Topological Spectral Winding in Non-Hermitian Systems

• Topological spectral winding is defined as the count of times complex energy bands encircle a reference point, serving as a robust invariant in non-Hermitian systems.
• It underpins the non-Hermitian bulk-boundary correspondence and skin effects, especially in systems with periodic and effectively closed boundaries.
• Experimental detection methods, such as phase-tunable perturbations and Green’s function responses, enable precise measurement of the winding number.

Topological spectral winding is a fundamental invariant characterizing spectra in non-Hermitian systems, where the complex energy bands traced as functions of crystal momentum wind nontrivially around reference points in the complex plane. This phenomenon fundamentally underpins non-Hermitian bulk-boundary correspondence, governs the emergence of skin effects, and enables quantized responses even in the absence of Hermitian spectral or eigenstate topology. While classically formulated for 1D periodic systems, extensions to higher-dimensional boundaries, multiband and Floquet systems, as well as disorder and real-space settings, have revealed a unifying framework capturing a wide range of topological phenomena in non-Hermitian and hybrid Hermitian–non-Hermitian lattices (Ou et al., 2022).

1. Formal Definition and General Properties

The spectral winding number is defined for a non-Hermitian lattice Bloch Hamiltonian $H(k)$, periodic in $k \in [0,2\pi)$, and a reference point $E_\mathrm{ref} \in \mathbb{C}$ that does not cross the energy bands:

$$W(E_{\mathrm{ref}}) = \frac{1}{2\pi i} \int_0^{2\pi} \partial_k \ln \det[H(k) - E_{\mathrm{ref}}]\, dk$$

Equivalently, $W(E_\mathrm{ref})$ counts the (signed) number of times the complex spectrum traces a closed loop encircling $E_\mathrm{ref}$ as $k$ sweeps the Brillouin zone. For each eigenvalue band $E_a(k)$, the corresponding winding is

$$W_a(E_{\mathrm{ref}}) = \frac{1}{2\pi} \int_0^{2\pi} \partial_k \arg[E_a(k) - E_{\mathrm{ref}}]\, dk \in \mathbb{Z}$$

In Hermitian systems, spectra are real, and $W(E_\mathrm{ref})$ vanishes for $k \in [0,2\pi)$0 off the real axis. Non-Hermitian systems, by contrast, generically exhibit point-gapped spectra—complex loops avoiding $k \in [0,2\pi)$1—allowing nonzero winding.

Spectral winding is protected under small continuous deformations that do not close the point gap at $k \in [0,2\pi)$2. Under open boundary conditions (OBC), the spectrum generally collapses to intervals or arcs, causing the winding to vanish (Ou et al., 2022).

2. Manifestations and Boundary Spectral Winding

While conventional spectral winding vanishes under OBC, boundary-localized modes in higher dimensions can inherit effective PBC topology. In 2D non-Hermitian systems such as the breathing Kagome lattice, the interplay between Hermitian boundary localization and non-reciprocal pumping drives the formation of chiral edge modes along the closed 1D perimeter. These edge modes are described by an effective 1D non-Hermitian chain with periodic boundaries, exhibiting nontrivial spectral winding even though the bulk Hamiltonian with OBC supports no spectral loop (Ou et al., 2022).

This edge winding can be sharply controlled by geometry: in a triangular flake, the perimeter remains a PBC chain with persistent winding, while a truncated trapezoidal geometry converts the edge into a nearly OBC chain, leading to hybrid skin-topological effects. Spectral winding thus encodes a topological boundary invariant robust to the imposition of OBC, conditional on the perimeter remaining effectively closed.

3. Physical Detection via Green’s Function Response

The presence of nonzero spectral winding can be diagnosed via the steady-state or local response to a tunable perturbation, providing a quantized signature without requiring full spectral measurement. For instance, inserting a phase-tunable hopping ($k \in [0,2\pi)$3) across a single boundary bond, the corresponding Green’s function $k \in [0,2\pi)$4 satisfies

$k \in [0,2\pi)$5

This quantized slope is integer-valued and directly proportional to the underlying spectral winding. For 2D systems, this scheme isolates edge or boundary winding by targeting physical edges and tracking the topological response domain as $k \in [0,2\pi)$6 is scanned (Ou et al., 2022, Li et al., 2020).

4. Extension to Higher Winding and Multiband Systems

Generalizations of spectral winding allow for higher absolute values, determined by the topology of multiband Hamiltonians and by the presence of longer-range hoppings or modulated drives. For example, an extended SSH model in an atomic superradiance lattice realizes a flat band with winding number two, detected via the multiplicity of superradiant spectral peaks (Li et al., 2023). In non-unitary Floquet systems, photonic quantum walks with multi-step protocols have experimentally demonstrated windings up to three and four, mapped via statistical moments of walker distributions (Xiao et al., 2018). The classification in these cases is always via the net loop winding of the spectrum in the complex plane, possibly evaluated as a product over band energies around the base point (Yang et al., 2024).

Topological transitions occur as spectral loops merge, bifurcate, or pass through reference energies, with analytically accessible phase boundaries in coefficient representations for broad classes of lattice models (Chang et al., 9 Jun 2026).

5. Relation to Non-Hermitian Skin Effect and Bulk–Boundary Correspondence

Nonzero spectral winding number in a periodic 1D non-Hermitian system predicts the accumulation of modes (skin effect) at a boundary upon opening the system. This non-Hermitian bulk–boundary correspondence is mediated by the winding: Only as long as the OBC boundary retains PBC-like connectivity (e.g., closed edge in a 2D flake) does the point-gap winding survive; the destruction of that topology (e.g., by cutting the chain) collapses the winding and triggers skin mode accumulation at a point (hybrid skin-topological effect) (Ou et al., 2022). In higher dimensions, Möbius-like real-space topologies produce multi-loop energy surfaces and discrete jumps in spectral winding, demonstrating that both real-space connectivity and complex spectral topology jointly control boundary phenomena (Shen et al., 30 May 2026).

In disordered systems, reentrant transitions between extended and localized phases accompany spectral transitions between trivial and non-trivial windings, tightly linking point-gap topology with localization properties (Sharma, 14 May 2026, Ghosh et al., 12 Jan 2026).

6. Real-Space and Statistical Frameworks

For non-periodic, random, or amorphous systems, real-space markers encode spectral winding locally. The spectral localizer provides a numerically efficient, Clifford-algebraic means of extracting the winding density at a given position and energy, reducing to the standard Bloch invariant in clean limits but maintaining quantization and spatial resolution in strongly disordered regimes (Jezequel et al., 31 Jul 2025). Random-matrix approaches admit exact results and universal statistics for winding number distributions and correlations, including central limit theorems and universal short-distance kernels, by mapping topological winding to spherical matrix ensembles (Guhr, 2023).

7. Experimental Signatures and Applications

Spectral winding is accessible through diverse measurement schemes. In circuit QED, photonic, or acoustical platforms, its presence is revealed by topologically quantized plateaus in the steady-state Green’s function as the boundary condition is interpolated from periodic to open (Li et al., 2020). In atomic and photonic systems, direct observation aligns the number of boundary or midgap resonances with the predicted winding (Li et al., 2023, Xiao et al., 2018). In solid-state systems, the single-particle spectral function as measured by ARPES or STS integrates to the winding or Chern number, recoverable from momentum-integrated and sign-weighted spectral intensities (Estake et al., 2024). In inelastic x-ray scattering, the harmonic modulation of intensity around Fermi points reflects the eigenvector winding tied to Dirac or Weyl crossings (Andriushin et al., 2023).

By combining spectral winding numbers with generalized Brillouin zone invariants, one achieves a complete, symmetry-independent topological classification applicable even in the presence of non-Hermitian localization and symmetry breaking, with direct ramifications for the design and manipulation of robust edge and skin modes for sensing, lasing, or energy-flow control (Yang et al., 2024).

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References:

• (Ou et al., 2022) Non-Hermitian boundary spectral winding
• (Li et al., 2023) Topological flat band with higher winding number in a superradiance lattice
• (Yang et al., 2024) Inverse Design of Winding Tuple for Non-Hermitian Topological Edge Modes
• (Chang et al., 9 Jun 2026) Higher-winding phases in one-dimensional non-Hermitian topological superconductors
• (Xiao et al., 2018) Higher winding number in a non-unitary photonic quantum walk
• (Sharma, 14 May 2026) Spin resolved spectral topology and re-entrant localization in a non Hermitian quasiperiodic SSH chain
• (Ghosh et al., 12 Jan 2026) Spectral Topology and Delocalization in Disordered Hatano-Nelson Chains
• (Li et al., 2020) Quantized classical response from spectral winding topology
• (Estake et al., 2024) Detecting winding and Chern numbers in topological matter using spectral function
• (Jezequel et al., 31 Jul 2025) Explicit equivalence between the spectral localizer and local Chern and winding markers
• (Guhr, 2023) Statistical Topology -- Distribution and Density Correlations of Winding Numbers in Chiral Systems
• (Andriushin et al., 2023) Phonon topology and winding of spectral weight in graphite
• (Shen et al., 30 May 2026) Möbius-like Real-Space Topology Reshapes Spectral Winding Topology in Hatano-Nelson Rings