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

TL;DR

This paper derives an exact relationship between eigenvalue discreteness and eigenstate localization in 1D SSH lattices, revealing a logarithmic correlation that applies to both Hermitian and non-Hermitian cases.

Contribution

It provides an exact analytical solution linking eigenvalue discreteness and eigenstate behavior in SSH lattices, extending the analysis to non-Hermitian systems with real eigenvalues.

Findings

Logarithmic relationship between eigenvalue discreteness and localization

Exact solution considering zero and non-zero modes in Hermitian case

Extension of the relationship to non-Hermitian systems with real eigenvalues

Abstract

Eigenstate localization and bulk-boundary correspondence are fundamental phenomena in one-dimensional (1D) Su-Schrieffer-Heeger (SSH) lattices. The eigenvalues discreteness and the eigenstates localization exhibit a high degree of consistency as system information evolve. We explore the relationship between the eigenvalue discreteness and the eigenstates behavior in 1D SSH lattices. The discreteness fraction and the inverse participation ratio (IPR) combined with a Taylor expansion are utilized to describe the relationship. In the Hermitian case, we employ the bulk-edge correspondence and the perturbation theory to derive an exact solution considering both zero and non-zero modes. We also extend our analysis to the non-Hermitian cases, assuming that eigenvalues remain purely real. Our findings reveal a logarithmic relationship between the degree of eigenvalue discreteness and eigenstates localization, which holds under both the Hermitian and non-Hermitian conditions. This result is fully consistent with the theoretical predictions.