Towards a theory of the integer quantum Hall transition: from the nonlinear sigma model to superspin chains

Abstract

A careful study of the supersymmetric version of Pruisken's nonlinear sigma model for the integer quantum Hall effect is presented. The lattice regularized model is cast in Hamiltonian form by taking the anisotropic limit and interpreting the topological density as an alternating sum of Wess-Zumino terms. It is argued that the relevant large-scale physics of the model is preserved by projection of the quantum Hamiltonian on its sector of degenerate strong-coupling ground states. For values of the Hall conductivity close to e^2/2h (mod e^2/h), where a delocalization transition occurs, this yields the Hamiltonian of a quantum superspin chain which is closely related to an anisotropic version of the Chalker-Coddington model. The relation implies that the ratio of magnetic length over potential correlation length is an irrelevant parameter at the transition. The superspin chain resembles a 1d isotropic antiferromagnet with spin 1/2. It has an alternating structure which however permits an invariance under translation by one site. The conductance coefficients of a quantum Hall system with N small contacts translate into N-superspin correlation functions which are governed by conformal invariance. The superspin formalism provides a framework for studying the crossover from classical to quantum percolation. It does not however encompass the frequency-dependent correlations of wave amplitudes at criticality.

64 pages, plain TEX, with 8 figures appended as uuencoded postscript files, to appear in Annalen der Physik (Germany)

References in corpus (3)

• Relation between the Correlation Dimensions of Multifractal Wavefunctions and Spectral Measures in Integer Quantum Hall Systems
• A Unified Model for Two Localisation Problems: Electron States in Spin-Degenerate Landau Levels, and in a Random Magnetic Field
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Cited by in corpus (58)

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• Scaling Theory of the Integer Quantum Hall Effect
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• Two-dimensional conformal field theory for disordered systems at criticality
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• Random-bond Ising model in two dimensions, the Nishimori line, and supersymmetry
• Supersymmetry for Systems with Unitary Disorder: Circular Ensembles
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• Multifractality of wavefunctions at the quantum Hall transition revisited
• Towards a Field Theory of the Plateau Transition
• Toward a theory of the integer quantum Hall transition: continuum limit of the Chalker-Coddington model
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• Point-Contact Conductances at the Quantum Hall Transition
• Continuum Limit of the Integrable sl(2/1) 3-\bar{3} Superspin Chain
• Numerical study of the localization length critical index in a network model of plateau-plateau transitions in the quantum Hall effect
• Generalization of the Haldane conjecture to SU(3) chains
• Scaling and Crossover Functions for the Conductance in the Directed Network Model of Edge States
• Localization and conductance fluctuations in the integer quantum Hall effect: Real--space renormalization group approach
• Disordered Dirac Fermions: the Marriage of Three Different Approaches
• Wave function statistics and multifractality at the spin quantum Hall transition
• Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions
• Supersymmetry and Localization in the Quantum Hall Effect
• Localization in the quantum Hall regime
• Theta renormalization, electron-electron interactions and super universality in the quantum Hall regime
• The Chalker-Coddington Network Model is Quantum Critical
• Monitored fermions with conserved $\mathrm{U}(1)$ charge
• Feynman's Propagator Applied to Network Models of Localization
• Evidence for the PSL(2$|$2) Wess-Zumino-Novikov-Witten model as a model for the plateau transition in Quantum Hall effect: Evaluation of numerical simulations
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• Point-Contact Conductances from Density Correlations
• Chiral Metal as a Ferromagnetic Super Spin Chain
• Random network models with variable disorder of geometry
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• Mesoscopic conductance and its fluctuations at non-zero Hall angle
• Localization length index in a Chalker-Coddington model: a numerical study
• Information about the Integer Quantum Hall Transition Extracted from the Autocorrelation Function of Spectral Determinants
• On The Random Vector Potential Model In Two Dimensions
• Anderson localization in fractional quantum Hall effect
• On the theory of plateau-plateau transitions in Quantum Hall Effect
• Disordered Electrons in a Strong Magnetic Field: Transfer Matrix Approaches to the Statistics of the Local Density of States
• Generalized multifractality at spin quantum Hall transition
• Generalized multifractality at metal-insulator transitions and in metallic phases of 2D disordered systems
• From Disordered Quantum Walk to Physics of Off-diagonal Disorder
• Dimensional Crossover of Localisation and Delocalisation in a Quantum Hall Bar
• Weakly chiral networks and 2D delocalized states in a weak magnetic field
• On osp(2|2) conformal field theories
• Localization in non-chiral network models for two-dimensional disordered wave mechanical systems
• Generalized surface multifractality in 2D disordered systems
• Super universality of the quantum Hall effect and the "large $N$ picture" of the $\vartheta$ angle
• Edge states and conformal boundary conditions in super spin chains and super sigma models
• Gauge Invariance and the Critical Properties of Quantum Hall Plateaux Transitions
• A solvable non-unitary fermionic long-range model with extended symmetry
• Weak Localization and Integer Quantum Hall Effect in a Periodic Potential
• The $D_3^{(2)}$ spin chain and its finite-size spectrum
• Edge state transmission, duality relation and its implication to measurements
• Super universality of dimerised $SU(N+M)$ spin chains
• The superspin approach to a disordered quantum wire in the chiral-unitary symmetry class with an arbitrary number of channels