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arXiv:cond-mat/9810334

Nonperturbative Renormalization Group Function for Quantum Hall Plateau Transitions Imposed by Global Symmetries

Nobuhiko Taniguchi

Published Oct 26, 1998• 6 citations
arXivHTMLPDF

Abstract

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 $ν$ and the irrelevant scaling index $y$ are obtained analytically and turn out to be irrational. Their values ($ν\approx 2.1$, $y\approx 0.3$) agree favorably well with experiments and numerics.

DOI
10.48550/arxiv.cond-mat/9810334

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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.
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 influence of short-range electron-electron interactions on scaling behavior near the integer quantum Hall plateau transitions. Short-range interactions are known to be irrelevant at the renormalization group fixed point which represents the transition in the noninteracting system. We find, nevertheless, that transport properties change discontinuously when interactions are introduced. Most importantly, in the thermodynamic limit the conductivity at finite temperature is zero without interactions, but nonzero in the presence of arbitrarily weak interactions. In addition, scaling as a function of frequency $\ensuremath{\omega}$ and temperature T is determined by the scaling variable $\ensuremath{\omega}{/T}^{p}$ (where p is the exponent for the temperature dependence of the inelastic scattering rate) and not by $\ensuremath{\omega}/T,$ as it would be at a conventional quantum phase transition described by an interacting fixed point. We express the inelastic exponent p and the thermal exponent zT{z}_{T}zT​ in terms of the scaling dimension $\ensuremath{-}\ensuremath{\alpha}<0$ of the interaction strength and the dynamical exponent z (which has the value z=2),z=2),z=2), obtaining $p=1+2\ensuremath{\alpha}/z$ and zT=2/p.{z}_{T}=2/p.zT​=2/p.
We construct an effective-field theory for the quantum Hall system which embodies both localization and fractional statistics. The latter involves a Chern-Simons interaction, while the former involves a generalization of conventional localization theory. The theory is invariant under ``complexified'' duality transformations of the conductivities which appear as effective parameters of the model. By exploiting these parameter space symmetries, as well as the conformal symmetry which appears at renormalization-group fixed points, we are able to extract a precise prediction for the whole scaling diagram. It exhibits both fractional and integer phases, the exact location of all fixed points, and universal scaling exponents. With a plausible identification of the universality class of the theory in the replica limit, the value of the critical exponent for the delocalization transition between plateaus in the Hall conductivity is found to be 7/3, in apparent agreement with available scaling experiments.
The phenomenological analysis of fully spin-polarized quantum Hall systems, based on holomorphic modular symmetries of the renormalization group (RG) flow, is generalized to more complicated situations where the spin or other ``flavors'' of charge carriers are relevant, and where the symmetry is different. We make the simplest possible ansatz for a family of RG potentials that can interpolate between these symmetries. It is parametrized by a single number aaa and we show that this suffices to account for almost all scaling data obtained to date. The potential is always symmetric under the main congruence group at level two, and when aaa takes certain values, this symmetry is enhanced to one of the maximal subgroups of the modular group. We compute the covariant RG $\ensuremath{\beta}$ function, which is a holomorphic vector field derived from the potential, and compare the geometry of this gradient flow with available temperature driven scaling data. The value of aaa is determined from experiment by finding the location of a quantum critical point, i.e., an unstable zero of the $\ensuremath{\beta}$ function given by a saddle point of the RG potential. The data are consistent with $a\ensuremath{\in}\mathbb{R}$, which together with the symmetry leads to a generalized semicircle law.
We suggest that a unified description of the integer and fractional phases of the quantum Hall system may be possible if the scaling diagram of transport coefficients is invariant under linear fractional (modular) transformations. In this model the hierarchy of states, as well as the observed universality of critical exponents, are consequences of a discrete SL(2,openZ) symmetry acting on the parameter space of an effective quantum-field theory. Available scaling data on the position of delocalization fixed points in the integer case and the position of mobility fixed points in the fractional case agree with the model within experimental accuracy.
It is suggested that it is not necessary to solve the full non-perturbative problem of two-dimensional charge transport in order to obtain global information about the phase and flow diagrams for the quantum Hall system and its relatives. It is argued that the effective quantum field theories encoding the macroscopic properties of these systems are invariant under a symmetry contained in the modular group. New phase and flow diagrams are obtained which exhibit a hierarchy of only odd or only even phases (Hall plateaus). The Kramers-Wannier-like symmetry determines the exact location of all renormalization group fixed points, which appear to be in agreement with available scaling experiments on the quantum Hall system. It also explains the observed 'super-universality', of the delocalization exponent.