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The Modular Symmetric Renormalization Group Flow of the Quantum Hall Effect

Henrik Sverre Limseth

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

A comprehensive introduction to the theory of modular symmetry in the renormalization group flow of the quantum Hall effect is given. The theory is tested with all available experimental scaling data, across a wide range of different materials. The data generally fits the scaling flow predicted from modular symmetry. Future experiments to further test the theory are suggested.

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A comprehensive introduction to the theory of modular symmetry in the renormalization group flow of the quantum Hall effect is given. The theory is tested with all available experimental scaling data, across a wide range of different materials. The data generally fits the scaling flow predicted from modular symmetry. Future experiments to further test the theory are suggested.
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.
The new states of matter and concomitant quantum critical phenomena revealed by the quantum Hall effect appear to be accompanied by an emergent modular symmetry. The extreme rigidity of this infinite symmetry makes it easy to falsify, but two decades of experiments have failed to do so, and the location of quantum critical points predicted by the symmetry is in increasingly accurate agreement with scaling experiments. The symmetry severely constrains the structure of the effective quantum field theory that encodes the low energy limit of quantum electrodynamics of 1010 charges in two dirty dimensions. If this is a non-linear σ-model the target space is a torus, rather than the more familiar sphere. One of the simplest toroidal models gives a critical (correlation length) exponent that agrees with the value obtained from numerical simulations of the quantum Hall effect.
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.
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.
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.