Characterizing all non-Hermitian degeneracies using algebraic approaches: Defectiveness and asymptotic behavior
Sharareh Sayyad, Grigory A. Starkov
Abstract
The presence of degeneracies plays a crucial role in describing the behavior of non-Hermitian (NH) systems. In these systems, there are two key types of degeneracies: -bolical degeneracies, which are analogous to Hermitian degeneracies, and various forms of exceptional points, each associated with different orders that correspond to sizes of the Jordan blocks. These types of degeneracies may coalesce at the same energy level, forming multi-block degeneracies. To understand how a multi-block degenerate NH system responds to perturbations, one should address how each types of involved degeneracies disperse. In this work, we systematically characterize the asymptotic behavior of all types of multi-block degeneracies in NH systems using a rigorous mathematical formulation. Through a range of examples, we demonstrate that our algebraic approach can facilitate the analysis of NH degeneracies in various settings relevant to experiments.
AI Impact Assessments
(3 models)Scientific Impact Assessment
1. Core Contribution
This paper provides a systematic algebraic framework for characterizing the asymptotic eigenvalue splitting behavior of *all* types of degeneracies in non-Hermitian (NH) systems — not just the commonly studied non-derogatory exceptional points (EPs), but also derogatory (multi-block) degeneracies where multiple Jordan blocks of varying sizes coexist. The key methodological contribution is the application of tropical geometry (specifically tropical polynomials and Newton polygons) to classify how eigenvalues disperse under both generic and non-generic perturbations.
The authors tropicalize the characteristic polynomial of perturbed NH matrices, mapping the problem of finding leading-order eigenvalue dispersions to finding roots of piecewise-linear tropical polynomials. This converts a potentially complex analytic problem into a combinatorial/geometric one. The framework is applied exhaustively to 2×2, 3×3, and 4×4 matrices, cataloguing all possible Jordan structures and their perturbative responses.
2. Methodological Rigor
The mathematical framework is well-grounded. The connection between Newton polygons, tropical polynomials, and the Lidskii-Vishik-Ljusternik theorem is clearly established. The authors demonstrate that tropical roots directly correspond to leading power exponents of eigenvalue dispersions, with multiplicities read off from slope changes — this is made rigorous by referencing Sturmfels' proof.
The exhaustive treatment of all Jordan forms for matrices up to size 4 is thorough. For each case, the authors systematically identify trivial perturbations (those arising from similarity transformations), quotient them out, and then analyze the tropical polynomial under various conditions on perturbation coefficients. The tables (I, II, III) summarizing generic and non-generic results provide a useful reference.
However, some aspects could be stronger. The treatment is restricted to one-parameter perturbations (complex linear subspaces), which the authors acknowledge but don't deeply explore the limitations of. The extension to arbitrary matrix sizes is gestured at but not fully developed — the paper primarily provides a recipe and examples rather than general theorems for arbitrary . The connection between tropical roots and physical observables like braiding/topology is mentioned but not deeply explored.
3. Potential Impact
The paper addresses a genuine need in NH physics. Most prior work on EPs focuses on non-derogatory cases (single Jordan block), while multi-block (derogatory) degeneracies are physically relevant but theoretically underdeveloped. The framework has concrete applications demonstrated across:
The breadth of applications suggests the framework could become a standard tool in NH physics analysis.
4. Timeliness & Relevance
The paper is highly timely. NH physics has seen explosive growth, with EPs playing central roles in sensing, topology, and open quantum systems. The community has largely focused on non-derogatory EPs, but recent work on multi-block/fragmented EPs [44-46] signals growing interest in more complex degeneracy structures. This paper provides the systematic framework that was missing. The connection to tropical geometry, while previously hinted at [50], is developed here into a complete and practical methodology.
The paper also addresses a practical bottleneck: understanding non-generic perturbations. In experiments, perturbations rarely align with "generic" directions, so knowing how different perturbation structures affect splitting is directly useful for sensor design and system characterization.
5. Strengths & Limitations
Strengths:
Limitations:
Overall Assessment:
This is a solid methodological contribution that provides a rigorous and practical framework for a problem of broad interest in NH physics. While not revolutionary in terms of new physical predictions, it fills an important gap by systematizing the analysis of multi-block degeneracies and non-generic perturbations. The diverse examples demonstrate genuine utility across subfields.
Generated Apr 20, 2026
Comparison History (37)
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