Fragmented Exceptional Points (FEPs)

• Fragmented Exceptional Points (FEPs) are non-Hermitian degeneracies where eigenvalues have algebraic multiplicity α and geometric multiplicity γ (with 1<γ<α), leading to partial eigenvector collapse.
• The algebraic detection framework employs a partial multiplicity decomposition and matrix trace–rank methods to quantify the extent of eigenvector fragmentation and extract the key parameter ℓ.
• FEPs exhibit tunable singular responses with eigenvalue splitting scaling as ε^(1/ℓ), making them pivotal for designing systems in non-Hermitian lattice and topological platforms.

Fragmented exceptional points are non-Hermitian spectral degeneracies in which an eigenvalue of algebraic multiplicity $\alpha_i$ has geometric multiplicity $\gamma_i$ satisfying $1<\gamma_i<\alpha_i$, so the eigenvectors are only partially degenerate. In the formulation introduced for finite-dimensional non-Hermitian Hamiltonians, the fragmentation is encoded by a partial multiplicity decomposition $\alpha_i=\sum_{j=1}^{\gamma_i} l_{i,j}$, where each $l_{i,j}$ records how many algebraically degenerate modes share the same eigenvector. This places fragmented exceptional points between fully defective exceptional points, for which all degenerate modes collapse into one eigenvector, and nondefective diabolic or “$n$-bolic” points, for which all degenerate modes retain independent eigenvectors. The resulting degeneracies exhibit intermediate singular response behavior and can be designed explicitly in bulk and edge spectra of lattice models, notably in non-Hermitian variants of a Lieb lattice and a higher-order topological Dirac semimetal (Bid et al., 29 Jul 2025).

1. Definition and classification

Consider a finite-dimensional non-Hermitian Hamiltonian $H$ with right and left eigenvalue equations

$$H\,\mathbf{u}_{i,j}=E_i\,\mathbf{u}_{i,j},\qquad \mathbf{v}_{i,j}\,H=E_i\,\mathbf{v}_{i,j}.$$

For each eigenvalue $E_i$, the algebraic multiplicity $\alpha_i$ is the order of the root of $\gamma_i$0, while the geometric multiplicity $\gamma_i$1 is the number of linearly independent right eigenvectors, equivalently left eigenvectors. The distinction between these two multiplicities organizes the classification of non-Hermitian degeneracies (Bid et al., 29 Jul 2025).

Degeneracy type	Multiplicity condition	Partial multiplicity pattern
Diabolic or “$\gamma_i$2-bolic” point	$\gamma_i$3	all $\gamma_i$4
Conventional $\gamma_i$5	$\gamma_i$6	$\gamma_i$7
Fragmented exceptional point	$\gamma_i$8	intermediate, e.g. $\gamma_i$9

For an FEP, the partial multiplicity decomposition

$1<\gamma_i<\alpha_i$0

specifies the fragmentation pattern. The list $1<\gamma_i<\alpha_i$1 characterizes how the eigenvectors are shared across algebraically degenerate modes. For $1<\gamma_i<\alpha_i$2, the pattern $1<\gamma_i<\alpha_i$3 is the simplest nontrivial example: two algebraically degenerate modes share one eigenvector, while the third retains an independent eigenvector. Physically, FEPs combine features of EPs, notably enhanced sensitivity, with residual orthogonality sectors.

2. Singular response and perturbative behavior

The response of a system near an FEP is governed not by the full algebraic multiplicity $1<\gamma_i<\alpha_i$4 alone, but by the maximal partial multiplicity

$1<\gamma_i<\alpha_i$5

Around such a degeneracy, the Green’s function

$1<\gamma_i<\alpha_i$6

obeys

$1<\gamma_i<\alpha_i$7

so the physical response diverges with an exponent set by $1<\gamma_i<\alpha_i$8 rather than necessarily by $1<\gamma_i<\alpha_i$9 (Bid et al., 29 Jul 2025).

A generic perturbation of strength $\alpha_i=\sum_{j=1}^{\gamma_i} l_{i,j}$0 splits the eigenvalues according to

$\alpha_i=\sum_{j=1}^{\gamma_i} l_{i,j}$1

where $\alpha_i=\sum_{j=1}^{\gamma_i} l_{i,j}$2 is a characteristic spectral response strength. A corresponding physical response strength $\alpha_i=\sum_{j=1}^{\gamma_i} l_{i,j}$3 appears in the residue of the Green’s function. Compared with a conventional $\alpha_i=\sum_{j=1}^{\gamma_i} l_{i,j}$4, which has $\alpha_i=\sum_{j=1}^{\gamma_i} l_{i,j}$5, an FEP with $\alpha_i=\sum_{j=1}^{\gamma_i} l_{i,j}$6 has weaker singularities: the physical response diverges only as $\alpha_i=\sum_{j=1}^{\gamma_i} l_{i,j}$7 and the spectral splitting scales as $\alpha_i=\sum_{j=1}^{\gamma_i} l_{i,j}$8. At the same time, FEPs retain non-orthogonality in subspaces of size $\alpha_i=\sum_{j=1}^{\gamma_i} l_{i,j}$9 while preserving residual orthogonality among the remaining eigenvectors. In this sense they interpolate between diabolic points, for which $l_{i,j}$0, and full exceptional points, for which $l_{i,j}$1.

3. Algebraic detection and partial-multiplicity extraction

A central development in the systematic study of FEPs is an algebraic framework that determines $l_{i,j}$2, $l_{i,j}$3, and the full partial-multiplicity structure directly from a given Hamiltonian, without carrying out a delicate Jordan decomposition. The construction uses the so-called modes of the adjugate matrix $l_{i,j}$4 for the shifted matrix

$l_{i,j}$5

with the coefficients $l_{i,j}$6 defined by the shifted characteristic polynomial $l_{i,j}$7 (Bid et al., 29 Jul 2025).

The modes satisfy the Faddeev–LeVerrier recursion

$l_{i,j}$8

The algebraic degeneracy condition

$l_{i,j}$9

guarantees that $n$0 is a root of multiplicity $n$1. Once $n$2 is known, the geometric partial multiplicities $n$3 are extracted from matrix ranks as

$n$4

Equivalently, the maximal partial multiplicity $n$5 is the smallest $n$6 for which $n$7 but $n$8 for all $n$9.

The same formalism yields explicit expressions for the response strengths,

$H$0

Because the procedure only involves traces and ranks of polynomials in $H$1, it applies to any explicit matrix representation of a non-Hermitian model. This makes the framework suitable both for diagnosis of existing degeneracies and for inverse design of systems with prescribed degeneracy structures and response characteristics.

4. Bulk realization in the non-Hermitian Lieb lattice

The non-Hermitian Lieb lattice provides a minimal two-dimensional setting in which bulk FEPs can be realized. The Hermitian Lieb lattice has a three-site unit cell $H$2–$H$3–$H$4 with symmetric nearest-neighbour hopping. Setting intra- and intercell hopping $H$5, its Bloch Hamiltonian is

$H$6

which has a flat band at zero energy and two dispersive bands meeting at a tribolic point with $H$7 and $H$8 (Bid et al., 29 Jul 2025).

A simple non-Hermitian variant preserving chiral symmetry is

$H$9

At $$H\,\mathbf{u}_{i,j}=E_i\,\mathbf{u}_{i,j},\qquad \mathbf{v}_{i,j}\,H=E_i\,\mathbf{v}_{i,j}.$$0, this model has $$H\,\mathbf{u}_{i,j}=E_i\,\mathbf{u}_{i,j},\qquad \mathbf{v}_{i,j}\,H=E_i\,\mathbf{v}_{i,j}.$$1 and $$H\,\mathbf{u}_{i,j}=E_i\,\mathbf{u}_{i,j},\qquad \mathbf{v}_{i,j}\,H=E_i\,\mathbf{v}_{i,j}.$$2, giving the partial multiplicity pattern $$H\,\mathbf{u}_{i,j}=E_i\,\mathbf{u}_{i,j},\qquad \mathbf{v}_{i,j}\,H=E_i\,\mathbf{v}_{i,j}.$$3, identified in the paper as a FEP $$H\,\mathbf{u}_{i,j}=E_i\,\mathbf{u}_{i,j},\qquad \mathbf{v}_{i,j}\,H=E_i\,\mathbf{v}_{i,j}.$$4. By contrast, the other degeneracies in the Brillouin zone are generic $$H\,\mathbf{u}_{i,j}=E_i\,\mathbf{u}_{i,j},\qquad \mathbf{v}_{i,j}\,H=E_i\,\mathbf{v}_{i,j}.$$5 points with $$H\,\mathbf{u}_{i,j}=E_i\,\mathbf{u}_{i,j},\qquad \mathbf{v}_{i,j}\,H=E_i\,\mathbf{v}_{i,j}.$$6 and $$H\,\mathbf{u}_{i,j}=E_i\,\mathbf{u}_{i,j},\qquad \mathbf{v}_{i,j}\,H=E_i\,\mathbf{v}_{i,j}.$$7.

The classification can be stated in closed form. Writing

$$H\,\mathbf{u}_{i,j}=E_i\,\mathbf{u}_{i,j},\qquad \mathbf{v}_{i,j}\,H=E_i\,\mathbf{v}_{i,j}.$$8

degeneracies at $$H\,\mathbf{u}_{i,j}=E_i\,\mathbf{u}_{i,j},\qquad \mathbf{v}_{i,j}\,H=E_i\,\mathbf{v}_{i,j}.$$9 occur when

$E_i$0

An $E_i$1, corresponding to the $E_i$2 scenario, occurs precisely if, for example, $E_i$3 while $E_i$4, or symmetrically if $E_i$5 while $E_i$6. All other nontrivial solutions of $E_i$7 with none of the pairs simultaneously vanishing produce $E_i$8 points. This establishes an explicit criterion for distinguishing partial and full eigenvector collapse directly at the Bloch-Hamiltonian level.

5. Bulk and hinge realizations in a higher-order topological Dirac semimetal

A three-dimensional realization is furnished by a higher-order Dirac semimetal. Its Hermitian parent is a 4-band model built by stacking 2D quadrupole insulators with criss-cross interlayer couplings. In momentum space it can be written in chiral block form

$E_i$9

Bulk Dirac points at $\alpha_i$0 are tetrabolic points with $\alpha_i$1 and $\alpha_i$2 (Bid et al., 29 Jul 2025).

Four minimal non-Hermitian variants are obtained by adding intracell nonreciprocal hopping terms of amplitude $\alpha_i$3. At those momenta where $\alpha_i$4, the four models realize distinct degeneracy patterns.

Model	Bulk degeneracy	Hinge degeneracy
1	generic $\alpha_i$5 coexisting with diabolic points $\alpha_i$6	hinge states approach an $\alpha_i$7
2	FEP with $\alpha_i$8	hinge states form an $\alpha_i$9 with $\gamma_i$00
3	FEP with $\gamma_i$01	hinge states form an $\gamma_i$02 with $\gamma_i$03
4	FEP with $\gamma_i$04	hinge states form an $\gamma_i$05 with $\gamma_i$06

In the topological phase with open $\gamma_i$07 boundaries, four nearly zero-energy hinge modes appear as functions of $\gamma_i$08. In the same four variants, these hinge modes bisect bulk degeneracy points and then hybridize into FEPs of the same type found in the bulk. The fragmentation can be confirmed either by the rank–trace formalism or by combining atomistic-limit corner-mode counting with non-Hermitian skin-effect arguments. The model therefore demonstrates that FEPs are not restricted to bulk band touchings; they can also characterize boundary spectra in higher-order topological systems.

6. Terminological extensions, detached EPs, and nonlinear fragmentation

Recent literature also uses the phrase “fragmented exceptional points” in senses that differ from the partial-multiplicity definition above. In a reciprocal non-Hermitian Lieb lattice under open boundary conditions, geometry-induced non-Bloch EPs arise from generalized Brillouin-zone geometry. There the open-boundary spectrum is obtained from

$\gamma_i$09

with $\gamma_i$10 and $\gamma_i$11. The discriminant

$\gamma_i$12

satisfies $\gamma_i$13 at EPs. Branch points are obtained from

$\gamma_i$14

while a Whitney cusp additionally satisfies $\gamma_i$15 and does not correspond to an EP because phase rigidity remains finite. By contrast, a saddle point on the generalized Brillouin-zone surface gives a true EP with a square-root transition, a gap scaling as $\gamma_i$16, vanishing phase rigidity, and vorticity $\gamma_i$17. These geometry-induced EPs lie off the non-Bloch Fermi arcs defined by $\gamma_i$18 with unequal imaginary parts, so they are described there as fragmented from the Fermi arcs rather than as partially defective degeneracies in the multiplicity sense (Zhao et al., 5 Jan 2026).

A different usage appears in a saturable-nonlinear non-Hermitian dimer. In the linear limit the model has a single second-order EP at

$\gamma_i$19

With non-equal saturable nonlinearities $\gamma_i$20, the nonlinear EP condition yields

$\gamma_i$21

In that setting, the unique linear EP at $\gamma_i$22 fragments into distinct EPs displaced in detuning, and the discriminant of the nonlinear polynomial in the population imbalance identifies the corresponding coalescences. The paper further notes that, in more elaborate multi-mode saturable-nonlinear systems, multiple real zeros of the discriminant could produce a true fan of FEPs. Here fragmentation refers to the splitting of a single linear EP into multiple nonlinear EPs in parameter space, not to partial eigenvector collapse at fixed $\gamma_i$23 and $\gamma_i$24 (Gu et al., 2024).

In the strict structural sense, FEPs enrich the palette of non-Hermitian topology, lead to exceptional surfaces containing both EPs and special FEP loci, and permit tunable sensitivity because the maximal partial multiplicity $\gamma_i$25 sets the power law of response versus perturbation. The same algebraic-mode framework that diagnoses these degeneracies also enables inverse design of non-Hermitian devices with prescribed response characteristics, including prospective bulk or edge realizations in photonic, acoustic, cold-atom, and electrical-circuit platforms (Bid et al., 29 Jul 2025).