Equality of the bulk and edge Hall conductances in a mobility gap

Abstract

We consider the edge and bulk conductances for 2D quantum Hall systems in which the Fermi energy falls in a band where bulk states are localized. We show that the resulting quantities are equal, when appropriately defined. An appropriate definition of the edge conductance may be obtained through a suitable time averaging procedure or by including a contribution from states in the localized band. In a further result on the Harper Hamiltonian, we show that this contribution is essential. In an appendix we establish quantized plateaus for the conductance of systems which need not be translation ergodic.

arXiv comment38 pages, LaTeX, uses svjour class. Corrected a number of typos and an error in proof of Lemma four. The latter correction appears as a separate erratum in the published version. Additional typos corrected in v3

References in corpus (7)

• The Non-Commutative Geometry of the Quantum Hall Effect
• Charge Deficiency, Charge Transport and Comparison of Dimensions
• Localization Bounds for an Electron Gas
• Simultaneous quantization of edge and bulk Hall conductivity
• Finite-Volume Fractional-Moment Criteria for Anderson Localization
• Equality of bulk and edge Hall conductance revisited
• Hofstadter butterfly as Quantum phase diagram

Cited by in corpus (50)

• Disordered Topological Insulators: A Non-Commutative Geometry Perspective
• New characterizations of the region of complete localization for random Schrödinger operators
• Anomalous bulk-edge correspondence in continuous media
• Bulk-Edge correspondence for two-dimensional Floquet topological insulators
• Continuous bulk and interface description of topological insulators
• The Bulk-Edge Correspondence for Disordered Chiral Chains
• The Noncommutative Index Theorem and the Periodic Table for Disordered Topological Insulators and Superconductors
• The $Z_2$ Index of Disordered Topological Insulators with Time Reversal Symmetry
• On the Bott index of unitary matrices on a finite torus
• The edge spectrum of Chern insulators with rough boundaries
• Spectral flows associated to flux tubes
• Mathematical theory for topological photonic materials in one dimension
• The bulk-edge correspondence for the quantum Hall effect in Kasparov theory
• Microlocal analysis of the bulk-edge correspondence
• Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases
• Orbital polarization and magnetization for independent particles in disordered media
• Strongly Disordered Floquet Topological Systems
• Time-dependent topological systems: A study of the Bott index
• Manifestations of dynamical localization in the disordered XXZ spin chain
• Persistence of spin edge currents in disordered quantum spin Hall systems
• Edge Currents for Quantum Hall Systems, I. One-Edge, Unbounded Geometries
• The Bulk-Edge Correspondence in Three Simple Cases
• Quantization of edge currents along magnetic barriers and magnetic guides
• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Characterization of edge states in perturbed honeycomb structures
• Chiral Flow in One-dimensional Floquet Topological Insulators
• Effective vacua for Floquet topological phases: A numerical perspective on switch-function formalism
• Spin Conductance and Spin Conductivity in Topological Insulators: Analysis of Kubo-like terms
• Computing edge states without hard truncation
• Fredholm Homotopies for Strongly-Disordered 2D Insulators
• Universal edge transport in interacting Hall systems
• Asymmetric transport computations in Dirac models of topological insulators
• Localization of generalized Wannier bases implies Chern triviality in non-periodic insulators
• Edge states in ordinary differential equations for dislocations
• Topology of 2D Dirac operators with variable mass and an application to shallow-water waves
• Two-Dimensional Time-Reversal-Invariant Topological Insulators via Fredholm Theory
• Multi-Channel Luttinger Liquids at the Edge of Quantum Hall Systems
• Bulk-edge correspondence for unbounded Dirac-Landau operators
• Wave-packet propagation in a finite topological insulator and the spectral localizer index
• Topological quantization of ensemble averages
• From charge to spin: analogies and differences in quantum transport coefficients
• Asymmetric transport for magnetic Dirac equations
• Absolutely Continuous Edge Spectrum of Hall Insulators on the Lattice
• Dynamics of a classical Hall system driven by a time-dependent Aharonov--Bohm flux
• From orbital magnetism to bulk-edge correspondence
• Boundary conditions and violations of bulk-edge correspondence in a hydrodynamic model
• Widths of the Hall Conductance Plateaus
• Scattering theory of topologically protected edge transport
• Equality of magnetization and edge current for interacting lattice fermions at positive temperature
• Fractional index of Bargmann-Fock space and Landau levels