Competition of non-Hermitian skin effect and topological localization of corner states observed in circuits

Exploring topological phases in non-Hermitian systems has attracted significant recent attention. One intriguing question is how topological edge states compete with the non-Hermitian skin effect. Here, we report the experimental observation of corner states in a two-dimensional non-reciprocal rhombus honeycomb electric circuit. We construct non-reciprocal and non-Hermitian circuits by introducing current-direction resolved capacitance between two nodes depends on the current direction. Skin effect thus emerges due to the non-reciprocity and prevails in dragging the corner state into the bulk. The non-Bloch winding number defined in generalized Brillouin zone is adopted to characterize the topological phase transition. Interestingly, we find that the non-Bloch $Z_2$ Berry phase can serve as an invariant to describe the non-Hermitian topology. By tuning the non-reciprocal parameter, we observe unbalanced distribution of corner states emerging on two acute angles of the rhombus lattice, with the localization length of the left corner state increasing exponentially with the degree of non-reciprocity.