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PreprintCC BY 4.0arXiv

Dirac Points in High-Contrast Dielectric Honeycomb Structures

  • Yi Huang
  • Bowen Li
  • Ping Liu
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Abstract

We study Dirac points for a two-dimensional scalar Helmholtz equation in a periodic dielectric medium with two penetrable inclusions per honeycomb cell. In a joint small-inclusion and high-contrast regime, we reduce the Bloch eigenvalue problem to a two-dimensional interaction matrix and derive a two-term asymptotic expansion for the two relevant eigenvalues, uniformly for quasi-momenta separated from the reciprocal lattice. At a vertex K of the Brillouin zone, rotation and inversion-conjugation symmetries force these eigenvalues to coincide. A Schur-complement analysis of the exact operator pencil, together with a nonvanishing lattice Green-function coefficient, proves that the resulting double eigenvalue has a nonzero conical splitting and gives an asymptotic formula for the conical slope in the scaled eigenvalue. For fixed sufficiently large contrast, symmetry-preserving changes of variables and analytic perturbation theory then continue this Dirac point as the inclusion size varies: it remains nondegenerate except at a relatively closed discrete set of admissible sizes.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026dirac,
  title = {{Dirac Points in High-Contrast Dielectric Honeycomb Structures}},
  author = {Yi Huang and Bowen Li and Ping Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19426},
  eprint = {2610.19426},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19426},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Dirac Points in High-Contrast Dielectric Honeycomb Structures
AU  - Yi Huang
AU  - Bowen Li
AU  - Ping Liu
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19426
UR  - https://arxiv.org/abs/2610.19426
AB  - We study Dirac points for a two-dimensional scalar Helmholtz equation in a periodic dielectric medium with two penetrable inclusions per honeycomb cell. In a joint small-inclusion and high-contrast regime, we reduce the Bloch eigenvalue problem to a two-dimensional interaction matrix and derive a two-term asymptotic expansion for the two relevant eigenvalues, uniformly for quasi-momenta separated from the reciprocal lattice. At a vertex K of the Brillouin zone, rotation and inversion-conjugation symmetries force these eigenvalues to coincide. A Schur-complement analysis of the exact operator pencil, together with a nonvanishing lattice Green-function coefficient, proves that the resulting double eigenvalue has a nonzero conical splitting and gives an asymptotic formula for the conical slope in the scaled eigenvalue. For fixed sufficiently large contrast, symmetry-preserving changes of variables and analytic perturbation theory then continue this Dirac point as the inclusion size varies: it remains nondegenerate except at a relatively closed discrete set of admissible sizes.
ER  -