Quantum Hall transitions: An exact theory based on conformal restriction

Abstract

We revisit the problem of the plateau transition in the integer quantum Hall effect. Here we develop an analytical approach for this transition, based on the theory of conformal restriction. This is a mathematical theory that was recently developed within the context of the Schramm-Loewner evolution which describes the stochastic geometry of fractal curves and other stochastic geometrical fractal objects in 2D space. Observables elucidating the connection with the plateau transition include the so-called point-contact conductances (PCCs) between points on the boundary of the sample, described within the language of the Chalker-Coddington network model. We show that the disorder-averaged PCCs are characterized by classical probabilities for certain geometric objects in the plane (pictures), occurring with positive statistical weights, that satisfy the crucial restriction property with respect to changes in the shape of the sample with absorbing boundaries. Upon combining this restriction property with the expected conformal invariance at the transition point, we employ the mathematical theory of conformal restriction measures to relate the disorder-averaged PCCs to correlation functions of primary operators in a conformal field theory (of central charge $c=0$). We show how this can be used to calculate these functions in a number of geometries with various boundary conditions. Since our results employ only the conformal restriction property, they are equally applicable to a number of other critical disordered electronic systems in 2D. For most of these systems, we also predict exact values of critical exponents related to the spatial behavior of various disorder-averaged PCCs.

Published version

References in corpus (24)

• Anderson Transitions
• 2D growth processes: SLE and Loewner chains
• A Guide to Stochastic Loewner Evolution and its Applications
• Critical exponent for the quantum Hall transition
• Multifractality at the quantum Hall transition: Beyond the parabolic paradigm
• Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory
• Numerical study of the localization length critical index in a network model of plateau-plateau transitions in the quantum Hall effect
• Boundary multifractality at the integer quantum Hall plateau transition: implications for the critical theory
• Density of quasiparticle states for a two-dimensional disordered system: Metallic, insulating, and critical behavior in the class D thermal quantum Hall effect
• Multifractality and Conformal Invariance at 2D Metal-Insulator Transition in the Spin-Orbit Symmetry Class
• Wave function multifractality and dephasing at metal-insulator and quantum Hall transitions
• SLE(κ,ρ) martingales and duality
• Percolation Crossing Formulas and Conformal Field Theory
• Boundary criticality and multifractality at the 2D spin quantum Hall transition
• Critical curves in conformally invariant statistical systems
• Reflected planar Brownian motions, intertwining relations and crossing probabilities
• Finite Size Scaling of the Chalker-Coddington Model
• 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
• An integrable modification of the critical Chalker-Coddington network model
• Quantum critical behaviour of the plateau-insulator transition in the quantum Hall regime
• Quantum-Hall plateau-plateau transition in top-gated epitaxial graphene grown on SiC (0001)
• Observation of the quantized Hall insulator in the quantum critical regime of the two-dimensional electron gas
• Corner Multifractality for Reflex Angles and Conformal Invariance at 2D Anderson Metal-Insulator Transition with Spin-Orbit Scattering
• Exact exponents for the spin quantum Hall transition in the presence of multiple edge channels

Cited by in corpus (12)

• Finite Size Effects and Irrelevant Corrections to Scaling near the Integer Quantum Hall Transition
• Loop models with crossings
• Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions
• Gaussian free fields at the integer quantum Hall plateau transition
• Devil's staircase phase diagram of the fractional quantum Hall effect in the thin-torus limit
• Pure scaling operators at the integer quantum Hall plateau transition
• Statistics of Conductances and Subleading Corrections to Scaling near the Integer Quantum Hall Plateau Transition
• On truncations of the Chalker-Coddington model
• Relevant perturbations at the spin quantum Hall transition
• Logarithmic Correlations in Quantum Hall Plateau Transitions
• Mesoscopic Fluctuations and Multifractality at and across Measurement-Induced Phase Transition
• Supersymmetry approach to delocalization transitions in a network model of the weak field quantum Hall effect and related models