Exact solution of the relationship between the eigenvalue discreteness and the behavior of eigenstates in Su–Schrieffer–Heeger lattices

Abstract The interplay between eigenvalue discreteness and eigenstate localization is a fundamental characteristic of one-dimensional Su–Schrieffer–Heeger (SSH) lattices. In this study, we investigate the relationship between the eigenvalue discreteness and the eigenstates behavior in 1D SSH lattices. The discreteness fraction ( D ) are introduced in combination with the inverse participation ratio to quantify this relationship. By employing the bulk-edge correspondence and perturbation theory, we derive an exact solution that accounts for both zero and non-zero modes. Our findings reveal a logarithmic relationship between the degree of eigenvalue discreteness and eigenstate localization in both the Hermitian and non-Hermitian conditions. This result provides a direct measure of edge-state localization strength in the topologically nontrivial phase.