Exceptional-Point Locus

• Exceptional-point locus is the set of parameters where non-Hermitian operators become defective, with eigenvalues and eigenvectors coalescing into nontrivial Jordan blocks.
• It organizes spectral behavior through diverse geometries—points, lines, surfaces—modulated by symmetry constraints, boundary conditions, and system dimensionality.
• This locus underpins observable phenomena like branch exchange, topological winding, and anomalous response in systems ranging from quantum optics to black-hole spectroscopy.

Searching arXiv for recent and foundational papers on “exceptional-point locus” and closely related EP manifold terminology. An exceptional-point locus is the subset of parameter space at which a non-Hermitian operator becomes defective: two or more eigenvalues coalesce, their eigenvectors also coalesce, and the operator acquires a nontrivial Jordan block. In its most compact form, it is the set

$$\mathcal E=\{\boldsymbol\lambda:\; H(\boldsymbol\lambda)\ \text{is defective}\},$$

and it may appear as isolated points, lines, surfaces, or higher-dimensional sets, depending on the dimension of parameter space, boundary conditions, symmetry constraints, and whether the problem is linear, nonlinear, Bloch, or non-Bloch (Downing et al., 23 Apr 2025, Tang et al., 2022). Across current non-Hermitian research, the exceptional-point locus functions both as a spectral singular set and as an organizing geometric object: it controls branch exchange, line-gap changes, topological winding, anomalous response, and, in several experimentally relevant models, directly observable spectral reconstruction (Silva, 20 Jun 2026, Cao et al., 21 Nov 2025).

1. Definition and local spectral structure

For a non-Hermitian matrix or operator $M(\mu)$, an exceptional point is a parameter value $\mu_{\rm EP}$ where eigenvalues and eigenvectors coalesce and the matrix is not diagonalizable. In the $2\times 2$ case, the Jordan normal form at the EP is

$J=\begin{pmatrix}\lambda_{\rm EP}&1\0&\lambda_{\rm EP}\end{pmatrix},$

and, generically, the local eigenvalue structure is a square-root branch point,

$$\lambda_\pm(\mu)\simeq \lambda_{\rm EP}\pm C\sqrt{\mu-\mu_{\rm EP}},$$

so that analytic continuation around the EP exchanges the two sheets of the Riemann surface (Downing et al., 23 Apr 2025).

Equivalent formulations are standard. If

$p(\lambda;\mu)=\det[\lambda I-M(\mu)],$

then an EP corresponds to a multiple root of $p$ for which the algebraic multiplicity exceeds the geometric multiplicity; for a quadratic polynomial this appears as vanishing of the discriminant (Downing et al., 23 Apr 2025). In practical two-level settings one often uses

$\det(H-EI)=0,\qquad \partial_E\det(H-EI)=0,$

supplemented by defectiveness or eigenvector coalescence, as the local EP condition (Egenlauf et al., 2024).

This local picture extends directly to loci. In a multi-parameter family $H(\boldsymbol\lambda)$, the exceptional-point locus is the set of all $M(\mu)$0 satisfying these degeneracy and defectiveness conditions. The locus is therefore not defined by eigenvalue coincidence alone; it is a subset of parameter space on which the operator fails to be diagonalizable (Tang et al., 2022).

A significant refinement arises in non-Bloch band theory. In Bloch settings, EPs are branch points of the eigenvalue manifold and sit at the endpoints of Fermi arcs. In the reciprocal non-Hermitian Lieb lattice under open boundary conditions, however, geometry-induced non-Bloch EPs are saddle points on generalized-Brillouin-zone surfaces, while branch points on the non-Bloch Riemann surface appear as Whitney cusps and are not themselves EPs (Zhao et al., 5 Jan 2026). This separates the notions of branch-point locus and exceptional-point locus.

2. Geometries of exceptional-point loci

The geometry of an exceptional-point locus depends on how many real control parameters are available and on what constraints remain after symmetry and boundary conditions are imposed. Concrete realizations span isolated points, lines, ellipses, surfaces, and embedded higher-order structures.

Geometry	Representative model	Defining relation
Isolated point	Resonant driven atom	$M(\mu)$1 at $M(\mu)$2 (Downing et al., 23 Apr 2025)
Curves in a plane	Rice–Mele PBC bulk EPs	$M(\mu)$3 (Martinez-Strasser et al., 23 Jun 2026)
Straight tracking line	Helicoidal OAM photonics	$M(\mu)$4, $M(\mu)$5 (Silva, 20 Jun 2026)
Continuous line in 3D	Black-hole spectroscopy	EL in $M(\mu)$6 space (Cao et al., 21 Nov 2025)
Surfaces with embedded lines	Synthetic 3D momentum space	EL3 embedded in ES2 (Tang et al., 2022)

In the driven two-level atom with spontaneous emission, the Liouvillian spectrum is governed by a cubic, and the EP condition is $M(\mu)$7. Rewritten in the $M(\mu)$8 plane, this gives a sextic equation in $M(\mu)$9. The resulting exceptional-point locus exists for

$\mu_{\rm EP}$0

where there are two positive real EP solutions $\mu_{\rm EP}$1; at $\mu_{\rm EP}$2 they merge, and for $\mu_{\rm EP}$3 no EPs exist (Downing et al., 23 Apr 2025).

In the generalized non-Hermitian Rice–Mele model under periodic boundary conditions, introducing $\mu_{\rm EP}$4 and $\mu_{\rm EP}$5 yields

$\mu_{\rm EP}$6

For generic $\mu_{\rm EP}$7 this is an ellipse; for $\mu_{\rm EP}$8 or $\mu_{\rm EP}$9 it degenerates into the four straight lines

$2\times 2$0

Thus the PBC exceptional-point locus is a family of ellipses and lines in $2\times 2$1 space (Martinez-Strasser et al., 23 Jun 2026).

In black-hole perturbation theory with a Gaussian bump added to the Regge–Wheeler potential,

$2\times 2$2

the EPs of the fundamental mode and first overtone organize into a continuous exceptional line in the three-dimensional parameter space $2\times 2$3 (Cao et al., 21 Nov 2025). In this setting the line realizes the generic codimension-two expectation for second-order EPs in a three-real-parameter family.

3. Boundary conditions, geometry, and locus control

Boundary conditions can change the exceptional-point locus qualitatively. In the generalized non-Hermitian Rice–Mele model, next-nearest-neighbor hopping $2\times 2$4 enters the Bloch Hamiltonian under periodic boundary conditions only through the identity term $2\times 2$5. As a result, the PBC EP locus in $2\times 2$6 is independent of $2\times 2$7; $2\times 2$8 shifts EP energies but not their positions in parameter space (Martinez-Strasser et al., 23 Jun 2026).

Under open boundary conditions the situation changes. The same model develops OBC EP loci that differ from the PBC ellipses and lines even for $2\times 2$9, and for $J=\begin{pmatrix}\lambda_{\rm EP}&1\0&\lambda_{\rm EP}\end{pmatrix},$0 the long-range hopping not only shifts existing EP branches but generates new ones. Numerically, some OBC branches approach the asymptotic line

$J=\begin{pmatrix}\lambda_{\rm EP}&1\0&\lambda_{\rm EP}\end{pmatrix},$1

which also signals an OBC topological gap closing that is absent in the PBC spectrum (Martinez-Strasser et al., 23 Jun 2026). This establishes that the exceptional-point locus is not solely a property of a bulk Bloch symbol; it can be boundary-sensitive even when the bulk spectrum remains conventional.

A distinct control mechanism appears in helicoidal orbital-angular-momentum photonics. There the geometric twist produces a real, signed, chiral splitting between $J=\begin{pmatrix}\lambda_{\rm EP}&1\0&\lambda_{\rm EP}\end{pmatrix},$2 and $J=\begin{pmatrix}\lambda_{\rm EP}&1\0&\lambda_{\rm EP}\end{pmatrix},$3 modes, while openness or gain/loss supplies the non-Hermitian part. In the reciprocal two-mode reduction, the EP condition is

$J=\begin{pmatrix}\lambda_{\rm EP}&1\0&\lambda_{\rm EP}\end{pmatrix},$4

with

$J=\begin{pmatrix}\lambda_{\rm EP}&1\0&\lambda_{\rm EP}\end{pmatrix},$5

Hence the exceptional-point locus in the $J=\begin{pmatrix}\lambda_{\rm EP}&1\0&\lambda_{\rm EP}\end{pmatrix},$6 plane is the straight line

$J=\begin{pmatrix}\lambda_{\rm EP}&1\0&\lambda_{\rm EP}\end{pmatrix},$7

or, in paraxial propagation language,

$J=\begin{pmatrix}\lambda_{\rm EP}&1\0&\lambda_{\rm EP}\end{pmatrix},$8

In this formulation the twist rate is the control parameter, not the source of non-Hermiticity (Silva, 20 Jun 2026).

The non-Bloch Lieb lattice adds a geometric perspective of a different kind. Changing the real-space sample from rectangular to parallelogram geometry induces a geometry-dependent skin effect, deforms the generalized Brillouin zone, removes Bloch EPs from the OBC spectrum, and creates new geometry-induced EPs. These non-Bloch EPs occur at saddle points on GBZ surfaces and are detached from the branch points of non-Bloch Fermi arcs (Zhao et al., 5 Jan 2026). This shows that the exceptional-point locus can be reshaped by boundary geometry even when the bulk Hamiltonian is held fixed.

4. Higher-order, embedded, and nonlinear exceptional loci

Higher-order exceptional geometries require additional structure. In a three-band acoustic system with a synthetic three-dimensional Brillouin zone, the exceptional geometry consists of order-2 exceptional surfaces and order-3 exceptional lines entirely embedded within them. The characteristic polynomial $J=\begin{pmatrix}\lambda_{\rm EP}&1\0&\lambda_{\rm EP}\end{pmatrix},$9 satisfies

$$\lambda_\pm(\mu)\simeq \lambda_{\rm EP}\pm C\sqrt{\mu-\mu_{\rm EP}},$$0

on the ES2, while EP$$\lambda_\pm(\mu)\simeq \lambda_{\rm EP}\pm C\sqrt{\mu-\mu_{\rm EP}},$$1 points require the simultaneous vanishing

$$\lambda_\pm(\mu)\simeq \lambda_{\rm EP}\pm C\sqrt{\mu-\mu_{\rm EP}},$$2

Defining

$$\lambda_\pm(\mu)\simeq \lambda_{\rm EP}\pm C\sqrt{\mu-\mu_{\rm EP}},$$3

with $$\lambda_\pm(\mu)\simeq \lambda_{\rm EP}\pm C\sqrt{\mu-\mu_{\rm EP}},$$4 and $$\lambda_\pm(\mu)\simeq \lambda_{\rm EP}\pm C\sqrt{\mu-\mu_{\rm EP}},$$5, one obtains a field whose zeros are precisely the EL3 and not the ES2. The corresponding resultant winding number detects the EL3 while ignoring the ES2 (Tang et al., 2022).

Response theory supplies another structural viewpoint. For an EP$$\lambda_\pm(\mu)\simeq \lambda_{\rm EP}\pm C\sqrt{\mu-\mu_{\rm EP}},$$6 embedded in a Hilbert space of dimension $$\lambda_\pm(\mu)\simeq \lambda_{\rm EP}\pm C\sqrt{\mu-\mu_{\rm EP}},$$7, the spectral response strength is

$$\lambda_\pm(\mu)\simeq \lambda_{\rm EP}\pm C\sqrt{\mu-\mu_{\rm EP}},$$8

where $$\lambda_\pm(\mu)\simeq \lambda_{\rm EP}\pm C\sqrt{\mu-\mu_{\rm EP}},$$9 is extracted from the highest-order pole of the resolvent near the EP. Along an exceptional surface of EP$p(\lambda;\mu)=\det[\lambda I-M(\mu)],$0 approaching a higher-order EP$p(\lambda;\mu)=\det[\lambda I-M(\mu)],$1, the response strengths satisfy the scaling

$p(\lambda;\mu)=\det[\lambda I-M(\mu)],$2

so $p(\lambda;\mu)=\det[\lambda I-M(\mu)],$3 diverges as the higher-order EP is approached (Wiersig, 2023). The same framework identifies $p(\lambda;\mu)=\det[\lambda I-M(\mu)],$4, with $p(\lambda;\mu)=\det[\lambda I-M(\mu)],$5 the Petermann factor, as the response strength of an isolated eigenvalue (Wiersig, 2023).

Nonlinear non-Hermitian systems produce yet another universal geometry. For a broad class of nonlinear perturbations of a linear EP2, the local EP locus in the three-dimensional control space $p(\lambda;\mu)=\det[\lambda I-M(\mu)],$6 is governed by the elliptic umbilic catastrophe. The linear EP2 becomes the apex of two cone-like surfaces with three-cusped deltoid cross sections; inside the cone there are four eigenvectors, outside there are two, smooth boundary segments are EP2 folds, and cusp lines correspond to nonlinear EP3 coalescences (Kwong et al., 26 Feb 2025). This makes the exceptional-point locus a bifurcation set rather than merely a set of defective matrices.

Sublattice symmetry constrains higher-order EPs differently. In fermionic systems with block-off-diagonal Hamiltonians,

$p(\lambda;\mu)=\det[\lambda I-M(\mu)],$7

eigenvalues appear in $p(\lambda;\mu)=\det[\lambda I-M(\mu)],$8 pairs, yet odd-order EPs can still occur at zero energy because EP order is determined by the size of the largest Jordan block, not by total algebraic multiplicity. In the $p(\lambda;\mu)=\det[\lambda I-M(\mu)],$9 case the paper exhibits an EP$p$0 with Jordan form

$p$1

and shows explicitly that it can be obtained both as a limit of EP$p$2 matrices and as a limit of EP$p$3 matrices (Yang et al., 2022). The exceptional-point locus in this symmetry class is therefore stratified by mixed-type intersections of different EP orders.

5. Diagnostics, topology, and localization methods

Encircling remains the canonical diagnostic of an exceptional-point locus. In the generic EP2 case, a loop around the EP exchanges the two eigenvalues and eigenvectors after one circuit and restores them only after the second (Downing et al., 23 Apr 2025, Egenlauf et al., 2024). In the black-hole exceptional line, loops that encircle the EL have vorticity

$p$4

and Berry phase

$p$5

whereas loops that do not encircle the line have $p$6 and $p$7 (Cao et al., 21 Nov 2025). This identifies the EL as a one-dimensional topological defect in the three-parameter space.

Several numerical diagnostics are now standard. In finite chains, EPs can be located by the condition number of the eigenvector matrix $p$8,

$p$9

whose logarithm diverges at an EP; Jordan decomposition is then used to confirm the order of the defect (Martinez-Strasser et al., 23 Jun 2026). Phase rigidity provides a complementary state-space diagnostic: in OAM photonics and in non-Bloch lattices, its collapse signals eigenvector coalescence, and in the OAM platforms the minimum rigidity falls to $\det(H-EI)=0,\qquad \partial_E\det(H-EI)=0,$0–$\det(H-EI)=0,\qquad \partial_E\det(H-EI)=0,$1 near the EP (Silva, 20 Jun 2026, Zhao et al., 5 Jan 2026).

For expensive spectra, surrogate and reduced models are effective. A Gaussian-process-regression method localizes EPs by modeling

$\det(H-EI)=0,\qquad \partial_E\det(H-EI)=0,$2

and performing a cheap root search for $\det(H-EI)=0,\qquad \partial_E\det(H-EI)=0,$3, with iterative retraining on selected exact eigenvalue pairs (Egenlauf et al., 2024). In hydrogenic and excitonic Rydberg systems in parallel electric and magnetic fields, the octagon method uses a local $\det(H-EI)=0,\qquad \partial_E\det(H-EI)=0,$4 matrix model with

$\det(H-EI)=0,\qquad \partial_E\det(H-EI)=0,$5

expanded to quadratic order in $\det(H-EI)=0,\qquad \partial_E\det(H-EI)=0,$6, and refines an EP from an avoided crossing by iteratively solving $\det(H-EI)=0,\qquad \partial_E\det(H-EI)=0,$7 (Feldmaier et al., 2016). Both methods treat the exceptional-point locus as a root set of analytic surrogates.

The pseudospectrum supplies an order diagnostic. For a degenerate eigenvalue with largest Jordan block of size $\det(H-EI)=0,\qquad \partial_E\det(H-EI)=0,$8, the $\det(H-EI)=0,\qquad \partial_E\det(H-EI)=0,$9-pseudospectrum contour size scales as

$H(\boldsymbol\lambda)$0

at the EP, in contrast to the linear $H(\boldsymbol\lambda)$1 scaling away from EPs (Cao et al., 21 Nov 2025). In black-hole spectroscopy this square-root scaling is verified numerically for the second-order exceptional line (Cao et al., 21 Nov 2025).

6. Physical consequences and conceptual caveats

The exceptional-point locus is physically consequential because it organizes qualitative changes in observables. In the driven two-level atom, crossing the EP locus transforms the Liouvillian spectrum from all-real damping exponents to a complex-conjugate pair, changes population dynamics from overdamped to underdamped, and reconstructs the emission spectrum from a Lorentzian-like singlet into a Mollow triplet; on resonance the critical point is

$H(\boldsymbol\lambda)$2

(Downing et al., 23 Apr 2025). In OAM photonics, the EP locus appears directly in input-output response, spectral maps, and three-dimensional Riemann-surface visualizations, with twist-controlled linear tracking of the EP position (Silva, 20 Jun 2026). In black-hole spectroscopy, the exceptional line marks a region of enhanced spectral instability, with pseudospectral bulging and $H(\boldsymbol\lambda)$3 sensitivity (Cao et al., 21 Nov 2025).

Topological and spectral boundaries need not coincide with the exceptional-point locus in every setting. In the generalized Rice–Mele model under OBC, EP branches cluster near topological boundaries and the line $H(\boldsymbol\lambda)$4 marks an OBC topological gap closing, but the EP locus is not identical to the topological phase boundary (Martinez-Strasser et al., 23 Jun 2026). More sharply, in nonlinear $H(\boldsymbol\lambda)$5-symmetric dimers the original linear system evolves along two distinct $H(\boldsymbol\lambda)$6-symmetric trajectories, each of which can possess an EP, yet the two trajectories can collide and vanish away from these EPs. The system is then left only with a $H(\boldsymbol\lambda)$7-broken phase, so the $H(\boldsymbol\lambda)$8-transition occurs away from an exceptional point (Ge, 2016). This shows that, in nonlinear problems, the phase boundary and the exceptional-point locus can separate.

Taken together, these works define the exceptional-point locus as a geometric and algebraic object with several layers. Locally it is the defective set of a non-Hermitian family. Globally it may be a line, ellipse, surface, deltoid cone, embedded network, or non-Bloch saddle set. In higher-order and symmetry-constrained settings it becomes stratified, with mixed-type intersections and order-changing endpoints. Its practical importance lies in the fact that it is simultaneously a singular set of the spectrum, a carrier of topological data, and a precise organizer of dynamical, transport, and spectroscopic phenomena (Tang et al., 2022, Wiersig, 2023).