Elliptic mirror of the quantum Hall effect

Abstract

Toroidal sigma models of magneto-transport are analyzed, in which integer and fractional quantum Hall effects automatically are unified by a {holomorphic modular symmetry}. By exploiting a quantum equivalence called \emph{mirror symmetry}, these models are mapped to tractable mirror models (also elliptic), in which topological protection is provided by more familiar winding numbers. Phase diagrams and scaling properties of elliptic models are compared to some of the experimental and numerical data accumulated over the past three decades. The geometry of scaling flows extracted from quantum Hall experiments is in good agreement with modular predictions, including the location of many quantum critical points. One conspicuous model %(arguably the simplest and most natural one) has a critical delocalization exponent ${\nu }_{\mathrm{tor}}=18\ln 2/({\pi }^2{G}^4)=2.6051\dots$ ($G$ is Gauss' constant) that is in excellent agreement with the value ${\nu }_{\mathrm{num}}=2.607\pm .004$ calculated in the numerical Chalker-Coddington model, suggesting that these models are in the same universality class. The real delocalization exponent may be disentangled from other scaling exponents in finite size scaling experiments, giving an experimental value ${\nu }_{\exp }=2.3\pm 0.2$. The modular model suggests how these theoretical and experimental results may be reconciled, but in order to determine if these theoretical models really are in the quantum Hall universality class, improved finite size scaling experiments are urgently needed.

arXiv comment31 pages, 12 figures

References in corpus (31)

• Entanglement in quantum critical phenomena
• Trajectory of Anomalous Hall Effect toward the Quantized State in a Ferromagnetic Topological Insulator
• Notes on Some Entanglement Properties of Quantum Field Theory
• Holomorphic Anomalies in Topological Field Theories
• Entanglement entropy: holography and renormalization group
• Goldstone and Pseudo-Goldstone Bosons in Nuclear, Particle and Condensed-Matter Physics
• Critical exponent for the quantum Hall transition
• Scaling in Plateau-to-Plateau Transition: A Direct Connection of Quantum Hall Systems with Anderson Localization Model
• Quantum information and physics: some future directions
• Fine-grained entanglement loss along renormalization group flows
• Evidence for charge-flux duality near the quantum Hall liquid to insulator transition
• Quantum Hall effect in a one-dimensional dynamical system
• Entanglement measures and their properties in quantum field theory
• Finite Size Effects and Irrelevant Corrections to Scaling near the Integer Quantum Hall Transition
• Conformal invariance, multifractality, and finite-size scaling at Anderson localization transitions in two dimensions
• Numerical study of the localization length critical index in a network model of plateau-plateau transitions in the quantum Hall effect
• D-Branes on K3-Fibrations
• One-Dimensional Flows in the Quantum Hall System
• Finite Size Scaling of the Chalker-Coddington Model
• Universal flow diagram for the magnetoconductance in disordered GaAs layers
• The Quantum Hall Effect in Graphene: Emergent Modular Symmetry and the Semi-circle Law
• Scaling flow diagram in the fractional quantum Hall regime of GaAs/AlGaAs heterostructures
• Renormalization group approach to energy level statistics at the integer quantum Hall transition
• D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories
• On the Implications of Discrete Symmetries for the Beta Function of Quantum Hall Systems
• Geometric scaling in the quantum Hall system
• Topology and Fractional Quantum Hall Effect
• Non-linear Yang-Mills instantons from strings are $\pi$-stable D-branes
• Experimental probes of emergent symmetries in the quantum Hall system
• On the origin of duality in the quantum Hall system
• An RG potential for the quantum Hall effects

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