High-Winding-Number Zero-Energy Edge States in Rhombohedral-Stacked Su-Schrieffer-Heeger Multilayers

TL;DR

This work demonstrates that rhombohedral-stacked $N$-layer SSH networks host $2N$ zero-energy edge states whose topological winding number scales linearly as $W=N$. Using an effective low-energy approach and Zak-phase calculations, the authors show that layer-by-layer interlayer coupling amplifies topology such that $W=N$, corresponding to the observed edge-state degeneracy. They introduce the Wigner distribution and a derived Wigner entropy $W_s$ as a phase-space diagnostic, finding that boundary states exhibit pronounced localization and significantly enhanced entropy relative to bulk states. This stacking strategy extends layer-dependent topology to 1D SSH systems and offers a potential route to high-winding-number topological insulators with relevance to quantum information processing and fractional charge phenomena.

Abstract

We study the topological properties of rhombohedral-stacked N-layer Su-Schrieffer-Heeger networks with interlayer coupling. We find that these systems exhibit $2N$-fold degenerate zero-energy edge states with winding number $W=N$, providing a direct route to high-winding-number topological phases where $W$ equals the layer number. Using effective Hamiltonian theory and Zak phase calculations, we demonstrate that the winding number scales linearly with $N$ through a layer-by-layer topological amplification mechanism. We introduce the Wigner entropy as a novel detection method for these edge states, showing that topological boundary states exhibit significantly enhanced Wigner entropy compared to bulk states. Our results establish rhombohedral stacking as a systematic approach for engineering high-winding-number topological insulators with potential applications in quantum information processing.

Figures (3)

• Figure 1: (Color online) Schematic illustration of the rhombohedral-stacked SSH multilayer system. Each unit cell (indicated by rounded rectangles) contains two sublattice sites: $A$ (red) and $B$ (blue). Within each SSH layer, the intra-cell and inter-cell hopping strengths are denoted by $\nu$ and $\omega$, respectively. The interlayer coupling is characterized by two distinct hopping parameters: $t'_{\perp}$ connects sublattice sites vertically between adjacent layers, while $t_{\perp}$ provides the skewed interlayer connections that establish the rhombohedral stacking geometry.
• Figure 2: (Color online) (a) Band structure under $N=1$. (b) Energy versus energy level under $N=1$ and $\omega=2\nu$. (c) Band structure under $N=5$. (d) Energy versus energy level under $N=5$ and $\omega=2\nu$. Other involved parameters are $t'_{\perp}=0.1\nu$, $t_{\perp}=\nu$, and $L=100$.
• Figure 3: (Color online) Distinct signatures of topological edge states versus bulk states in Wigner phase space. (a) Wigner distribution $W (x, p)$ for the 1000-th eigenstate in the topological phase ($\omega=1.5 \nu$), showing pronounced spatial localization at the boundaries along the $x$-axis. (b) Corresponding Wigner entropy $W_s$ as a function of energy level index, with significantly enhanced values for in-gap states (highlighted in inset), providing a clear phase-space diagnostic for topological edge states. (c), (d) Analogous results for the trivial phase ($\omega=0.5 \nu$), where the Wigner distribution remains delocalized throughout the bulk (c) and the Wigner entropy maintains consistently low values across all states (d). The other parameters used are $t_{\perp}^{\prime}=0.1 \nu, t_{\perp}=\nu, N=5$, and $L=200$.