Exceptional Lines in Non-Hermitian Systems

• Exceptional lines are one-dimensional manifolds where non-Hermitian Hamiltonians become defective due to the coalescence of eigenvalues and eigenvectors.
• They exhibit quantized Berry phases and fractional vorticity, providing robust topological invariants that classify phenomena like the non-Hermitian skin effect.
• Exceptional lines are realized in diverse systems such as nodal-line semimetals, photonic crystals, and atomic spin systems, enabling advanced quantum sensing and spectral control.

An exceptional line (EL) is a one-dimensional manifold—typically a curve or ring—in parameter or momentum space across which a non-Hermitian operator, such as a Hamiltonian, fails to be diagonalizable due to the coalescence of two or more eigenvalues and their eigenvectors. ELs generalize the notion of exceptional points (EPs), which are isolated degeneracies with defective Jordan block structure. In the mathematical context of algebraic geometry, “exceptional line” also refers to line bundles forming part of exceptional collections in the derived category of coherent sheaves. In physics, ELs constitute the structural backbone of a range of non-Hermitian topological phases, notably in open quantum systems, condensed-matter settings, and black hole quasinormal-mode spectra. They are characterized by distinct topological invariants, manifest unique spectral and eigenvector degeneracies, and play key roles in phenomena from the non-Hermitian skin effect to sensitivity enhancements in quantum devices.

1. Formal Definition and Occurrence of Exceptional Lines

For a non-Hermitian operator $H(\lambda)$ depending analytically on a set of $d$ real parameters $\lambda=(\lambda_1,\ldots,\lambda_d)$, a $k$th-order exceptional point (EP$_k$) is a point at which $k$ eigenvalues and their eigenvectors coalesce, and $H$ admits a single $k\times k$ Jordan block. In the generic case, the dimension of the locus of EP$_k$ in parameter space is $d-2(k-1)$. Thus in $d=3$ parameter space, EP$_2$ (second-order) generically form one-dimensional manifolds: exceptional lines (ELs) (Zhang et al., 2018, Cao et al., 21 Nov 2025, Moors et al., 2018).

Consider a two-level Hamiltonian: $$H(\boldsymbol\lambda) = \alpha \sigma_x + (\beta + i\gamma) \sigma_y, \qquad \boldsymbol\lambda = (\alpha, \beta, \gamma)$$ with eigenvalues $E = \pm \sqrt{\alpha^2 + (\beta + i\gamma)^2}$. The ELs are the solution set $\beta=0,\, \gamma=\pm\alpha$ in $(\alpha, \beta, \gamma)$ space, along which both eigenvalues and eigenvectors coalesce and $H$ reduces to a Jordan block (Zhang et al., 2018).

In band theory, for a generic non-Hermitian Bloch Hamiltonian $H(\mathbf{k})$, such coalescence occurs along one-dimensional manifolds in $\mathbf{k}$-space, termed exceptional lines, which can manifest as rings, chains, knots, or links, depending on symmetry and model specifics (Yang et al., 2018, Zhang et al., 2022, Hu et al., 2024).

2. Algebraic, Topological, and Geometric Structure

Algebraic Characterization

At a point on an EL, the characteristic polynomial $p(E;\lambda) = \det[H(\lambda)-E\,\mathbf{I}]$ admits a multiple root with singular Jordan structure: $$p(E_{\rm EL};\lambda_{\rm EL}) = 0, \quad \frac{\partial p}{\partial E}(E_{\rm EL};\lambda_{\rm EL}) = 0$$ for EP$_2$, and, for EP$_3$,

$$p(E_{\rm EL};\lambda_{\rm EL}) = p'(E_{\rm EL};\lambda_{\rm EL}) = p''(E_{\rm EL};\lambda_{\rm EL}) = 0$$

with discriminant conditions on the coefficients yielding the EL as a codimension-one locus (Wu et al., 2024).

Topological Properties

Encircling an EL results in half-integer vorticity of eigenvalues, signaled by a $2\pi$ phase jump in the relative argument of the coalesced eigenvalues: $$\nu = \frac{1}{2\pi} \oint d\mathbf{k} \cdot \nabla_{\mathbf{k}} \arg(E_+ - E_-)$$ yielding $\nu = \pm \tfrac{1}{2}$ per EP$_2$ in parameter or momentum space (Cao et al., 21 Nov 2025, Moors et al., 2018, Zhang et al., 2022). The associated Berry phase for loops encircling the EL acquires a value (modulo $2\pi$) diagnostic of the Jordan structure—$\pi$ for EP$_2$, $2\pi/3$ for EP$_3$ (Cao et al., 21 Nov 2025, Wu et al., 2024, Zhang et al., 2018).

Geometrically, ELs act as “vortex filaments” in parameter space, with the directionality (handedness) determined by the sign of the Berry curvature flux threading an infinitesimal loop around the EL (Zhang et al., 2018).

3. Physical Realizations and Model Systems

Condensed-Matter and Metamaterials

In three-dimensional non-Hermitian band structures, ELs arise as one-dimensional loci of defective degeneracies. For instance, in non-Hermitian nodal-line semimetals, weak anti-Hermitian perturbations generically split Hermitian nodal rings into one or more ELs. These ELs may form nontrivial links (e.g., Hopf-links), with their topology classified by linking numbers derived from Chern-Simons 3-forms in momentum space (Yang et al., 2018).

Disorder and tilting in nodal-line semimetals can generically produce pairs of ELs delimiting a “bulk Fermi ribbon”—an open orientable surface of vanishing real energy bounded by the ELs. Control parameters include the directionality of tilt and properties of the disorder potential (Moors et al., 2018).

Experimental platforms include phononic crystals engineered to realize pairs of exceptional rings (ERs) of opposite wave-function and spectral topology, producing robust drumhead surface states and geometry-dependent skin effects (Hu et al., 2024). Photonic and cold-atom lattices permit the observation of linked ELs and their associated Fermi ribbons (Yang et al., 2018, Hu et al., 2024).

Quantum Sensing and Atomic Platforms

The full realization of third-order ELs (EL$_3$) in a single spin-1 nitrogen-vacancy (NV) center demonstrates that high-order EL geometries, protected by symmetries (PT and pseudo-chirality), can be engineered at the atomic scale (Wu et al., 2024). The parameter locus $\gamma^2 - h^2 = 1$ in drive–dissipation space realizes a continuous EL$_3$, with quantized threefold state permutation and enhanced cubic-root sensitivity scaling.

Black Hole Perturbation Theory

Non-Hermitian perturbation operators in black hole quasinormal-mode (QNM) spectra exhibit ELs in external parameter space—e.g., the amplitude, location, and width of a Gaussian bump in the Regge–Wheeler potential, or the spin and mass in Kerr–de Sitter and Myers–Perry–de Sitter black holes (Cao et al., 21 Nov 2025, Nakamoto et al., 2 Jan 2026). These ELs delineate continuous sets of exceptional points marking the double-coalescence of QNM eigenfrequencies and shape early-time ringdown transients with universal linear-in-time growth near the EL.

4. Topological Invariants and Physical Signatures

Vorticity and Berry Phase

Loops in parameter space encircling an EL yield fractional vorticity ($\pm\tfrac12$ for EP$_2$, $\pm\tfrac13$ for EP$_3$) and quantized Berry phase ($\pi$ for second-order, $2\pi/3$ for third-order) (Cao et al., 21 Nov 2025, Wu et al., 2024). These invariants distinguish topological (boundary) ELs from non-topological (bulk or accidental) ELs.

Linking Number and Source-Free Principle

In momentum space, ELs can form knots and links (e.g., Hopf-links) characterized by integer linking numbers: $$\nu_{\mathrm{link}} = \frac{1}{4\pi^2} \int_{\mathrm{BZ}} \mathrm{CS}_3(A)$$ with $\mathrm{CS}_3(A)$ the Chern–Simons 3-form, encoding the global linking of ELs (Yang et al., 2018, Zhang et al., 2022). The “source-free” principle, analogous to the fermion doubling theorem, dictates that EL junctions must conserve their net discriminant number: the number entering equals the number exiting at every node (Zhang et al., 2022).

Spectral and Response Features

The scaling of the $\epsilon$-pseudospectrum around an EL, governed by the size $q$ of the largest Jordan block, follows $|\delta z| \sim \epsilon^{1/q}$: thus, for second-order ELs $|\delta z| \sim \epsilon^{1/2}$, signaling enhanced spectral sensitivity compared with non-EP points ($|\delta z| \sim \epsilon$) (Cao et al., 21 Nov 2025). In physical observables, this manifests as transient linear-in-time growth and destructive interference in response functions near the EL, ensuring boundedness in the total observable even as components diverge (Nakamoto et al., 2 Jan 2026).

5. Symmetry Protection and Morphology

ELs can be stabilized, oriented, and morphed by spatial or antiunitary symmetries. Mirror, mirror-adjoint, and $C_2\mathcal{T}$ (twofold rotation with time reversal) symmetries can confine ELs to high-symmetry planes and enforce their pairwise or chain connectivity (“exceptional chains”). The interplay between symmetries dictates possible EL linkages and allowed reconnection scenarios, with photonic crystals providing concrete platforms for the systematic realization and observation of these symmetry-protected nodal morphologies (Zhang et al., 2022).

6. Exceptional Collections of Line Bundles: Algebraic-Geometric Interpretation

In algebraic geometry, an “exceptional line bundle” on a smooth projective variety $X$ is a line bundle $L \in \mathrm{Pic}(X)$ that is exceptional as an object of the bounded derived category: $\mathrm{Ext}^k_X(L, L) = 0$ for $k\neq 0$ and $\mathrm{End}_X(L) = \mathbb{C}$ (Anderson, 2023, Coughlan, 2014). A strong exceptional collection of line bundles is a sequence $(L_1, ..., L_n)$ with $\mathrm{Ext}_X^k(L_i, L_j) = 0$ for all $k>0$ and all $i, j$, and which generates the derived category $D^b(X)$. Such collections are known explicitly for toric Fano surfaces (e.g., $\mathbb{P}^2$, blowups at points) via the cellular or Bayer–Popescu–Sturmfels diagonal resolution, and for toric Fano threefolds via the Hanlon–Hicks–Lazarev construction. These collections are crucial in the quiver description of $D^b(X)$ and in the study of homological mirror symmetry (Anderson, 2023, Coughlan, 2014).

7. Experimental and Computational Realizations

ELs can be engineered and observed in photonic and phononic crystals via tailored gain/loss profiles and geometric arrangements (Hu et al., 2024, Zhang et al., 2022), in atomic spin systems via parameter control and Hamiltonian dilation (Wu et al., 2024), and in black hole or gravitational wave spectroscopy by numerically scanning parameter-modified perturbation operators (Cao et al., 21 Nov 2025, Nakamoto et al., 2 Jan 2026). Numerical computations employ finite-difference, spectral, or pseudo-spectral discretizations, perturbation theory for pseudospectrum analysis, and direct diagonalization to capture exceptional degeneracies and their associated topological phenomena.

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References:

• (Yang et al., 2018) Non-Hermitian Hopf-Link Exceptional Line Semimetals
• (Moors et al., 2018) Disorder-driven exceptional lines and Fermi ribbons in tilted nodal-line semimetals
• (Zhang et al., 2018) Classical correspondence of the exceptional points in the finite non-Hermitian system
• (Zhang et al., 2022) Symmetry-protected topological exceptional chains in non-Hermitian crystals
• (Hu et al., 2024) Observation of exceptional line semimetal in three-dimensional non-Hermitian phononic crystals
• (Wu et al., 2024) Third-order exceptional line in a nitrogen-vacancy spin system
• (Cao et al., 21 Nov 2025) Exceptional line and pseudospectrum in black hole spectroscopy
• (Nakamoto et al., 2 Jan 2026) Exceptional Lines and Excitation of (Nearly) Double-Pole Quasinormal Modes: A Semi-Analytic Study in the Nariai Black Hole
• (Anderson, 2023) Exceptional Collections of Line Bundles for Smooth Toric Fano Surfaces and Threefolds
• (Coughlan, 2014) Enumerating exceptional collections of line bundles on some surfaces of general type