Scattering description of edge states in Aharonov-Bohm triangle chains

TL;DR

This paper investigates how scattering theory can identify topological phases in disordered systems using a triangle chain model, revealing that transmission peaks alone do not reliably indicate topological phase transitions.

Contribution

It demonstrates that reflection amplitudes indicate edge states regardless of topology, and shows that transmission peaks are not definitive markers of topological phase transitions.

Findings

Reflection amplitudes reveal edge states in all cases.

Transmission peaks occur at phase transitions and at non-topological edge states.

Transmission peaks depend on lead coupling, not solely on topology.

Abstract

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 scattering matrix that ``flows" in parameter space among several members of the topology classification scheme. Our results show that the reflection amplitudes (from both ends of a sufficiently long chain) do indicate the appearance of edge states in {\it all} (topological and non-topological) cases. For the topological cases, the transmission has a peak at the topological phase transition, which happens at the Fermi energy. A peak still exists as one moves into the non-topological `trivial' regions, in which another transmission peak may occur at nonzero energy, at which a relevant edge state appears in the isolated chain. For finite chains, the peak in the transmission strongly depends on their coupling of the leads, and {\it not} on the phase transition of the isolated chain. In any case, {\it the appearance of a peak in the transmission is not sufficient to conclude that the system undergoes a topological phase transition.}