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arXiv:cond-mat/9906193 · Journal of Physics A Mathematical and General

Gamma(2) modular symmetry, renormalization group flow and the quantum hall effect

Yvon Georgelin, Thierry Masson, Jean-Christophe Wallet

Published Dec 15, 1999• 16 citations
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Abstract

We construct a family of holomorphic -functions whose renormalization group (RG) flow preserves the (2) modular symmetry and reproduces the observed stability of the Hall plateaus. The semicircle law relating the longitudinal and Hall conductivities that has been experimentally observed is obtained from the integration of the RG equations for any permitted transition which can be identified from the selection rules encoded in the flow diagram. The generic scale dependence of the conductivities is found to agree qualitatively with the present experimental data. The existence of a crossing point occurring in the crossover of the permitted transitions is discussed.

DOI
10.1088/0305-4470/33/1/303

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Γ\GammaΓ(2) modular symmetry, renormalization, group flow and the quantum hall effect
We construct a family of holomorphic β\betaβ-functions whose RG flow preserves the Γ(2)\Gamma(2)Γ(2) modular symmetry and reproduces the observed stability of the Hall plateaus. The semi-circle law relating the longitudinal and Hall conductivities that has been observed experimentally is obtained from the integration of the RG equations for any permitted transition which can be identified from the selection rules encoded in the flow diagram. The generic scale dependance of the conductivities is found to agree qualitatively with the present experimental data. The existence of a crossing point occuring in the crossover of the permitted transitions is discussed.
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.
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.
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.