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Defect Modes for Dislocated Periodic Media Drouot, A.; Fefferman, C. L.; Weinstein, M. I.
Communications in mathematical physics,
08/2020, Letnik:
377, Številka:
3
Journal Article
Recenzirano
Odprti dostop
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
.
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