Fig. 4: Phononic crystals (PCs) for the topological protection selected based on the Jaccard index.
From: Robust separation of topological in-plane and out-of-plane waves in a phononic crystal
a Feasible PCs S00, S10, S01, and S11 in the superposed phase diagram with c = 0.25. S00, S10, S01, and S11 are named after their superposed topological phase. Black dashed triangles have vertices S10, S01, and S11, and used in (e). b Jaccard index between in-plane polarized (IPP) bandgap and out-of-plane polarized (OPP) bandgap of the PCs with c = 0.25. Line L1 and L2 indicates perturbation sets possessing large Jaccard index approximately, where L1: 3(b1/B1) − 5(b2/B2) = 0 and L2: 7(b1/B1) + 3(b2/B2) = 0. c Jaccard index between intersected bandgaps (intersection of IPP and OPP bandgap) of L1 and L2. Part of the graph which can be drawn by symmetry is omitted. The red arrow indicates the perturbation intensity hierarchy of the PC pairs having large Jaccard indices. d Energy band structures of the PCs S10 and S11. The overall common bandgap is from 29.0 to 30.7 kHz, so the width of the bandgap is 5.7%. e Superposed phase diagram of PCs on the black dashed triangle shown in (a). Only IPP (OPP) phase transits between S01 and S11 (S11 and S10) owing to the accidental Dirac degeneracy while both IPP and OPP phase transits between S10 and S01, mirror-symmetry pair.
