhibit critical phenomena described by universal critical exponents. Numerous numerical finite size scaling studies have focused on the critical exponent ν for the correlation length. In such studies, it is important to take proper account of irrelevant corrections to scaling. Recently, Dresselhaus et al. proposed a new scaling ansatz, which they applied to the two terminal conductance of the Chalker–Coddington model. Herein, their proposal is applied to the previously reported data for the Lyapunov exponents of that model using both polynomial fitting and Gaussian process fitting.
We study a network model on the kagome lattice (NMKL). This model generalizes the Chalker-Coddington network model for the integer quantum Hall transition. Unlike random network models we studied earlier, the geometry of the kagome lattice is regular. Therefore, we expect that the critical behavior of the NMKL should be the same as that of the Chalker-Coddington model. We numerically compute the localization length index $\ensuremath{\nu}$ in the NKML. Our result $\ensuremath{\nu}=2.658\ifmmode\pm\else\textpm\fi{}0.046$ is close to Chalker-Coddington model values obtained in a number of recent papers. We also map the NMKL to the Dirac fermions in random potentials and in a fixed periodic curvature background. The background turns out irrelevant at long scales. Our numerical and analytical results confirm our expectation of the universality of critical behavior on regular network models.
We show that the localization transition in the integer quantum Hall effect as described by the Chalker-Coddington network model is quantum critical. We first map the anisotropic network model to the problem of diagonalizing a one-dimensional non-Hermitian noncompact supersymmetric lattice Hamiltonian of interacting bosons and fermions. Its behavior is investigated numerically using the density matrix renormalization group method, and critical behavior is found at the plateau transition. This result is confirmed by a generalization of the Lieb-Schultz-Mattis theorem.
We study the critical properties of the noninteracting integer quantum Hall to insulator transition (IQHIT) in a "dual" composite-fermion (CF) representation. A key advantage of the CF representation over electron coordinates is that at criticality CF states are delocalized at all energies. The CF approach thus enables us to study the transition from a new vantage point. Using a lattice representation of CF mean-field theory, we compute the critical and multifractal exponents of the IQHIT. We obtain ν=2.56±0.02 and η=0.51±0.01, both of which are consistent with the predictions of the Chalker-Coddington network model formulated in the electron representation.
We introduce and analyze a class of model systems to study transitions in the integer quantum Hall effect (IQHE). Even without disorder our model exhibits an IQHE transition as a control parameter is varied. We find that the transition is in the two-dimensional Ising universality class and compute all associated exponents and critical transport properties. The fixed point has time-reversal, particle-hole, and parity invariance. We then consider the effect of quenched disorder on the IQHE transition and find the following. (i) Randomness in the control parameter (which breaks all the above symmetries) translates into bond randomness in the Ising model and is hence marginally irrelevant. The transition may equally well be viewed as a quantum percolation of edge states localized on equipotentials. The absence of random-phase factors for the edge states is responsible for the nongeneric (Ising) critical properties. (ii) For a random magnetic field (which preserves particle-hole symmetry in every realization) the model exhibits an exactly solvable fixed line, described in terms of a product of a Luttinger liquid and an SU(n) spin chain. While exponents vary continuously along the fixed line, the longitudinal conductivity is constant due to a general conformal sum rule for Kac-Moody algebras (derived here), and is computed exactly. We also obtain a closed expression for the extended zero-energy wave function for every realization of disorder and compute its exact multifractal spectrum f(\ensuremath{\alpha}) and the exponents of all participation ratios. One point on the fixed line corresponds to a recently proposed model by Gade and Wegner. (iii) The model in the presence of a random on-site potential scales to a strong disorder regime, which is argued to be described by a symplectic nonlinear-sigma-model fixed point. (iv) We find a plausible global phase diagram in which all forms of disorder are simultaneously considered. In this generic case, the presence of random-phase factors in the edge-state description indicates that the transition is described by a Chalker-Coddington model, with a so far analytically inaccessible fixed point.
Extreme-value fluctuations at quantum critical points remain poorly understood in the presence of strong correlations and openness. At the integer quantum Hall transition in the open Chalker--Coddington network, we show that the maximal wave-function amplitude separates into a global gain and an intrinsic extreme component, ∣ψ∣max⁡=A ∣ψ~∣max⁡|ψ|_{\max}=A\,|\tildeψ|_{\max}∣ψ∣max​=A∣ψ~​∣max​. We introduce extreme-moment scaling for ∣ψ∣max⁡|ψ|_{\max}∣ψ∣max​ and observe an approximately parabolic exponent function τmax⁡(q)τ_{\max}(q)τmax​(q) over moderate qqq, while ln⁡∣ψ∣max⁡\ln|ψ|_{\max}ln∣ψ∣max​ displays an almost Gaussian bulk over the studied sizes. The gain factor is close to log-normal and largely controls the raw extremes. Gain normalization reorganizes the statistics: τ~max⁡(q)\tildeτ_{\max}(q)τ~max​(q) changes qualitatively and ∣ψ~∣max⁡|\tildeψ|_{\max}∣ψ~​∣max​ does not support a single-parameter generalized extreme-value collapse under standard centering/scaling in the accessible size window. Extreme observables thus provide a robust probe of correlated criticality in open quantum systems.