Spectral flows associated to flux tubes

Abstract

When a flux quantum is pushed through a gapped two-dimensional tight-binding operator, there is an associated spectral flow through the gap which is shown to be equal to the index of a Fredholm operator encoding the topology of the Fermi projection. This is a natural mathematical formulation of Laughlin's Gedankenexperiment. It is used to provide yet another proof of the bulk-edge correspondence. Furthermore, when applied to systems with time reversal symmetry, the spectral flow has a characteristic $Z_2$ signature, while for particle-hole symmetric systems it leads to a criterion for the existence of zero energy modes attached to half-flux tubes. Combined with other results, this allows to explain all strong invariants of two-dimensional topological insulators in terms of a single Fredholm operator.

final version, to appear in Ann. H. Poincare

References in corpus (8)

• Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions
• Spin-charge Separated Solitons in a Topological Band Insulator
• Spin Charge Separation in the Quantum Spin Hall State
• Equality of the bulk and edge Hall conductances in a mobility gap
• Classification of "Real" Bloch-bundles: Topological quantum systems of type AI
• Z_2 indices and factorization properties of odd symmetric Fredholm operators
• Spectral flows of dilations of Fredholm operators
• Classification of "Quaternionic" Bloch-bundles: Topological Quantum Systems of type AII

Cited by in corpus (18)

• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• The $K$-theoretic bulk-edge correspondence for topological insulators
• Limits of Quantum Graph Operators With Shrinking Edges
• 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
• Anderson-Kitaev spin liquid
• Z_2 indices and factorization properties of odd symmetric Fredholm operators
• Spectral flows of dilations of Fredholm operators
• Local formula for the $Z_2$ invariant of topological insulators
• Topology in shallow-water waves: A spectral flow perspective
• The KO-valued spectral flow for skew-adjoint Fredholm operators
• Bulk-edge correspondence and the cobordism invariance of the index
• The non-commutative topology of two-dimensional dirty superconductors
• On ${\mathbb{Z}}_2$-indices for ground states of fermionic chains
• Parity flow as ${\mathbb{Z}}_2$-valued spectral flow
• From orbital magnetism to bulk-edge correspondence
• Dynamical abelian anyons with bound states and scattering states
• The index of a pair of pure states and the interacting integer quantum Hall effect