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arXiv:cond-mat/0303349 · Physical review. B, Condensed matter

Action formulation of the network model of plateau-plateau transitions in the quantum Hall effect

A. Sedrakyan

Published Dec 29, 2003• 14 citations
arXivHTMLPDF

Abstract

The Lagrangian (action) formulation of the Chalker-Coddington network model for plateau-plateau transitions in the quantum Hall effect is presented based on a model of fermions hopping on Manhattan lattice (ML). The dimensionless Landauer resistance is considered and its average is calculated over the random U(1) phases with constant distribution on the circle. The Lagrangian of the resultant model on ML is found and the corresponding R matrix is written down. It appeared, that this model is integrable, raising hope to investigate physics of plateau- plateau transitions by the exact method of powerful algebraic Bethe ansatz.

DOI
10.1103/physrevb.68.235329

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The Lagrangian (action) formulation of the Chalker-Coddington network model for plateau-plateau transitions in the quantum Hall effect is presented based on a model of fermions hopping on Manhattan lattice (ML). The dimensionless Landauer resistance is considered and its average is calculated over the random U(1) phases with constant distribution on the circle. The Lagrangian of the resultant model on ML is found and the corresponding R matrix is written down. It appeared, that this model is integrable, raising hope to investigate physics of plateau- plateau transitions by the exact method of powerful algebraic Bethe ansatz.
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 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.
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The scaling behavior near the transition between plateaus of the Integer Quantum Hall Effect (IQHE) has traditionally been interpreted on the basis of a two-parameter renormalization group (RG) flow conjectured from Pruisken's non-linear sigma model (NL σ M). Yet, the conformal field theory (CFT) describing the critical point remained elusive, and only fragments of a quantitative analytical understanding existed up to now. In the present paper we carry out a detailed analysis of the current-current correlation function for the conductivity tensor, initially in the Chalker-Coddington network model for the IQHE plateau transition and then in its exact reformulation as a supersymmetric vertex model. We develop a heuristic argument for the continuum limit of the non-local conductivity response function at criticality and thus identify a non-Abelian current algebra at level n = 4 . Based on precise lattice expressions for the CFT primary fields we predict the multifractal scaling exponents of critical wavefunctions to be Δ q = q ( 1 − q ) / 4 . The Lagrangian of the RG-fixed point theory for r retarded and r advanced replicas is proposed to be the GL ( r | r ) n = 4 Wess-Zumino-Witten model deformed by a truly marginal perturbation. The latter emerges from the NL σ M by a natural scenario of spontaneous symmetry breaking.
We study the spin quantum Hall effect and transitions between Hall plateaus in quasi-two-dimensional network models consisting of several coupled layers. Systems exhibiting the spin quantum Hall effect belong to class C in the symmetry classification for Anderson localization, and for network models in this class there is an established mapping between the quantum problem and a classical one involving random walks. This mapping permits numerical studies of plateau transitions in much larger samples than for other symmetry classes, and we use it to examine localization in systems consisting of nnn weakly coupled layers. Standard scaling ideas lead one to expect nnn distinct plateau transitions, but in the case of the unitary symmetry class this conclusion has been questioned. Focusing on a two-layer model, we demonstrate that there are two separate plateau transitions, with the same critical properties as in a single-layer model, even for very weak interlayer coupling.