One-dimensional extended Su-Schrieffer-Heeger models as descendants of a two-dimensional topological model

Abstract

The topological phase diagrams and finite-size energy spectra of onedimensional extended Su-Schrieffer-Heeger models with long-range hoppings on the trimer lattice are investigated in detail. Due to the long-range hoppings, the band structure of the original Su-Schrieffer-Heeger model becomes more complicated and new phases with the large Zak phase can emerge. Furthermore, a seeming violation of bulk-edge correspondence occurs in the one-dimensional topological system whose band topology stems from the inversion symmetry. The one-dimensional models are mapped onto a two-dimensional topological model when a parameter of the one-dimensional models is regarded as an additional degree of freedom. As Fourier components of the derived two-dimensional model, phase boudaries and the finite-size spectra of onedimensional models can be recovered from the model in the higher spatial dimensions. Then the origin of edge modes of one-dimensional models can be understood from two dimensions and we give a reasonable explanation of the violation of bulk-edge correspondence in one spatial dimension. In fact, we may give a general perspective that the topological properties of one-dimensional (lower-dimensional) systems can be found their origin from two-dimensional (higher-dimensional) systems.