---
title: 'Exceptional Points (EPs): Definition, Properties, and Significance'
url: https://www.emergentmind.com/topics/exceptional-points
type: topic
---

# Exceptional Points (EPs): Definition, Properties, and Significance

Exceptional points (EPs) are singularities of parameter-dependent non-Hermitian spectral problems at which eigenvalues and their associated eigenvectors coalesce, rendering the operator defective and non-diagonalizable. For an EP of order two, the repeated eigenvalue has algebraic multiplicity two but geometric multiplicity one; equivalently, the operator contains a $2\times2$ Jordan block. This distinguishes an EP from an ordinary degeneracy, or diabolic point, at which eigenvalues coincide while linearly independent eigenvectors remain available. EPs generate square-root branch points, nontrivial eigenvector monodromy, self-orthogonality in biorthogonal formulations, anomalous time evolution, and strong responses to parameter variation. They occur in optical, acoustic, elastic, atomic, molecular, nuclear, condensed-matter, hydrodynamic, many-body, and open quantum systems.

## 1. Mathematical structure and spectral signatures

Let $H(\lambda)$ be a parameter-dependent operator. An order-two EP at $\lambda=\lambda_{\mathrm{EP}}$ satisfies
\[
H(\lambda_{\mathrm{EP}})|\phi\rangle
=
E_{\mathrm{EP}}|\phi\rangle,
\]
with algebraic multiplicity two, geometric multiplicity one, and
\[
\det[H(\lambda_{\mathrm{EP}})-E_{\mathrm{EP}}I]=0,
\]
\[
\left.
\frac{\partial}{\partial E}
\det[H(\lambda_{\mathrm{EP}})-EI]
\right|_{E=E_{\mathrm{EP}}}=0.
\]
The one-dimensional eigenspace is accompanied by a generalized eigenvector satisfying
\[
[H(\lambda_{\mathrm{EP}})-E_{\mathrm{EP}}I]
|\psi_{\mathrm{assoc}}\rangle
=
|\phi\rangle.
\]
In an appropriate basis, the operator has Jordan form
\[
H(\lambda_{\mathrm{EP}})
=
S
\begin{pmatrix}
E_{\mathrm{EP}}&1\\
0&E_{\mathrm{EP}}
\end{pmatrix}
S^{-1}.
\]

For right and left eigenvectors,
\[
H|\psi_k\rangle=E_k|\psi_k\rangle,
\qquad
\langle\widetilde{\psi}_k|H
=
E_k\langle\widetilde{\psi}_k|,
\]
the coalesced state is self-orthogonal:
\[
\langle\widetilde{\phi}|\phi\rangle=0.
\]
This vanishing biorthogonal norm is distinct from ordinary Hilbert-space normalization and is a central algebraic signature of an EP. The associated spectral decomposition becomes singular.

For a two-level Hamiltonian, the eigenvalues can be written as
\[
E_{\pm}(\lambda)
=
\frac{\operatorname{tr}H(\lambda)}{2}
\pm
\frac12
\sqrt{
\bigl(H_{11}(\lambda)-H_{22}(\lambda)\bigr)^2
+
4H_{12}(\lambda)H_{21}(\lambda)
}.
\]
The EP condition is the vanishing of the discriminant,
\[
\bigl(H_{11}-H_{22}\bigr)^2+4H_{12}H_{21}=0.
\]
Near an isolated EP,
\[
E_{\pm}(\lambda)
\simeq
E_{\mathrm{EP}}
\pm
C\sqrt{\lambda-\lambda_{\mathrm{EP}}},
\]
where $C$ is nonzero. Thus the eigenvalues lie on two Riemann sheets joined at a square-root branch point, and
\[
\frac{dE_{\pm}}{d\lambda}
\sim
\frac{1}{\sqrt{\lambda-\lambda_{\mathrm{EP}}}}.
\]
The corresponding eigenvectors exhibit analogous singular behavior. In the normalization discussed in “The physics of exceptional points,” normalized eigenvectors diverge approximately as
\[
|\psi_k\rangle
\sim
(\lambda-\lambda_{\mathrm{EP}})^{-1/4}.
\]
These singularities explain why small parameter changes can produce large spectral and observable effects [1210.7536].

An ordinary Hermitian degeneracy has
\[
E_1=E_2
\]
but two linearly independent eigenvectors. The operator remains diagonalizable. At an EP, eigenvalue coincidence is accompanied by eigenvector coalescence and a loss of geometric multiplicity. A repeated root of a characteristic equation therefore identifies a candidate degeneracy, but defectiveness must also be established.

The resolvent exhibits a higher-order pole at an EP:
\[
(E-H)^{-1}
\sim
\frac{P}{E-E_{\mathrm{EP}}}
+
\frac{Q}{(E-E_{\mathrm{EP}})^2}.
\]
The second-order pole modifies scattering amplitudes, resonance line shapes, interference, and decay laws. Jordan-block time evolution contains a polynomial factor,
\[
|\Psi(t)\rangle
\sim
e^{-iE_{\mathrm{EP}}t/\hbar}
\left(
|\phi\rangle
-
\frac{it}{\hbar}
|\psi_{\mathrm{assoc}}\rangle
\right).
\]
The linear factor in $t$ is the dynamical counterpart of the second-order resolvent pole.

## 2. Encircling, topology, and higher-order exceptional points

The two eigenvalues of an order-two EP occupy two Riemann sheets. One counterclockwise circuit around the EP exchanges them:
\[
E_1\longrightarrow E_2,
\qquad
E_2\longrightarrow E_1.
\]
Eigenvectors have a richer monodromy. With the normalization used in the cited analysis,
\[
|\psi_1\rangle
\longrightarrow
-|\psi_2\rangle
\longrightarrow
-|\psi_1\rangle
\longrightarrow
|\psi_2\rangle
\longrightarrow
|\psi_1\rangle.
\]
Four circuits are required to restore the original normalized eigenvector. Clockwise and counterclockwise circuits can produce different signs, yielding orientation-dependent or chiral mode switching. These effects underlie state exchange, geometric phases, and parameter-sensitive mode conversion.

Higher-order EPs involve larger Jordan blocks. An EP of order $n$ generically requires $n-1$ complex eigenvalue-coalescence conditions, or
\[
2n-2
\]
real constraints. For an $n\times n$ matrix, these conditions can be expressed using the determinant and traces of powers of the matrix. After shifting away the trace, an EP of order $n$ requires the remaining characteristic-polynomial coefficients to vanish.

For example, a generic EP3 requires four real constraints, whereas a generic EP4 requires six. Local unitary or antiunitary symmetries can reduce these requirements by constraining characteristic-polynomial coefficients to be real, imaginary, or identically zero. Pseudo-Hermiticity, $\mathcal{PT}$ symmetry, pseudo-anti-Hermiticity, and related symmetries can therefore make higher-order EPs accessible in lower-dimensional parameter spaces [2202.07009].

In three-band $\mathcal{PT}$-symmetric systems, the characteristic polynomial can be reduced to a depressed cubic with real coefficients $p$ and $q$. The EP3 conditions become
\[
p=0,\qquad q=0,
\]
which are two real conditions rather than four. The generic local dispersion is
\[
E-E_{\mathrm{EP}}\sim(k-k_0)^{1/3}.
\]
The discriminant-like quantity
\[
p^3+q^2
\]
organizes surrounding EP2 lines and regions in which two eigenvalues form a complex-conjugate pair.

Generalized chiral or sublattice symmetry produces a different EP3 structure. In an odd-dimensional system, the spectral relation
\[
\{E_i\}=\{-E_i\}
\]
forces one eigenvalue to remain at zero. A three-band model can then exhibit
\[
E=0,\qquad E_{\pm}=\pm\sqrt{bd+fh},
\]
with an EP3 at $bd+fh=0$. The resulting dispersion is
\[
E\sim k^{1/2},
\]
rather than the generic cube-root law. The flat band participates in eigenvector coalescence even though it remains pinned at zero. Non-Hermitian Lieb lattices realize this structure together with real-energy Fermi arcs, imaginary-energy arcs, and crossed arcs.

Higher-order EPs are often embedded in lower-order exceptional manifolds. Similarity-induced exceptional structures in four-, six-, eight-, and nine-band systems can contain EP2 surfaces, EP3 arcs, EP4 arcs, and maximal EPs connected by these lower-order sets. Consequently, counting the equations defining an EP$n$ is insufficient: the same algebraic conditions may be satisfied by several lower-order Jordan blocks or by symmetry-related degeneracies at distinct eigenvalues. This hierarchy is developed in “The analytically tractable zoo of similarity-induced exceptional structures” [2508.02565].

## 3. Symmetry breaking, quasi-Hermiticity, and non-Hermitian continuation

A central realization of EP physics occurs in $\mathcal{PT}$-symmetric systems. A non-Hermitian Hamiltonian satisfying
\[
[H,\mathcal{PT}]=0
\]
can possess a real spectrum. In the unbroken phase, eigenstates are also $\mathcal{PT}$ symmetric:
\[
\mathcal{PT}|\psi_E\rangle=c|\psi_E\rangle.
\]
At a critical parameter, two real eigenvalues and eigenvectors coalesce at an EP. Beyond the EP, $\mathcal{PT}$ symmetry is spontaneously broken and the eigenvalues generally form a complex-conjugate pair.

A representative balanced gain-loss model is
\[
H=
\begin{pmatrix}
\omega+i\gamma&\kappa\\
\kappa&\omega-i\gamma
\end{pmatrix},
\]
with
\[
E_{\pm}
=
\omega\pm\sqrt{\kappa^2-\gamma^2}.
\]
For $|\gamma|<|\kappa|$, the spectrum is real. At $|\gamma|=|\kappa|$, the levels coalesce at an EP. For $|\gamma|>|\kappa|$, the spectrum becomes complex.

A related notion is quasi-Hermiticity. A non-Hermitian operator is quasi-Hermitian if a bounded positive metric $\Theta$ exists such that
\[
\Theta H=H^\dagger\Theta.
\]
With $S=\sqrt{\Theta}$,
\[
h_S=SHS^{-1}
\]
is Hermitian, and the modified scalar product is
\[
\langle\phi|\psi\rangle_{\Theta}
=
\langle\phi|\Theta|\psi\rangle.
\]
The spectrum remains real while the metric exists. At the EP, the metric becomes singular or ceases to exist. The EP therefore marks the breakdown of the Hilbert-space structure that made the non-Hermitian operator equivalent to a Hermitian one.

Even a Hermitian problem can reveal EPs after analytic continuation of a real parameter into the complex plane. For a Hermitian Hamiltonian depending on a real parameter, coupled levels generally repel on the real axis; the EPs associated with the avoided crossing occur in the complexified parameter plane. By contrast, genuinely non-Hermitian systems can possess EPs at real parameter values.

In open systems, non-Hermiticity may arise from radiation, absorption, gain-loss engineering, effective boundary conditions, or eliminated reservoirs. A topological photonic waveguide coupled to two counter-propagating whispering-gallery modes, for example, has an overall Hermitian Hamiltonian, but eliminating the waveguide produces an effective cavity Hamiltonian
\[
H_{\mathcal{PT}}
=
\begin{pmatrix}
\omega_c-i\Gamma/2&g\\
g&\omega_c+i\Gamma/2
\end{pmatrix}.
\]
Its EP occurs at $\Gamma=2g$. The two transmission minima coalesce at the transition from unbroken to broken $\mathcal{PT}$ symmetry, although no active gain material is required [2007.05900].

A recent extension concerns non-Bloch systems under open boundary conditions. In geometry-induced skin systems, branch points of the unrestricted energy Riemann surface need not be physical EPs on the open-boundary spectrum. A branch point lying on the generalized Brillouin zone can appear as a Whitney cusp without eigenvector coalescence, whereas a saddle point of the GBZ-adhered energy surface can generate a genuine non-Bloch EP. This distinguishes
\[
\text{non-Bloch branch point}
\neq
\text{non-Bloch EP}.
\]
The distinction requires simultaneous analysis of the GBZ, phase rigidity, eigenvectors, line-gap closing, and eigenvalue-knot evolution [2601.01886].

## 4. Physical realizations and experimental methodologies

EPs occur in both quantum and classical systems because their defining structure is non-Hermitian spectral defectiveness rather than specifically quantum dynamics. Classical coupled oscillators, electromagnetic systems, fluid stability, hydrodynamics, microwave cavities, optical waveguides, and plasmonic resonators all exhibit EP phenomena.

Microwave cavities provide direct realizations in which two real control parameters simulate the complex parameter plane. Encircling an EP produces resonance exchange, four-cycle eigenvector behavior, characteristic phase relations, and modifications under explicit time-reversal-symmetry breaking. Coupled electronic circuits and open cavity-atom systems provide analogous settings.

In plasmonic nanostructures, three coupled gold nanobars support even and odd collective modes. Symmetry isolates two even modes whose resonance frequencies and radiative linewidths can be tuned independently through lateral displacement and vertical separation. The effective Hamiltonian contains intrinsic absorption and coupling to radiation. An EP requires simultaneous equality of both resonance frequencies and linewidths. Numerical full-wave simulations identify an EP near
\[
dx\simeq345~\mathrm{nm},
\qquad
dz\approx61\text{--}61.5~\mathrm{nm},
\]
at approximately $212~\mathrm{THz}$. Near the EP, individual scattering residues diverge while their sum remains finite, a signature of the singular modal decomposition [1609.02276].

Hydrogen-like Rydberg systems in parallel electric and magnetic fields provide another numerical and experimental setting. Complex scaling converts outgoing resonances into discrete complex eigenvalues. An avoided crossing supplies an initial estimate, after which a local $2\times2$ model is fitted using the sum and squared difference of the two resonance energies. The octagon method uses nine quantum calculations per iteration and solves
\[
\eta(\gamma_{\mathrm{EP}},f_{\mathrm{EP}})=0
\]
for the EP. For hydrogen, the required fields are experimentally inaccessible in the reported examples, whereas cuprous-oxide excitons reach corresponding dimensionless EPs at fields below $10~\mathrm{T}$ in several cases. Proposed signatures include avoided crossings, resonance coalescence, exchange under field-space encircling, exchanged nodal patterns, anomalous linewidths, and square-root splitting [1602.00909].

Thermoacoustic EPs arise from delayed flame-response dynamics. The characteristic equation is a nonlinear dispersion relation
\[
D(\omega;n,\tau)=0.
\]
An algebraic double root satisfies
\[
D=0,
\qquad
\frac{\partial D}{\partial\omega}=0,
\qquad
\frac{\partial^2D}{\partial\omega^2}\neq0.
\]
The interacting branches may originate from a conventional acoustic mode and an intrinsic thermoacoustic mode. At the EP, both eigenvalues and pressure eigenfunctions coalesce. Contour-integral methods are useful because fixed-point and mode-tracking algorithms become ill-conditioned near defective eigenvalues [1807.08793].

In open elastic spheres, outgoing radiation produces complex scattering poles without explicit material gain or loss. A depth-first search in a two-dimensional material-geometry parameter space identifies EPs by tracking pole permutations around rectangular loops. Two EPs were reported in the coupled P–SV sector for angular order $n=4$, at
\[
(R_1,\rho_1)=(0.516002,8.64215)
\]
and
\[
(R_1,\rho_1)=(0.74472,6.9567).
\]
The pole exchange under one encirclement and the coalescence of resonant displacement fields provide the numerical evidence for EPs [2409.08560].

In inverse scattering, exceptional points are defined through nontrivial solutions of the homogeneous Faddeev problem. For the zero-energy two-dimensional problem, a complex parameter $k$ is exceptional if the homogeneous problem has a nonzero solution. Boundary-operator formulations give the exact criterion
\[
k\text{ exceptional}
\quad\Longleftrightarrow\quad
\ker\bigl(F_n-F^{\mathrm{out}}(k)\bigr)\neq\{0\}.
\]
Equivalently, the operator
\[
I+S_k(F_n-F_0)
\]
loses invertibility. Absorbing potentials with $\operatorname{Im}n(x)\geq\delta>0$ have no EPs in a sufficiently small punctured neighborhood of $k=0$, while small positive perturbations of conductive potentials can generate an exceptional set exponentially close to the origin [1407.1548].

## 5. Dynamical, many-body, and topological consequences

EPs modify not only spectra but also time-dependent evolution. In a non-Hermitian Kibble–Zurek protocol, a ramp through an EP produces a defect density
\[
n(\tau)
\sim
\tau^{-\frac{(d+z)\nu}{z\nu+1}},
\]
rather than the Hermitian result
\[
n_{\mathrm H}(\tau)
\sim
\tau^{-\frac{d\nu}{z\nu+1}}.
\]
The additional exponent reflects decay of the defect component back toward the ground state near the EP. The same dynamics can be interpreted through a Lindblad master equation and continuous monitoring, where no-jump evolution induces effective non-unitary attraction toward the coalesced state [1812.08668].

Floquet systems support specialized EP structures. Floquet $\pi$-EPs occur at quasienergy $\pi$ and are characterized by an eigenvector that rotates once on the Bloch sphere during one driving period and accumulates a $\pi$ geometric phase. Separated order-$1/2$ Floquet $\pi$-EPs exhibit square-root phase-rigidity scaling and exchange eigenstates under encirclement. Two EPs with the same dynamical rotation structure can merge into an order-$1$ Floquet $\pi$-EP, whereas opposite dynamical structures cannot merge. At the merger, the quasienergy band becomes gapless, and the system can exhibit perfect transmission with strongly asymmetric reflection [2407.13789].

In a driven dissipative two-level atom, the EP belongs to the Liouvillian rather than to the closed Hermitian Hamiltonian. At resonance,
\[
\lambda_1=-\frac{\gamma}{2},
\qquad
\lambda_{2,3}
=
-\frac{3\gamma}{4}
\pm
i\widetilde{\Omega},
\]
where
\[
\widetilde{\Omega}
=
\sqrt{(2\Omega)^2-\left(\frac{\gamma}{4}\right)^2}.
\]
The EP occurs at
\[
\Omega_{\mathrm{EP}}=\frac{\gamma}{8}.
\]
Below the EP, relaxation is overdamped and nonoscillatory. Above it, relaxation contains damped Rabi oscillations. At the EP, the Jordan block produces $t e^{-3\gamma t/4}$ terms. The fluorescence spectrum changes from a singlet-like structure to the Mollow triplet as the EP is crossed [2504.16888].

Many-body interactions can preserve, propagate, create, and annihilate EPs. In symmetry-protected interacting fermion systems, pre-existing single-particle EPs spread into exceptional fans in the enlarged parameter space. Interactions can also convert diagonalizable degeneracies into EPs. Compatible second-order EP lines can terminate at a third-order EP, whereas EPs in different total-momentum sectors cannot hybridize. Exceptional perturbation theory constructs effective Hamiltonians in the generalized eigenspaces and accounts for Jordan chains explicitly [2204.05340].

In free parafermion chains, complexifying the magnetic field produces a finite-size set of
\[
N(L-1)
\]
EPs. They arise when two quasienergy roots of the finite-size momentum equation merge. The EP positions satisfy
\[
\sin([2L+1]k_{\mathrm{EP}})
-
(2L+1)\sin(k_{\mathrm{EP}})
=0,
\]
with corresponding field values determined by
\[
\lambda_{\mathrm{EP}}^{\,N}
=
\left[
\frac{-\sin([L+1]k_{\mathrm{EP}})}
{\sin(Lk_{\mathrm{EP}})}
\right]^2.
\]
As system size increases, the EPs approach an approximately uniform distribution on the unit circle in the complex $\lambda$ plane, excluding the trivial $N$th roots of unity. Their proximity to the positive real critical point influences fidelity susceptibility and finite-size spectral response [2301.11031].

Topological defect modes can also be formulated through EPs in complex momentum. In chiral topological superconductors, zero-mode decay lengths are determined by
\[
\det\mathcal A(i\boldsymbol z)=0.
\]
The analytically continued momentum conditions
\[
\det\mathcal A(\boldsymbol k)=0
\quad\text{or}\quad
\det\mathcal A^\dagger(\boldsymbol k)=0
\]
produce defective zero-energy Hamiltonians when only one off-diagonal block has a null vector. The signs of the imaginary parts of complex momenta determine whether solutions decay or grow away from a defect. A sign change therefore signals the appearance or disappearance of a localized chiral Majorana mode. The resulting count agrees with winding-number descriptions of bulk-defect correspondence [1503.03839].

## 6. Applications, generalized definitions, and limitations

The conventional definition of an EP emphasizes simultaneous eigenvalue and eigenvector coalescence. A broader proposal introduces **general exceptional points** (GEPs), defined by defectiveness of the matrix basis under a non-Hermitian similarity transformation. Under this definition, a singular basis can occur even when eigenvalues remain distinct or eigenvectors do not coalesce. Matrix-type, basis-type, and hybrid-type GEPs are distinguished according to which similarity transformation becomes singular. In nonreciprocal SSH edge-state subspaces, basis-type GEPs arise from the non-Hermitian skin effect, and a hidden transition between noncoalescing and coalescing subclasses can occur without an ordinary spectral degeneracy [2109.05980].

The distinction between conventional EPs and generalized singular structures is consequential. A conventional EP requires a defective operator and a nontrivial Jordan block. A GEP, by contrast, concerns collapse of a representation basis and can retain equal algebraic and geometric multiplicities under the authors’ generalized criterion. The broader terminology should therefore not be used interchangeably with the standard spectral definition.

Hadronic systems provide a proposed QCD realization. Analytically continuing a magnetic field as
\[
B\longrightarrow iB_I
\]
turns magnetic couplings into imaginary interactions. In pseudoscalar-vector meson systems, the effective mass matrix has antisymmetric off-diagonal terms and is pseudo-Hermitian. For the $\eta_c$–$J/\psi$ system, the lower-field EP occurs near
\[
eB_I^{\mathrm{EP}}\simeq0.18~\mathrm{GeV}^2.
\]
Below the EP, the masses exhibit level attraction; at the EP, eigenvalues and eigenvectors coalesce; above it, the masses form a complex-conjugate pair. A constituent-quark description attributes the EP to imaginary Zeeman mixing between the spin-singlet and longitudinal spin-triplet states. The same model predicts an inverted transverse potential and eventual field-induced deconfinement at stronger fields [2606.31697].

Potential applications of EPs include resonance sensing, mode switching, light-matter interaction, subwavelength sensing, communication and imaging, acoustic and elastic-wave routing, radiation-loss engineering, inverse reconstruction, and control of thermoacoustic or quantum dynamics. These applications depend on the observable consequences of defectiveness: square-root spectral response, mode switching, enhanced parameter sensitivity, anomalous linewidths, singular residues, nontrivial scattering, and Jordan-block time dependence.

Several qualifications are essential. A repeated eigenvalue is not sufficient evidence for an EP; eigenvector coalescence or an equivalent defectiveness criterion must be verified. Enhanced eigenvalue sensitivity does not automatically imply improved measurement precision, because loss, disorder, modal overlap, fabrication error, noise, and finite resolution can obscure the splitting. Numerical pole tracking can fail when resonances approach one another, and contour-integral or encircling methods are often required. Results obtained for absorbing media, perturbative conductive potentials, specific symmetry classes, or particular Floquet protocols are generally local or model-dependent rather than global classifications.

The central unifying principle is
\[
\text{eigenvalue coalescence}
+
\text{eigenvector coalescence}
+
\text{non-diagonalizability}.
\]
From this structure follow square-root and higher-order Puiseux laws, self-orthogonality, Riemann-sheet exchange, geometric phases, Jordan-block dynamics, second-order poles, anomalous scattering and decay, symmetry breaking, instability, unusual Kibble–Zurek scaling, and limitations of approximation schemes. EPs consequently provide a common spectral framework linking non-Hermitian quantum theory, classical waves, open-system dynamics, topological band structures, many-body physics, inverse scattering, hadronic models, and geometry-dependent non-Bloch phenomena.

Source: https://www.emergentmind.com/topics/exceptional-points