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].
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics