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Mathematical Theory for Photonic Hall Effect in Honeycomb Photonic Crystals

Wei Li, Junshan Lin, Jiayu Qiu, Hai Zhang

TL;DR

This work develops a rigorous mathematical framework for the photonic valley (Hall) edge states in honeycomb photonic crystals. By formulating the problem with a periodic elliptic operator $\mathcal{L}(\varepsilon,\delta)$, analyzing Dirac points at $K$ and $K'$, and lifting degeneracy via perturbations, the authors connect bulk Berry curvature with interface modes. They introduce a boundary-integral approach on an infinite strip, derive asymptotics for layer potentials, and obtain precise conditions for the existence and multiplicity of interface modes across joined crystals with opposite valley invariants, thereby establishing a bulk-edge correspondence in this photonic setting. The results have implications for designing magneto-optical valley Hall devices and robust edge-guided photonic channels, providing a rigorous counterpoint to experimental valley-Hall implementations. The methods—layer potentials, Green function asymptotics, and generalized Rouche arguments—offer a versatile toolkit for topological PDE analysis beyond the specific honeycomb geometry.

Abstract

In this work, we develop a mathematical theory for the photonic Hall effect and prove the existence of guided electromagnetic waves at the interface of two honeycomb photonic crystals. The guided wave resembles the edge states in electronic systems: it is induced by the topological Hall effect, and the wave propagates along the interface but not in the bulk media. Starting from a symmetric honeycomb photonic crystal that attains Dirac points at the high-symmetry points of the Brillouin zone, $K$ and $K'$, we introduce two classes of perturbations for the periodic medium. The perturbations lift the Dirac degeneracy, forming a spectral band valley at the points $K$ and $K'$ with well-defined topological phase that depends on the sign of the perturbation parameters. By employing the layer potential techniques and spectral analysis, we investigate the existence of guided wave along an interface when two honeycomb photonic crystals are glued together. In particular, we elucidate the relationship between the existence of the interface mode and the nature of perturbations imposed on the two periodic media separated by the interface.

Mathematical Theory for Photonic Hall Effect in Honeycomb Photonic Crystals

TL;DR

This work develops a rigorous mathematical framework for the photonic valley (Hall) edge states in honeycomb photonic crystals. By formulating the problem with a periodic elliptic operator , analyzing Dirac points at and , and lifting degeneracy via perturbations, the authors connect bulk Berry curvature with interface modes. They introduce a boundary-integral approach on an infinite strip, derive asymptotics for layer potentials, and obtain precise conditions for the existence and multiplicity of interface modes across joined crystals with opposite valley invariants, thereby establishing a bulk-edge correspondence in this photonic setting. The results have implications for designing magneto-optical valley Hall devices and robust edge-guided photonic channels, providing a rigorous counterpoint to experimental valley-Hall implementations. The methods—layer potentials, Green function asymptotics, and generalized Rouche arguments—offer a versatile toolkit for topological PDE analysis beyond the specific honeycomb geometry.

Abstract

In this work, we develop a mathematical theory for the photonic Hall effect and prove the existence of guided electromagnetic waves at the interface of two honeycomb photonic crystals. The guided wave resembles the edge states in electronic systems: it is induced by the topological Hall effect, and the wave propagates along the interface but not in the bulk media. Starting from a symmetric honeycomb photonic crystal that attains Dirac points at the high-symmetry points of the Brillouin zone, and , we introduce two classes of perturbations for the periodic medium. The perturbations lift the Dirac degeneracy, forming a spectral band valley at the points and with well-defined topological phase that depends on the sign of the perturbation parameters. By employing the layer potential techniques and spectral analysis, we investigate the existence of guided wave along an interface when two honeycomb photonic crystals are glued together. In particular, we elucidate the relationship between the existence of the interface mode and the nature of perturbations imposed on the two periodic media separated by the interface.
Paper Structure (29 sections, 22 theorems, 171 equations, 6 figures)

This paper contains 29 sections, 22 theorems, 171 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Periodic elliptic operators
  5. The interface modes for the joint operators
  6. An example of the material fulfilling the assumptions
  7. Outline
  8. Band structure for the periodic operators and Berry curvature
  9. Floquet-Bloch theory
  10. Dirac point of the operator $\mathcal{L}_0$
  11. Lifting of Dirac degeneracy for the operator $\mathcal{L}(\varepsilon, \delta)$ and Berry curvature
  12. Bandgap opening near the Dirac points for $\mathcal{L}(\varepsilon,\delta)$
  13. Berry curvature at $K$ and $K'$
  14. Smooth parametrization of the dispersion relations
  15. The Green functions in the infinite strip
  16. ...and 14 more sections

Key Result

Theorem 6

Let $k_{\parallel}^*:=K\cdot\mathbf e_2$ and $\mathfrak d$ be an arbitrary constant in $(0,1)$. Suppose Assumption lem:assNoFold hold along $\boldsymbol\beta_1$, and the two constants $t_1$ and $t_2$ defined in Proposition lem:Tderiv are nonzero. All the above eigenvalues satisfy $\lambda= \lambda_* + o(\max(|\varepsilon|,|\delta|)$.

Figures (6)

  • Figure 1: The fundamental cell $\mathcal{C}_z$ (left) and the Brillouin zone $\mathcal{B}$ (right).
  • Figure 2: The spectral band of $\mathcal{L}_0$ when $\mathbf p \in \{K+\ell\boldsymbol\beta$, $\ell\in [-\pi, \pi]\}$ for a photonic crystal consisting of an infinite array of inclusions shown in Figure \ref{['fig:periodic_media']} (left). The material tensor $A(0,0;\mathbf x) = a\, \chi_D(\mathbf x)$ for $\mathbf x\in\mathcal{C}_z$, where $D\in\mathcal{C}_z$ is a Lipchitz domain that in invariant under $R$ and $F$. Top: $\boldsymbol\beta=\boldsymbol\beta_1:=(\frac{1}{\sqrt3},-1)^T$; Bottom: $\boldsymbol\beta=\boldsymbol\beta_1^a:=(0, -2)^T$. The no-fold condition holds for the spectrum in the first two columns when $a=\frac{1}{30}$ and $\frac{1}{100}$ respectively. The no-fold condition does not hold for the spectrum in the last column when $a=\frac{1}{15}$.
  • Figure 3: Left: The medium for the elliptic operator $\mathcal{L}_0:=\mathcal{L}(0,0)$, for which the tensor $A_0(\mathbf x):=A(0,0,;\cdot)$ satisfies Assumptions \ref{['ass:symFull']} and \ref{['ass:symA']}. Right: The medium of the joint honeycomb structure over which the elliptic operator $\mathcal{L}^{\text{int}}(\varepsilon_L,\delta_L,\varepsilon_R,\delta_R)$ is defined. The interface $\gamma$ of two periodic media is along the $\mathbf e_2$ direction. The perturbations of $\mathcal{L}_0$, $(\varepsilon_L,\delta_L)$ and $(\varepsilon_R,\delta_R)$, are on either sides of of the interface.
  • Figure 4: The medium of the joint honeycomb structure with an armchair interface along the $\mathbf e_2^a$ direction.
  • Figure 5: The hexagon $E$ represents another fundamental domain of the honeycomb lattice $\Lambda$ (cf. Figure \ref{['fig:periodic_cell']}). The vertices of the hexagon $E$ are given by $\frac{\sqrt{3}}{3}(\cos\theta_j, \sin\theta_j)$, wherein $\theta_j=\frac{j\pi}{3}$ for $j= 1, \cdots, 6$. $E$ can be decomposed as $E=C_1\sqcup C_2\sqcup C_3$ shown above, wherein $C_j=R^{-(j-1)}C_1$. It is clear that $E$ invariant under the rotation map $R$.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Remark 4
  • Theorem 6
  • Corollary 7
  • Lemma 8
  • Theorem 10
  • Proposition 11
  • proof
  • Proposition 13
  • Proposition 14
  • Proposition 15
  • ...and 28 more