Topological edge states and disorder robustness in one-dimensional off-diagonal mosaic lattices

Abstract

We investigate topological edge states in one-dimensional off-diagonal mosaic lattices, where nearest-neighbor hopping amplitudes are modulated periodically with period $κ>1$. Analytically, we demonstrate that discrete edge states emerge at energy levels $E=ε+2t\cos(πi/κ)$ ($i=1,\cdots,κ-1$), extending the Su-Schrieffer-Heeger model to multi-band systems. Numerical simulations show that these edge states are robustly localized and display characteristic nodal structures, with their existence being strongly dictated by the specific edge arrangement of long and short bonds. We further examine their stability under off-diagonal disorder, where the hopping amplitudes $β$ fluctuate randomly at intervals of $κ$. Using the inverse participation ratio as a localization measure, we show that these topological edge states remain robust over a broad range of disorder strengths. In contrast, additional $β$-dependent edge states that appear for $κ\ge 4$ are fragile and vanish even under relatively weak disorder. These findings highlight a rich interplay between topology, periodic modulation, and disorder, offering insights for engineering multi-gap topological phases and their realization in synthetic quantum and photonic systems.

Figures (11)

• Figure 1: (a) Energy $E$ versus wavenumber $q$ for $\kappa=2$ and $\beta = 1.5$ under periodic boundary conditions. A band gap opens at $q = \pm\pi/2$ when $\beta \ne 1$. (b, c) Energy $E$ versus state index under open boundary conditions for $\beta = 1.5$ and $\beta = 0.5$, with $N = 200$ and $m = 0$. Two zero-energy edge states, localized at the left and right boundaries, appear for $\beta = 1.5$ but are absent for $\beta = 0.5$.
• Figure 2: Energy $E$ versus wavenumber $q$ ($-\pi/3 \leq q \leq \pi/3$) for $\kappa=3$ and $\beta = 1.5$ under periodic boundary conditions. For nonzero $\beta$, band gaps open at $q = 0$ and $q = \pm \pi/3$.
• Figure 3: Energy $E$ versus state index for $\kappa = 3$ and $\beta = 1.5$ under open boundary conditions. The six configurations listed in Table \ref{['table1']} are shown separately. Red dots indicate doubly degenerate edge states localized at both boundaries, while blue dots denote single edge states localized at the left boundary.
• Figure 4: Energy $E$ versus wavenumber $q$ ($-\pi/4 \leq q \leq \pi/4$) for $\kappa=4$ and $\beta = 1.5$ under periodic boundary conditions. For nonzero $\beta$, band gaps open at $q = 0$ and $q = \pm \pi/4$.
• Figure 5: Energy $E$ versus state index for $\kappa = 4$ and $\beta = 1.5$ under open boundary conditions. The ten configurations listed in Table \ref{['table2']} are shown separately. Red dots indicate doubly degenerate edge states localized at both boundaries, while blue and violet dots represent single edge states localized at the left and right boundaries, respectively.