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Edge states in super honeycomb structures with PT-symmetric deformations

Ying Cao, Yi Zhu

TL;DR

This work analyzes edge states in a two-dimensional Schrödinger operator with a super honeycomb lattice under PT-symmetric, folding-symmetry breaking perturbations. By introducing a folding-symmetry breaking domain-wall edge model, it rigorously proves the existence of two gapped edge states near the Dirac energy $E_D$, with a gap scaling as $O(\delta^2)$, derived through near-energy and multiscale analyses and Lyapunov–Schmidt reduction. The study reveals how a fourfold Γ-point degeneracy (double Dirac cone) splits into topologically protected edge modes when the folding symmetry is broken, and it connects bulk symmetries to edge behavior via effective local topological charges and parity considerations. The results provide a rigorous foundation for helix-like edge states in super honeycomb systems and offer insights into bulk-edge correspondence in PT-preserving, folding-breaking contexts, complemented by numerical demonstrations.

Abstract

The existence of edge states is one of the most vital properties of topological insulators. Although tremendous success has been accomplished in describing and explaining edge states associated with PT symmetry breaking, little work has been done on PT symmetry preserving cases. Two-dimensional Schroedinger operators with super honeycomb lattice potentials always have double Dirac cones at the Gamma point - the zero momentum point on their energy bands due to C6 symmetry, PT symmetry, and the "folding" symmetry - caused by an additional translation symmetry. There are two topologically different ways to deform such a system by PT symmetry preserving but folding symmetry breaking perturbations. Interestingly, there exist two gapped edge states on the interface between such two kinds of perturbed materials. In this paper, we illustrate the existence of such PT preserving edge states rigorously for the first time. We use a domain wall modulated Schroedinger operator to model the phenomenon under small perturbations and rigorously prove the existence of two gapped edge states. We also provide a brief interpretation from the point of view of "topology" by the parities of degenerate bulk modes. Our work thoroughly explains the existence of "helical" like edge states in super honeycomb configurations and lays a foundation for the descriptions of topologies of such systems.

Edge states in super honeycomb structures with PT-symmetric deformations

TL;DR

This work analyzes edge states in a two-dimensional Schrödinger operator with a super honeycomb lattice under PT-symmetric, folding-symmetry breaking perturbations. By introducing a folding-symmetry breaking domain-wall edge model, it rigorously proves the existence of two gapped edge states near the Dirac energy , with a gap scaling as , derived through near-energy and multiscale analyses and Lyapunov–Schmidt reduction. The study reveals how a fourfold Γ-point degeneracy (double Dirac cone) splits into topologically protected edge modes when the folding symmetry is broken, and it connects bulk symmetries to edge behavior via effective local topological charges and parity considerations. The results provide a rigorous foundation for helix-like edge states in super honeycomb systems and offer insights into bulk-edge correspondence in PT-preserving, folding-breaking contexts, complemented by numerical demonstrations.

Abstract

The existence of edge states is one of the most vital properties of topological insulators. Although tremendous success has been accomplished in describing and explaining edge states associated with PT symmetry breaking, little work has been done on PT symmetry preserving cases. Two-dimensional Schroedinger operators with super honeycomb lattice potentials always have double Dirac cones at the Gamma point - the zero momentum point on their energy bands due to C6 symmetry, PT symmetry, and the "folding" symmetry - caused by an additional translation symmetry. There are two topologically different ways to deform such a system by PT symmetry preserving but folding symmetry breaking perturbations. Interestingly, there exist two gapped edge states on the interface between such two kinds of perturbed materials. In this paper, we illustrate the existence of such PT preserving edge states rigorously for the first time. We use a domain wall modulated Schroedinger operator to model the phenomenon under small perturbations and rigorously prove the existence of two gapped edge states. We also provide a brief interpretation from the point of view of "topology" by the parities of degenerate bulk modes. Our work thoroughly explains the existence of "helical" like edge states in super honeycomb configurations and lays a foundation for the descriptions of topologies of such systems.
Paper Structure (23 sections, 10 theorems, 137 equations, 3 figures)

This paper contains 23 sections, 10 theorems, 137 equations, 3 figures.

Table of Contents

  1. Introduction and notations
  2. Introduction
  3. Notations
  4. Bulk and edge operators
  5. Super honeycomb lattice potentials and the folding symmetry
  6. Bulk operators
  7. Domain wall modulated edge operator
  8. Near-energy approximation
  9. Order $O(\Vert{\bf k}\Vert^2)$ bifurcation matrix
  10. Near-energy approximation along certain direction
  11. Multiscale expansions of edge states
  12. Order $O(\delta)$ terms
  13. Order $O(\delta^2)$ terms
  14. Order $O(\delta^n)$ terms
  15. Rigorous formulation of two gapped edge states
  16. ...and 8 more sections

Key Result

Theorem 2.4

(Fourfold degeneracy at the $\Gamma$ point) The following is true for energy surfaces of ${\mathcal{H}}^{(\epsilon)}=-\Delta + \epsilon V({\bf x})$ with $V({\bf x})$ a super honeycomb lattice potential for all $\epsilon \in {\mathbb R} \setminus A$, where $A$ is a discrete subset of ${\mathbb R}$ :

Figures (3)

  • Figure 1: Numerical simulations of edge states curves of a limiting domain wall model. (a) the figure of the piecewise constant domain wall potential. On one side of the edge, the hexagons of the super honeycomb lattice are shrunk, and on the other side, they are expanded. (b) the figure of the edge states energy curves along $k_{\parallel}\Tilde{\bm{l}}_1$. $E_{_D}$ is the Dirac point's energy. The parts of two red curves near the $\Gamma$ point correspond to two edge states.
  • Figure 2: (a) The lattice, and (b) the dual lattice. The blue lattice corresponds to a honeycomb lattice, with periods ${\bf u}_1$ and ${\bf u}_2$ and dual periods ${\bf k}_1$ and ${\bf k}_2$. The black lattice corresponds to the associated super honeycomb lattice, with periods ${\bf v}_1$ and ${\bf v}_2$ and dual periods ${\bf q}_1$ and ${\bf q}_2$.
  • Figure 3: Figures of edge and bulk modes. (a) and (b) the figures of numerical solutions of two edge states at $k_{\parallel} =0$ of the limiting domain wall model in Figure \ref{['figure-energy-curve']}. (c), (d), (e), and (f) the figures of numerical solutions of fourfold degenerate bulk modes of corresponding unperturbed bulk operators. The eigenstates in figure (a) and figure (b) are modulations of the eigenstates in figure (c) and figure (d), respectively.

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 12 more