Bulk-boundary correspondence in extended trimer Su-Schrieffer-Heeger model

We consider an extended trimer Su-Schrieffer-Heeger (SSH) tight-binding Hamiltonian keeping up to next-nearest-neighbor (NNN)-hopping terms and on-site potential energy. The Bloch Hamiltonian can be expressed in terms of all the eight generators (i.e., Gell-Mann matrices) of the SU(3) group. We provide exact analytical expressions of three dispersive energy bands and the corresponding eigenstates for any choices of the system parameters. The system lacks full chiral symmetry since the energy spectrum is not symmetric around zero, except at isolated Bloch wave vectors. We explore parity, time reversal, and certain special chiral symmetries for various system parameters. We discuss the bulk-boundary correspondence by numerically computing the Zak phase for all the bands and the boundary modes in the open boundary condition. There are three different kinds of topological phase transitions, which are classified based on the gap closing points in the Brillouin zone (BZ) while tuning the nearest-neighbor (NN)- and NNN-hopping terms. We find that quantized changes (in units of $\pi$) in two out of three Zak phases characterize these topological phase transitions. We propose another bulk topological invariant, namely the sublattice winding number, which also characterizes the topological phase transitions changing from ${\nu }^{\alpha }=0\leftrightarrow 2$ and ${\nu }^{\alpha }=0\leftrightarrow 1\leftrightarrow 2$ ($\alpha$: sublattice index). The sublattice winding number not only provides a relatively simple analytical understanding of topological phases but also successfully establishes bulk-boundary correspondence in the absence of inversion symmetry, which may help in characterizing the bulk-boundary correspondence of systems without chiral and inversion symmetry.