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We demonstrate how introducing an edge to periodic potentials creates localized edge states. These topologically protected states are robust and can be realized in photonic waveguides, mimicking graphene structures.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Mechanics
  • Photonics

Background:

  • Periodic Schrödinger operators exhibit Dirac points, crucial for understanding electronic and photonic band structures.
  • Topologically protected edge states offer robustness against defects and disorder, making them highly desirable for applications.

Purpose of the Study:

  • To investigate the emergence and properties of spatially localized edge states in periodic systems with domain walls.
  • To establish a theoretical model for topologically protected edge states applicable to both quantum systems and photonic structures.

Main Methods:

  • Analysis of periodic Schrödinger operators on ℝ with adiabatic modulation to introduce domain walls.
  • Derivation of edge states from the zero-energy mode of an asymptotic one-dimensional Dirac operator.
  • Mapping the theoretical model to robust transverse-magnetic electromagnetic modes in photonic waveguides.

Main Results:

  • Demonstrated the bifurcation of spatially localized edge states upon introducing an edge via domain wall modulation.
  • Constructed bound states corresponding to topologically protected zero-energy modes.
  • Showcased the realization of these states as robust electromagnetic modes in photonic waveguides with phase defects.

Conclusions:

  • The study successfully models topologically protected edge states, analogous to those in 2D bulk structures like graphene.
  • The proposed model provides a pathway for designing robust photonic devices utilizing topological principles.
  • Confirms the broad applicability of topological concepts across different physical domains, from quantum mechanics to optics.