Photonic topological Weyl degeneracies and ideal type-I Weyl points in the gyromagnetic metamaterials

As a new topological phase in three-dimensional momentum space, Weyl semimetals extend the repertoire of exotic topological states. In this paper, we find the existence of photonic Weyl points in electromagnetic continuous gyromagnetic metamaterials. The dispersion of the longitudinal mode of one of the two bands forming the Weyl point is flat; thus, the Weyl degeneracy point of the system is exactly at the critical transition between the type-I and type-II Weyl points. The Fermi arc connects the projections of the Weyl points with opposite chirality at the interface between the gyromagnetic metamaterial and vacuum. Full-wave simulation results show that surface waves can bypass sharp defects and achieve robust, unidirectional optical transmission without reflection. Furthermore, taking the spatial nonlocal effects into account, the ideal type-I Weyl points can exist in this class of metamaterials. This paper sheds light on fundamental understanding of topological physics, and the study of Weyl points in metamaterials will greatly enrich the field of topological photonics.

Figure 1

The dispersion relation of the gyromagnetic Weyl system at (a) $g=1.2$ and (b) $g=1.0$. Weyl points are highlighted by the coloured dots (red for $C=+1$ and blue for $C=-1$). The other parameters are ${\omega }_p=1, {\epsilon }_t=1$, and ${\mu }_t=1.2$.

Figure 2

The dispersion relation of transverse and longitudinal modes of Weyl degenerate points with different parameters at (a) $g=0.8$, (b) $g=1.0$, (c) $g=1.2$, and (d) $g=1.4$. The other parameters are ${\omega }_p=1, {\epsilon }_t=1$, and ${\mu }_t=1.2$. The dispersion is along the $z$ direction. The bold orange (black horizontal) lines represent the dispersion of transverse modes (longitudinal modes).

Figure 3

The dispersion relation of the gyromagnetic Weyl system. (a) and (c) The band dispersions in the $k_x-k_y$ plane at $k_z=1.55/-1.55$. (b) and (d) The band dispersion in the $k_y-k_z$ plane at $k_x=0$. Weyl points are highlighted by the colored dots (red for $+1$ and blue for $-1$).

Figure 4

Spatial dispersion of the Fermi arc (shown as a black curve) between the gyromagnetic metamaterials and vacuum at (a) $\omega =0.99{\omega }_p$, (b) $\omega =1.00{\omega }_p$, and (c) $\omega =1.01{\omega }_p$. The $z$-direction permittivity parameters are (a) ${\epsilon }_z=-0.02$, (b) ${\epsilon }_z=0$, and (c) ${\epsilon }_z=0.02$. The other parameters are ${\epsilon }_t=1, {\mu }_t=1.2, g=1.2$, and ${\omega }_p=1$. The gray dotted lines represent vacuum states, and the green solid lines represent the bulk states of the gyromagnetic metamaterial. The details of the gray shaded region will be analyzed later.

Figure 5

COMSOL simulation results for our model. (a) The black dotted line represents the Fermi arc connecting the Weyl points. The values of three different propagation constants $k_z$ at points $A, B$, and $C$ are (b) $k_z=0.95$, (c) $k_z=1.20$, and (d) $k_z=1.60$, respectively. (b)–(e) Full-wave simulation of the topologically protected Fermi arc surface states. Black stars indicate the line sources. (e) When the direction of the magnetic field is flipped (along the $z$ direction), the propagation of the surface wave is also switched to the opposite direction.

Figure 6

Band dispersion and Weyl points, considering the nonlocal effect in the gyromagnetic metamaterials. The relevant parameters are ${\omega }_p=1,{\epsilon }_t=1, {\mu }_t=1.2, g=1.2, {\epsilon }_z=1-{\omega }_p^2/{\omega }^2+\beta {k_z}^2$, and $\beta =0.05$. (a) Band dispersion in momentum space; Weyl points are highlighted by the colored dots (red for $+1$ and blue for $-1$). (b) The dispersion along the $z$ direction, where the green (black) lines correspond to the transverse (longitudinal) modes. (c) The band dispersion in the $k_x-k_y$ plane at $k_z=1.47$. (d) The band dispersion in the $k_y-k_z$ plane at $k_x=0$.

Figure 7

The Fermi arcs (black solid lines) at the boundary between vacuum and metamaterial while considering the nonlocal effect at (a) $\omega =0.92{\omega }_p$, (b) $\omega =0.95{\omega }_p$, and (c) $\omega =0.98{\omega }_p$. The gray dotted (green solid) line represents the bulk states of vacuum (the gyromagnetic metamaterial).