Analytical estimations of edge states and extended states in large finite-size lattices

TL;DR

The work develops an analytical framework to characterize edge states and extended states in large finite-size diatomic lattices, clarifying how lattice length and boundary conditions govern bulk–boundary correspondence. Using asymptotic methods, it derives explicit conditions for edge-state existence in semi-infinite and long finite chains, and provides near-band-edge frequency patterns with two characteristic regimes. It further shows how these linear insights enable nonlinear continuations to nonlinear edge/middle-localized states and extends the approach to multi-layer and two-dimensional diatomic lattices, highlighting the framework's universality and practical relevance for topological and nonlinear localized phenomena.

Abstract

The bulk-boundary correspondence, one of the most significant features of topological matter, theoretically connects the existence of edge modes at the boundary with topological invariants of the bulk spectral bands. However, it remains unspecified in realistic examples how large the size of a lattice should be for the correspondence to take effect. In this work, we employ the diatomic chain model to introduce an analytical framework to characterize the dependence of edge states on the lattice size and boundary conditions. In particular, we apply asymptotic estimates to examine the bulk-boundary correspondence in long diatomic chains as well as reveal the finite-size regimes where it fails. Moreover, under our framework the eigenfrequencies near the band edges can be well approximated where two special patterns are detected. These estimates on edge states and eigenfrequencies in linear diatomic chains can be further extended to nonlinear chains to investigate the emergence of new nonlinear edge states and other nonlinear localized states. In addition to one-dimensional diatomic chains, examples of more complicated and higher-dimensional lattices are provided to show the universality of our analytical framework.

Figures (15)

• Figure 1: Here we illustrate the dependence of the number of edge states on the bulk topology and boundary conditions in diatomic chains, where different colors represent different numbers of edge states. Specifically, green represents $0$, red represents $2$, yellow represents $1$ and blue represents $4$. The panel (a) represents the number of edge states in the semi-infinite chain \ref{['eq:semi_infinite']}, while panels (b) to (d) correspond to finite chain \ref{['eq:finite']}. In panel (a), we show the number of edge states with varying $k_{3,1}$ and $k_{2}$ where the whole area is divided into three parts with two lines, namely $k_{3,1}=2k_{2}$ and $k_{2}=1$. The panels (b) to (d) are similarly plotted for varying $k_{3,1}$ and $k_2$ but with different assumptions on $k_{3,2}$.
• Figure 2: Here panel (a) shows the eigenfrequencies of a linear diatomic chain \ref{['eq:finite']} with $k_1=1$, $k_2=2.3$ and $n=100$ under different boundary stiffness settings. The crosses in each column of panel (a) correspond to the eigenfrequencies under a setting with a different pair of $(k_{3,1}, k_{3,2})$, ranging among $\{ (1.3, 4.6), (8.3, 4.6), (1.3, 3.5), (11.1, 10.5), (1, 3.8), (4.6, 4.6) \}$, from left to right. It can be observed that no more than two eigenfrequencies are outside the bands where blue dashed lines represent the band edges. Panel (b) just zooms in panel (a) on the region near the lower edge of the optical band where red solid and black dot-dash auxiliary lines are added to better demonstrate the two patterns of the eigenfrequencies. We find that the near-band-edge eigenfrequencies in the first two columns are almost the same while those in the last four columns are very close.
• Figure 3: Here we plot some representative eigenstates in a linear diatomic chain \ref{['eq:finite']} with $n=50$ (length $100$), $k_{1}=1$ and $k_{2}=2.3$. The panel (a) shows a left edge state for the generic case with $k_{3,1}=1.3$ and $k_{3,2}=3.5$ while the panel (b) shows an eigenstate localized at both ends when $k_{3,1}=k_{3,2}=1.5$. In the panel (c), we plot a left edge state with $k_{3,1}=k_{2}$ and $k_{3,2}=1.2$. The panel (d) illustrates an eigenstate decays "slowly" with $a\approx-0.9910345\approx-1$ when $4.58=k_{3,1}\approx 2k_{2}\approx k_{3,2}=4.62$. The panel (e) in the lower left (panel (f) in the lower right) shows an eigenstate with $\omega^{2}=2k_{2}$ and $k_{3,1}=2k_{2}=k_{3,2}$ ($\omega^{2}=2k_{1}$, $k_{3,1}=2k_{2}$ and $k_{3,2}=\frac{2k_{1}k_{2}}{(2n-1)(k_{2}-k_{1})+k_{2}}$).
• Figure 4: Here panel (a) shows the eigenfrequencies in the linear diatomic chain \ref{['eq:finite']} with $\{ 2n=50, k_{1}=1, k_{2}=1.3, k_{3,1}=1.6 \}$ over varying stiffness constant $k_{3,2}$. Panel (b) zooms in on the region near $k_{3,1}=k_{3,2}$ in panel (a).
• Figure 5: Here we plot the schematic diagram to demonstrate the correspondence between $k_{3,2}$ and $\omega^{2}$ as $a$ approaches $\tilde{a}$ for \ref{['eq:finite']}, where panel (a) is for generic case ($2k_{2}\not\approx k_{3,1}\not\approx k_{2}$) and panel (b) is for special case ($k_{3,1}=k_{2}$). The horizontal black dashed line represents $\omega^{2}(\tilde{a})$ corresponding to $a=\tilde{a}$, while the green solid curves and the blue bold solid curves illustrate $\omega^{2}$ corresponding to variations in $k_{3,2}$. The bold blue curves represent $\omega^{2}$ of left edge states, while the green curves correspond to right edge states. The two red vertical dashed lines indicate the regions where the edge states are more "two-sided" with $k_{3,2}\approx k_{3,1}$. The cyan vertical dashed line in panel (a) represents the special value $k_{3,2}=\tilde{k}_{3,2}$ such that $\omega^2=\omega^2(\tilde{a})$.

Theorems & Definitions (28)

• Remark 3.1
• Remark 3.2
• Lemma 4.1
• Corollary 4.2
• Lemma 4.3
• Lemma 4.4
• Remark 4.5
• Remark 4.6
• Remark 4.7
• Remark 4.8