Derogatory Exceptional Points in Non-Hermitian Systems

• Derogatory exceptional points are non-Hermitian degeneracies where a repeated eigenvalue is split among multiple Jordan blocks, leading to unique sensitivity and eigenvector coalescence phenomena.
• Their perturbative response follows Puiseux scaling laws, with the eigenvalue splitting determined by the size of the largest Jordan block, thereby controlling non-analytic sensitivity.
• Infinitesimal perturbations can convert a derogatory EP into another with a different Jordan structure while preserving degeneracy, offering a tunable platform for engineering non-Hermitian devices.

Searching arXiv for the cited papers and closely related work on derogatory exceptional points, fragmented exceptional points, and similarity-/symmetry-constrained non-Hermitian degeneracies. Derogatory exceptional points are non-Hermitian degeneracies in which a coalesced eigenvalue is represented by multiple Jordan blocks rather than a single block. In the framework developed for non-Hermitian degeneracies, an exceptional point (EP) occurs when a Hamiltonian $H(\mathbf{p})$ becomes defective, several eigenvectors coalesce, and $H$ is not diagonalizable; if the coalesced eigenvalue $\lambda$ has algebraic multiplicity $m$ and geometric multiplicity $g$, then its Jordan normal form contains $g$ Jordan blocks of sizes $\{r_1,\dots,r_g\}$ with $\sum_i r_i=m$. A nonderogatory EP has one block, whereas a derogatory EP has $g>1$ blocks sharing the same eigenvalue (Starkov et al., 17 Apr 2026). Recent work has shown that such degeneracies are not static classifications: under carefully chosen infinitesimal but nontrivial perturbations, a derogatory EP can convert into another EP of the same total order but different Jordan structure, often with a larger largest block and therefore stronger non-analytic sensitivity (Starkov et al., 17 Apr 2026). Closely related literature also studies the same phenomenon under the name fragmented exceptional points (FEPs), emphasizing partial eigenvector coalescence and mode fragmentation (Bid et al., 29 Jul 2025).

1. Definitions, Jordan structure, and algebraic data

For a finite-dimensional non-Hermitian matrix, the local structure of a degenerate eigenvalue is encoded by its Jordan decomposition. Near an EP at parameters $\mathbf{p}_0$ with eigenvalue $H$0, one may pass to a basis where the $H$1-subspace takes the form

$H$2

with $H$3, where $H$4 has ones on the superdiagonal and zeros elsewhere. The characteristic polynomial factorizes as $H$5, with $H$6, and defectiveness is expressed by chains of generalized eigenvectors satisfying

$H$7

for each block (Starkov et al., 17 Apr 2026).

A central invariant is the size $H$8 of the largest Jordan block. Equivalently, the factor attached to $H$9 in the minimal polynomial is

$\lambda$0

so the degree of the minimal polynomial at the degenerate eigenvalue is $\lambda$1, not the full algebraic multiplicity $\lambda$2 unless the EP is nonderogatory (Starkov et al., 17 Apr 2026). In the terminology of fragmented exceptional points, the fragmentation pattern is the ordered partition $\lambda$3, with $\lambda$4 the geometric multiplicity and $\lambda$5 (Bid et al., 29 Jul 2025).

Several algebraic diagnostics distinguish the Jordan content beyond the mere existence of a repeated root. One route uses derivatives of the characteristic polynomial: a root $\lambda$6 has multiplicity $\lambda$7 iff $\lambda$8 and $\lambda$9, but fixing the Jordan structure requires additional rank tests on $m$0 (Montag et al., 4 Aug 2025). Another route uses kernel-growth identities. For a fragmentation pattern $m$1,

$m$2

so the nullity increments determine the block sizes directly (Bid et al., 29 Jul 2025). In sublattice-symmetric settings this kernel-growth viewpoint becomes an explicit algebraic classification method based on image and kernel relations of the off-diagonal blocks $m$3 and $m$4 (Yang et al., 2022).

A terminological caveat is necessary. In the usage of (Starkov et al., 17 Apr 2026, Montag et al., 4 Aug 2025, Yang et al., 2022), and (Bid et al., 29 Jul 2025), derogatory EPs remain defective degeneracies with multiple Jordan blocks at the same eigenvalue. By contrast, (Sayyad et al., 2022) identifies “derogatory exceptional points” with what it calls non-defective EPs, namely diagonalizable repeated-eigenvalue degeneracies protected by symmetry. This reflects a difference between the linear-algebraic adjective “derogatory” and the more specific non-Hermitian-physics usage centered on partial eigenvector coalescence (Sayyad et al., 2022).

2. Sensitivity, Puiseux scaling, and partial coalescence

The leading perturbative sensitivity of a non-Hermitian degeneracy is controlled by the largest Jordan block. Under a small perturbation $m$5, taken in any parameterization transverse to the degeneracy manifold, the generic eigenvalue splitting obeys a Puiseux expansion

$m$6

so the sensitivity scales as $m$7 (Starkov et al., 17 Apr 2026). For $m$8, the nonderogatory type $m$9 has $g$0 and exhibits $g$1, whereas the derogatory type $g$2 has $g$3 and $g$4, and $g$5 has $g$6 (Starkov et al., 17 Apr 2026).

This scaling law has an immediate structural interpretation: for a derogatory EP with partition $g$7, each Jordan block of size $g$8 behaves like an isolated nonderogatory $g$9 under generic perturbation, splitting with exponent $g$0 (Starkov et al., 17 Apr 2026). The same point is emphasized in the FEP language, where the perturbation response is controlled by the largest partial multiplicity $g$1, with

$g$2

while the Green’s function exhibits a super-Lorentzian divergence

$g$3

For non-derogatory EPs one has $g$4 and $g$5, whereas for fragmented or derogatory EPs $g$6 and $g$7 and $g$8 differ, reflecting the multi-mode geometry (Bid et al., 29 Jul 2025).

Partial coalescence also modifies monodromy. Encircling an $g$9 associated with a single block produces an $\{r_1,\dots,r_g\}$0-cycle permutation of Riemann sheets, while a decomposition $\{r_1,\dots,r_g\}$1 yields a direct product of cycles of lengths $\{r_1,\dots,r_g\}$2. Thus the monodromy at a nonderogatory $\{r_1,\dots,r_g\}$3 differs from that of a derogatory $\{r_1,\dots,r_g\}$4, where only one pair of sheets is exchanged and one sheet remains fixed (Montag et al., 4 Aug 2025).

In sublattice-symmetric systems, eigenvector coalescence can be strongly path dependent. A mixed-type $\{r_1,\dots,r_g\}$5 at $\{r_1,\dots,r_g\}$6, realized as $\{r_1,\dots,r_g\}$7, may behave along one path as an EP4-like fourfold collapse, with all four eigenvectors converging to the same EP eigenvector, and along another path as an EP2-like collapse with two spectators. This is quantified in (Yang et al., 2022) using the quantum distance

$\{r_1,\dots,r_g\}$8

which shows that different subsets of eigenvectors may converge to different limiting zero modes depending on the approach direction (Yang et al., 2022).

3. Degeneracy manifolds, codimension, and infinitesimal conversion

The geometric organization of derogatory EPs is expressed in terms of degeneracy manifolds. For an $\{r_1,\dots,r_g\}$9 matrix with a single eigenvalue $\sum_i r_i=m$0, the set of matrices realizing a given Jordan type forms the conjugacy manifold

$\sum_i r_i=m$1

where $\sum_i r_i=m$2, $\sum_i r_i=m$3, and $\sum_i r_i=m$4 (Starkov et al., 17 Apr 2026). Perturbations tangent to this orbit are similarity variations and are therefore trivial; perturbations transverse to the orbit are nontrivial and can change the Jordan type or lift the degeneracy (Starkov et al., 17 Apr 2026).

For nonderogatory EPs, Arnold’s miniversal deformation places a complete set of nontrivial parameters entirely in the last row of the Jordan block. In that case, any nontrivial perturbation generically lifts the degeneracy (Starkov et al., 17 Apr 2026). Derogatory EPs have a larger number of nontrivial parameters, i.e. higher codimension, and this excess of transverse directions over the number of independent constraints in the characteristic polynomial makes it possible to remain degenerate while changing Jordan type (Starkov et al., 17 Apr 2026).

The dimension and codimension of these manifolds are explicit. For fixed trace, the ambient space of $\sum_i r_i=m$5 matrices has dimension $\sum_i r_i=m$6. The orbit dimension is

$\sum_i r_i=m$7

and for nilpotent Jordan types

$\sum_i r_i=m$8

where $\sum_i r_i=m$9 are the column lengths of the Young diagram. Hence

$g>1$0

The number of independent nontrivial perturbation parameters equals this codimension (Starkov et al., 17 Apr 2026).

The mechanism of conversion appears already in low dimensions. In the $g>1$1 traceless case, the derogatory type $g>1$2 admits a 4-parameter nontrivial perturbation, while the characteristic polynomial depends only on two combinations. Imposing the pair of constraints

$g>1$3

keeps $g>1$4 and converts $g>1$5 with an infinitesimal perturbation (Starkov et al., 17 Apr 2026). The same logic scales to higher order: a derogatory EP can be kept on the degeneracy manifold while merging blocks and increasing $g>1$6 (Starkov et al., 17 Apr 2026).

An explicit $g>1$7 example realizes $g>1$8. Starting from

$g>1$9

one adds the structured perturbation $\mathbf{p}_0$0 and sets $\mathbf{p}_0$1. Then $\mathbf{p}_0$2 but $\mathbf{p}_0$3, so the nilpotency index is $\mathbf{p}_0$4, and $\mathbf{p}_0$5 is similar to $\mathbf{p}_0$6 for any $\mathbf{p}_0$7. The conversion increases $\mathbf{p}_0$8 from $\mathbf{p}_0$9 to $H$00, and the leading sensitivity strengthens from $H$01 to $H$02 (Starkov et al., 17 Apr 2026).

4. Hierarchies of Jordan types and closure relations

The possible conversions among EPs of fixed algebraic multiplicity are governed by orbit closure. A conversion $H$03 is possible if and only if $H$04 lies in the closure of $H$05, so every neighborhood of a $H$06-point intersects $H$07 (Starkov et al., 17 Apr 2026). This closure criterion induces a partial order on partitions of $H$08, identified with Young-diagram dominance: $H$09 for $H$10. Equivalently, in terms of the conjugate partition $H$11,

$H$12

for all $H$13 (Starkov et al., 17 Apr 2026).

Within this order, $H$14 is the top element and $H$15 the bottom. Moving upward corresponds to infinitesimal perturbations that merge blocks and increase $H$16; moving downward corresponds to splitting blocks (Starkov et al., 17 Apr 2026). For $H$17, the hierarchy is

$H$18

For $H$19, it is

$H$20

Thus $H$21 and $H$22 or directly $H$23 are allowed in principle (Starkov et al., 17 Apr 2026).

The hierarchy is not totally ordered. For $H$24, some EP types are incomparable; for example, $H$25 and $H$26 do not dominate each other (Starkov et al., 17 Apr 2026). This suggests that conversion engineering is constrained not only by order $H$27 but by the fine combinatorics of the partition lattice.

Related work on similarity-induced exceptional structures emphasizes a complementary viewpoint. In multiband systems subject to generalized similarities, multifold $H$28 generically emerge on manifolds of lower-order $H$29, and the reduced codimension caused by spectral symmetries means that “simply counting the number of constraints defining the $H$30s is not sufficient” (Montag et al., 4 Aug 2025). In that setting, lower-order EP manifolds can satisfy the full $H$31 constraints without naively raising codimension, generating a hierarchical network of $H$32 manifolds terminating on $H$33 (Montag et al., 4 Aug 2025). A plausible implication is that the closure hierarchy of (Starkov et al., 17 Apr 2026) and the symmetry-induced manifold hierarchy of (Montag et al., 4 Aug 2025) describe two compatible facets of the same geometric organization.

5. Symmetry constraints, pseudo-Hermiticity, and signed dominance

Pseudo-Hermitian symmetry imposes additional restrictions on which Jordan types are allowed and which conversions are accessible. For

$H$34

with parameter-independent, invertible Hermitian metric $H$35, admissible similarity transformations must preserve $H$36, so $H$37, where $H$38 and $H$39 count the positive and negative eigenvalues of $H$40 (Starkov et al., 17 Apr 2026). A canonical simultaneous form exists in which the Jordan decomposition contains real-eigenvalue blocks $H$41 and complex-eigenvalue pairs $H$42, while the metric decomposes blockwise with signs $H$43 on real-energy blocks and neutral forms on complex pairs (Starkov et al., 17 Apr 2026).

The signature constraint

$H$44

restricts which Jordan block sizes and signs can coexist at real energies (Starkov et al., 17 Apr 2026). In particular, not all partitions are compatible with a fixed $H$45, nonderogatory EPs of maximal order $H$46 are possible only when $H$47, and formation of EPs at real energies requires Krein collisions between eigenvectors of opposite $H$48-sign; collisions among equal-sign modes do not produce EPs (Starkov et al., 17 Apr 2026).

The natural combinatorial objects in this setting are signed Young diagrams. Each row of length $H$49 carries an alternating sign pattern whose rightmost sign is $H$50, and the total numbers of $H$51 and $H$52 boxes equal $H$53. Allowed conversions are governed by signed dominance: if $H$54 is obtained by removing the first $H$55 columns of a signed diagram, and $H$56 denote the numbers of $H$57 boxes remaining, then $H$58 iff

$H$59

for all $H$60 (Starkov et al., 17 Apr 2026). These are strict sub-hierarchies of the unsigned dominance order.

Concrete examples illustrate the selection rules. In signature $H$61, there is no $H$62; the top signed type at $H$63 is $H$64, and types like $H$65 are absent. Hence a fourth-order nonderogatory EP is symmetry-forbidden, and conversions can at best reach $H$66 (Starkov et al., 17 Apr 2026). In $H$67, the unsigned set matches the $H$68 partitions, but the signs prune the allowed paths and types (Starkov et al., 17 Apr 2026).

This pseudo-Hermitian hierarchy dovetails with the broader generalized-similarity program. Pseudo-Hermiticity enforces spectral symmetry $H$69, pseudo anti-Hermiticity enforces $H$70, and self skew-similarity enforces $H$71; any two imply the third (Montag et al., 4 Aug 2025). These symmetries reduce codimensions of $H$72, but they also forbid certain lower-order manifolds outright. Examples include the prohibition of EP3 in 4-band self-skew-symmetric systems and EP5 in 6-band systems with multiple similarities (Montag et al., 4 Aug 2025). The resulting exceptional structures therefore deviate from naive constraint counting, and derogatory manifolds appear only where symmetry permits their block content and spectral placement (Montag et al., 4 Aug 2025).

6. Model realizations, diagnostics, and related formulations

Several explicit models realize derogatory EPs and their conversions. In a non-Hermitian Lieb lattice with Bloch Hamiltonian

$H$73

the spectrum exhibits $H$74-dependent triple degeneracies: for $H$75 they are EP3, along $H$76 or $H$77 they become derogatory $H$78 EPs, and at $H$79 they reduce to a tribolical point (Starkov et al., 17 Apr 2026). Closely related Lieb-lattice constructions identify a minimal FEP of type $H$80 at $H$81, with cusp-like real and imaginary band structures and anisotropic energy contours distinct from a non-derogatory EP3 (Bid et al., 29 Jul 2025).

The FEP framework gives directly evaluable algebraic criteria for such models. Writing a general chiral $H$82 Hamiltonian as

$H$83

the algebraic degeneracy condition at $H$84 is $H$85. The Jordan type is then determined by the ranks of adjugate modes $H$86: EP3 occurs if $H$87, type $H$88 occurs if $H$89 but not all $H$90 vanish, and the tribolical point occurs when $H$91 (Bid et al., 29 Jul 2025).

Liouvillian superoperators provide a second major realization. For a non-Hermitian Hamiltonian $H$92, the no-jump Liouvillian

$H$93

is pseudo-Hermitian with respect to a swap metric $H$94 exchanging the two tensor factors (Starkov et al., 17 Apr 2026). If $H$95 is tuned to a nonderogatory $H$96, then $H$97 at the coalesced Liouvillian eigenvalue $H$98 exhibits a natural derogatory EP with the odd ladder

$H$99

This yields a generic, symmetry-protected source of high-order derogatory EPs (Starkov et al., 17 Apr 2026).

Two concrete Liouvillian examples are spelled out. For an effective dissipative qubit, the $\lambda$00 $\lambda$01 at the $\lambda$02-EP2 point has type $\lambda$03, and the signature is $\lambda$04; since the signed hierarchy forbids $\lambda$05, no $\lambda$06-pseudo-Hermitian perturbation can merge it into a 4-block (Starkov et al., 17 Apr 2026). For an effective dissipative qutrit, the $\lambda$07 $\lambda$08 at the $\lambda$09-EP3 point has type $\lambda$10 and signature $\lambda$11; the signed hierarchy admits an upward conversion to $\lambda$12, so appropriately engineered jump terms could in principle merge the 5- and 3-chains into a 7-chain, changing the sensitivity exponent from $\lambda$13 to $\lambda$14 (Starkov et al., 17 Apr 2026).

Bulk and edge realizations also appear in a non-Hermitian higher-order Dirac semimetal. With suitable intracell nonreciprocal couplings, bulk degeneracies realize fragmentation patterns $\lambda$15, $\lambda$16, and $\lambda$17, while under open boundaries the hinge-state branches form fragmented exceptional lines whose coalescence patterns are controlled by the non-Hermitian skin effect and generalized symmetries (Bid et al., 29 Jul 2025). This establishes that derogatory EPs are not restricted to isolated bulk points but can organize extended edge and hinge structures.

A distinct but related line of work studies symmetry-protected non-defective EPs. In two-band and four-band PT-, CP-, pseudo-Hermitian-, and TRS$\lambda$18-symmetric models, the Hamiltonian can be diagonalizable exactly at an isolated repeated-eigenvalue point, while becoming nondiagonalizable on surrounding exceptional manifolds. In that language, “non-defective EPs” are identified with “derogatory EPs” in a linear-algebraic sense, and the Jordan decomposition becomes unstable along certain approach directions even though the matrix at the degeneracy is diagonalizable (Sayyad et al., 2022). This usage should be distinguished from the defective multi-block notion employed in the conversion, hierarchy, and FEP literature.

7. Conceptual significance and engineering outlook

The recent theory of derogatory EPs shifts attention from isolated maximal Jordan blocks to families of degenerate structures connected by infinitesimal perturbations. The core result is that a derogatory EP can be converted into another EP of the same total algebraic multiplicity without lifting the degeneracy, provided the perturbation is chosen along nontrivial transverse directions that satisfy the relevant characteristic-polynomial constraints (Starkov et al., 17 Apr 2026). Because the leading sensitivity is determined by $\lambda$19, these conversions provide a controlled way to tune non-analytic response while keeping $\lambda$20 fixed (Starkov et al., 17 Apr 2026).

From an engineering perspective, the operative principles are explicit. One targets a desired $\lambda$21, works in the Jordan basis, restricts to nontrivial perturbation directions such as bottom-row and inter-block couplings, and uses codimension to count how many independent controls are needed to stay on the degeneracy manifold while changing type (Starkov et al., 17 Apr 2026). Derogatory EPs are advantageous precisely because their larger codimension provides more knobs than nonderogatory EPs (Starkov et al., 17 Apr 2026). Symmetry can then be exploited or mildly broken depending on the target: pseudo-Hermiticity imposes signed selection rules and forbids some endpoints, but it can also stabilize desired structures and restrict the search space (Starkov et al., 17 Apr 2026).

A broader implication, suggested jointly by the hierarchy-by-closure framework and the similarity-induced manifold picture, is that non-Hermitian degeneracies are best regarded as stratified geometric objects rather than isolated singularities. In this view, nonderogatory EPs, derogatory EPs, and even symmetry-protected diagonalizable degeneracies occupy different strata, with closures, seams, and selection rules determined by Jordan combinatorics and spectral symmetry (Montag et al., 4 Aug 2025). The resulting landscape includes odd ladders in Liouvillians, embedded FEP points on exceptional rings and lines, path-dependent eigenvector collapse in chiral systems, and symmetry-forbidden maximal blocks in pseudo-Hermitian classes (Starkov et al., 17 Apr 2026).

In this sense, derogatory exceptional points are not merely intermediate or incomplete degeneracies. They are structurally rich non-Hermitian singularities whose block partitions, codimensions, closure relations, and symmetry constraints furnish a systematic vocabulary for designing and converting exceptional structures across bulk, boundary, and open-system settings (Starkov et al., 17 Apr 2026).