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

A remarkable feature of the trimer Su-Schrieffer-Heeger (SSH3) model is that it supports localized edge states. However, in contrast to the dimer version of the model, a change in the total number of edge states in SSH3 without mirror-symmetry is not necessarily associated with a phase transition, i.e., a closing of the band gap. As such, the topological invariant predicted by the 10-fold way classification does not always coincide with the total number of edge states present. Moreover, 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 noncommensurate 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 straightforwardly generalizable, and invokes a chiral-like symmetry present in this model.