One of the major advancements in condensed matter physics in the 21st century is the discovery of topological insulators. These materials are insulating within their bulk crystal structure but exhibit conductivity on their surfaces or interfaces. The name "topological" insulators comes from the use of mathematics known as topology to analyze the electronic band structure.
Subsequently, it was discovered that similar phenomena could be achieved not only with electrons but also with classical wave phenomena, such as light and sound, by using artificially structured materials known as photonic and phononic crystals. These are referred to as topological photonic crystals and topological phononic crystals. In these artificial structures, wave propagation is restricted within the crystal structure for frequency ranges corresponding to photonic and phononic band gaps; waves can only travel along the crystal's surfaces or interfaces. Topological waveguides utilizing this feature are known to exhibit lower backscattering at bends and structural irregularities than conventional, non-topological waveguides that rely on lattice defects.
We developed an expanded two-dimensional wave machine and conducted demonstration experiments on topological waveguides.
In this video, the bar indicated by the arrow at the bottom is excited at a constant frequency. On the left, with a frequency of 4.20 Hz, which is outside the phononic band gap, vibrations spread into the periodic interior outside of the waveguide. In contrast, on the right, at a frequency of 4.66 Hz, which falls within the phononic band gap, the topological vibrations propagate along the waveguide section.
A video of vibrations propagating along the topological waveguide in a two-dimensional wave machine. To animate click the image (3 MB movie).
The original one-dimensional wave machine was developed by John Shive, who was with Bell Labs in the United States. With its characteristic of slow, large-amplitude movements that allow wave motion to be observed with the naked eye, it has been widely used in physics education demonstrations worldwide. The speed of the torsional waves traveling along the thin central rod of the wave machine is slowed by the effect of the long bars attached periodically. These bars also serve to make the wave amplitudes more visible.
Our two-dimensional wave machine was created by using a laser cutter to cut a thin stainless steel sheet into a two-dimensional lattice pattern, to which we attached commercially available round rods and weights with screws. It is used by suspending it from above, like a curtain. By adjusting the placement of the weights, we created a topological waveguide at the interface between two phononic crystal regions that share the same band gap.
The physics of topological materials can be a challenging field to approach, as it introduces unfamiliar concepts, such as the Berry phase in quantum mechanics and solid-state physics, even for those who studied physics in the past. However, realizing that even the propagation of vibrations, as demonstrated with this device, relates to topology may help lower the barriers to studying this area. For more details, see 'Phononic band calculations and experimental imaging of topological boundary modes in a hexagonal flexural wave machine,' H. Takeda, R. Minami, O. Matsuda, O.B. Wright, M. Tomoda, Appl. Phys. Express 17, 017004 (2024).