Higher-order Dirac semimetal in a photonic crystal

The recent discovery of higher-order topology has largely enriched the classification of topological materials. Theoretical and experimental studies have unveiled various higher-order topological insulators that exhibit topologically protected corner or hinge states. More recently, higher-order topology has been introduced to topological semimetals. Thus far, realistic models and experimental verifications on higher-order topological semimetals are still very limited. Here we design and demonstrate a three-dimensional photonic crystal that realizes a higher-order Dirac semimetal phase. Numerical results on the band structure show that the designed three-dimensional photonic crystal is able to host two fourfold Dirac points, which are connected in the momentum-space projections via higher-order hinge states localized at the hinge. The higher-order topology can be characterized by the topological invariant ${\chi }^{\left(6\right)}$ at different values of $k_z$. An experiment at microwave frequencies is also presented to measure the hinge state dispersion. Our work demonstrates the physical realization of a higher-order Dirac semimetal phase and paves the way to explore higher-order topological semimetal phases in three-dimensional photonic systems.

Figure 1

Structure and bulk dispersion of the photonic crystal exhibiting a higher-order Dirac semimetal phase. (a) Schematic of the unit cell of a deformed honeycomb lattice with lattice constant $a$. The lattice comprises gray rods sandwiched by metallic plates. The rods have height $h$, diameter $d_1$, and are separated by a distance $c_1$. The two metallic plates at the top and bottom are drilled to couple the neighboring layers. The holes have depth $d$, diameter $d_2$, and are separated by a distance $c_2$. (b) The first Brillouin zone of the crystal. (c) Simulated bulk dispersion of the photonic crystal. The colored region from $\mathrm{\Gamma }$ to A indicates the topological transition when $k_z$ goes through the Dirac point at $k_z=2\pi /5a_z$. (d) Schematic of phase transition in the entire Brillouin zone for different values of $k_z$. The upper and lower regions host the higher-order topological insulator (HOTI) phase, while the central part maintains the normal insulator (NI) phase. The red dots signify the two Dirac points.

Figure 2

The ${\chi }^{\left(6\right)}$ topological invariants and the band structures for the model in Fig. 1 with various $k_z$ values. (a) The ${\chi }^{\left(6\right)}=(\left[\mathbf{M}\right],\left[\mathbf{K}\right])$ topological invariants obtained from the tight-binding model in the first Brillouin zone demonstrate the phase transition. (b–d) The bulk band structures for that in Fig. 1 with $k_z=0, 2\pi /5a_z$ and $\pi /a_z$, respectively. The area shaded in green in (b) and (d) depicts the band gaps when $k_z$ deviates from the Dirac point.

Figure 3

Experimental measurement on the higher-order hinge states. (a) Photograph of the experimental sample. The inset shows a detailed view of the deformed lattice rods placed on each layer with the top plate removed. (b) The schematic of the experimental setup. The red line represents the source pin inserted from the air hole at the top. The red star represents the source dipole fixed at half of the height of the sample, next to the hard boundaries. The blue line marks the position of the probe. The purple lines designate the side surfaces with perfect electric conductors, and the measurement is performed around the obtuse angle consisting of these two surfaces. (c) Simulation result of a $20\times 20$ lattice. The black, blue, and red dots represent the bulk, surface, and hinge states. (d) The measured energy dispersion. The green curve is the hinge state reconstructed from (c). (e) The measured field distribution demonstrates the hinge state centered on the hard boundary of the sample at 6.4 GHz. (f) The measured field distribution revealing the bulk state localization at 6.2 GHz.