Topological complex-energy braiding of non-Hermitian bands

Effects connected with the mathematical theory of knots(1) emerge in many areas of science, from physics(2,3) to biology(4). Recent theoretical work discovered that the braid group characterizesthe topology of non-Hermitian periodic systems(5), where the complex band energies can braid in momentum space. However, such braids of complex-energy bands have not been realized or controlled experimentally. Here, we introduce a tight-binding lattice model that can achieve arbitrary elements in the braid group of two strands B-2. We experimentally demonstrate such topological complex-energy braiding of non-Hermitian bands in a synthetic dimension(6,7). Our experiments utilize frequency modes in two coupled ring resonators, one of which undergoes simultaneous phase and amplitude modulation. We observe a wide variety of two-band braiding structures that constitute representative instances of links and knots, including the unlink, the unknot, the Hopflink and the trefoil. We also show that the handedness of braids can be changed. Our results provide a direct demonstration of the braid-group characterization of non-Hermitian topology and open a pathway for designing and realizing topologically robust phases in open classical and quantum systems.