Scattering description of edge states in Aharonov-Bohm triangle chains

Scattering theory has been suggested as a convenient method to identify topological phases of matter, in particular of disordered systems for which the Bloch band-theory approach is inapplicable. Here we examine this idea, employing as a benchmark a one-dimensional triangle chain whose versatility yields a system that "flows"in parameter space among several members of the topology classification scheme. Our results show that the reflection amplitudes (from both ends of long chains) indicate the appearance of edge states in all (topological and nontopological) cases. For the topological cases, the transmission has a peak at the topological phase transition, located at the Fermi energy. A peak still exists as one moves into the nontopological regions, where another transmission peak may occur at nonzero energy, at which an edge state appears in the isolated chain. For finite chains, the transmission peak depends strongly on their coupling with the leads, and not on the phase transition of the isolated chain. In any case, the appearance of a transmission peak is insufficient to conclude that the system undergoes a topological phase transition.