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A fractal structure can support topologically protected transport along its many edges.

Topological insulators are found in diverse systems, including electronics, acoustics, and cold atoms. (See, for example, “Topological insulators: from graphene to gyroscopes,” Physics Today online, 27 November 2018.) Their unique behavior is topological because it derives from the connectedness of the band structure. More simply, the phase is characterized by an insulating bulk and robust conducting edge states that travel in one direction (see the article by Xiao-Liang Qi and Shou-Cheng Zhang, Physics Today, January 2010, page 33). The number of edge states is dictated by the topology of the bulk’s band structure—an idea known as the bulk–boundary correspondence.

But can you get topologically protected transport without an insulating bulk? Matthias Heinrich and Alexander Szameit of the University of Rostock in Germany and their colleagues have shown that you can in a photonic lattice with a fractal structure.

The lattice of coupled waveguides forms a Sierpinski triangle, a well-known fractal with a fractal dimension of about 1.585. The self-similar lattice is all edges and no bulk and thus an unlikely candidate to support topological edge modes. On its own, it’s topologically trivial—it has a value of zero for a quantity known as the real-space Chern number (see the article by Arthur Ramirez and Brian Skinner, Physics Today, September 2020, page 30). But when periodically driven, the Chern number becomes nonzero in multiple regions. The result is states that circle the outer boundary in one direction and the inner boundaries in the other direction.

Heinrich and his colleagues investigated the transport in fractal and conventional honeycomb lattices. For the Sierpinski-triangle lattice, they tracked light as it propagated in one direction around the outer edge, including around corners (shown in the left three panels of the image), and the other direction along an inner edge (right two panels).

The researchers also investigated how the behavior changes with energy. For wavepackets with energies higher than the bandgap that opens in the topological state, the light penetrated both lattice types. For energies in the energy gap, the light traveled exclusively along the edges. The topological gaps were about the same for the two lattices, but the fractal edge transport was around 11% faster than that of the hexagonal lattice. That velocity boost was likely because there were no bulk lattice sites to slow the edge states down.

Next up could be other fractals with other noninteger dimensionalities, such as the Cantor cube or Sierpinski tetrahedron. But according to the researchers’ calculations, some fractals, such as 1.26-dimensional triflakes, have fractal dimensions too low to support topological states. (T. Biesenthal et al., Science 376, 1114, 2022.)