Modular Invariance, Self-Duality and The Phase Transition Between Quantum Hall Plateaus

Abstract

We investigate the problem of the superuniversality of the phase transition between different quantum Hall plateaus. We construct a set of models which give a qualitative description of this transition in a pure system of interacting charged particles. One of the models is manifestly invariant under both Duality and Periodic shifts of the statistical angle and, hence, it has a full Modular Invariance. We derive the transformation laws for the correlation functions under the modular group and use them to derive symmetry constraints for the conductances. These allow us to calculate exactly the conductivities at the modular fixed points. We show that, at least at the modular fixed points, the system is critical. Away from the fixed points, the behavior of the model is determined by extra symmetries such as Time Reversal. We speculate that if the natural connection between spin and statistics holds, the model may exhibit an effective analyticity at low energies. In this case, the conductance is completely determined by its behavior under modular transformations.

44 pages, Revtex. Corrected manuscript, Submitted to Nucl. Phys. B

References in corpus (6)

• Mott Transition in An Anyon Gas
• Phase Diagram of the Dissipative Hofstadter Model
• Intersections forms and the geometry of lattice Chern-Simons theory
• Chern_simons Theory of the Anisotropic Quantum Heisenberg Antiferromagnet on a Square Lattice
• Critical Theories of the Dissipative Hofstadter Model
• Driving Operators Relevant: A Feature of Chern-Simons Interaction

Cited by in corpus (57)

• Seeing the magnetic monopole through the mirror of topological surface states
• Deconfined quantum critical points: symmetries and dualities
• Quantum critical transport, duality, and M-theory
• Non-zero temperature transport near quantum critical points
• Quantum Hall Physics - hierarchies and CFT techniques
• Ohm's Law at strong coupling: S duality and the cyclotron resonance
• Particle-vortex duality of 2d Dirac fermion from electric-magnetic duality of 3d topological insulators
• Duality between $(2+1)d$ Quantum Critical Points
• Holographic Quantum Critical Transport without Self-Duality
• SL(2,Z) Action on Three-Dimensional CFTs and Holography
• Effects of dissipation on quantum phase transitions
• Bosonization and Mirror Symmetry
• Disorder Operators and their Descendants
• Global Phase Diagram of a Dirty Weyl Liquid and Emergent Superuniversality
• Unifying the Anderson Transitions in Hermitian and Non-Hermitian Systems
• Symmetry and duality in bosonization of two-dimensional Dirac fermions
• Unconventional Fusion and Braiding of Topological Defects in a Lattice Model
• One-Dimensional Flows in the Quantum Hall System
• Non-commutative Chern-Simons for the Quantum Hall System and Duality
• AdS/QHE: Towards a Holographic Description of Quantum Hall Experiments
• Duality and Universal Transport in a Mixed-Dimension Electrodynamics
• SL(2,Z) Action on AdS/BCFT and Hall Conductivities
• Fractional $S$-duality, Classification of Fractional Topological Insulators and Surface Topological Order
• Duality Near Quantum Hall Transitions
• Effective hydrodynamic field theory and condensation picture of topological insulators
• Composite Fermion Metals from Dyon Black Holes and S-Duality
• Monte Carlo Study of a U(1)xU(1) system with π-statistical Interaction
• Modular Groups, Visibility Diagram and Quantum Hall Effect
• Non-Abelian bosonization and modular transformation approach to superuniversality
• Loop Models, Modular Invariance, and Three Dimensional Bosonization
• A series of (2+1)d Stable Self-Dual Interacting Conformal Field Theories
• Superuniversality of topological quantum phase transition and global phase diagram of dirty topological systems in three dimensions
• Quantum generated vortices, dual singular gauge transformation and zero temperature transition from d-wave superconductor to underdoped regime
• Theory of oblique topological insulators
• Particle-vortex symmetric liquid
• $\Gamma (2)$ modular symmetry, renormalization, group flow and the quantum Hall effect
• Line of continuous phase transitions in a three dimensional U(1) model with 1/r^2 current-current interactions
• Reflection Symmetry and Quantized Hall Resistivity near Quantum Hall Transition
• An experimental study on $\Gamma$(2) modular symmetry in the quantum Hall system with a small spin-splitting
• Universal Transverse Conductance between Quantum Hall Regions and (2+1)D Bosonization
• Network of edge states: random Josephson junction array description
• Phases and phase transitions in a $U(1)\times U(1)$ system with $\theta =2\pi /3$ mutual statistics
• Lattice realization of a bosonic integer quantum Hall state - trivial insulator transition and relation to the self-dual line in the easy-plane NCCP1 model
• Monte Carlo Study of a $U(1)\times U(1)$ Loop Model with Modular Invariance
• Relativistic non-Fermi liquid from interacting birefringent fermions: A robust superuniversality
• Global symmetries of quantum Hall systems: lattice description
• Lattice Construction of Duality with Non-Abelian Gauge Fields in 2+1D
• Renormalization of modular invariant Coulomb gas and Sine-Gordon theories, and quantum Hall flow diagram
• Duality Web on a 3D Euclidean Lattice and Manifestation of Hidden Symmetries
• N=2 Supersymmetric Yang-Mills and the Quantum Hall Effect
• Boson-fermion duality in a gravitational background
• Holomorphic and anti-holomorphic conductivity flows in the quantum Hall effect
• Wiedemann-Franz laws and $Sl(2,\mathbb{Z})$ duality in AdS/CMT holographic duals and one-dimensional effective actions for them
• The point-splitting regularization of (2+1)d parity breaking models
• Quantization of a Self-dual Conformal Theory in (2+1) Dimensions
• Gauge Invariance and the Critical Properties of Quantum Hall Plateaux Transitions
• Visibility diagrams and experimental stripe structure in the quantum Hall effect