Defect Modes for Dislocated Periodic Media

• Published in: Communications in mathematical physics Vol. 377; no. 3; pp. 1637 - 1680
• Main Authors: Drouot, A., Fefferman, C. L., Weinstein, M. I.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-08-2020; Springer Nature B.V
• Subjects: Bifurcations; Classical and Quantum Gravitation; Complex Systems; Defects; Dispersion curve analysis; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-020-03787-0

We study defect modes in a one-dimensional periodic medium perturbed by an adiabatic dislocation of amplitude
δ
≪
1
. If the periodic background admits a Dirac point—a linear crossing of dispersion curves—then the dislocated operator acquires a gap in its essential spectrum. For this model (and its honeycomb analog) Fefferman et al. (Proc Natl Acad Sci USA 111(24):8759–8763, 2014, Mem Am Math Soc 247(1173):118, 2017, Ann PDE 2(2):80, 2016, 2D Mater 3:1, 2016) constructed (at leading order in
δ
) defect modes with energies within the gap. These bifurcate from the eigenmodes of an effective Dirac operator. Here we address the following open problems:
Do
all
defect modes arise as bifurcations from the Dirac operator eigenmodes?
Do these modes admit expansions to all order in
δ
?
We respond positively to both questions. Our approach relies on (a) resolvent estimates for the bulk operators; (b) scattering theory for highly oscillatory potentials [
Dr18a
,
Dr18b
,
Dr18c
]. It has led to an understanding of the topological stability of defect states in continuous dislocated systems—in connection with the bulk-edge correspondence [
Dr18d
].