Zero-Winding Skin Modes in Non-Hermitian Systems

• Zero-winding skin modes are boundary-localized eigenstates in non-Hermitian systems that arise despite a zero net spectral winding, enabled by symmetry, geometry, and interference mechanisms.
• Research reveals that multi-channel, Floquet, and higher-order effects can induce these modes, leading to scale-free localization and a pronounced dependence on system geometry.
• Experimental realizations in photonic, mechanical, and electronic metamaterials demonstrate practical applications such as enhanced sensing, directional energy transport, and tunable signal localization.

Zero-winding skin modes are boundary-localized eigenstates in non-Hermitian systems that emerge in the absence of a global, nonzero spectral winding number. Traditionally, the non-Hermitian skin effect (NHSE) is associated with nontrivial spectral winding and manifests as an extensive accumulation of bulk eigenstates at one or more boundaries under open boundary conditions. However, recent theoretical and experimental advances have revealed a variety of mechanisms by which skin modes can arise even when the net winding vanishes. These zero-winding skin modes are distinguished by their dependence on symmetry, geometry, multi-channel interference, dynamical (Floquet) topology, or higher-order effects, and often deviate from the canonical bulk–boundary correspondence and localization characteristics found in one-dimensional nonreciprocal systems.

1. Fundamental Mechanisms of Skin Modes and Spectral Winding

In non-Hermitian lattices, the spectral winding number,

$$W(E_b) = \frac{1}{2\pi i} \int_{BZ} \frac{d}{dk} \log \det[H(k) - E_b I]~dk,$$

quantifies the topological winding of the complex band structure around a base energy $E_b$ as momentum traverses the Brillouin zone (Zhang et al., 2019, Lee et al., 2018, Kong et al., 21 Jun 2024). A nonzero $W(E_b)$ is generically sufficient (and, for one-dimensional single-band systems, necessary) for the existence of skin modes, as it reflects “charge pumping” of bulk states toward a boundary upon opening the system. However, several classes of systems can host skin-localized modes despite a zero net winding:

• Multi-component and spinful models: Interference between distinct subspaces or spin sectors can yield bi-localized or scale-restricted skin modes where the total winding over all channels cancels (Sanahal et al., 8 May 2025).
• Floquet (periodically driven) systems: Skin effects can emerge due to a dynamical (Floquet) topological invariant, even when the static (Floquet) Hamiltonian under PBC forms a point (zero winding) in the complex plane (Liu et al., 2023).
• Higher-order and spatially symmetric models: Spatial symmetries can enforce vanishing 1D winding along all principal axes, but symmetry-protected higher-dimensional invariants or local winding projections can generate edge, corner, or even defect-localized skin effects (Kawabata et al., 2020, Zhang et al., 2023, Xiong et al., 1 Jul 2024).
• Multi-channel interference and geometry-induced effects: By engineering nonconservative couplings with competing spectral winding channels, one can realize skin modes through interference—even with zero net winding (Kong et al., 21 Jun 2024, Liang et al., 14 Jul 2024).

In these cases, skin modes defy the conventional winding–skin mode correspondence and require new diagnostic frameworks (Sanahal et al., 8 May 2025, Xiong et al., 1 Jul 2024).

2. Classification and Taxonomy: Geometry, Symmetry, and Order

A unified taxonomy of NHSE in arbitrary $d$-dimensional systems identifies two main categories (Xiong et al., 1 Jul 2024):

• Nonreciprocal NHSE: At least one net winding number is nonzero along a system cut direction; skin modes are exponentially localized with finite, system-size–independent decay lengths, and the spectra are robust (geometrically insensitive) in the thermodynamic limit.
• Critical (zero-winding) NHSE: All net winding numbers vanish—that is,

$\forall\, j:\;\bar{w}_j(E) = 0,$

where

$$\bar{w}_j(E) = \int\frac{d^dk}{i(2\pi)^d}\partial_{k_j}\log\det[H(\vec{k}) - E]$$

for each lattice direction $j$. Here skin modes possess scale-free localization, with decay lengths scaling with the system size $L$, and pronounced geometric or shape dependence in finite-size systems (Xiong et al., 1 Jul 2024). Weak disorder, boundary deformations, or small perturbations can destabilize these modes and drive the spectra toward an "Amoeba" (universal) form, highlighting their sensitivity to geometry and the non-exchangeability of the thermodynamic and zero-perturbation limits.

Higher-order skin effects arise when line (1D) winding invariants vanish by symmetry (e.g., inversion or fourfold rotation), but nontrivial 2D or 3D topological invariants (such as a $\mathbb{Z}_2$ quantized Wess–Zumino term) protect $O(L)$ corner modes (second order) or higher (Kawabata et al., 2020).

Spinful or multi-channel systems can host “unconventional” skin effects—for instance, scale-restricted, bi-localized, or spin-polarized modes that persist despite $w = 0$, as found in the Hatano–Nelson model with spin-dependent complex hopping phases (Sanahal et al., 8 May 2025). An external field coupling the channels may then induce a crossover to critical skin phenomena, signaling a deep relationship between multi-channel interference, symmetry, and the emergence of zero-winding skin effects.

3. Mathematical Frameworks and Diagnostic Methods

• Generalized Brillouin Zone (GBZ) and Non-Bloch Theory: For both 1D and higher dimensions, open boundary modes are constructed by replacing the Bloch phase $e^{ik}$ with a complexified variable $\beta = r e^{ik}$ (with $|r|\ne 1$ generically in the NHSE). The GBZ is the set of all such $\beta$ for which the characteristic equation $\det[H(\beta) - E] = 0$ appropriately matches the number of left/right boundary conditions (Yao et al., 2018, Lee et al., 2018, Xiong et al., 1 Jul 2024). For higher-order and critical cases, the minimization of the spectral potential,

$$\Phi(E) = \min_{\mu_j} \int \frac{d^dk}{(2\pi)^d} \log | \det[ H(e^{ik_1+\mu_1}, ..., e^{ik_d+\mu_d}) - E ] |$$

yields the geometry-adaptive energy spectrum and density of states (Xiong et al., 1 Jul 2024).

• Winding–Defect Correspondence and Green’s Function Diagnostics: The existence and localization of skin or defect states can be characterized by whether the spectral winding number $W$ about a reference energy surpasses a critical threshold determined by the defect structure, e.g., $|W| > N_P$ for $N_P$ periodic boundary components in multi-channel chains (Liang et al., 14 Jul 2024). The appearance of Green’s function poles and amplified local responses are physical signatures of these localized states (Liang et al., 14 Jul 2024).
• Topological Invariants Beyond the Winding Number: In several scenarios, generalized invariants such as the non-Abelian Chern–Simons index of the time evolution operator (in Floquet NHSE (Liu et al., 2023)), a quantized Wess–Zumino term in two or higher dimensions (Kawabata et al., 2020), or field-theoretic geometric response actions (e.g., Wen–Zee terms (Sun et al., 2021)) supply the necessary protection for skin-localized states where winding is identically zero.

4. Physical Realizations and Experimental Signatures

Zero-winding skin modes have been predicted and observed in various platforms:

• Photonic Lattices and Reciprocal Crystals: Geometry-induced NHSE can be realized without nonreciprocal couplings by exploiting the interplay between eigenvalue topology of exceptional points (EPs) and macroscopic boundary geometry in reciprocal photonic crystals. Skin modes emerge at interfaces if the eigenvalue discriminant number is nonzero for the relevant $k$-loop, while “zero-winding” configurations along high-symmetry boundaries fail to produce skin-localized states (Fang et al., 2022).
• Mechanical and Electronic Metamaterials: Experiments on nonreciprocal mechanical metamaterials have directly observed zero-winding skin effects, with extreme biorthogonal sensitivity and diverging response quantified by the Petermann factor (Schomerus, 2019). Such systems may be engineered for robust, highly sensitive sensors.
• Quasiperiodic or Disordered Systems: The non-Hermitian Anderson skin effect (NHASE) can arise upon introducing disorder into a model with trivial clean-system winding. Here, disorder can induce a finite winding, causing the selective emergence and destruction of skin phases as a function of disorder strength; this process is tracked by real-space winding numbers, inverse participation ratios, and chiral currents (Sarkar et al., 2022).
• Periodically Driven (Floquet) Lattices: Zero-winding skin effects in Floquet systems are dynamically protected and can produce second-order skin mode localization (e.g., at corners) in parameter regimes where the Floquet Hamiltonian forms a trivial point under PBCs (Liu et al., 2023).
• Spinful and Multi-band Models: The use of synthetic gauge fields with spin/two-component structure allows for the systematic tuning between conventional NHSE (with nonzero winding) and scale-restricted, spin-polarized, zero-winding NHSE. Application of an external magnetic field modifies symmetry protection and can induce a transition to a critical, size-dependent NHSE (Sanahal et al., 8 May 2025).
• Continuum Systems with Parametric Drive: Nonlocal parametric pairing processes, stabilized by damping, can lead to tilted exceptional-point lines and a bulk anomaly in current response; the resulting skin effect and boundary anomalies are present even though the drive itself can, in the absence of dissipation, be Hermitian (Bestler et al., 5 May 2025).

5. Physical Origin, Bulk–Boundary Correspondence, and Instabilities

In all cases, the physical mechanism underlying the emergence (or suppression) of skin modes is closely linked to the topology of the complex energy bands—specifically, their winding and the presence or absence of persistent current flow in the periodic system (Zhang et al., 2019). For standard NHSE, the skin effect and the persistent current vanish together as the net winding number goes to zero.

In higher dimensions or for critical (zero-winding) NHSE, the localization is “scale-free,” with decay lengths scaling with system size rather than being fixed. This results in pronounced geometric dependence and a nonconverging spectrum in the thermodynamic limit (Xiong et al., 1 Jul 2024). The presence (or absence) of skin modes can usually be deduced from the bulk projection criterion or via mappings to Hermitian zero-energy edge states (Zhang et al., 2023). In these nonreciprocal or critical regimes, the bulk-boundary correspondence may become geometry-sensitive or be governed by higher-order topological invariants.

A distinctive property of zero-winding (critical) skin modes is their spectral instability in the presence of weak disorder or boundary perturbations, reflecting the essential non-exchangeability of the zero-perturbation and infinite-size limits. In contrast, skin modes associated with nonzero net winding are geometrically robust and spectrally stable (Xiong et al., 1 Jul 2024).

6. Theoretical and Practical Implications

Zero-winding skin modes extend the landscape of non-Hermitian topology beyond canonical winding-based classifications (Xiong et al., 1 Jul 2024, Kawabata et al., 2020, Sanahal et al., 8 May 2025). Their paper reveals new forms of boundary and defect localization, multi-channel–driven or scale-restricted phenomena, and dynamical or geometry-dependent protection, all with practical implications for device design:

• Enhanced sensitivity and signal localization: Applications in sensors, photonic and mechanical circuits, and directional energy transport can leverage the geometry and symmetry-dependent accumulation of eigenstates.
• Experimental observation and control: Synthetic gauge fields, gain/loss engineering, Floquet (periodic) drives, and boundary geometry tuning are viable routes to realizing and manipulating zero-winding skin effects (Sanahal et al., 8 May 2025, Fang et al., 2022, Liu et al., 2023).
• Challenges and open questions: The role of disorder, the development of higher-dimensional generalizations of non-Bloch theory, and the classification of critical NHSEs point to rich directions for future research (Xiong et al., 1 Jul 2024).

7. Comparative Table: Types of NHSE/Zero-Winding Skin Modes

Category	Spectral Winding	Localization Scaling	Diagnostic Invariant
Standard nonreciprocal NHSE	$W\neq 0$	Exponential, O(1)	1D/line net winding number
Critical (zero-winding)	$W=0$ (all cuts)	Scale-free, $\sim O(L)$	Higher-order, Wess–Zumino, or geometry-proj.
Floquet/dynamical	$W_{\text{PBC}}=0$	Edge/corner, robust	Dynamical (Chern–Simons) invariant
Spinful scale-restricted	$W=0$ (total)	Bi-localized, scale-restricted	Channel-resolved winding/interference
Geometry-induced	$W=0$ (symmetric cuts)	Interface-selective	Local (e.g., EP) discriminant number

This summary delineates the major types of zero-winding skin modes, their mathematical and physical origins, diagnostic tools, and representative physical settings. The phenomenon of zero-winding skin modes illustrates the deep and subtle interplay between topology, geometry, symmetry, drive, and interference in non-Hermitian systems, representing a continuing frontier in the paper of boundary phenomena and topological matter.