Non-Hermitian Hopf-Bundle Skin Effects in Electric Circuits

For matters describable with line and loop excitations, a topological phase can be classified by a linking and braiding structure. Recently, it was proven that the topological phase of one dimensional non-Hermitian system can be characterized by winding and braiding of complex eigenenergy strings. While nontrivial winding of complex eigenenergy strings has recently been demonstrated in a one-synthetic-dimensional photonic system, a topological phase transition and its corresponding physical manifestation in real and momentum spaces have not been visualized yet. Moreover, in higher spatial dimensions, it is found that every string in space can be mutually linked by all other strings, exhibiting a more intricate linking structure. A non-Hermitian Hopf bundle, as such a phase of linked strings, has not been experimentally observed yet. Here, we report the visualization of linking and unlinking of the Hopf bundle in momentum space of a two-dimensional non-Hermitian circuit by introducing an eigenspinor string. We observe that the eigenspinor strings take the form of the nontrivial Hopf bundle in the topological phase. Furthermore, we discover the bulk-boundary correspondence between the nontrivial Hopf-bundle in momentum space and a higher-order skin defect mode in real space. By utilizing controllable capacitance switching, we experimentally confirm the bulk-boundary correspondence by explicitly imaging linking and unlinking topological transitions of the Hopf bundle in momentum space.