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

TL;DR

This paper investigates how long-range hopping in 1D topological insulators affects topological invariants, revealing the importance of site indexing and symmetry considerations for understanding edge states and phase transitions.

Contribution

It demonstrates that long-range hoppings alter topological properties and edge states, emphasizing the need to consider indexing and symmetry changes in 1D topological models.

Findings

Long-range hopping affects topological invariants.

Site indexing influences symmetry and topological classification.

Edge states reflect transitions between different chiral symmetries.

Abstract

In this paper, we show that the introduction of long-range hoppings in 1D 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 SSH 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 edge-like states with different localization lengths and reflect the gradual transition between these different chiral symmetries.