Fig. 1: Topological defects in a hybrid metasurface.

a Schematics of a topological point defect created by stitching domains with three different choices of the unit cell denoted as \(\lambda={\mathrm{1,2,3}}\); b zigzag domain wall between expanded and shrunken hexamers vs the line defect separating the off-set domains. c Numerically simulated electromagnetic energy spectra revealing frequency bands and differing dispersion of edge states at the zig-zag domain wall and line defect interfaces. Magenta and white dotted lines show edge-state dispersion calculated based on the effective Dirac model. Magenta solid lines trace the spectral position of the phonon mode in hBN. The dark-gray line marks the frequency of the bound state at the point defect, whose simulated spatial structure is depicted in d squared absolute value of the out-of-plane magnetic field component distribution \({\left|{H}_{z}\right|}^{2}\) in a silicon slab is overlaid with a transparent contour plot \({|u|}\) obtained from the Dirac model within the continuum approximation. Bar insert visualizes photonic and phononic fractions in the HOT point defect mode. e Optical and SEM images of the fabricated metasurface (the triangle side is \(s=1668\) nm, a spatial period is \(a=4220\) nm, the clustering parameter \({{\rm{C}}}=\frac{3{{\rm{R}}}}{{{\rm{a}}}}=0.935\), where \(R\) is the distance between the centers of the nearest triangles. f, g Experimental real-space images of the fields confined to the line defect (f) and the point defect (g) at wavelengths 6.80 μm and 6.95 μm respectively.