Inverse Design of Strongly Localized Topological $π$ Modes in One-Dimensional Nonperiodic Systems

Abstract

This study investigates the spatial confinement of topological $π$-modes in one-dimensional chiral-symmetric systems. In conventional periodic and quasiperiodic structures, edge-mode wave functions inevitably penetrate the bulk. To suppress this, inverse design of a potential sequence is performed using a generative model under a global topological constraint. The generated sequence reveals a characteristic structure consisting of a topological boundary layer and a macroscopic S-dense domain, leading to enhanced confinement ($ξ=0.85$) while preserving topology. Based on the physical principle extracted from this result, a minimal heterostructure composed of only two S-blocks is manually constructed, which further reduces the localization length to $ξ=0.75$. These results provide a compact design principle for strongly localized topological states.

Figures (3)

• Figure 1: Spatial distributions of the coin parameters and decay curves of the probability density $|\psi|^2$ for the $\pi$-mode for (a) the periodic lattice and (b) the Fibonacci lattice, with enlarged views of the rightmost 28 sites. The quantity $|\psi|^2$ is plotted on a logarithmic scale, and the dashed lines represent the slopes of the fitting curves indicating exponential decay. The binary coin parameters correspond to the rotation angles $\theta_\mathrm{W}=0.1\pi$ and $\theta_\mathrm{S}=0.65\pi$. The brackets shown at the top of each panel indicate the spatial range and the corresponding number of S blocks required for the probability density to reach $10^{-15}$.
• Figure 2: Spatial distributions of the generated coin parameters and the corresponding quasienergy spectrum. (a) Raw output sequence from the generative model. (b) Sequence binarized from (a) with a 0.5 threshold. Region [A] denotes the hierarchical edge heterostructure. (c) Enlarged view of the rightmost 50 sites of (b). Regions [B] and [C] indicate the topological host layer and the internal S-dense domain, respectively. (d) Quasienergy spectrum. The zero and $\pi$ modes, highlighted by open square ($\square$) and open circle ($\bigcirc$), are clearly isolated from the dense bulk continuum. The total number of eigenvalues is $2N$, due to the two internal coin states.
• Figure 3: Decay curves of the probability density $|\psi|^2$ for the $\pi$-mode localized at the right boundary. The panels show the results for the (a) generated sequence, and (b) principle-guided sequence constructed manually based on the generated design. The format of the figure is identical to that in Fig. \ref{['fig:periodic_fibonacci']}.