An elementary rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model

TL;DR

This paper provides a rigorous mathematical proof of the bulk-boundary correspondence in a generalized SSH model with long-range hopping, confirming the topological nature of zero-energy edge modes.

Contribution

It introduces an elementary, rigorous proof of bulk-boundary correspondence in a generalized SSH model, including cases with arbitrary long-range hopping and disorder.

Findings

Zero-energy edge modes correspond to the winding number.

Nonzero-energy edge modes are unstable under deformations.

Zero-energy modes remain robust under small disorder.

Abstract

We generalize the Su-Schrieffer-Heeger (SSH) model with the inclusion of arbitrary long-range hopping amplitudes, providing a simple framework to investigate arbitrary adiabatic deformations that preserve the chiral symmetry upon the bulk energy bands with any arbitrary winding numbers. Using only elementary techniques of solving linear difference equations and applying Cauchy's integral formula, we obtain a mathematically rigorous and physically transparent proof of the bulk-boundary correspondence for the generalized SSH model. The multiplicity of robust zero-energy edge modes is shown to be identical to the winding number. On the other hand, nonzero-energy edge modes, if any, are shown to be unstable under adiabatic deformations and not related to the topological invariant. Furthermore, under deformations of small spatial disorder, the zero-energy edge modes remain robust.