Microlocal Analysis of the Bulk-Edge Correspondence
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Communications in
Mathematical Physics
Microlocal Analysis of the Bulk-Edge Correspondence Alexis Drouot Columbia University, New York, USA. E-mail: [email protected] Received: 29 April 2020 / Accepted: 30 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: The bulk-edge correspondence predicts that interfaces between topological insulators support robust currents. We prove this principle for PDEs that are periodic away from an interface. Our approach relies on semiclassical methods. It suggests novel perspectives for the analysis of topologically protected transport.
1. Introduction In solid state physics, insulators are materials modeled by Hamiltonians with spectral gaps. They are topologically classified through winding properties of their spectral projections. Gluing together topologically distinct insulators generate materials with a different electronic behavior: robust currents emerge along interfaces. Strikingly, the existence of these currents depends on the bulk structure rather than on the interface. This phenomenon is called the bulk-edge correspondence. It is a universal principle that reaches beyond electronics, for instance in accoustics [YGS15], photonics [HR07,RH08], fluid mechanics [DMV17,PDV19] and molecular physics [F19]. While bulk and edge indices were introduced as early as [H82,TKN82,BES94], the mathematical formulation of the bulk-edge correspondence started with [H93]. It has been the object of various developments, covering Landau Hamiltonians [KRS02,EG02,KS04a, KS04b], strong disorder [EGS05,GS18,T14], Z2 -topological insulators [GP13,ASV13], K-theoretic aspects [BKR17,K17,BR18,B19] and periodic forcing [GT18,ST19]. In this work, we define and derive the bulk-edge correspondence for a class of PDEs that are periodic away from the interface. The most important characteristics of our approach is the use of microlocal techniques in a field traditionally dominated by Ktheory and functional analysis. It opens two promising perspectives: • The quantitative analysis of topologically protected transport; • The geometric calculation of bulk/edge indices in terms of eigenvalue crossings.
A. Drouot
1.1. Setting and main result. We study the Schrödinger evolution of electrons in a twodimensional material, i∂t ψ = Pψ. The Hamiltonian P is an elliptic selfadjoint second order differential operator on L 2 (R2 ): def def 1 ∂ P = , α = (α1 , α2 ) ∈ N2 . aα (x)Dxα , aα (x) ∈ Cb∞ (R2 , C), Dx = i ∂x |α|≤2
(1.1) The space Cb∞ refers to bounded functions together with all their derivatives; see Sect. 1.5. The class (1.1) models for instance Schrödinger operators with a potential V (x) ∈ Cb∞ (R2 , R) and a (transverse) magnetic field ∂x1 A2 − ∂x2 A1 , where A(x) ∈ Cb∞ (R2 , R2 ): 2 − ∇R2 + i A(x) + V (x). (1.2) It also includes the stationary form of the wave equation that appears in photonics and meta-material realizations of topological insulators [HR07,RH08,KMT13,LWZ18]: − divR2 σ (x) · ∇R2 , σ (x) ∈ Cb∞ R2 , M2 (C) Hermitian-valued. (1.3) In relatio
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