The Su-Schrieffer-Heeger model on a one-dimensional lattice: Analytical wave functions of topological edge states

Abstract: The one-dimensional Su-Schrieffer-Heeger (SSH) model is a prototype model in the field of topological condensed matter physics, and the existence and characteristics of its topological edge states are crucial for revealing the topological essence of low-dimensional systems. This paper focuses on the analytical wave functions of topological edge states in the SSH model, systematically sorting out the fundamentals of model construction. Based on the tight-binding approximation, it derives the matrix form of the SSH model Hamiltonian and the band structureand clarifies the role of chiral symmetry in the classification of topological phases. By solving the Schrodinger equation of the finite-length lattice, the mathematical expression of the analytical wave function of topological edge states is fully derived, verifying its localized feature of exponential decay. Moreover, it quantitatively analyzes the influence laws of lattice parameters (hopping integrals, lattice constants), electron-electron interactions, and external field perturbations on the wave function amplitude, decay coefficient, and energy. Furthermore, it establishes the quantitative correlation between the analytical wave function and electronic transport (conductivity) as well as optical properties (light absorption coefficient), and discusses its application prospects in the design of topological qubits and the development of new topological materials. The research results provide theoretical support for an in-depth understanding of the physical essence of low-dimensional topological states and lay a foundation for the performance optimization of topology-related devices.