Fractional Charges in the Su-Schrieffer-Heeger Model

TL;DR

This paper clarifies the relationship between fractional charges and topological phases in the SSH model, revealing that polarization depends solely on positive charge distribution, resolving a longstanding paradox.

Contribution

It demonstrates that the polarization in the SSH model depends only on positive charge distribution, challenging previous interpretations based on the Zak phase and Berry connection.

Findings

Polarization depends only on positive charge distribution.

Fractional boundary charges are not directly linked to topological phases.

The Berry connection-based polarization cannot characterize topology in the 2D SSH extension.

Abstract

The Su-Schrieffer-Heeger(SSH) model has been widely used to study the topological property of 1D systems. It is claimed that there is fractional charge at the boundary of the nontrivial phase while none at that of trivial phase. However, this conclusion is in direct contradiction to the modern theory of polarization(MTP). We solve this paradox by showing that the polarization of SSH model depends only on the distribution of the positive charges and is irrelevant to the Zak phase defined in previous works. Thus the distribution of positive charges alone determines whether the SSH chain is topological or not. Similarly, we show the polarization defined by Berry connection can not be used to characterize topological property of a 2D generalization of SSH model.