Winding number and Zak phase in multi-band SSH models

TL;DR

This paper presents a method for accurately calculating the Zak phase and winding number in multi-band SSH models, validated through comparisons with edge state analyses and numerical eigenvalue solutions.

Contribution

It introduces a new prescription for computing topological invariants in multi-band systems and verifies its effectiveness across several SSH models.

Findings

The proposed method accurately predicts winding numbers.

Bulk edge correspondence is confirmed in multi-band models.

The method can be applied to other 1D multi-band systems.

Abstract

We use multi-band SSH models to demonstrate a prescription for calculating the correct Zak phase and winding number of multi-band systems. We verify our prescription by comparing the resultant winding number of a four-band SSH model with the edge states of a semi-infinite chain which we find by solving the equation of motion. We then also carry out an extensive comparison with the numerical results obtained by solving the matrix eigenvalue problem of finite chains with various number of sites. As a double check of our prescription, we also confirm the bulk edge correspondence in a six-band SSH model. Similar to the usual SSH model, the winding numbers associated to the left and right boundaries in a finite chain may be different if space inversion symmetry is violated in the system. We believe the prescription we propose here may also be applied to other 1D multi-band systems.