Fragile topology based helical edge states in two-dimensional moon-shaped photonic crystals

We study topological phases in a two-dimensional photonic crystal in a hexagonal lattice composed of an all-dielectric moon-shaped column (MSC) array, which preserves time-reversal symmetry, by calculating band structures, the Wilson loop, edge states, and topological-protected helical surface waves. Breaking the traditional way of changing the position or size of the cylinder, instead we simply rotate the MSCs to achieve topological phase transitions. Furthermore, the Wilson-loop method is used to directly calculate the topological properties of the bulk bands. By observing the eigenvalues of the Wilson loop, we demonstrate that bands in our system display fragile topology behavior. We also simulate the electromagnetic (EM) wave propagation in the designed photonic crystals with a point source at different operating frequencies, showing that the EM waves can propagate unidirectionally along the interface between topological trivial and nontrivial MSCs. Finally, the edge states at different frequencies can be easily tuned by adjusting the size of the MSCs. Our work provides a platform to study fragile topology and achieve frequency-tunable edge states, which may have potential applications for optical devices.

Figure 1

(a) Schematic representation of a triangular photonic crystal composed of six moon-shaped dielectric cylinders in air, where $a$ is the lattice constant. (b) Enlarged view of the unit cell in (a), with $d=m·R$ and ${\varepsilon }_d=11.7$, where $R$ is the distance between the center of the lattice and the MSCs and tunable parameter $m$ can change the size of the MSCs. The shape of the moon is obtained by subtracting the black cylinder from the blue cylinder. Here, rotation angle $\theta$ is assigned to each of the MSCs such that they possess $C_6$ symmetry. (c) The first Brillouin zone (BZ) of a hexagon lattice with labeled high-symmetry points, where ${\mathbf{b}}_{\mathbf{1}}$ and ${\mathbf{b}}_{\mathbf{2}}$ are primitive vectors of the reciprocal lattice.

Figure 2

The band structures of moon-shaped photonic crystals with $m=0.2$ for rotation angles (a) $\theta =0^{\circ }$, (b) $\theta ={90}^{\circ }$, and (c) $\theta ={180}^{\circ }$, where the red and green curves represent $d$-type and $p$-type bands, respectively. The cyan (yellow) shading in (a) and (c) represents the topological nontrivial (trivial) bulk band gap, and the point marked by yellow shading in (b) represents the double Dirac cone. (d) Topological phase diagram as a function of $\theta$ recorded at the $\mathrm{\Gamma }$ point. The normalized frequency of $p$-type and $d$-type bands is plotted by green and red curves, respectively. Insets show the electric field $E_z$ of $\theta =0^{\circ }$ and $\theta ={180}^{\circ }$.

Figure 3

Wilson loops of Fig. 2. The Wilson loop of the first band (a) and the second and the third bands (b) when the rotation angle is ${180}^{\circ }$. Both (a) and (b) show trivial topology. In contrast, (c) is a fragile topology case when the rotation angle is $0^{\circ }$.

Figure 4

(a) Edge band diagram of a supercell formed by MSCs of $\theta =0^{\circ }$ and $\theta ={180}^{\circ }$. The bulk states are filled with gray, and the red and the green bands represent the edge states with opposite pseudospin, respectively. (b) Electric field $E_z$ and phase $arg\left(E_z\right)$ distribution of the supercell at point $A$ and point $B$ in (a); the black arrows are time-averaged Poynting vectors $\vec{S}$.

Figure 5

Numerical simulation of EM wave propagation in PCs consisting of MSCs with $\theta =0^{\circ }$ (topological nontrivial) and $\theta ={180}^{\circ }$ (trivial). Calculated $\left|\mathbf{E}\right|$ distribution with the point-source setting (a) $S_0=H_0{e}^{i\omega t}\widehat{x}$, (b) $S_{-}=H_0{e}^{i\omega t}(\widehat{x}-i\widehat{y})$, and (c) $S_{+}=H_0{e}^{i\omega t}(\widehat{x}+i\widehat{y})$, where the position is marked by a yellow star. The upper region is the $\theta =0^{\circ }$ region, and the lower region is the $\theta ={180}^{\circ }$ region. The time-averaged Poynting vector distribution shown in the inset in (c) is similar to Fig. 4. All these numerical simulations have the same frequency, $\omega =0.67(2\pi c/a)$. (d) Intensity as a function of the distance from the point source, which corresponds to (b).

Figure 6

(a) The band structures with $(m,\theta )=(0.2,90.6^{\circ }), (0.3,93.4^{\circ })$, and $(0.4,96.1^{\circ })$. (b) The black and red curves represent the rotation angle ${\theta }_D$ and the frequency $f_D$, respectively, which are recorded at the double Dirac point as $m$ varies from 0.1 to 0.4. The region below the black curve is the topological nontrivial region, while the upper region is the trivial region.

Figure 7

The edge states with the size parameter (a) $m=0.3$ and (b) $m=0.4$; the rest of the parameters and settings are the same as in Fig. 4. (c) and (d) are numerical simulations of $\left|\mathbf{E}\right|$ distribution with LCP point sources (yellow stars) $S_{-}=H_0{e}^{i\omega t}(\widehat{x}-i\widehat{y})$, which operate at (c) $\omega =0.59(2\pi c/a)$ and (d) $\omega =0.55(2\pi c/a)$, respectively.