# Dimensional Crossover of Topological Edge States in Su-Schrieffer-Heeger Model

## TL;DR

This paper analytically derives the full energy spectrum and wave functions of the 2D SSH model, revealing the dimensional crossover of topological phase transition points and providing detailed insights into edge states and topological properties.

## Contribution

It provides explicit analytical wave functions and spectrum for the 2D SSH model, elucidating the dimensional crossover of topological transitions.

## Key findings

- Analytical wave functions for 2D SSH model obtained.
- Dimensional crossover of topological transition point demonstrated.
- Wave functions reveal parity, localization, and critical points.

## Abstract

Su-Schrieffer-Heeger (SSH) model on two-dimensional square lattice exhibits a topological phase transition, which is related to the Zak phase determined by bulk band topology. The strong modulation of electron hopping causes nontrivial charge polarization even in the presence of inversion symmetry. The energy band structures and topological edge states have been calculated numerically in previous studies. Here, however, full energy spectrum and explicit form of wave functions for two-dimensional bulk and one-dimensional ribbon geometries of SSH model are analytically derived using wave mechanics approach. Explicit analytic representations of wave functions provide the information of parity for each subband, localization length and critical point of topological phase transition in SSH ribbon. It is also shown that the dimensional crossover of topological transition point for SSH model from one to two-dimension.