Nontrivial structural cells induce topologically nontrivial phases under a certain symmetry

Abstract In this work, we demonstrate that topologically nontrivial phases can be induced by nontrivial structural cells under certain symmetries. Specifically, we investigate the topological properties of incomplete Su-Schrieffer-Heeger (SSH) chains with varying numbers of nontrivial structural cells (for chiral symmetry, the corresponding nontrivial structural cell is an alternating coupled bipartite lattice). We construct incomplete SSH chains by replacing some trivial structural lattice cells in a one-dimensional trivial chain with nontrivial structural cells. Through numerical simulations, we show that the winding numbers of these incomplete SSH chains are not strictly one or zero, but are related to the number of nontrivial structural cells. However, in the nontrivial and trivial phases, their winding numbers are greater than and less than a reference value, respectively, and they have the same phase transition point. Additionally, the distribution of topological edge states is no longer purely exponential but is related to the distribution of nontrivial structural cells in the incomplete SSH chain. In some special incomplete SSH chains, the entire parameter region exhibits topologically nontrivial phases, and the edge states are no longer localized at the ‘edges’. The incomplete SSH chain is topologically protected and robust against parameter errors. Our findings indicate that topological phases originate from nontrivial structural cells, and each nontrivial structural cell contributes to the topological phase and affects the distribution of edge states. This provides theoretical and technical guidance for the study of new non-periodic topological materials.