Topologically Protected States in One-Dimensional Systems
Abstract
We study a class of periodic Schrödinger operators, which in distinguished cases can be proved to have linear band-crossings or "Dirac points". We then show that the introduction of an "edge", via adiabatic modulation of these periodic potentials by a domain wall, results in the bifurcation of spatially localized "edge states". These bound states are associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. Our model captures many aspects of the phenomenon of topologically protected edge states for two-dimensional bulk structures such as the honeycomb structure of graphene. The states we construct can be realized as highly robust TM- electromagnetic modes for a class of photonic waveguides with a phase-defect.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2014
- DOI:
- 10.48550/arXiv.1405.4569
- arXiv:
- arXiv:1405.4569
- Bibcode:
- 2014arXiv1405.4569F
- Keywords:
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- Mathematical Physics;
- Condensed Matter - Mesoscale and Nanoscale Physics;
- Mathematics - Analysis of PDEs;
- Quantum Physics
- E-Print:
- To appear in Memoirs of the American Mathematical Society -- 100+ pages, 10 figures