Multipole higher-order topology in a multimode lattice

We put forward a two-dimensional multipolar higher-order topological insulator whose properties originate from the interference of the degenerate modes of the meta-atoms. This system features disorder-robust corner modes and cannot be reduced to the known crystalline topological phases, providing the example of multipolar topology in a ${C}₃$-symmetric lattice. The multimode nature of the lattice gives rise to flat bands and corner states with extreme localization. We assemble the designed structure, observe multipolar topological corner states experimentally, and demonstrate their coherent control.