# Topological characterizations of an extended Su-Schrieffer-Heeger model

## TL;DR

This paper experimentally realizes the extended SSH4 model using ultracold atoms, verifies its topological winding number, maps phase transitions, and observes edge states, advancing understanding of higher-dimensional topological systems.

## Contribution

It provides the first experimental verification of the topological properties of the SSH4 model with ultracold atoms, including measurement of the winding number and edge states.

## Key findings

- Winding number verified via mean chiral displacement measurement.
- Topological phase transition mapped in the SSH4 system.
- Topological edge states observed through quench dynamics.

## Abstract

The Su-Schrieffer-Heeger (SSH) model perhaps is the easiest and the most basic model for topological excitations. Many variations and extensions of the SSH model have been proposed and explored to better understand both fundamental and novel aspects of topological physics. The SSH4 model has been proposed theoretically as an extended SSH model with higher dimension (the internal dimension changes from two to four). It has been proposed that the winding number in this system can be determined through a higher-dimensional extension of the mean chiral displacement measurement, however this has not yet been verified in experiment. Here we report the realization of this model with ultracold atoms in a momentum lattice. We verify the winding number through measurement of the mean chiral displacement in a system with higher internal dimension, we map out the topological phase transition in this system, and we confirm the topological edge state by observation of the quench dynamics when atoms are initially prepared at the system boundary.