Möbius-like Real-Space Topology Reshapes Spectral Winding Topology in Hatano-Nelson Rings

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 Möbius 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 Möbius 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.

• The paper introduces a novel mechanism where Möbius real-space topology induces quantized spectral winding transitions in Hatano-Nelson rings.
• It employs analytical self-consistent maps and numerical validation to predict distinct winding-number jumps with 1% accuracy.
• The study reveals scale-free skin localization, opening new avenues for non-Hermitian device design and high-sensitivity sensing.

Möbius-like Real-Space Topology Reshapes Spectral Winding Topology in Hatano-Nelson Rings

Overview

The paper "Möbius-like Real-Space Topology Reshapes Spectral Winding Topology in Hatano-Nelson Rings" (2606.00627) introduces a novel paradigm for non-Hermitian topological physics by demonstrating that global real-space topology—specifically Möbius connectivity—can independently restructure spectral winding in Hatano-Nelson (HN) ring lattices. Traditional non-Hermitian systems have relied on local parameter tuning to modify spectral winding numbers, which govern skin modes and bulk-boundary correspondence. This work reveals that topological reconnection, while conserving microscopic parameters (hopping, on-site energies), induces striking global alterations in spectral and localization properties.

Möbius Reconstruction: Model and Topological Transition

Starting from two parallel HN rings with asymmetric hopping ($t_R$, $t_\ell$) and reciprocal inter-ring coupling ($t_{mo}$), the authors enact a Möbius-like real-space recombination by cutting and cross-linking the rings with a half-twist. This construction yields a single Möbius ring system with unchanged local Hamiltonian terms but fundamentally distinct global boundary conditions.

The Möbius topology transforms the periodic boundary condition (PBC) spectrum from two disjoint ellipses into a multi-petalled rose curve, signaling a nontrivial restructuring of the effective Brillouin zone. Crucially, while the conventional two-ring reference system remains topologically rigid as $t_{mo}$ varies, the Möbius ring displays discrete winding-number jumps at analytically predictable critical couplings. These results demonstrate that real-space connectivity alone can be a topological control parameter, decoupled from bulk non-Hermitian bias.

Non-Hermitian Skin Effect and Scale-Free Localization

Both the two-ring and Möbius systems exhibit the NHSE (macroscopic localization of eigenstates under open boundary conditions, OBC). However, their localization profiles are qualitatively distinct:

• In the Möbius ring, OBC eigenstates display heterogeneous decay lengths that are broadly and continuously distributed—a scale-free localization regime where no single characteristic length emerges. This persists as system size increases, indicating that the behavior is intrinsic and not a finite-size artifact.
• By contrast, in the parallel ring, all eigenstates are sharply localized at the boundary with essentially uniform spatial envelopes, consistent with standard single-chain HN physics.

The Möbius topology thus induces a fundamentally new kind of skin effect, where eigenstates are localized at boundary sites across multiple scales, leading to enhanced sensitivity in localization characteristics.

Quantized Winding Number Jumps: Analytical Predictability

Spectral winding, encoded by the integer winding number of energy loop trajectories in the complex plane, is shown to undergo quantized transitions in the Möbius ring as $t_{mo}$ is tuned. Each jump corresponds to a topological phase transition in the PBC spectrum, with the number of rose-curve petals increasing discretely.

The critical coupling strengths for winding number jumps are analytically predicted by iterative self-consistent maps of critical momenta and amplitude conditions. Numerical results confirm the analytic thresholds to within 1% accuracy. This establishes a direct mapping from tunable real-space parameters to bulk topological invariants and measurable skin mode localization properties.

Practical and Theoretical Implications

The paper posits real-space topology, specifically Möbius connectivity, as an independent and tunable foundation for manipulating non-Hermitian topology. This provides:

• Enhanced design flexibility for topological devices, by decoupling the control of winding numbers from local energy/hopping bias.
• A platform for high-sensitivity sensing, as the scale-free localization and abrupt winding-number jumps could yield threshold-detectable physical observables.
• New concepts in reconfigurable topological systems, where a simple real-space twist—without modification of local parameters—induces dramatic changes in spectral and localization behavior.

The scale-free nature of localization also suggests potential utility in engineering non-reciprocal devices and robust topological sensors, with localization behavior customizable via real-space connectivity. The extension of Möbius concepts to higher-dimensional non-Hermitian lattices and synthetic platforms (circuits, waveguides, metamaterials) may yield further novel topological phenomena.

Future Directions

Several open questions and avenues arise from these results:

• Identification of physical observables that detect winding-number jumps directly, leading to new classes of non-Hermitian sensors with threshold sensitivity.
• Exploration of Möbius connectivity in interacting, time-dependent, or disorder-driven non-Hermitian platforms.
• Generalization to multicomponent or higher-dimensional Möbius-like arrangements, potentially combining with synthetic gauge fields or PT symmetry.
• Application to device-level implementations in photonics, acoustics, and quantum circuits, exploiting tunable topology for robust and reconfigurable functionalities.

Conclusion

This work establishes Möbius-like real-space topology as a powerful, independent tool for spectral and localization control in non-Hermitian HN ring systems. Unlike conventional parameter tuning, topological reconnection generates quantized winding-number transitions, scale-free skin localization, and rich spectral features. It unlocks new routes for non-Hermitian bulk-boundary engineering, sensing schemes, and topological device design, highlighting the importance of global connectivity in future developments of artificial quantum matter and topological photonics.