Fig. 3: Superposed valley Hall phase and phase diagram.
From: Robust separation of topological in-plane and out-of-plane waves in a phononic crystal
a In-plane polarized (IPP), out-of-plane polarized (OPP), and superposed valley Hall phase of the phononic crystal (PC). Frequency order between p− and q+ states and ideal valley Chern numbers at valley K linked to each IPP or OPP phase are denoted. Different colors assigned to each valley Hall phases are shown, for example, yellow for the phase i0o0. b Superposed phase diagram of PCs with c = 0 and restricted perturbation b1/B1 + b2/B2 = 0.6. Blue and red solid lines represent the IPP and OPP band-edge frequencies at point K, and symbols on the lines represent the Bloch states. Insets show the energy distributions of the Bloch states at (b1/B1, b2/B2) = (0.1, 0.5) (empty star) and (b1/B1, b2/B2) = (0.5, 0.1) (filled star). c Superposed phase diagram of PCs with c = 0. A solid violet line indicates the deterministic Dirac degeneracy or phase transition line. The Black dashed line indicates the restricted perturbations used in (b). d IPP, OPP, and superposed phase diagram of PCs with c = 0 and 1. Blue (red) lines and arrows in IPP (OPP) phase diagram represents the phase transition line and direction of the phase transition line at origin (b1/B1, b2/B2) = (0, 0), respectively. A superposed phase diagram is nothing but the superposition of IPP and OPP phase diagram; the violet line is a superposition of blue and red lines. e IPP, OPP, and superposed phase diagram of PCs with c = 0.25. All superposed phases appear, different from the diagram of (d) c = 0 and 1.
