Fig. 3: Robustness of a kagome corner state with respect to SubSy perturbations and the LRHS.
a, Sketch of the rhombic BKL with three sublattices. b, Illustration of the LRHS condition expressed in equation (4), for which the coupling between two sites indicated with red links must be equal, and the same for the coupling indicated with two blue links, and so on. c–e, Eigenvalue spectra of the rhombic BKL flake with 29 lattice sites along one edge for different perturbations. c, Ensemble of spectra for a set of 70 randomly chosen A-SubSy-preserving perturbations H′ = HBB + HCC + HBC of various strengths quantified by δ′, which leaves the zero-energy mode (red cross) intact. d, Bandgap structure of the unperturbed rhombic BKL flake (t1 = 0.1, t2 = 1). The rhombic BKL flake has a single corner state shown in the inset. e, Spectra for a set of 70 perturbations H′ + H″, which respect the A-SubSy and the LRHS (H″ = HAC + HAB). The magnitude of perturbations H′ is fixed at δ′ = 0.05, whereas that of the H″ perturbations, δ″ is varied. The zero-energy mode (red cross) is protected despite the presence of long-range hopping. In c and e, we calculate 70 spectra for 70 different perturbations, which are plotted one on top of the other. What appears as a single red cross at zero value indicates that for any perturbation, the zero mode is protected. A slight spread of red crosses for δ″ > 0.12 in e indicates that the zero mode becomes adjacent to the band modes (blue crosses) when finite-size effects become relevant. f, FCA and the mode densities (ρ and σ) calculated for perturbations from e (Methods).
