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Generalized Honeycomb-structured Materials in the Subwavelength Regime

Borui Miao, Yi Zhu

Abstract

Honeycomb structures lead to conically degenerate points on the dispersion surfaces. These spectral points, termed as Dirac points, are responsible for various topological phenomena. In this paper, we investigate the generalized honeycomb-structured materials, which have six inclusions in a hexagonal cell. We obtain the asymptotic band structures and corresponding eigenstates in the subwavelength regime using the layer potential theory. Specifically, we rigorously prove the existence of the double Dirac cones lying on the 2nd-5th bands when the six inclusions satisfy an additional symmetry. This type of inclusions will be referred to as super honeycomb-structured inclusions. Two distinct deformations breaking the additional symmetry, contraction and dilation, are further discussed. We prove that the double Dirac cone disappears, and a local spectral gap opens. The corresponding eigenstates are also obtained to show the topological differences between these two deformations. Direct numerical simulations using finite element methods agree well with our analysis.

Generalized Honeycomb-structured Materials in the Subwavelength Regime

Abstract

Honeycomb structures lead to conically degenerate points on the dispersion surfaces. These spectral points, termed as Dirac points, are responsible for various topological phenomena. In this paper, we investigate the generalized honeycomb-structured materials, which have six inclusions in a hexagonal cell. We obtain the asymptotic band structures and corresponding eigenstates in the subwavelength regime using the layer potential theory. Specifically, we rigorously prove the existence of the double Dirac cones lying on the 2nd-5th bands when the six inclusions satisfy an additional symmetry. This type of inclusions will be referred to as super honeycomb-structured inclusions. Two distinct deformations breaking the additional symmetry, contraction and dilation, are further discussed. We prove that the double Dirac cone disappears, and a local spectral gap opens. The corresponding eigenstates are also obtained to show the topological differences between these two deformations. Direct numerical simulations using finite element methods agree well with our analysis.
Paper Structure (18 sections, 22 theorems, 115 equations, 4 figures)

This paper contains 18 sections, 22 theorems, 115 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main Results and Contributions
  3. Outlines
  4. Problem Formulation and Preliminaries
  5. Generalized Honeycomb-structured Inclusions
  6. Problem Formulation
  7. Green's Functions and Layer Potentials
  8. Integral Equation Formulation
  9. Asymptotic Behavior of Band Structure
  10. Asymptotic Expansion of Green's Function
  11. Asymptotic Behavior of Band Structure
  12. Structure of Periodic Capacitance Matrix
  13. Band Structure of Super Honeycomb-structured Materials
  14. Double Dirac Cone Structure
  15. Numerical Simulation
  16. ...and 3 more sections

Key Result

Theorem 3.1

\newlabelthm:EigValAsympt0 The value $\{\omega_{n}^2(\alpha,\sigma,\delta)\}_{n=1}^{6}$ are approximated by uniformly for $|\alpha|< \min(k_1^2,k_0^2)$. Here $|D|$ is the area of the inclusions in the unit cell, and $\lambda_{n}$ are the eigenvalues of the periodic capacitance matrix $\mathbf{C}^{0}$, which are arranged in ascending order.

Figures (4)

  • Figure 1: Left: honeycomb-structured inclusions. The unit cell $Y$ is enclosed in black bold line. Right: illustration of the unit cell $Y$. The region $\tilde{Y}$ is enclosed in blue dash lines.
  • Figure 1: Top Panel: we plot the first six bands $\{\omega_n(\alpha,\sigma,\delta)\}_{n=1}^6$ on the line segment $\overline{M_1M_2}$ in the dual unit cell. Bottom Panel: In these figures we plot the second to fifth dispersion surfaces $\{\omega_n(\alpha,\sigma,\delta)\}_{n=1}^6$ near $\alpha=0$. Here $\delta = 1/50$.
  • Figure 2: Left: illustrations of the generalized honeycomb-structured inclusions and the unit cell when the inclusions are contracted. Here $\sigma$ is negative. Right: illustration of the generalized honeycomb-structured inclusions and the unit cell when the inclusions are dilated. Here $\sigma$ is positive. The corresponding super honeycomb-structured inclusions are plotted in dash lines.
  • Figure 2: In the top panel we plot the second to fifth eigenfunction when the inclusions are contracted. In the bottom panel we plot the second to fifth eigenfunction when the inclosions are dilated. By 'second', 'third', 'fourth', 'fifth' we mean the corresponding eigenfunction.

Theorems & Definitions (42)

  • Remark 1.1
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.2
  • Proof 1
  • Lemma 3.3
  • Proof 2
  • Proposition 3.4
  • Proof 3
  • Remark 3.5
  • ...and 32 more