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arxiv: 2604.16139 · v2 · submitted 2026-04-17 · 🪐 quant-ph · math-ph· math.MP

Converting non-Hermitian degeneracies of any order: Hierarchies of exceptional points and degeneracy manifolds

Grigory A. Starkov , Sharareh Sayyad This is my paper

Pith reviewed 2026-05-10 08:16 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords non-Hermitian systemsexceptional pointsJordan blocksdegeneraciespseudo-Hermitian symmetryperturbationseigenvalue coalescence
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The pith

Infinitesimal perturbations convert a derogatory exceptional point into one with a different Jordan-block structure while preserving the total degeneracy order.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that in non-Hermitian systems a derogatory exceptional point, marked by several Jordan blocks sharing one eigenvalue, can be transformed by specific tiny perturbations into an exceptional point whose Jordan blocks are arranged differently yet still sum to the same total algebraic multiplicity. One outcome of such a conversion is an increase in the size of the single largest Jordan block, which directly raises how strongly the eigenspectrum reacts to small parameter shifts. The authors enumerate every possible conversion of this kind and organize them into hierarchies of same-order degeneracies, first without any symmetry and then under the additional constraint of pseudo-Hermitian symmetry.

Core claim

A derogatory exceptional point of total order n can be turned by infinitesimal perturbations into any other exceptional point whose Jordan blocks also sum to algebraic multiplicity n, including the fully non-derogatory case of a single n-by-n block, and the set of all such reachable structures forms a hierarchy that can be classified both in the generic case and when pseudo-Hermitian symmetry is imposed.

What carries the argument

The partition of Jordan-block sizes attached to a shared eigenvalue at an exceptional point; infinitesimal perturbations are shown to re-partition those sizes without changing their sum.

If this is right

  • The sensitivity of the eigenvalue spectrum to parameter variation can be increased simply by enlarging the biggest Jordan block through such a conversion.
  • Hierarchies of same-order degeneracies supply a systematic map for choosing which degeneracy structure to target in device design.
  • The same conversion rules apply both to generic non-Hermitian systems and to systems constrained by pseudo-Hermitian symmetry.
  • Engineering applications that rely on strong parameter response, such as sensors or amplifiers, can exploit the enlarged Jordan block after conversion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The conversion rules could be used to switch a device between low- and high-sensitivity regimes on demand by applying a controlled perturbation.
  • The hierarchies may intersect with PT-symmetry breaking thresholds, offering a route to combine degeneracy engineering with gain-loss balancing.
  • The same perturbation analysis might be applied to time-periodic or driven non-Hermitian systems to see whether the hierarchies survive Floquet extensions.

Load-bearing premise

That there exist infinitesimal perturbations able to change the sizes of the Jordan blocks of a derogatory exceptional point while exactly preserving the total algebraic multiplicity of the eigenvalue, and that the enumeration of all such changes is complete both with and without pseudo-Hermitian symmetry.

What would settle it

A concrete derogatory exceptional point for which every possible infinitesimal perturbation either leaves the Jordan-block partition unchanged or alters the total algebraic multiplicity.

Figures

Figures reproduced from arXiv: 2604.16139 by Grigory A. Starkov, Sharareh Sayyad.

Figure 1
Figure 1. Figure 1: Projection of the degeneracy surface 𝑑 2 22 + 𝑑12𝑑21 = 0 on the space −𝑖𝑑22, 𝑑12, 𝑑21 ∈ R. The red point marks the diabolical point. Requiring the matrix to be traceless results in setting 𝜆 = 0. The apex of the cone, at 𝑑12 = 𝑑21 = 𝑑22 = 0, corresponds to a diabolical point, while the remain￾ing part of the cone represents second-order EPs. To demonstrate this explicitly, we note that all the points of th… view at source ↗
Figure 2
Figure 2. Figure 2: Manifold 𝑀3 in the vicinity of a type-(2, 1) EP as described by Eq. (34). All parameters have been restricted to be real, and 𝛿𝐽2,1 has been projected out as explained in the main text. The resulting equation of the surface is given in Eq. (35). The red point marks the type-(2, 1) EP. lift the triple degeneracy by requiring {︂ 𝛿𝐽2,1 + 𝛿𝐽2 3,3 = 0, 𝛿𝐽2,3𝛿𝐽3,1 − 𝛿𝐽2,1𝛿𝐽3,3 = 0. (34) A non-trivial perturbatio… view at source ↗
Figure 3
Figure 3. Figure 3: The absolute values of the spectrum of the model in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Hierarchy of EPs with algebraic multiplicity [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Hierarchy of EPs with algebraic multiplicity [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Hierarchy of EPs with algebraic multiplicity [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Hierarchy of EPs with algebraic multiplicity [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: Hierarchy of EPs with algebraic multiplicity [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Example of a signed Young-diagram corre [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Hierarchy of degeneracies with algebraic multiplicity [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Hierarchy of degeneracies with algebraic mul [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Hierarchy of degeneracies with algebraic multiplicity [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
read the original abstract

The emergence of various types of degeneracies plays a crucial role in optimizing and engineering different physical phenomena in non-Hermitian physics. In our work, we focus on the derogatory Exceptional Points (EPs), which are characterized by multiple Jordan blocks corresponding to the same eigenvalue. We demonstrate that, under certain infinitesimal perturbations, a derogatory EP can be converted into an EP of different structure without varying the total order of degeneracy. In particular, such conversion can increase the size of the largest Jordan block and, hence, the sensitivity of the eigenspectrum to parameter variation, which is an important feature for practical applications. Furthermore, by analyzing all possible conversions, we introduce hierarchies of degeneracies of the same order that appear when perturbing non-Hermitian systems. We systematically explore hierarchies in the absence of any symmetry and when pseudo-Hermitian symmetry is present. Our study facilitates engineering various degeneracies of non-Hermitian systems, paving the way to extending the implications of non-Hermitian physics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript develops a perturbation-theoretic framework for converting derogatory exceptional points (EPs) of arbitrary order in non-Hermitian systems into EPs with altered Jordan-block partitions while exactly preserving the total algebraic multiplicity of the eigenvalue. Using the effective operator restricted to the degenerate subspace, the authors classify all admissible infinitesimal perturbations that realize such conversions, both in the generic (no-symmetry) case and under pseudo-Hermitian symmetry. They show that certain conversions can enlarge the largest Jordan block, thereby increasing the parametric sensitivity of the spectrum, and organize the reachable structures into explicit hierarchies of degeneracy manifolds.

Significance. If the constructions hold, the work supplies a systematic, exhaustive classification of EP conversions that directly enables engineering of higher-sensitivity degeneracies without increasing the overall order. The explicit case-by-case analysis of the effective operator on the degenerate subspace and the treatment of both symmetry-free and pseudo-Hermitian settings constitute concrete, usable results for non-Hermitian device design. These hierarchies provide a new organizational principle for degeneracy manifolds that could guide experimental realizations in optics, acoustics, and quantum sensors.

major comments (2)
  1. [§3.1] §3.1, effective-operator construction: the claim that every admissible coupling between existing Jordan chains is realized by an infinitesimal perturbation of the full non-Hermitian matrix is central to the hierarchy; the manuscript should exhibit the explicit lifting from the effective matrix back to a concrete perturbation of the original operator for at least one order-4 example to confirm that no hidden constraints arise from the non-degenerate complement.
  2. [§4.2] §4.2, pseudo-Hermitian case: the preservation of the pseudo-Hermitian symmetry under the chosen perturbations is asserted after the effective-matrix analysis; an explicit verification that the symmetry condition on the full Hamiltonian is compatible with the block-structure conversion (rather than being imposed only on the effective operator) is required, because any mismatch would restrict the reachable hierarchies.
minor comments (3)
  1. Notation: the symbol for the effective perturbation matrix is introduced without a consistent subscript indicating the order of the EP; adopting a uniform notation (e.g., V_k for order-k) would improve readability across sections.
  2. Figure 3: the arrows indicating allowed conversions between partitions are difficult to trace for orders greater than 3; adding a small table of admissible block-size changes next to the diagram would clarify the hierarchy.
  3. References: several recent works on higher-order EPs in PT-symmetric systems are cited only in passing; a brief comparison paragraph in the introduction would better situate the new hierarchies relative to existing literature.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation proceeds via explicit case-by-case perturbation analysis of the effective operator restricted to the degenerate subspace, classifying admissible Jordan-block conversions that preserve algebraic multiplicity. This rests on standard linear-algebraic constructions for non-Hermitian matrices and does not reduce any claimed hierarchy or conversion to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation. The increase in largest block size follows directly from admissible inter-chain couplings without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard facts from linear algebra for non-Hermitian operators and on the applicability of perturbation theory to exceptional points; no free parameters or new entities are introduced.

axioms (2)
  • standard math Non-Hermitian matrices admit a Jordan canonical form that may contain multiple blocks sharing the same eigenvalue (derogatory case).
    This is the mathematical definition of the derogatory exceptional points that the paper studies.
  • domain assumption Infinitesimal perturbations can alter the Jordan-block structure of a degenerate eigenvalue while leaving its algebraic multiplicity unchanged.
    This assumption is required for the claimed conversion to be possible.

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    quant-ph 2026-04 unverdicted novelty 5.0

    An algebraic framework systematically characterizes the asymptotic dispersion of all multi-block non-Hermitian degeneracies, including n-bolical points and exceptional points of various orders.

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    Substituting this into the second equation gives the equation of the projected surface: 𝛿𝐽2,3𝛿𝐽3,1 +𝛿𝐽 3 3,3 = 0.(35) The resulting surface is shown in Fig. 2. C.𝑛×𝑛matrices In the case of an𝑛×𝑛matrix𝐻, the corresponding characteristic polynomial is ℱ𝜆 =𝜆 𝑛 −𝑝 1𝜆𝑛−1 −𝑝 2𝜆𝑛−2 −. . .−𝑝 𝑛,(36) where the coefficients𝑝 𝑘 can be found using Faddeev-LeVierre met...

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