Zero-energy Edge States of Tight-Binding Models for Generalized Honeycomb-Structured Materials

TL;DR

This work analyzes zero-energy edge states in tight-binding models on generalized honeycomb lattices for two interface geometries. By applying Floquet-Bloch reduction and transfer-matrix techniques, it derives necessary and sufficient conditions for zero-energy edge states at $k=0$ for Type I and Type II interfaces, proving the existence of two spin-like edge states when present and showing non-tangential crossings of the edge-state curves at $k=0$. It characterizes the local dispersion near $k=0$ as $E_{ m I, }(k)= \pm E^{(1)}_{ m I} |k| + O(k^2)$ and $E_{ m II, }(k)= \pm E^{(1)}_{ m II} |k| + O(k^2)$, implying bi-directional wave-packet propagation, with numerical spectra and dynamics validating the predictions. The results show that Type II edge states exist only between topologically distinct materials, while Type I edge states require parameter constraints and can occur even for identical materials, offering insight for designing robust bi-directional waveguides in generalized honeycomb systems.

Abstract

Generalized honeycomb-structured materials have received increasing attention due to their novel topological properties. In this article, we investigate zero-energy edge states in tight-binding models for such materials with two different interface configurations: type-I and type-II, which are analog to zigzag and armchair interfaces for the honeycomb structure. We obtain the necessary and sufficient conditions for the existence of such edge states and rigorously prove the existence of spin-like zero-energy edge states. More specifically, type-II interfaces support two zero-energy states exclusively between topologically distinct materials. For type-I interfaces, zero-energy edge states exist between both topologically distinct and identical materials when hopping coefficients satisfy specific constraints. We further prove that the two energy curves for edge states exhibit strict crossing. We numerically simulate the dynamics of edge state wave packets along bending interfaces, which agree with the topologically protected motion of spin-like edge states in physics.

Figures (9)

• Figure 1: Generalized Honeycomb Structure. The dashed lines denote the edge of unit hexagonal cell.
• Figure 2: Left Panel: An example of type I interface. The cells in red denote the unit cell of type I interface. Right Panel: An example of type II interface. The cells in green denote unit cell of type II interface. In both cases, the unit cell is periodic in $\mathbf{v}_{\alpha,\mathrm{I}}(\mathbf{v}_{\alpha,\mathrm{II}})$ direction, while extending in $\mathbf{v}_{\beta,\mathrm{I}}(\mathbf{v}_{\beta,\mathrm{II}})$ direction. The bold black lines denote the interface between two types of materials.
• Figure 3: The above figures display the calculated spectra of the Hamiltonian operator $\widehat{H}_{\mathrm{I}}(k)$ for $k\in [-\pi,\pi]$, when the hopping coefficients $c$ across the boundary are chosen properly. The point spectrum near zero are shown in bold blue line in each figure. There are crossings at $k=0$ in each figure. Near $k=0$, the slope of $E_{\mathrm{I}}(k)$ is nonzero.
• Figure 4: The above figures display the calculated spectra of the Hamiltonian operator $\widehat{H}_{\mathrm{I}}(k)$ for $k\in [-\pi,\pi]$, when the hopping coefficient $c=50$. The point spectrum near zero are shown in bold blue line in each figure. There are no crossing at $k=0$ in any figure.
• Figure 5: The above figures display the calculated spectra of the Hamiltonian operator $\widehat{H}_{\mathrm{II}}(k)$ for $k\in [-\pi,\pi]$, when the hopping coefficient $c=50$. The point spectrum near zero are shown in bold blue line in each figure. There is a crossing at $k=0$ when $\delta_{+}\delta_{-}<0$. And the two bands cross at $k=0$. However, it does not exist when $\delta_{+}\delta_{-}>0$.

Theorems & Definitions (33)

• Remark 1
• Remark 2
• Proposition 1
• Proposition 2
• Theorem 3
• Lemma 1
• Theorem 4
• proof
• Theorem 5
• Lemma 2