Chiral flow in one-dimensional Floquet topological insulators

Abstract

We propose a bulk topological invariant for one-dimensional Floquet systems with chiral symmetry which quantifies the particle transport on each sublattice during the evolution. This chiral flow is physically motivated, locally computable, and improves on existing topological invariants by being applicable to systems with disorder. We derive a bulk-edge correspondence which relates the chiral flow to the number of protected dynamical edge modes present on a boundary at the end of the evolution. In the process, we introduce two real-space edge invariants which classify the dynamical topological boundary behaviour at various points during the evolution. Our results provide the first explicit bulk-boundary correspondence for Floquet systems in this symmetry class. arXiv:1806.00026v1 [cond-mat.str-el] 31 May 2018

Topological Floquet interface states in optical fiber loops

“…In this paper, we will experimentally demonstrate a coexisting pair of anomalous Floquet interface states, which naturally arise in the context of recent studies [42,43] on the fully temporal analogue of the Discrete Photonic Quantum Walk [17]. Furthermore, we will experimentally reveal a decisive role of chiral symmetry for their topological robustness since in 1D this aspect, supporting the above mentioned bulk-edge correspondence for disordered Floquet systems [35], has been to date lacking an experimental approval. In particular, we will show that initially topologically protected interface states may leave their midgapped energy positions and may even finally dissolve in the bulk if the symmetry is locally broken.…”

“…It is worth mentioning, that in the special case of chiral one-dimensional Floquet insulators as well as in many other classes the bulk-edge correspondence had been strictly proven only for bulks with translational symmetry, as it was still based on winding numbers. However, recent theoretical advances [35] rigorously show that a generalization of the winding number to inhomogeneous systems is possible, implying that the bulk-edge correspondence remains in force also for inhomogeneous systems and systems with disorder.…”

Topological chiral modes in random scattering networks

“…Essentially, the flow operatorΦ is defined with a projector P that selects the half infinite region P of the system toward which the flow is directed, and readsΦ = U −1 PU − P [38,45,46]. When considering a flow along an infinite boundary, the trace ofΦ is an integer that corresponds to the number of boundary modes flowing along the edge.…”

Topological chiral modes in random scattering networks

“…Essentially, the flow operator Φ is defined with a projector P that selects the half infinite region P of the system toward which the flow is directed, and reads Φ = U −1 PU − P [45,38,46]. When considering a flow along an infinite boundary, the trace of Φ is an integer that corresponds to the number of boundary modes flowing along the edge.…”