Continuous Topological Insulators Classification and Bulk-Edge Correspondence

Guillaume Bal ()
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Guillaume Bal: University of Chicago, Departments of Statistics and Mathematics and CCAM

Chapter 107 in Operator Theory, 2026, pp 3301-3354 from Springer

Abstract: Abstract This chapter reviews recent results on the classification of partial differential operators modeling bulk and interface topological insulators in Euclidean spaces. Our main objective is the mathematical analysis of the unusual, robust-to-perturbations, asymmetric transport that necessarily appears at interfaces separating topological insulators in different phases. The central element of the analysis is an interface current observable describing this asymmetry. We show that this observable may be computed explicitly by spectral flow when the interface Hamiltonian is explicitly diagonalizable. We review the classification of bulk phases for Landau and Dirac operators and provide a general classification of elliptic interface pseudo-differential operators by means of domain walls and a corresponding bulk-difference invariant (BDI). The BDI is simple to compute by the Fedosov-Hörmander formula implementing in a Euclidean setting an Atiyah-Singer index theory. A generalized bulk-edge correspondence then states that the interface current observable and the BDI agree on elliptic operators, whereas this is not necessarily the case for non-elliptic operators.

Keywords: Topological insulators; Topological classification; Index theory; Bulk-difference invariant; Bulk-edge correspondence; Spectral calculus; Pseudo-differential calculus.; 35Q40; 35S35; 47A53; 47A60; 47G30; 81Q10 (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16356-1_76

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DOI: 10.1007/978-3-032-16356-1_76

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