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Journal of Physics A Mathematical and General

Global phase diagrams for charge transport in two dimensions

C.A. Lütken

Published Sep 7, 1993• 24 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/26/17/011

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•Jan 1, 2017•0 citations
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.
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.
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 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.