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Symmetry-protected Interface Modes Bifurcated from Double Dirac Cones

Habib Ammari, Jiayu Qiu

Abstract

We rigorously prove the existence of interface modes in a sharp interface model, which bifurcate from the double Dirac cone as a consequence of the band inversion induced by super-symmetry breaking. The exact number of interface modes are determined. The proof is based on a discrete version of the layer-potential framework. Moreover, we prove that such interface modes are symmetry-protected against perturbations that respect the reflection symmetry.

Symmetry-protected Interface Modes Bifurcated from Double Dirac Cones

Abstract

We rigorously prove the existence of interface modes in a sharp interface model, which bifurcate from the double Dirac cone as a consequence of the band inversion induced by super-symmetry breaking. The exact number of interface modes are determined. The proof is based on a discrete version of the layer-potential framework. Moreover, we prove that such interface modes are symmetry-protected against perturbations that respect the reflection symmetry.
Paper Structure (23 sections, 21 theorems, 255 equations, 6 figures)

This paper contains 23 sections, 21 theorems, 255 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Outline
  4. Setup and Main Results
  5. Double Dirac Cone at the $\Gamma$ Point
  6. Preliminaries
  7. Proof of Proposition \ref{['prop_local_flat_double_eigenvalue']}
  8. Proof of Theorem \ref{['thm_double_cone']}
  9. Band-gap Opening and Asymptotic Expansions of Floquet-Bloch Eigenpairs
  10. Proof of Theorem \ref{['thm_asymptotics_eigenpairs']}
  11. Existence of Interface Modes
  12. Preliminaries
  13. Analytic Labelling of Floquet-Bloch Eigenpairs
  14. Limiting Absorption Principle and the Physical Green Operator
  15. Discrete Energy Flux of Floquet-Bloch Modes at the Double Dirac Cone
  16. ...and 8 more sections

Key Result

Proposition 2.4

Assume that $\lambda_*\in \text{Spec}(\mathcal{H}_b(\bm{0}))$ with multiplicity two. Let $\lambda_1(\bm{\kappa})$ and $\lambda_2(\bm{\kappa})$ be two branches of dispersion surfaces such that $\lambda_*=\lambda_1(\bm{\kappa}_*)=\lambda_2(\bm{\kappa}_*)$, and let $u_{n}(\cdot;\bm{\kappa})$ ($n=1,2$)

Figures (6)

  • Figure 1: A super-symmetric periodic structure, which contains six sublattices (illustrated by the six sites in a single unit cell). The numbers in the first cell label the ordering of the sublattices.
  • Figure 2: (a) $\tilde{\mathcal{T}}-$breaking structures by dislocating the sublattices (upper panel) outward with a distance $d_+>0$ or (lower panel) inward with a distance $d_->0$. In that case, the dislocated lattice is no longer translational invariant with respect to the $\mathbb{Z}\tilde{\bm{\ell}}$ translation. (b) a zigzag interface model: aside the interface, the $\tilde{\mathcal{T}}-$symmetry is broken by dislocating the sublattice oppositely, where the letter $d_+$ ($d_-$, resp.) in each cell indicates the first (second, resp.) type of deformation in Figure \ref{['fig_unit_structure']} is implemented. We note that this interface structure is translational invariant along $\mathbb{Z}\bm{\ell}_2$ (the zigzag interface) and reflectional symmetric about the $x-$axis.
  • Figure 3: Two examples of perturbed interface structures. (a) a compact defect, (b) a line defect. Both perturbations are $\mathcal{F}_x$ symmetric and localized in the longitudinal direction.
  • Figure 4: Dispersion surfaces of $\mathcal{H}_{\pm\delta}$ (sliced along $\kappa_2=0$). A common band gap (the bold part on the $\lambda-$axis) is opened for both $\mathcal{H}_{\delta}$ and $\mathcal{H}_{-\delta}$. The switching of colours indicates the appearance of band inversion.
  • Figure 5: (a) Increasing labeling. Each coloured curve represents a single branch of Floquet-Bloch eigenvalue $\lambda_{n,\sharp}$. Clearly, the derivatives of $\lambda_{n,\sharp}$ jump at $\kappa_1=0$ due to the conic structure. (b) Analytic labeling. Each colored curve is smooth in $\kappa_1$. Note that this rearrangement of eigenvalues can be written as, for example, $\mu_{1,\sharp}(\kappa_1)=\lambda_{1,\sharp}(\kappa_1)$ for $\kappa_1\leq 0$ and $\mu_{1,\sharp}(\kappa_1)=\lambda_{4,\sharp}(\kappa_1)$ for $\kappa_1> 0$.
  • ...and 1 more figures

Theorems & Definitions (47)

  • Example 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Theorem 2.7
  • Remark 2.8
  • Remark 2.9
  • Theorem 2.11: Existence of interface modes
  • Corollary 2.12
  • Remark 2.13
  • Remark 2.14
  • ...and 37 more