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Physical review. B, Condensed matter

Delocalization, duality, and scaling in the quantum Hall system

C.A. Lütken, G.G. Ross

Published Jul 15, 1993• 96 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.48.2500

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arXiv:cond-mat/9810334•Oct 26, 1998•6 citations
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.
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.
Quantum Hall inter-plateau transitions are physical exemplars of quantum phase transitions. Near each of these transitions, the measured electrical conductivity scales with the same correlation length and dynamical critical exponents, i.e., the critical points are superuniversal. In apparent contradiction to these experiments, prior theoretical studies of quantum Hall phase transitions within the framework of Abelian Chern-Simons theory coupled to matter found correlation length exponents that depend on the value of the quantum critical Hall conductivity. Here, we use non-Abelian bosonization and modular transformations to theoretically study the phenomenon of superuniversality. Specifically, we introduce a new effective theory that has an emergent U(N)U(N)U(N) gauge symmetry with any N>1N>1N>1 for a quantum phase transition between an integer quantum Hall state and an insulator. We then use modular transformations to generate from this theory effective descriptions for transitions between a large class of fractional quantum Hall states whose quasiparticle excitations have Abelian statistics. We find the correlation length and dynamical critical exponents are independent of the particular transition within a controlled 't Hooft large NNN expansion, i.e., superuniversal! We argue that this superuniversality could survive away from this controlled large NNN limit using recent duality conjectures.
The quantum Hall effect is one of the most challenging problems in condensed matter physics in which many rare physical phenomena like fractional charges and generally the fractional statistics are involved. In the integer quantum Hall effect the role of the disorder is more than others. The important point in describing these systems is that their bulk states should be gapful and in this sense the system should be incompressible. For describing the electron condensates in an arbitrary filling fraction with the odd denominator one needs the topological U(1) Chern-Simons effective field theory. Our motivation for using these field theories is the emergence of quantized conductivity. This effective field theory also has the ability of constructing the hierarchical fractional condensates with various filling factors. The excitations of these theories have fractional statistics which are investigated in this review. The non-Abelian Chern-Simons for quantum Hall systems will be also investigated. The interesting point is that the edge states of these systems are gapless and their excitations are equipped with the Kac-Moody algebra. The connection to the conformal field theories can be established using this point. The investigation of this correspondence is one of the aims of the present thesis. The connection to the logarithmic conformal field theories (e.g. for the Haldane-Rezaei wave function) is also reported in some works which will be addressed in this thesis.
A fundamental property of the quantum Hall effect is the presence of edge states at the boundary of the sample, robust against localization and responsible for the perfect quantization of the Hall conductance. The transition between integer quantum Hall plateaus is a delocalization transition, which can be identified as a strong-coupling fixed point of a 1 + 1-dimensional supersymmetric sigma model with topological theta-term. The conformal field theory describing this transition displays unusual features such as non-unitarity, and resisted any attempt of solution so far. In this thesis we investigate the role of edge states at quantum Hall transitions using lattice discretizations of super sigma models. Edge states correspond to twisted boundary conditions for the fields, which can be discretized as quantum spin chains or geometrical (loop) models. For the spin quantum Hall effect, a counterpart of the integer quantum Hall effect for spin transport (class C), our techniques allow the exact computation of critical exponents of the boundary conformal field theories describing higher plateaus transitions. Our predictions for the mean spin conductance are validated by extensive numerical simulations of the related localization problems. Envisaging applications to transport in network models of 2 + 1-dimensional disordered electrons, and to quenches in one-dimensional gapless quantum systems, in this thesis we have also developed a new formalism for dealing with partition functions of critical systems in rectangular geometries. As an application, we derive formulas for probabilities of self-avoiding walks.
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.