Topological Spectral Winding

• Topological spectral winding number is a quantized invariant that counts how many times complex functions encircle a reference point, thereby enforcing bulk–edge correspondence.
• It is computed through momentum-space, real-space, and Green’s-function methods, offering robust predictions for edge, skin, and defect-localized modes.
• Applications include predicting zero modes in chiral Hamiltonians, guiding experimental photonic protocols, and analyzing disorder effects in topological systems.

A topological spectral winding number is a quantized invariant encoding the global topological properties of non-interacting Hermitian and non-Hermitian systems, as well as classical analogs. It counts the net number of times specified functions (such as the off-diagonal block of a chiral Bloch Hamiltonian, energy loops in the complex plane, or determinants of Hamiltonians minus a reference energy) wind around a point or the origin as a system parameter (often crystal momentum or a complexified version thereof) traverses a closed contour. Its integer (or half-integer) value underpins bulk–edge correspondence: it predicts the presence and number of robust zero modes or edge states under open boundary conditions. The spectral winding number has precise algebraic, geometric, and analytic manifestations in chiral symmetric systems, non-Hermitian lattices, Floquet (periodically driven) phases, and higher-dimensional setups, and can be accessed through real-space, momentum-space, or Green’s-function-based formulations.

1. Formal Definitions: Spectral and Momentum-Space Winding Numbers

The canonical definition is for a 1D chiral-symmetric Bloch Hamiltonian in basis diagonalizing the chiral operator $\Gamma=\sigma_3$: $$H(k) = \begin{pmatrix} 0 & q(k) \ q^\dagger(k) & 0 \end{pmatrix},$$ with $q(k)$ analytic and complex-valued. The integer winding number,

$$\nu = \frac{1}{2\pi i}\int_0^{2\pi}\!dk\,\partial_k \ln \det q(k) = \frac{1}{2\pi}\int_0^{2\pi}\!dk\,\partial_k \arg \det q(k),$$

counts encirclings of the origin by $q(k)$ as $k$ traverses the Brillouin zone (Chen et al., 2019).

More abstractly, for any complex-analytic family $H(\lambda)$ with $\lambda$ parameterizing a closed loop $\gamma$ in the complex plane, the spectral winding number is

$$W = \frac{1}{2\pi i}\oint_\gamma d\lambda\,\partial_\lambda \ln \det H(\lambda),$$

which, by the argument principle, equals the difference between the number of zeros and poles of $\det H(\lambda)$ inside $\gamma$ (Guhr, 2023).

In non-Hermitian systems, for a fixed reference energy $E_r$, the spectral winding counts how often the complex spectrum $E(k)$ winds around $E_r$: $$\nu(E_{r}) = \frac{1}{2\pi i}\oint_{\rm BZ} dk\,\frac{d}{dk} \ln \det[H(k)-E_{r}]$$ (Li et al., 2020, Liang et al., 2024, Yang et al., 2024).

These definitions hold for both single-band and multi-band models, and generalize to higher dimensions.

2. Physical Interpretations and Bulk–Edge Correspondence

The topological spectral winding number encodes robust, quantized features of physical systems. In Hermitian chiral chains (SSH, Kitaev), $|\nu|$ predicts the number of zero-modes bound to an edge (Chen et al., 2019, Guhr, 2023); $\nu=0$ (trivial), $\nu=1$ (one edge mode), or $\nu=2$ (two edge modes). This quantization signifies that during adiabatic deformations which do not close the gap, the number of edge-localized zero modes remains invariant—an explicit realization of bulk–edge correspondence.

In non-Hermitian systems, the spectral winding is similarly predictive: it controls the emergence of skin modes, defect-localized states, and the amplification directionality in classical or active lattices (Li et al., 2020, Liang et al., 2024). In mixed boundary conditions, a generalized “threshold” applies: right-localized skin modes appear if and only if $|W|>N_P$, where $N_P$ is the periodic-boundary segment dimension (Liang et al., 2024). The difference of winding numbers for left and right groupings ($\nu_L$, $\nu_R$) can differ in generalized SSH-type models, directly controlling the number of modes on each boundary (Chen et al., 2019).

In higher dimensions, a precise link exists between the 2D Chern number and differences in 1D windings calculated on inversion-symmetric slices $p_2=0,\pi$: $C = \nu(p_2=\pi) - \nu(p_2=0),$ illuminating the microscopic relationship between 1D and 2D topological invariants (Chen et al., 2019).

3. Spectral Winding in Hermitian, Non-Hermitian, and Floquet Systems

In Hermitian 1D systems (class AIII), $H(k)$ can always be brought to an off-diagonal, chiral-symmetric form, and the winding of its off-diagonal block defines the topological invariant (Chen et al., 2019). In non-Hermitian cases, the winding number may take half-integer values due to exceptional points (EPs), and decomposes as $\nu=(\nu_1+\nu_2)/2$, where $\nu_{1,2}$ count windings about two EPs. The difference $(\nu_1-\nu_2)/2$ is the “energy vorticity” (Yin et al., 2018).

In non-Hermitian settings the spectral winding can be evaluated as

$$\nu(E_r) = \frac{1}{2\pi}\sum_\alpha \int_{0}^{2\pi} dk\,\partial_k \arg[E_\alpha(k) - E_r]$$

with $E_\alpha(k)$ the complex bands (Liang et al., 2024).

For Floquet systems, winding numbers $W_s$ in the quasienergy gaps (at $s=0$,$\pi$) are defined through time-ordered evolution operators and determine the number of (anomalous) edge modes traversing each gap: $$W_s = \frac{1}{8\pi^2} \int_{T} dt \int_{\rm BZ} d^2k\; \mathrm{Tr}[u_s^{-1}\partial_t u_s \cdot (u_s^{-1}\partial_{k_x} u_s,\ u_s^{-1}\partial_{k_y} u_s)]$$ (Shi et al., 2024). Large values $|W_s|>1$ are found in multi-step or nontrivial protocols, leading to multiple pairs of edge modes in a single gap.

4. Measurement and Computation: Real-Space, Spectral-Function, and Experimental Probes

The winding number admits multiple computation schemes:

• Momentum-space integration: Direct evaluation via $\partial_k \arg q(k)$ along BZ.
• Spectral function-based: For ARPES-accessible systems, the winding $W$ can be extracted from the single-particle spectral function $A_{\alpha\beta}(k,\omega)$ as the winding of the phase $\theta(k)$ in $F(k) = A_{12}(k,\omega_0) + iA_{21}(k,\omega_0)$ at an energy $\omega_0$ inside the bulk gap (Estake et al., 2024).
• Green’s function formalism: The formula

$$W(\omega_0) = \frac{1}{2\pi i}\int_{BZ} dk\ \mathrm{Tr}\left[G^R(k,\omega_0) \partial_k G^R(k,\omega_0)^{-1}\right]$$

relates the topological index to poles and zeros of the Green’s function (Estake et al., 2024).

• Spectral localizer (real-space): The index is

$$I_{\rm SL}(\kappa) = \frac{1}{2}\Sigma[L(\kappa)] = \frac{1}{2} \operatorname{Tr}(L(\kappa)[L(\kappa)^2]^{-1/2})$$

for suitable choices of operator, with the small-$\kappa$ limit recovering the real-space winding marker $w(x)$ (Jezequel et al., 31 Jul 2025).

• Single-shot photonic protocols: The weighted output intensity difference in a broadband-excited SSH waveguide/fiber array directly yields the winding via simple summation and normalization (Roberts et al., 2023).
• Electric field spectroscopy: The winding of Wannier–Stark ladder branches under applied fields offers direct bulk measurement of Chern or $\mathbb{Z}_2$ invariants (Lee et al., 2015).
• Superradiant and cold-atom systems: Counting spectral peaks (flat-band modes) or analyzing quench/stroboscopic dynamics resolves winding in topological superradiance lattices or Floquet setups (Li et al., 2023, Shi et al., 2024).

5. Extensions: Long-Range Couplings, Defect/Fractal States, and Disordered Systems

Long-range couplings or additional hopping terms can cause qualitative changes in spectral winding properties. In SSH-type chains with beyond-nearest-neighbor hoppings, the total winding number $W$ can fail to predict the number of domain-wall or defect modes when multiple Dirac points/EPs appear. Analysis of the derivative of arg$\,h(k)$ (Berry connection) as a function of $k$ identifies local topological transitions: each peak of area 1 in $\Delta B(k)$ signals a Jackiw–Rebbi zero mode, even when $W$ remains unchanged (Alisepahi et al., 2023). In non-Hermitian multiband systems, the threshold condition $|W|>N_P$ exactly predicts the emergence of defect-localized (“skin”) modes—robust even under disorder unless the bulk winding falls below the threshold, in which case modes delocalize (Liang et al., 2024).

For statistical ensembles, the full distribution $P_N(W)$ and two-point correlators are analytically accessible, exhibiting universal crossover as ensemble size increases; fluctuations become Gaussian with $\mathrm{Var}(W)\sim 2\sqrt{N/\pi}$ in large-$N$ chains (Guhr, 2023).

Disordered, quasicrystalline, or amorphous samples can be analyzed using the real-space winding marker via the spectral localizer; as $\kappa\to0$, this reproduces the momentum-space value even in systems lacking translation symmetry (Jezequel et al., 31 Jul 2025).

6. Applications and Experimental Realizations

The topological spectral winding number underpins:

• Prediction and engineering of edge, skin, and defect-localized states for electronic, photonic, magnonic, and mechanical metamaterials (Chen et al., 2019, Li et al., 2020, Liang et al., 2024).
• Design and control of directional signal amplification and quantized response in classical non-Hermitian circuits and photonics, with measurable Green’s-function plateaus controlled by spectral winding (Li et al., 2020).
• Single-shot measurement protocols in photonic fibers and waveguide arrays, enabling extraction of topology from integrated output signals without spectral or $k$-resolved scans (Roberts et al., 2023).
• Real-time detection and manipulation of Floquet phases with large winding numbers in cold-atom lattices and time-modulated systems, revealing richer edge mode structures than conventional static topological insulators (Shi et al., 2024).
• Extraction of winding and Chern numbers in spectroscopic experiments (ARPES, STS), translating measured spectral data into topological invariants (Estake et al., 2024).
• Tuning and “inverse design” of topological mode localization (left, right, both, none) in non-Hermitian lattices via choice of spectral winding tuple $(w_{\rm GBZ},w_{\rm BZ})$ (Yang et al., 2024).
• Analysis of disorder effects and real-space topological profiles via the spectral localizer marker, with precise small-$\kappa$ expansion connecting to bulk invariants (Jezequel et al., 31 Jul 2025).

7. Exemplary Systems and Generalizations

Topological spectral winding numbers pervade a wide class of physical models:

System Class	Key Invariant	Winding Formula
Hermitian SSH, Kitaev	$W$ (integer winding)	$W = \frac{1}{2\pi}\int \partial_k \arg q(k)dk$
Non-Hermitian SSH	$\nu$ (possibly half-integer)	$\nu = (\nu_1+\nu_2)/2$, where $\nu_{1,2}$ enclose EPs
Multi-band/Non-Hermitian	$W(E_r)$ (by complex bands)	$$W(E_r) = \sum_\alpha \frac{1}{2\pi}\int \partial_k \arg[E_\alpha(k) - E_r]dk$$
Floquet (driven)	$W_s$ (gap winding)	$$W_s = \tfrac{1}{8\pi^2}\int dt\,d^2k\,\mathrm{Tr}[...]$$
Disordered/amorphous	$I_{\rm SL}$ (localizer)	$I_{\rm SL} = \sum_x w(x)$, $$w(x) = -\mathrm{Tr}[\hat C \hat H_F [\hat H_F, \hat x]]$$

This versatility supports bulk–boundary correspondence, quantized response in classical and quantum systems, and generalizes to higher dimensions: Chern numbers in 2D as winding of 1D invariants, Weyl-point charges in 3D as jumps in 2D Chern integrals across momentum slices (Chen et al., 2019, Lee et al., 2015). In summary, the topological spectral winding number is a unifying, quantized descriptor of bulk topology manifest across broad classes of Hermitian, non-Hermitian, and time-periodic/engineered systems.