Group Theoretical Route to Deterministic Weyl Points in Chiral Photonic Lattices

Topological phases derived from point degeneracies in photonic band structures show intriguing and unique behavior. Previously identified band degeneracies are based on accidental degeneracies and subject to engineering on a case-by-case basis. Here we show that deterministic pseudo Weyl points with nontrivial topology and hyperconic dispersion exist at the Brillouin zone center of chiral cubic symmetries. This conceivably allows realization of topologically protected frequency isolated surface bands in 3D and $n=0$ properties as demonstrated for a nanoplasmonic system and a photonic crystal.

Figure 1

Illustration of the $P{2}_{1}3$ sphere packing. (a) The glass cube shows the simple cubic unit cell, that is centred at the position of one of the spheres, whose 6 nearest neighbors lie on the cubes’ facets. The thick Cartesian rods, and the thin connection rods are shown for illustration purposes only. (b) The same cube shown from the [111] direction. (c) Projection of (a) onto the [001] plane, with spatial unit $a/8$ and $z$ coordinate in the respective sphere. (d) Same as (c), but with crystallographic choice of origin [21].

Figure 2

Band structure of the plasmonic sphere packing illustrated in Fig. 1 for $x/a=0.175$, $k_{p}a/(2\pi )=0.1$, and $\rho /a=0.1$. (a) All $N=12$ branches, corresponding to the solutions of Eq. (2). (b) The isolated triplet states that meet at $K_1=0.9992$ and $K_2=1.0076$ show a particularly clear and isotropic Weyl hypercone (red) and a flat dark mode (black) in between, even for relatively large $k\approx \pi /(5a)$. The blue (red) boxes highlighted in (a) correspond to the subfigures of the same color in (b).

Figure 3

Surface modes close to a PWP frequency. (a) Surface band structure for a supercell made of, respectively, 12 unit cells of two enantiomorphic sphere packings. Topologically protected (dark red) and unprotected (pale red) surface bands are present within the partial gap of the projected bulk band structure (blue). The main graph shows the band structure along the irreducible BZ boundary (red path, left inset), whereas the inset on the right follows a small semicircle at the $\mathrm{\Gamma }$ point (green path, left inset). The individual paths intersect at two points $u$ and $v$. (b) Field energy distribution (arbitrary units) corresponding to points of the same color in (a). Dielectric spheres are shaded for illustration. The interfaces between the right handed (RH) and the left handed (LH) crystal are at the center (black line) and at the end of the unit cell.

Figure 4

Isofrequency plot at $k_{0}a/(2\pi )=0.51$ for the same supercell configuration as in Fig. 3. Bulk modes (blue area) exist only in the direct vicinity of the PWP at $\mathrm{\Gamma }$ and the double Weyl cone at $A$. Eight Fermi arclike protected surface bands (red points) emanate from the PWP with topological charge of $C=2$ and connect it with the double Weyl cone that carries the opposite charge $C=-2$.