Fig. 6.

Band structures of the unperturbed double gyroid for different waves. (A) Dispersive photonic band structure of a metallic double-gyroid structure made of a Drude metal with the plasma frequency of gold standing in vacuum. (B) Phononic band structure for an elastic double gyroid in steel embedded in an epoxy elastic matrix. (C) Acoustic band structure for sound in air confined outside of a double gyroid with hard wall boundary conditions. In the dispersive photonic and phononic band structures (A and B), a threefold degeneracy (highlighted by gray circles) is found. As such, we expect such systems to exhibit Weyl points when strained. In A, ${\omega }_0=2\pi c/a$, where $c$ is the speed of light in vacuum. We use the plasma frequency of gold, ${\omega }_{\text{p}}/2\pi \simeq 2.19\times {10}^{15}\text{Hz}$ (108), and $a\simeq 500\text{nm}$. The loss term $\mathrm{\Gamma }$ is initially set to 0, and the results show no significant deviations from the case computed with the tabulated value $\mathrm{\Gamma }/2\pi =5.79\times {10}^{12}\text{Hz}$ (108). In B, ${\omega }_0=2\pi c_{\text{t}}/a$, where $c_{\text{t}}$ is the speed of transverse waves in epoxy. The values assumed for the longitudinal and transverse speeds of sound in steel and epoxy are obtained from the components of elastic tensor $C_{IJ}$ as $c_{\text{t}}^2=C_{44}/\rho$ and $c_{\mathrm{\ell }}^2=C_{11}/\rho$ from the values in refs. 109–111, namely ${\rho }^{\text{epoxy}}=1180{\text{kg m}}^{-3}$, $C_{11}^{\text{epoxy}}=7.61\text{GPa}$, and $C_{44}^{\text{epoxy}}=1.59\text{GPa}$ and ${\rho }^{\text{steel}}=7780{\text{kg m}}^{-3}$, $C_{11}^{\text{steel}}=264\text{GPa}$, and $C_{44}^{\text{steel}}=81\text{GPa}$. In C, ${\omega }_0=2\pi c_{\text{air}}/a$, where $c_{\text{air}}$ is the speed of sound in air. All computations are performed with a $48\times 48\times 48$ grid. More details on the model and computation are in SI Appendix.