Continuous bulk and interface description of topological insulators

Abstract

We analyze continuous partial differential models of topological insulators in the form of systems of Dirac equations. We describe the bulk and interface topological properties of the materials by means of indices of Fredholm operators constructed from the Dirac operators by spectral calculus. We show the stability of these topological invariants with respect to perturbations by a large class of spatial heterogeneities. These models offer a quantitative tool to analyze the interplay between topology and spatial fluctuations in topological phases of matter. The theory is first presented for two-dimensional materials, which display asymmetric (chiral) transport along interfaces. It is then generalized to arbitrary dimensions with the additional assumption of chiral symmetry in odd spatial dimensions.

54 pages; long version of paper in J.M.P. (60), 2019; some typos corrected

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• Mathematical theory for topological photonic materials in one dimension
• Microlocal analysis of the bulk-edge correspondence
• Optical $N$-invariant of graphene's viscous Hall fluid
• Odd elasticity and topological waves in active surfaces
• Bulk-interface correspondences for one dimensional topological materials with inversion symmetry
• Asymmetric transport computations in Dirac models of topological insulators
• Edge states in ordinary differential equations for dislocations
• Bulk-edge correspondence for unbounded Dirac-Landau operators
• Linear and nonlinear electromagnetic waves in modulated honeycomb media
• Complete absorption of topologically protected waves
• Generalized topological bulk-edge correspondence in bulk-Hermitian continuous systems with non-Hermitian boundary conditions
• Asymmetric transport for magnetic Dirac equations
• From orbital magnetism to bulk-edge correspondence
• Boundary conditions and violations of bulk-edge correspondence in a hydrodynamic model
• Scattering theory of topologically protected edge transport
• Topological edge states of continuous Hamiltonians