Critical Exceptional Points in Open Systems

• Critical exceptional points (CEPs) are nonequilibrium critical points where multiple excitation modes coalesce, forming a defective Jordan block at zero gap.
• Nonequilibrium O(N) field theories use Langevin dynamics and linear stability analysis to reveal unique q⁻⁴ fluctuations and fluctuation-induced first-order transitions below four dimensions.
• In dissipative collective-spin systems, CEPs reorganize quantum fluctuations into anisotropic squeezing, signaling the transition from static order to rotating states.

Critical exceptional points (CEPs) are a specific class of nonequilibrium critical points in open systems for which criticality and non-Hermitian defectiveness coincide. In the strongest explicit formulation, a CEP is a nonequilibrium critical point of an open many-body system at which multiple collective excitation modes coalesce, so that the critical sector of the linearized stability matrix develops a Jordan block at zero eigenvalue (Nakanishi, 27 May 2026). In nonequilibrium $O(N)$ field theories, the same notion appears as a transition where a mode becomes gapless at the same time that two dynamical modes coalesce, with vanishing friction but finite noise, so that the renormalized retarded response satisfies $\bar\gamma(0)=\bar r(0)=0$ and $\Gamma^R(0,\omega)=-\omega^2$ (Zelle et al., 2023). The term is, however, not uniform across the literature: some neighboring works study exceptional points with clear critical behavior but do not use the CEP label, while other fields use “CEP” for “chiral exceptional point” or “critical end point” instead (Hashemi et al., 2021, Lacey, 2015).

1. Terminology and scope

The phrase “critical exceptional point” is explicit in the nonequilibrium many-body literature, but the acronym “CEP” is used differently in adjacent domains. The most stable usage for the present topic is the open-system many-body definition just described, especially in nonequilibrium $O(N)$ models and dissipative collective-spin systems (Zelle et al., 2023, Nakanishi, 27 May 2026).

Usage of CEP	Meaning	Representative papers
Critical exceptional point	Nonequilibrium critical point with mode coalescence	(Zelle et al., 2023, Nakanishi, 27 May 2026)
Chiral exceptional point	Directional EP in traveling-wave or chiral photonic settings	(Hashemi et al., 2021, Dong et al., 17 Jun 2026)
Critical end point	Thermodynamic endpoint in QCD or nuclear matter phase diagrams	(Lacey, 2015, Ferreira et al., 2017, Wang, 2019)

Several papers central to the broader EP-criticality discussion are relevant without adopting the term itself. Examples include the generalization of Kibble-Zurek scaling to ramps through exceptional points (Dóra et al., 2018), the approach to higher-order EPs along exceptional manifolds (Wiersig, 2023), the geometric treatment of singular Berry potentials at EPs and quantum phase transitions (Ju et al., 2024), radiation-loss-induced EPs in passive dielectric photonics (Abdrabou et al., 2019), complex-momentum EPs tied to topological Majorana modes (Mandal, 2015), and fluctuation-induced Green-function EPs near density-wave formation (Fang et al., 26 Nov 2025). This suggests that CEPs are best understood as one sharply defined subset within a larger landscape of critical non-Hermitian singularities.

2. Nonequilibrium $O(N)$ field theories

The most systematic field-theoretic treatment of CEPs is the nonequilibrium $O(N)$ program of (Zelle et al., 2023). The starting point is a Langevin equation for an $O(N)$ order parameter $\boldsymbol\phi$ with inertia, damping, diffusion, and nonconservative nonlinear damping,

$$\Big(\partial_t^2+(2\gamma-Z\nabla^2+u\rho(x,t))\partial_t+r+\lambda\rho(x,t)-v^2\nabla^2 +u'\partial_t\rho(x,t)\Big)\boldsymbol{\phi}(x,t)+\xi(x,t)=0,$$

with $\rho(x,t)=\boldsymbol{\phi}^T\boldsymbol{\phi}$ and Gaussian white noise

$\bar\gamma(0)=\bar r(0)=0$0

The couplings $\bar\gamma(0)=\bar r(0)=0$1 and $\bar\gamma(0)=\bar r(0)=0$2 are nonconservative nonequilibrium damping terms, and their role is decisive: they allow vanishing effective friction in the presence of finite noise, a situation excluded by equilibrium fluctuation-dissipation balance (Zelle et al., 2023).

In the ordered phase, the theory supports both a static ordered state and a rotating or limit-cycle state. The static state breaks $\bar\gamma(0)=\bar r(0)=0$3. The rotating state has

$\bar\gamma(0)=\bar r(0)=0$4

and breaks $\bar\gamma(0)=\bar r(0)=0$5, which yields an enhanced number of $\bar\gamma(0)=\bar r(0)=0$6 Goldstone modes (Zelle et al., 2023). The transition between these two ordered phases is the CEP. In the damped-oscillator language used by the paper, an ordinary EP satisfies $\bar\gamma(0)=\bar r(0)=0$7, whereas a CEP is the zero-gap version

$\bar\gamma(0)=\bar r(0)=0$8

For $\bar\gamma(0)=\bar r(0)=0$9, this becomes $\Gamma^R(0,\omega)=-\omega^2$0, so the exceptional degeneracy is itself critical (Zelle et al., 2023).

The Gaussian infrared structure differs sharply from ordinary equilibrium criticality. Near the CEP, the retarded poles behave as

$\Gamma^R(0,\omega)=-\omega^2$1

while the equal-time Keldysh correlator scales as

$\Gamma^R(0,\omega)=-\omega^2$2

The paper identifies $\Gamma^R(0,\omega)=-\omega^2$3 as the upper critical dimension and $\Gamma^R(0,\omega)=-\omega^2$4 as the correlation-length exponent of the Gaussian CEP (Zelle et al., 2023). The $\Gamma^R(0,\omega)=-\omega^2$5 structure is stronger than the usual $\Gamma^R(0,\omega)=-\omega^2$6 critical singularity and reflects the coexistence of finite noise with vanishing friction.

That enhanced fluctuation sector destabilizes naive continuous CEP behavior below four dimensions. The paper’s central conclusion is that for $\Gamma^R(0,\omega)=-\omega^2$7 the divergent fluctuations universally either destroy a preexisting static order or drive a fluctuation-induced first-order transition (Zelle et al., 2023). In other words, the mean-field CEP is not generically the final infrared fixed point in low dimensions. The resulting phase structure is organized by the competition between a symmetry-restoration scale and a fluctuation-induced first-order scale.

3. Collective-spin CEPs and quantum fluctuations

A distinct but closely aligned realization appears in dissipative collective-spin systems (Nakanishi, 27 May 2026). There a CEP is again defined as a nonequilibrium critical point of an open many-body system at which multiple collective excitation modes coalesce. The dynamics is generated by a GKSL master equation for collective spin operators $\Gamma^R(0,\omega)=-\omega^2$8, with intensive variables $\Gamma^R(0,\omega)=-\omega^2$9. Linearizing the mean-field dynamics gives a stability matrix $O(N)$0. At the CEP, the critical sector of $O(N)$1 becomes defective: the paper writes

$O(N)$2

so the zero-eigenvalue sector contains a nontrivial Jordan block $O(N)$3, and the coalescing direction is $O(N)$4 (Nakanishi, 27 May 2026).

The order parameter is the steady-state polarization $O(N)$5. In the $O(N)$6-broken phase one has $O(N)$7, and approaching the CEP means $O(N)$8. The paper’s main result is that this defective criticality reorganizes steady-state quantum fluctuations into a singularly anisotropic covariance ellipse. For the Kitagawa-Ueda squeezing parameter,

$O(N)$9

the universal scaling near the CEP is

$O(N)$0

Thus the optimally squeezed variance vanishes linearly in $O(N)$1, while the anti-squeezed variance diverges inversely (Nakanishi, 27 May 2026).

The same work identifies a geometric fingerprint of defectiveness: the anti-squeezed axis aligns asymptotically with the coalescing eigenvector of the stability matrix. In the rotated transverse frame, the principal directions satisfy

$O(N)$2

so the large fluctuation axis locks to the CEP mode itself (Nakanishi, 27 May 2026). The scaling remains robust against dephasing channels generated by spin components orthogonal to the coalesced critical collective mode; by contrast, dephasing directly along that mode changes the asymptotics and cuts off the squeezing enhancement.

A representative exactly solvable model in the paper uses

$O(N)$3

with critical point $O(N)$4. On the stable $O(N)$5-broken branch, the transverse covariance is

$O(N)$6

which realizes the universal CEP scaling in closed form (Nakanishi, 27 May 2026).

4. Critical behavior adjacent to CEPs

Several works sharpen the critical structure associated with exceptional points even when they do not formulate CEPs explicitly. The most direct dynamical analogue is the Kibble-Zurek analysis of ramps through EPs (Dóra et al., 2018). There EPs are treated as the non-Hermitian counterpart of conventional critical points, and the defect density obeys

$O(N)$7

rather than the Hermitian Kibble-Zurek law $O(N)$8. The modification originates in nonorthogonality: near an EP the defect component can decay back toward the surviving state, so defect production is suppressed relative to Hermitian critical dynamics (Dóra et al., 2018). While this paper does not define CEPs, it supplies a universal slow-drive scaling framework for EP-critical dynamics.

A complementary static viewpoint appears in the analysis of exceptional surfaces approaching higher-order EPs (Wiersig, 2023). That work studies motion along an exceptional manifold of fixed-order EPs toward a point where the EP order increases. The spectral response strength $O(N)$9 of the lower-order EP obeys

$O(N)$0

so $O(N)$1 can diverge even though the eigenvalues themselves remain finite (Wiersig, 2023). This is a precise mathematical realization of CEP-like criticality: a singular response coefficient develops as an order-changing exceptional merger is approached.

A geometric reformulation is given by the parameter-space transport theory of (Ju et al., 2024). There the evolution generator $O(N)$2 in the parameter direction produces Berry potentials that are singular at EPs and at critical points associated with quantum phase transitions. The paper’s central distinction is between the singular adiabatic or eigenstate bundle and the full Hilbert-space bundle, for which transport can remain smooth. This suggests a useful CEP reading: singular reduced geometric data need not imply singular full-state evolution (Ju et al., 2024).

Fluctuation-induced electronic EPs near density-wave formation provide another nearby case (Fang et al., 26 Nov 2025). In that setting the retarded Green’s function

$O(N)$3

develops a higher-order pole when

$O(N)$4

Because $O(N)$5 and $O(N)$6, these EPs are induced by critical density-wave fluctuations, although the paper explicitly presents them as ordinary second-order EPs rather than CEPs (Fang et al., 26 Nov 2025).

5. Neighboring singularities that are not CEPs proper

The CEP label is often invoked too broadly in discussions of non-Hermitian criticality. Several important examples in the dataset illustrate why sharper distinctions matter. In passive photonics, resonant states of finite clusters of dielectric cylinders exhibit second- and third-order EPs produced entirely by radiation leakage, with square-root and cubic-root Puiseux behavior and explicit topological sheet permutations, but the study does not introduce a separate CEP concept (Abdrabou et al., 2019). These are parameter-tuned exceptional degeneracies in a radiation-loss-induced nonlinear eigenvalue problem, not nonequilibrium many-body CEPs.

In topological superconductors with chiral symmetry, complexified transverse momenta can host EPs associated with Majorana zero modes. At a topological phase transition, the imaginary part of the relevant complex momentum vanishes, the bulk gap closes at real momentum, and the EP disappears because the Hamiltonian becomes diagonalizable again (Mandal, 2015). The paper does not use the term CEP, but it does present a clear EP-to-critical-boundary correspondence in complex momentum space.

Boundary-sensitive non-Bloch band theory supplies another counterexample. In a reciprocal non-Hermitian Lieb lattice, geometry can induce an open-boundary EP located at a saddle point of the generalized Brillouin-zone manifold, with square-root gap scaling $O(N)$7 (Zhao et al., 5 Jan 2026). The paper is highly relevant to critical singularity discussions, but it explicitly distinguishes the resulting EP from the branch points of non-Bloch Fermi arcs, which appear instead as Whitney cusps. Here the singularity is geometry-tuned and boundary-induced, not a CEP in the open-many-body sense.

A different boundary case is the Hermitian critical-point sensor of (Tang et al., 24 Jan 2026). There a chiral Hermitian cavity exhibits a square-root response in the spectral-extremum splitting,

$O(N)$8

despite having orthogonal eigenvectors, no defectiveness, no Jordan block, and $O(N)$9. The paper therefore describes an EP-like critical response without a true exceptional point (Tang et al., 24 Jan 2026). This is especially useful for excluding a common misconception: sublinear square-root sensitivity alone does not define a CEP.

6. Acronym collisions: chiral exceptional points and critical end points

Outside the nonequilibrium criticality literature, “CEP” often denotes something else entirely. In traveling-wave and discrete photonics, CEP commonly means chiral exceptional point. In that usage, a CEP is an EP at which clockwise and counterclockwise modes coalesce into a single traveling mode with a preferred direction (Hashemi et al., 2021). The reduced Hamiltonian has asymmetric effective backscattering, and the necessary ingredient is dissipative asymmetry rather than critical softening. The same acronym is used in whispering-gallery microcavities with two-photon light-matter coupling, where chiral exceptional points produce strongly direction-dependent photon blockade and nonreciprocal photon statistics (Dong et al., 17 Jun 2026). These are important EP structures, but “critical” there means chirality-driven mode selectivity only insofar as the acronym is concerned.

In nuclear and QCD matter, CEP instead means critical end point. Finite-size scaling analyses of heavy-ion data have been interpreted as indicating a second-order critical end point in the $O(N)$0 plane, with extracted location $O(N)$1, $O(N)$2, and static exponents near the $O(N)$3D Ising values (Lacey, 2015). Other QCD studies discuss multiple CEPs in magnetized three-flavor quark matter or layer- and flavor-dependent CEPs generated by the chiral magnetic effect (Ferreira et al., 2017, Wang, 2019). None of these works concerns non-Hermitian defectiveness.

The terminological consequence is straightforward. In the non-Hermitian many-body sense, a critical exceptional point is a critical open-system singularity where the critical dynamical sector is defective and mode coalescence occurs at zero gap (Zelle et al., 2023, Nakanishi, 27 May 2026). In chiral photonics and in QCD thermodynamics, the same acronym denotes different objects. Any technical use of “CEP” therefore requires immediate domain specification.