Random Network Models and Quantum Phase Transitions in Two Dimensions

Abstract

An overview of the random network model invented by Chalker and Coddington, and its generalizations, is provided. After a short introduction into the physics of the Integer Quantum Hall Effect, which historically has been the motivation for introducing the network model, the percolation model for electrons in spatial dimension 2 in a strong perpendicular magnetic field and a spatially correlated random potential is described. Based on this, the network model is established, using the concepts of percolating probability amplitude and tunneling. Its localization properties and its behavior at the critical point are discussed including a short survey on the statistics of energy levels and wave function amplitudes. Magneto-transport is reviewed with emphasis on some new results on conductance distributions. Generalizations are performed by establishing equivalent Hamiltonians. In particular, the significance of mappings to the Dirac model and the two dimensional Ising model are discussed. A description of renormalization group treatments is given. The classification of two dimensional random systems according to their symmetries is outlined. This provides access to the complete set of quantum phase transitions like the thermal Hall transition and the spin quantum Hall transition in two dimension. The supersymmetric effective field theory for the critical properties of network models is formulated. The network model is extended to higher dimensions including remarks on the chiral metal phase at the surface of a multi-layer quantum Hall system.

176 pages, final version, references corrected

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• Real Space Renormalization of the Chalker-Coddington Model
• Weak levitation of 2D delocalized states in a magnetic field.
• The Chalker-Coddington Network Model is Quantum Critical
• Landau level mixing and spin degeneracy in the quantum Hall effect
• Transitions Between Hall Plateaus and the Dimerization Transition of a Hubbard Chain
• Conductance distribution in quasi-one-dimensional disordered quantum wires
• Network models for localisation problems belonging to the chiral symmetry classes
• Anderson transitions in three-dimensional disordered systems with randomly varying magnetic flux
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• Connecting polymers to the quantum Hall plateau transition
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Cited by in corpus (117)

• Anderson Transitions
• Detection of phase transition via convolutional neural network
• The Quantum Hall Transition in Real Space: From Localized to Extended States
• Critical exponent for the quantum Hall transition
• Critical exponent for the Anderson transition in the three dimensional orthogonal universality class
• Universal Scaling of Quantum Anomalous Hall Plateau Transition
• Network Models of Photonic Floquet Topological Insulators
• Measurement of a topological edge invariant in a microwave network
• Nonunitary Scaling Theory of Non-Hermitian Localization
• Observation of the zero Hall plateau in a quantum anomalous Hall insulator
• Effect of Disorder in a Three-Dimensional Layered Chern Insulator
• Two-dimensional spin-filtered chiral network model for the Z_2 quantum spin-Hall effect
• Quantum Hall effect in a one-dimensional dynamical system
• Finite difference method for transport properties of massless Dirac fermions
• 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
• Theory of Disordered $\nu =5/2$ Quantum Thermal Hall State: Emergent Symmetry and Phase Diagram
• The integer quantum Hall plateau transition is a current algebra after all
• Effective mass and tricritical point for lattice fermions localized by a random mass
• Half-integer quantum Hall effect of disordered Dirac fermions at a topological insulator surface
• Quantum phase transition of chiral Majorana fermion in the presence of disorder
• Drawing Phase Diagrams of Random Quantum Systems by Deep Learning the Wave Functions
• Critical Exponent of the Anderson Transition using Massively Parallel Supercomputing
• Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions
• Topological quantum number and critical exponent from conductance fluctuations at the quantum Hall plateau transition
• Unraveling quantum Hall breakdown in bilayer graphene with scanning gate microscopy
• Integer Quantum Magnon Hall Plateau-Plateau Transition in a Spin Ice Model
• Effects of random domains on the zero Hall plateau in quantum anomalous Hall effect
• Phase Structure of the Topological Anderson Insulator
• Topological phase transitions in glassy quantum matter
• Finite Size Scaling of the Chalker-Coddington Model
• In-Plane Magnetic-Field-Induced Quantum Anomalous Hall Plateau Transition
• Spectral Stability of Unitary Network Models
• Localization length exponent in two models of quantum Hall plateau transitions
• Even-odd flux quanta effect in the Fraunhofer oscillations of an edge-channel Josephson junction
• Spin-directed network model for the surface states of weak three-dimensional ${\mathbb{Z}}_2^{}$ topological insulators
• Modification and Control of Topological Insulator Surface States Using Surface Disorder
• Finite Size Scaling Analysis of the Anderson Transition
• Calculation of the conductance of a graphene sheet using the Chalker-Coddington network model
• Chiral quantum optics in the bulk of photonic quantum Hall systems
• Criticality of two-dimensional disordered Dirac fermions in the unitary class and universality of the integer quantum Hall transition
• Criticality in amorphous topological matter -- beyond the universal scaling paradigm
• Thermal metal-insulator transition in a helical topological superconductor
• Microscopics of disordered two-dimensional electron gases under high magnetic fields: Equilibrium properties and dissipation in the hydrodynamic regime
• Compressibility stripes for mesoscopic quantum Hall samples
• Finite size properties of staggered $U_q[sl(2|1)]$ superspin chains
• The Z_2 network model for the quantum spin Hall effect: two-dimensional Dirac fermions, topological quantum numbers, and corner multifractality
• Scaling collapse of longitudinal conductance near the integer quantum Hall transition
• Spectral Transition for Random Quantum Walks on Trees
• Numerical evidence for marginal scaling at the integer quantum Hall transition
• Unconventional conductance plateau transitions in quantum Hall wires with spatially correlated disorder
• Anisotropic Topological Anderson Transitions in Chiral Symmetry Classes
• Analytical approach to directed sandpile models on the Apollonian network
• Localization Properties of the Chalker-Coddington Model
• Field theory of non-Hermitian disordered systems
• Multicriticality of Two-dimensional Class D Disordered Topological Superconductors
• Chalker-Coddington model described by an S-matrix with odd dimensions
• Electron correlations and single-particle physics in the Integer Quantum Hall Effect
• Dynamical Localization of the Chalker-Coddington Model far from Transition
• Network model for periodically strained graphene
• Exact exponents for the spin quantum Hall transition in the presence of multiple edge channels
• Conductance distributions in disordered quantum spin-Hall systems
• Marginal CFT perturbations at the integer quantum Hall transition
• Generalized multifractality at spin quantum Hall transition
• Generalized multifractality at metal-insulator transitions and in metallic phases of 2D disordered systems
• Spontaneous Layer Polarization and Conducting Domain Walls in the Quantum Hall Regime of Bilayer Graphene
• Integer Quantum Hall Effect: Disorder, temperature, floating, and plateau width
• Effective Floquet model for minimally twisted bilayer graphene
• Network model for higher-order topological phases
• Quantum site percolation on triangular lattice and the integer quantum Hall effect
• Ferromagnetism and glassiness on the surface of topological insulators
• Tachyonic Dispersion in Coherent Networks
• Weakly chiral networks and 2D delocalized states in a weak magnetic field
• Statistics of Conductances and Subleading Corrections to Scaling near the Integer Quantum Hall Plateau Transition
• Generalized surface multifractality in 2D disordered systems
• Levitation and percolation in quantum Hall systems with correlated disorder
• Crossover from diffusive to strongly localized regime in two-dimensional systems
• Continuous and reversible tuning of the disorder-driven superconductor-insulator transition in bilayer graphene
• Conductance plateau transitions in quantum Hall wires with spatially correlated random magnetic fields
• Classical percolation fingerprints in the high-temperature regime of the integer quantum Hall effect
• Geometry of random potentials: Induction of 2D gravity in Quantum Hall plateau transitions
• A simple microscopic description of quantum Hall transition without Landau levels
• Hartree-Fock Interactions in the Integer Quantum Hall Effect
• Point-Contact Conductance in Asymmetric Chalker-Coddington Network Model
• On Fermionic walkers interacting with a correlated structured environment
• On truncations of the Chalker-Coddington model
• Transport properties in network models with perfectly conducting channels
• Diagrammatic Approach for the High-Temperature Regime of Quantum Hall Transitions
• Irrelevant corrections at the quantum Hall transition
• Scattering theory of higher order topological phases
• Transmission coefficient through a saddle-point electrostatic potential for graphene in the quantum Hall regime
• Thouless conductances of a three-dimensional quantum Hall system
• Critical behavior at the integer quantum Hall transition in a network model on the Kagome lattice
• The integer quantum Hall transition: an $S$-matrix approach to random networks
• Quantum Hall criticality in an amorphous Chern insulator
• Transport via classical percolation at quantum Hall plateau transitions
• Proposal for realizing anomalous Floquet insulators via Chern band annihilation
• Lower Bounds on the Localisation Length of Balanced Random Quantum Walks
• Disordered Quantum Transport in Quantum Anomalous Hall Insulator-Superconductor Junctions
• Extended critical phase in quasiperiodic quantum Hall systems
• Network model for magnetic higher-order topological phases
• Spin-mixing-tunneling network model for Anderson transitions in two-dimensional disordered spinful electrons
• Symmetry and Topology of Monitored Quantum Dynamics
• Real-space renormalisation approach to the Chalker-Coddington model revisited: improved statistics
• Engineering stable quantum currents at bulk boundaries
• Killing the Hofstadter butterfly, one bond at a time
• Random Magnetic Field and the Dirac Fermi Surface
• Effect of surface disorder on the chiral surface states of a three-dimensional quantum Hall system
• Edge state critical behavior of the integer quantum Hall transition
• Chirality induced Interface Currents in the Chalker Coddington Model
• Supermetal-insulator transition in a non-Hermitian network model
• Tunable Topological Phases in Quantum Kirigamis
• Generalized multifractality at the spin quantum Hall transition: Percolation mapping and pure-scaling observables
• Extreme-Value Criticality and Gain Decomposition at the Integer Quantum Hall Transition
• Examples for stable quantum currents
• Network model and four-terminal transport in minimally twisted bilayer graphene
• Quantum Critical Dynamics Induced by Topological Zero Modes