M\"obius-like Real-Space Topology Reshapes Spectral Winding Topology in Hatano-Nelson Rings

arXiv:2606.00627v1 Announce Type: new Abstract: The spectral winding number serves as a bulk topological invariant in non-Hermitian systems, governing the emergence of skin modes and encoding the non-Hermitian bulk-boundary correspondence. However, most existing studies are built on conventional lattice geometries such as linear chains, rings, or planar arrays, leaving the role of real-space topological connectivity as an independent degree of freedom largely unexplored. Here, we construct a M\"obius ring system by cutting two parallel Hatano-Nelson (HN) rings and reconnecting them with a half-twist, without altering any local hopping parameter. This topological reconstruction transforms the periodic-boundary spectrum from two disjoint ellipses into a multi-petalled rose curve, and leads to distinct decay lengths for different eigenstates under open boundary conditions. Moreover, the spectral winding number can be driven through discrete winding-number jumps by tuning the coupling strength, with critical values obtained analytically. Our results demonstrate that real-space M\"obius connectivity, mediated by the coupling strength, provides an independent and tunable foundation for the systematic control of non-Hermitian topology, with implications for the design of topological devices and sensing schemes.