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arXiv:cond-mat/0702611 · Physical Review B

Evidence for the PSL(2∣2) Wess-Zumino-Novikov-Witten model as a model for the plateau transition in the quantum Hall effect: Evaluation of numerical simulations

Alexei M. Tsvelik

Published May 2, 2007• 29 citations
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Abstract

In this paper, I revise arguments in favor of the PSL(2\ensuremath{\mid}2) Wess-Zumino-Novikov-Witten (WZNW) model as a theory of the plateau transition in integer quantum Hall effect. I show that all available numerical data (including the correlation length exponent $\ensuremath{\nu}$) are consistent with the predictions of such WZNW model with the level $k=8$.

DOI
10.1103/physrevb.75.184201

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In this paper, I revise arguments in favor of the PSL(2\ensuremath{\mid}2) Wess-Zumino-Novikov-Witten (WZNW) model as a theory of the plateau transition in integer quantum Hall effect. I show that all available numerical data (including the correlation length exponent \ensuremathν\ensuremath{\nu}\ensuremathν) are consistent with the predictions of such WZNW model with the level k=8k=8k=8.
A careful study of the supersymmetric version of Pruisken’s nonlinear σ model for the integer quantum Hall effect is presented. The lattice regularized model is cast in Hamiltonian form by taking the anisotropic limit and interpreting the topological density as an alternating sum of Wess-Zumino terms. It is argued that the relevant large-scale physics of the model is preserved by projection of the quantum Hamiltonian on its sector of degenerate strong-coupling ground states. For values of the Hall conductivity close to e 2 /2h (mod e 2 /h), where a delocalization transition occurs, this yields the Hamiltonian of a quantum superspin chain which is closely related to an anisotropic version of the Chalker-Coddington model. The relation implies that the ratio of magnetic length over potential correlation length is an irrelevant parameter at the transition. The superspin chain resembles a 1d isotropic antiferromagnet with spin 1/2. It has an alternating structure which however permits an invariance under translation by one site. The conductance coefficients of a quantum Hall system with N small contacts translate into N-superspin correlation functions which are governed by conformal invariance. The superspin formalism provides a framework for studying the crossover from classical to quantum percolation. It does not however encompass the frequency-dependent correlations of wave amplitudes at criticality.
As a unified theory of integer and fractional quantum Hall plateau transitions, a nonperturbative theory of the two-parameter scaling renormalization group function is presented. By imposing global symmetries known as ``the law of corresponding states'', we seek a possible form of renormalization group flows. Asking for consistency with the result from weak-localization perturbation theory, such restriction is so intense that we can analytically determine its concrete form. Accordingly, the critical exponent ννν and the irrelevant scaling index yyy are obtained analytically and turn out to be irrational. Their values (ν≈2.1ν\approx 2.1ν≈2.1, y≈0.3y\approx 0.3y≈0.3) agree favorably well with experiments and numerics.
Correlation functions in the Wess-Zumino-Novikov-Witten (WZNW) theory satisfy a system of Knizhnik-Zamolodchikov (KZ) equations, which involve constants of motion of an exactly solvable model, known as Gaudin magnet. We show that modified KZ equations, where the Gaudin operators are replaced by constants of motion of the closely related pairing Hamiltonian, give rise to a deformed WZNW model that contains terms breaking translational symmetry. This boundary WZNW model is identified and solved. The solution establishes a connection between the WZNW model and the pairing Hamiltonian in the theory of superconductivity. We also argue and demonstrate on an explicit example that our general approach can be used to derive exact solutions to a variety of dynamical systems.
We study the SU ( 2 ) k Wess–Zumino–Novikov–Witten (WZNW) theory perturbed by the trace of the primary field in the adjoint representation, a theory governing the low-energy behavior of a class of strongly correlated electronic systems. While the model is non-integrable, its dynamics can be investigated using the numerical technique of the truncated conformal spectrum approach combined with numerical and analytical renormalization groups (TCSA + RG). The numerical results so obtained provide support for a semiclassical analysis valid at k ≫ 1 . Namely, we find that the low energy behavior is sensitive to the sign of the coupling constant, λ . Moreover, for λ > 0 this behavior depends on whether k is even or odd. With k even, we find definitive evidence that the model at low energies is equivalent to the massive O(3) sigma model. For k odd, the numerical evidence is more equivocal, but we find indications that the low energy effective theory is critical.
We study the effects of the electron-electron interaction on the critical properties of the plateau transitions in the integer quantum Hall effect. We find the renormalization group dimension associated with short-range interactions to be $\ensuremath{-}0.66\ifmmode\pm\else\textpm\fi{}0.04$. Thus the noninteracting fixed point (characterized z=2z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2z=2 and $\ensuremath{\nu}\ensuremath{\approx}2.3$) is stable. For the Coulomb interaction, we find the correlation effect is a marginal perturbation at a Hartree-Fock fixed point ( z=1z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1z=1, $\ensuremath{\nu}\ensuremath{\approx}2.3$) by dimension counting. Further calculations are needed to determine its stability upon loop corrections.