From knots to exceptional points: Emergence of topological features in non-Hermitian systems with long-range coupling

We present a study of complex energy braiding in a one-dimensional non-Hermitian system with $n\mathrm{th}$-order long-range asymmetrical coupling. Our work highlights the emergence of novel topological phenomena in such systems beyond the conventional nearest-neighbor interaction. The modified SSH model displays $n$ distinct knot and link combinations in the complex energy-momentum space under periodic boundary conditions, which can be controlled by varying the coupling strengths. A topological invariant, namely the braiding index, is introduced to characterize the different complex energy braiding profiles, which depends on the zeros and poles of the characteristic polynomials. Furthermore, we demonstrate that the non-Hermitian skin effect can be localized at one or both ends, signifying conventional or bipolar localization, depending on the sign of the braiding index. Phase transitions between different braiding phases with the same (opposite) sign of the topological invariant occur at Type-1 (Type-2) exceptional points, with Type-1 (Type-2) phase transitions accompanied by single (multiple) exceptional points. We propose an experimental setup to realize the various braiding schemes based on the RLC circuit framework, which provides an accessible avenue for implementation without recourse to high-dimensional momentum space required in most other platforms.