Exceptional Rings in Non-Hermitian Systems

• Exceptional Rings (ERs) are closed curves in momentum space where non-Hermitian degeneracies lead to coalescing eigenvalues and defective eigenstates.
• They emerge via mechanisms such as gain/loss imbalance, radiation leakage, and symmetry-induced codimension reduction in photonic, atomic, electronic, and mechanical systems.
• ERs display distinct topological invariants and dynamic signatures that partition spectral regimes and govern unusual transport and wave propagation behaviors.

Searching arXiv for papers on exceptional rings to ground the article in the literature. Exceptional rings (ERs) are closed loci of exceptional points in parameter or momentum space of a non-Hermitian problem, where both eigenvalues and eigenvectors coalesce and the governing operator becomes defective rather than merely degenerate. In contemporary usage, the term most commonly denotes a momentum-space ring of non-Hermitian degeneracies in photonic, atomic, electronic, and mechanical band structures, including Weyl exceptional rings and symmetry-protected exceptional rings (Xu et al., 2016, Zhen et al., 2015, Isobe et al., 2021). Distinct algebraic meanings also exist in ring theory, where “exceptional” may refer to prime rings in a հատուկ characteristic-and-dimension class, to low-rank algebras with standard involution, or to finite commutative rings studied through exceptional units [(Lee, 3 Aug 2025); (Voight, 2010); (Hu et al., 2016)]. The dominant physics literature treats ERs as codimension-reduced non-Hermitian singularities that organize complex spectra, topological invariants, and observable dynamics.

1. Definition and basic non-Hermitian structure

In non-Hermitian band theory, an exceptional point is a singularity at which two modes not only share the same complex eigenvalue but also lose linear independence. An exceptional ring is the one-dimensional extension of this phenomenon: a closed curve of exceptional points in momentum space or in a synthetic parameter space (Zhen et al., 2015, Xu et al., 2016). The defining feature is defectiveness of the effective Hamiltonian, Green’s-function pole matrix, or generalized eigenvalue problem at every point on the ring (Tao et al., 2021, Isobe et al., 2021).

A minimal mechanism appears whenever a Hermitian band touching is perturbed by unequal decay, gain, or other lifetime effects so that the spectrum acquires a square-root branch structure. In the photonic-crystal-slab model of a Dirac cone deformed by radiation loss, the effective Hamiltonian

$$H_{\rm eff}= \begin{pmatrix} \omega_0 & v_g k\ v_g k & \omega_0-i\gamma_d \end{pmatrix}$$

has eigenvalues

$$\omega_{\pm} = \omega_0-\frac{i\gamma_d}{2} \pm v_g\sqrt{k^2-k_c^2}, \qquad k_c=\frac{\gamma_d}{2v_g},$$

so the exceptional set is the circle $k=k_c$ in $(k_x,k_y)$ space (Zhen et al., 2015). Inside the ring the real parts are degenerate while the imaginary parts split; outside the ring the converse holds; on the ring both coincide and the eigenvector coalesces (Zhen et al., 2015).

The same structure reappears in Weyl-type problems. For the non-Hermitian Weyl model

$$H(\mathbf{k})=\sum_{\nu=x,y,z} k_\nu \sigma_\nu + i\gamma \sigma_z,$$

the degeneracy condition becomes

$k_z=0,\qquad k_x^2+k_y^2=\gamma^2,$

producing a Weyl exceptional ring in the $k_z=0$ plane (Xu et al., 2016). This construction is paradigmatic: a pointlike Hermitian singularity is “inflated” by non-Hermiticity into a closed exceptional loop.

A recurrent misconception is to treat an ER as an ordinary nodal ring with complex broadening. That is inaccurate. Ordinary Hermitian nodal rings are diagonalizable degeneracies; ERs are defective and therefore carry branch-point singularity structure absent in Hermitian line nodes (Xu et al., 2016, Zhen et al., 2015).

2. Formation mechanisms across physical platforms

Several distinct microscopic routes to ERs are established in the literature. A common theme is that a Hermitian or effectively Hermitian degeneracy is converted into a non-Hermitian defective locus by dissipation, gain/loss imbalance, radiation leakage, or finite-temperature lifetime asymmetry.

In photonics, radiation loss alone can suffice. In the photonic crystal slab of “Spawning rings of exceptional points out of Dirac cones,” an accidental Dirac cone at the $\Gamma$ point is deformed into an exceptional ring because dipole and quadrupole resonances radiate at dissimilar rates (Zhen et al., 2015). In bulk semiconductor polaritons, absorption converts Weyl points into Weyl exceptional rings in a magnetic field applied along the crystal $[001]$ axis (Guinness et al., 2020). There the key step is to complexify exciton frequencies,

$\omega_{0,x}\to \omega_{0,x}-i\gamma_{0,x},$

so that damping differences balance angle-dependent light–matter coupling at a critical angle

$$\omega_{\pm} = \omega_0-\frac{i\gamma_d}{2} \pm v_g\sqrt{k^2-k_c^2}, \qquad k_c=\frac{\gamma_d}{2v_g},$$0

and the parent Weyl point expands into a ring around the field axis (Guinness et al., 2020).

In ultracold atoms, spin-dependent gain/loss can generate Weyl exceptional rings directly. The dissipative cold-atom proposal of 2016 showed that a Weyl point in a three-dimensional optical-lattice setting becomes an ER under a term $$\omega_{\pm} = \omega_0-\frac{i\gamma_d}{2} \pm v_g\sqrt{k^2-k_c^2}, \qquad k_c=\frac{\gamma_d}{2v_g},$$1, with the resulting ring carrying both a Chern number and a Berry phase on a Riemann surface (Xu et al., 2016). A related Floquet route starts from a non-Hermitian nodal-line semimetal,

$$\omega_{\pm} = \omega_0-\frac{i\gamma_d}{2} \pm v_g\sqrt{k^2-k_c^2}, \qquad k_c=\frac{\gamma_d}{2v_g},$$2

and uses circularly polarized light to engineer a driven effective Hamiltonian whose ER structure can undergo a transition from a concentric pair to a dipolar pair (He et al., 2020).

In correlated and disordered electronic systems, non-Hermiticity arises from the self-energy rather than from externally imposed gain/loss. In the heavy-fermion periodic Anderson model on a zincblende lattice, finite temperature makes the self-energy complex, and inversion-symmetry breaking produces different quasiparticle lifetimes on the two sublattices. Near a Weyl point, this yields ERs in the complex poles of the Green’s function and bounded bulk Fermi surfaces (Tao et al., 2021). In disordered Weyl semimetals, finite lifetime from intra-valley multiple scattering generates a non-Hermitian self-energy whose imaginary part drives Weyl exceptional rings generically away from the unitarity limit (Matsushita et al., 2019).

In classical mechanics, homogeneous friction renders Newtonian dynamics effectively non-Hermitian. Mechanical graphene exhibits symmetry-protected exceptional rings because the dissipative first-order formulation inherits an extended chiral symmetry; the ER condition becomes

$$\omega_{\pm} = \omega_0-\frac{i\gamma_d}{2} \pm v_g\sqrt{k^2-k_c^2}, \qquad k_c=\frac{\gamma_d}{2v_g},$$3

where $$\omega_{\pm} = \omega_0-\frac{i\gamma_d}{2} \pm v_g\sqrt{k^2-k_c^2}, \qquad k_c=\frac{\gamma_d}{2v_g},$$4 is the friction coefficient and $$\omega_{\pm} = \omega_0-\frac{i\gamma_d}{2} \pm v_g\sqrt{k^2-k_c^2}, \qquad k_c=\frac{\gamma_d}{2v_g},$$5 are the undamped frequencies (Yoshida et al., 2019, Najima et al., 2021).

A broader methodological point is that ERs need not require an explicitly non-Hermitian starting matrix. In generalized eigenvalue problems

$$\omega_{\pm} = \omega_0-\frac{i\gamma_d}{2} \pm v_g\sqrt{k^2-k_c^2}, \qquad k_c=\frac{\gamma_d}{2v_g},$$6

with Hermitian $$\omega_{\pm} = \omega_0-\frac{i\gamma_d}{2} \pm v_g\sqrt{k^2-k_c^2}, \qquad k_c=\frac{\gamma_d}{2v_g},$$7 and Hermitian but indefinite $$\omega_{\pm} = \omega_0-\frac{i\gamma_d}{2} \pm v_g\sqrt{k^2-k_c^2}, \qquad k_c=\frac{\gamma_d}{2v_g},$$8, the problem maps to an effective non-Hermitian matrix $$\omega_{\pm} = \omega_0-\frac{i\gamma_d}{2} \pm v_g\sqrt{k^2-k_c^2}, \qquad k_c=\frac{\gamma_d}{2v_g},$$9, and symmetry-protected exceptional rings can emerge even though the original matrices are Hermitian (Isobe et al., 2021).

3. Symmetry, codimension reduction, and protection

The prevalence of ring-shaped rather than pointlike exceptional degeneracies is often a symmetry effect. For a generic $k=k_c$0 non-Hermitian matrix written as

$k=k_c$1

exceptional points satisfy two real conditions,

$k=k_c$2

In two dimensions, this generically gives isolated EPs, not rings (Najima et al., 2021, Yoshida et al., 2018). A ring appears when symmetry makes one condition automatic, leaving a single constraint in a two-dimensional Brillouin zone.

This codimension reduction underlies symmetry-protected exceptional rings (SPERs). In correlated two-dimensional systems with chiral symmetry, the effective non-Hermitian Hamiltonian obeys an extended chiral relation at zero frequency, so one of the exceptional-point conditions is identically satisfied. The remaining condition defines a closed one-dimensional manifold, i.e. a ring (Yoshida et al., 2018). The same logic holds in frictional mechanical systems, where the emergent symmetry

$k=k_c$3

protects SPERs without requiring finely balanced gain and loss (Yoshida et al., 2019). In the generalized-eigenvalue framework, pseudo-Hermiticity

$k=k_c$4

plays the analogous stabilizing role (Isobe et al., 2021).

Rotational symmetry can further force a closed circular locus. In the bulk-semiconductor polariton problem, the system is rotationally symmetric about the magnetic field axis, so any exceptional point occurring at nonzero polar angle extends over all azimuthal angles and therefore becomes a ring (Guinness et al., 2020). This suggests a general principle: once the local exceptional condition is angularly independent around a symmetry axis, the non-Hermitian degeneracy becomes a geometrically literal ring.

Protection may also be formulated through topological changes in a zeroth Chern number. For two-dimensional chiral systems, exceptional degeneracies occur on boundaries where the zero-th Chern number changes; in momentum space such boundaries are rings (Yoshida et al., 2018). Mechanical SPERs admit an especially simple version,

$k=k_c$5

so the SPER is the boundary between regions with different $k=k_c$6 (Yoshida et al., 2019).

4. Topological characterization

ERs are topological objects, but the appropriate invariants depend on dimensionality, symmetry class, and whether the exceptional structure descends from Dirac, Weyl, or higher-order semimetal physics.

For Weyl exceptional rings, a central result is the coexistence of two invariants. In the dissipative cold-atom model, the ring carries a quantized Chern number

$k=k_c$7

when the enclosing surface $k=k_c$8 wraps the whole ring, and a quantized Berry phase

$k=k_c$9

with the loop traversed twice because the state returns to itself only on the associated Riemann surface (Xu et al., 2016). This dual characterization distinguishes a Weyl exceptional ring from an ordinary Weyl nodal ring, which is associated primarily with Berry phase, and from a Weyl point, which is associated primarily with Chern charge.

Higher-order non-Hermitian semimetals enrich this picture. In the higher-order Weyl-exceptional-ring semimetal, each ring is characterized by both a spectral winding number,

$(k_x,k_y)$0

and a Chern number defined biorthogonally on a surface enclosing the ring (Liu et al., 2021). For small dissipation the four rings carry $(k_x,k_y)$1 and $(k_x,k_y)$2 (Liu et al., 2021).

Disorder-induced Weyl exceptional rings can also be described through vorticity of the complex-energy phase, while the flat band inside the ring inherits non-Hermitian topological meaning through that vorticity (Matsushita et al., 2019). By contrast, in certain merged Dirac-semimetal hybrid rings the vorticity cancels even though a non-diagonalizable ring-like structure remains (Matsushita et al., 2019).

Recent work has introduced more composite and higher-order ER structures. A composite exceptional ring may consist of a third-order exceptional ring together with multiple Weyl exceptional rings, with the total structure classified by band Chern numbers that control braiding periodicity and dynamical mode transfer (Lei et al., 2024). Nonlinear planar optical microcavities further show that a linear ER can split into two concentric nonlinear ERs, with the outer one forming a ring of third-order exceptional points; both rings carry $(k_x,k_y)$3 depending on the eigenvalue surface (Wingenbach et al., 9 Jul 2025).

A plausible implication is that ER topology is no longer well described by a single invariant once multiband, nonlinear, or higher-order coalescences are present. The literature increasingly treats ERs as loci supporting several simultaneously relevant structures: point-gap winding, Berry phase, Chern number, and branch-sheet connectivity (Xu et al., 2016, Liu et al., 2021, Lei et al., 2024, Wingenbach et al., 9 Jul 2025).

5. Spectral consequences, boundary phenomena, and dynamics

ERs divide momentum space into regions with qualitatively different spectral behavior. This partition is one of their most direct consequences.

In the photonic crystal slab, the exceptional ring separates a region with degenerate real parts and split imaginary parts from a region with split real parts and degenerate imaginary parts (Zhen et al., 2015). In the dissipative Weyl model, the ER is the boundary between purely real and purely imaginary spectral sectors in the $(k_x,k_y)$4 plane (Xu et al., 2016). In mechanical graphene, the SPER separates underdamped, critically damped, and overdamped dynamical regimes: outside the ring the system oscillates, on the ring it is critically damped, and inside the ring it is overdamped (Najima et al., 2021).

ERs also generate unconventional zero-energy manifolds. In the heavy-fermion semimetal, a single Weyl exceptional ring bounds a bulk Fermi disk, and at higher temperature merged rings can bound a bulk Fermi tube (Tao et al., 2021). In disordered Weyl semimetals, the real part of the spectrum becomes flat inside the ring, producing a non-Hermitian flat band (Matsushita et al., 2019). In the photonic-crystal example, the complex eigenvalues define a two-dimensional flat band enclosed by an exceptional ring even though reflection peaks still trace the Hermitian part of the system (Zhen et al., 2015).

Boundary states remain important but are modified by non-Hermiticity. Higher-order Weyl-exceptional-ring semimetals support both surface Fermi arcs and hinge Fermi arcs, each bounded by the projection of the ERs (Liu et al., 2021). Dissipation can move, merge, and annihilate rings with opposite topological charges, driving transitions between phases with both surface and hinge arcs and phases with hinge arcs alone (Liu et al., 2021). In non-reciprocal Haldane models, ERs on the torus coexist with non-chiral edge states on a cylinder that terminate at exceptional points and delocalize into the bulk (Prabhudesai et al., 13 Feb 2026).

Dynamically, ERs can govern wavepacket and mode evolution. In frictional mechanical graphene, Fourier-transformed displacements retain sign much longer inside the SPER than outside, providing a direct dynamical signature in $(k_x,k_y)$5-space (Najima et al., 2021). In photonic honeycomb lattices, Dirac-point conical diffraction is replaced by non-Hermitian broadening near an ER, and a transverse non-Hermitian drift appears in reciprocal space due to eigenstate non-orthogonality and the biorthogonal quantum metric (Zhang et al., 2024). This drift requires the biorthogonal metric

$(k_x,k_y)$6

and is not captured by the standard right-right metric alone (Zhang et al., 2024).

These results collectively indicate that ERs are not merely spectral defects. They organize transport, damping, localization, and beam dynamics across a wide range of non-Hermitian media (Najima et al., 2021, Zhang et al., 2024, Tao et al., 2021).

6. Detection and experimental signatures

ERs are typically inferred from angle-, momentum-, or time-resolved probes that resolve the interchange between frequency splitting and linewidth splitting, or that identify bounded zero-energy regions and defective-state dynamics.

In bulk semiconductor polaritons, proposed signatures include angle-resolved reflectivity and photoluminescence, with the central hallmark being the critical-angle interchange: before the ring only linewidths split, while after it resonance frequencies split (Guinness et al., 2020). The authors note that the ring may be difficult to distinguish from a Weyl point when the damping contrast is too small (Guinness et al., 2020).

In the photonic crystal slab, angle-resolved reflection spectroscopy along $(k_x,k_y)$7 and $(k_x,k_y)$8 reveals the underlying Dirac-cone dispersion, while the complex poles inferred from the non-Hermitian model exhibit the exceptional-ring structure (Zhen et al., 2015). A subtle but important point is that reflectivity peaks do not directly track the complex eigenvalues of the open system; temporal coupled-mode theory is required to interpret the mismatch (Zhen et al., 2015).

In heavy-fermion systems and disordered semimetals, ARPES is the natural probe. The heavy-fermion paper predicts that bulk Fermi disks appear as bright finite-area zero-energy regions and bulk Fermi tubes as tubular zero-energy structures bounded by ERs (Tao et al., 2021). The disorder study similarly identifies ARPES and quasiparticle interference as probes of ring-like spectral broadening and non-Hermitian flat bands (Matsushita et al., 2019).

Cold-atom proposals rely on quench dynamics. In the dissipative Weyl-gas scheme, one loads spin-up atoms, suddenly turns on spin-orbit coupling and dissipation, and monitors spin populations. Outside the ring the normalized spin-down number oscillates, whereas inside the ring the oscillation disappears because $(k_x,k_y)$9 (Xu et al., 2016). Floquet-engineered ERs can also be characterized through long-time averaged spin textures and a dynamical winding number in a quantum-spin simulation with an ancilla-based Hermitian dilation (He et al., 2020).

Mechanical systems admit especially direct real-time observation. The SPER can be reconstructed from the continuous Fourier transform of displacement fields in a finite sample before reflected waves return from the boundaries (Najima et al., 2021). This suggests that classical metamaterials offer an experimentally clean route to observing ER-governed dynamical phase separation.

A plausible implication is that ER detection is often indirect: one observes not the defectiveness itself but a switch in spectral regime, bounded zero-energy manifolds, anomalous transport, or encircling dynamics. The defective character is then inferred from the consistency of these signatures with a branch-point singularity model.

7. Alternative mathematical meanings of “exceptional ring”

Outside non-Hermitian physics, “exceptional ring” has established meanings in algebra, but these are unrelated to exceptional-point rings in momentum space.

In prime-ring theory, an exceptional prime ring is defined by

$$H(\mathbf{k})=\sum_{\nu=x,y,z} k_\nu \sigma_\nu + i\gamma \sigma_z,$$0

where $$H(\mathbf{k})=\sum_{\nu=x,y,z} k_\nu \sigma_\nu + i\gamma \sigma_z,$$1 is the extended centroid of the prime ring $$H(\mathbf{k})=\sum_{\nu=x,y,z} k_\nu \sigma_\nu + i\gamma \sigma_z,$$2 (Lee, 3 Aug 2025). This exceptional case is precisely where classical Lie-ideal theorems of Herstein and their prime-ring extensions require special treatment. The resulting structure is governed by the small commutator algebra $$H(\mathbf{k})=\sum_{\nu=x,y,z} k_\nu \sigma_\nu + i\gamma \sigma_z,$$3, the equivalence

$$H(\mathbf{k})=\sum_{\nu=x,y,z} k_\nu \sigma_\nu + i\gamma \sigma_z,$$4

and a dichotomy between Type I and Type II nonabelian Lie ideals (Lee, 3 Aug 2025).

In the theory of low-rank algebras with standard involution, an exceptional ring is an $$H(\mathbf{k})=\sum_{\nu=x,y,z} k_\nu \sigma_\nu + i\gamma \sigma_z,$$5-algebra $$H(\mathbf{k})=\sum_{\nu=x,y,z} k_\nu \sigma_\nu + i\gamma \sigma_z,$$6 of rank $$H(\mathbf{k})=\sum_{\nu=x,y,z} k_\nu \sigma_\nu + i\gamma \sigma_z,$$7 with a decomposition $$H(\mathbf{k})=\sum_{\nu=x,y,z} k_\nu \sigma_\nu + i\gamma \sigma_z,$$8 such that multiplication on $$H(\mathbf{k})=\sum_{\nu=x,y,z} k_\nu \sigma_\nu + i\gamma \sigma_z,$$9 factors through a linear map $k_z=0,\qquad k_x^2+k_y^2=\gamma^2,$0, equivalently $k_z=0,\qquad k_x^2+k_y^2=\gamma^2,$1 for $k_z=0,\qquad k_x^2+k_y^2=\gamma^2,$2 (Voight, 2010). In rank $k_z=0,\qquad k_x^2+k_y^2=\gamma^2,$3, exceptional rings are exactly the algebras with a standard involution (Voight, 2010).

In finite commutative ring theory, the phrase does not denote a named class of “exceptional rings” in the paper itself, but rings are analyzed through their exceptional units

$k_z=0,\qquad k_x^2+k_y^2=\gamma^2,$4

with exact criteria for when a finite commutative ring is generated by its exceptional units (Hu et al., 2016). Here the terminology is about units, not non-Hermitian spectral singularities.

These distinct usages share only the adjective “exceptional.” They belong to separate mathematical traditions and should not be conflated with the ERs of non-Hermitian band theory.

8. Current directions and open structure

Recent literature indicates an expansion of ER research beyond the original two-band, second-order setting. Nonlinearity can split a single ER into multiple concentric ERs, including rings of third-order exceptional points with tunable perturbation response quantified by the signal-enhancement factor

$k_z=0,\qquad k_x^2+k_y^2=\gamma^2,$5

for which a maximum $k_z=0,\qquad k_x^2+k_y^2=\gamma^2,$6 is reported in a nonlinear planar microcavity (Wingenbach et al., 9 Jul 2025). Composite exceptional rings combine third-order and Weyl exceptional rings into a multiband object whose quasistatic and dynamical encircling behavior is predicted by Chern numbers (Lei et al., 2024).

Other recent work emphasizes geometric and dynamical aspects. The biorthogonal quantum metric of ERs controls reciprocal-space beam drift in photonic honeycomb lattices (Zhang et al., 2024). In non-reciprocal Haldane models, ERs act as Berry-curvature flux tubes and coexist with nested cascades of exceptional-point pairs and dynamically stabilized but ultimately ephemeral edge states (Prabhudesai et al., 13 Feb 2026). Dissipative nodal-ring models can split a Hermitian nodal ring into two ERs that behave as vortex filaments of opposite circulation in a spectral phase field, with an experimentally simpler realization in the parameter space of a one-dimensional spinful Rice–Mele chain (Ma et al., 10 Feb 2026).

This suggests that the study of ERs is moving from isolated classification toward a broader program encompassing multiband exceptional composites, nonlinear singularity theory, quantum geometry, and experimentally accessible synthetic dimensions. What remains stable across these developments is the core definition: an ER is a closed non-Hermitian defective locus whose topology and dynamics cannot be reduced to those of an ordinary Hermitian band crossing (Xu et al., 2016, Zhen et al., 2015, Liu et al., 2021).