Long-range hopping and indexing assumption in one-dimensional topological insulators

In this paper, we show that the introduction of long-range hoppings in one-dimensional topological insulator models implies that different possibilities of site indexing must be considered when determining the bulk topological invariants in order to avoid the existence of hidden symmetries. The particular case of the extended Su-Schrieffer-Heeger chain is addressed as an example where such behavior occurs. In this model, the introduction of long-range hopping terms breaks the bipartite property and a band inversion occurs in the band structure as the relative values of the hopping terms change, signaling a crossover between hopping parameter regions of ``influence'' of different chiral symmetries. Furthermore, edge states become a linear combination of edgelike states with different localization lengths and reflect the gradual transition between these different chiral symmetries.