Sub-symmetry-protected topological states

Abstract

A hallmark of symmetry-protected topological phases are topological boundary states, which are immune to perturbations that respect the protecting symmetry. It is commonly believed that any perturbation that destroys such a topological phase simultaneously destroys the boundary states. However, by introducing and exploring a weaker sub-symmetry requirement on perturbations, we find that the nature of boundary state protection is in fact more complex. Here we demonstrate that the boundary states are protected by only the sub-symmetry, using Su–Schrieffer–Heeger and breathing kagome lattice models, even though the overall topological invariant and the associated topological phase can be destroyed by sub-symmetry-preserving perturbations. By precisely controlling symmetry breaking in photonic lattices, we experimentally demonstrate such sub-symmetry protection of topological states. Furthermore, we introduce a long-range hopping symmetry in breathing kagome lattices, which resolves a debate on the higher-order topological nature of their corner states. Our results apply beyond photonics and could be used to explore the properties of symmetry-protected topological phases in the absence of full symmetry in different physical contexts.

Realization of Topological Corner States in Tailored Photonic Graphene

“…The degeneracy of the corner states indicates that the three corners in a finite-sized HCL can be decoupled with each other, thus one corner state can occupy only one corner. However, the three corners in a finite-sized BKL tend to couple with each other due to the finite-size effect, which breaks the zero-energy mode degeneracy and leads to mode beating and “fractionalization” of light to all three corners. ,,, Briefly speaking, the eigenstates of gapless zero-energy corner modes in finite-sized HCLs only distribute in one corner, while those in finite-sized BKLs distribute in all three corners, leading to corner mode localization at one corner in the HCLs, but not in the BKLs after long-distance propagation.…”

Observation of Topologically Distinct Corner States in “Bearded” Photonic Kagome Lattices

“…Instead, the OPCSs merge into the bulk continuum before the gap closes and form BICs. For perturbations maintaining the C 3 and mirror symmetries but not the generalized chiral symmetry (broken by next‐nearest‐neighbor couplings between A–A, B–B, and C–C sites), [ 35 ] both types of the corner states vanish before the gap closing takes place (Figure 4f). A direct confirmation of this robustness test is provided by calculating the IPR curves of the perturbed bearded BKL for any δ values, as shown in Supporting Information.…”

“…[ 49,54 ] Our results not only provide a versatile platform to study topological corner states but also bring about new opportunities to explore fundamental physics arising from the interplay between momentum‐space and real‐space topology. Moreover, with growing interest in various realistic materials with (breathing) Kagome arrangements, [ 17,22–24,35–38 ] topologically distinct corner states presented here may be applicable to other systems. Our work may also prove relevant to future photonic device applications including for example topological corner‐state lasing [ 55 ] and error filtering for quantum qubits.…”

“…[ 32–34 ] Recently, topological protection of corner states in BKLs has been studied by using the concept of sub‐symmetry and long‐range hopping symmetry. [ 35 ]…”

“…[ 31 ] A downward‐pointing unit cell is illustrated in Figure a. According to the boundary configurations of the resulting lattice, a finite‐sized BKL can be cut with “flat” [ 21–24,35–38 ] or “bearded” edges [see Figure 1a], depending on the appropriate choice of unit cells. Thus far, much of the previous exploitations of HOTI corner states in BKLs have focused on its flat‐edge structures, while little attention has been devoted to the bearded‐edge ones.…”