Bulk-edge correspondence in the trimer Su-Schrieffer-Heeger model

TL;DR

This paper introduces a new bulk invariant based on sublattice Zak's phase for the SSH3 model, establishing a clear bulk-edge correspondence even without mirror symmetry, and provides exact finite-size corrections.

Contribution

It proposes a simple, gauge-invariant bulk invariant for SSH3 that remains valid without mirror symmetry and predicts finite-size corrections.

Findings

The sublattice Zak's phase is a valid bulk invariant.

The invariant takes integer values and relates to edge states.

Finite chain corrections are explicitly derived.

Abstract

A remarkable feature of the trimer Su-Schrieffer-Heeger (SSH3) model is that it supports localized edge states. Although Zak's phase remains quantized for the case of a mirror-symmetric chain, it is known that it fails to take integer values in the absence of this symmetry and thus it cannot play the role of a well-defined bulk invariant in the general case. Attempts to establish a bulk-edge correspondence have been made via Green's functions or through extensions to a synthetic dimension. Here we propose a simple alternative for SSH3, utilizing the previously introduced sublattice Zak's phase, which also remains valid in the absence of mirror symmetry and for non-commensurate chains. The defined bulk quantity takes integer values, is gauge invariant, and can be interpreted as the difference of the number of edge states between a reference and a target Hamiltonian. Our derivation further predicts the exact corrections for finite open chains, is straightforwadly generalizable, and invokes a chiral-like symmetry present in this model.