Supersymmetric correspondence in spectra on a graph and its line graph: From circuit theory to spoof plasmons on metallic lattices

We investigate the supersymmetry (SUSY) structures for inductor-capacitor circuit networks on a simple regular graph and its line graph. We show that their eigenspectra must coincide (except, possibly, for the highest eigenfrequency) due to SUSY, which is derived from the topological nature of the circuits. To observe this spectra correspondence in the high-frequency range, we study spoof plasmons on metallic hexagonal and kagomé lattices. The band correspondence between them is predicted by a simulation. Using terahertz time-domain spectroscopy, we demonstrate the band correspondence of fabricated metallic hexagonal and kagomé lattices.

Figure 1

(a) Example of simple 3-regular directed graph. (b) Inductor-capacitor circuit network on the graph. (c) Line graph of the graph shown in (a). (d) Inductor-capacitor circuit network on the line graph (c).

Figure 2

(a) Hexagonal lattice. (b) Kagomé lattice. (c) Dispersion relation for a hexagonal inductor-capacitor circuit network. (d) Dispersion relation for a kagomé inductor-capacitor circuit network. (e) Eigenmodes of the higher band at the $\mathrm{\Gamma }$ and $\text{M}$ points for the hexagonal lattice. (f) Eigenmodes of the middle band at the $\mathrm{\Gamma }$ and $\text{M}$ points for the kagomé lattice.

Figure 3

Designs for (a) metallic hexagonal lattice and (b) metallic kagomé lattice. The following parameters are used: $d=10\mu \mathrm{m}, r=150\mu \mathrm{m}, b=800/\sqrt{3}\mu \mathrm{m}$, and thickness $h=30\mu \mathrm{m}$.

Figure 4

Dispersion relations obtained by simulation for a (a) metallic hexagonal lattice and (b) metallic kagomé lattice. The fitting parameters for the theoretical models discussed in Appendix pp2 are given by ${\omega }_0^{\text{hex}}=2\pi \times 0.107\mathrm{THz}, {\eta }_0^{\text{hex}}=0.0916$ for the hexagonal lattice and ${\omega }_0^{\text{kag}}=2\pi \times 0.101\mathrm{THz}$ and ${\eta }_0^{\text{kag}}=0.142$ for the kagomé lattice.

Figure 5

Electric flux density $D_z$ on $z=h/2$. (a) Eigenmodes of the higher band at the $\mathrm{\Gamma }$ and M points for the metallic hexagonal lattice and (b) eigenmode of the middle band at the $\mathrm{\Gamma }$ point for the metallic kagomé lattice.

Figure 6

Microphotographs of the (a) metallic hexagonal and (b) metallic kagomé lattices.

Figure 7

Power transmission spectra mapped on the wave-vector-frequency plane for a (a) metallic hexagonal lattice and (b-1),(b-2) metallic kagomé lattice. For (a) and (b-1), the incident waves are set as transverse electric modes in the $\mathrm{\Gamma }-\mathrm{K}$ scan and transverse magnetic modes in the $\mathrm{\Gamma }-\mathrm{M}$ scan. On the other hand, in (b-2), we use transverse magnetic excitation in the $\mathrm{\Gamma }-\mathrm{K}$ scan and transverse electric excitation for the $\mathrm{\Gamma }-\mathrm{M}$ scan. The calculated eigenfrequencies are depicted as circles.

Figure 8

Comparison between eigenfrequencies for a tuned metallic kagomé lattice with $d=28\mu \mathrm{m}$ and metallic hexagonal lattice with $d=10\mu \mathrm{m}$. The other parameters are $r=150\mu \mathrm{m}, b=800/\sqrt{3}\mu \mathrm{m}$, and thickness $h=30\mu \mathrm{m}$.