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arXiv:2604.16140

Characterizing all non-Hermitian degeneracies using algebraic approaches: Defectiveness and asymptotic behavior

Sharareh Sayyad, Grigory A. Starkov

quant-phmath-phSubmitted Apr 17, 2026• Announced Apr 20, 2026• 0 citations
v1 · Apr 17, 2026v2 · Apr 21, 2026v3 · Jul 10, 2026
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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: $n$-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.

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23 pages, 11 figures

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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: nnn-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.
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.
An exceptional point (EP) is exclusive for non-Hermitian systems and distinct from that at a degeneracy point (DP), supporting intriguing dynamics which can be utilized to probe quantum phase transitions and prepare eigenstates in a Hermitian many-body system. In this paper, we investigate the transition from a DP for a Hermitian system to an EP driven by non-Hermitian terms. We present a theorem on the existence of a transition between a DP and EP for a general system. Specifically, one of twofold degenerate eigenstates of a Hermitian system becomes a coalescing state when a selected non-Hermitian term is added. The obtained EP is robust to the strength of non-Hermitian terms. We illustrate the theorem by an exactly solvable quasi-one-dimensional model, which allows for the existence of a transition between fully degenerate and exceptional spectra driven by non-Hermitian tunnelings in real and kkk spaces, respectively. We also study the EP dynamics for generating coalescing edge modes in Su-Schrieffer-Heeger-like models. This finding reveals the ubiquitous connection between DPs and EPs.
We develop a systematically general theory of one-dimensional (1D) non-Hermitian systems, elaborating on the energy bands, the band degeneracy, and the defectiveness of eigenstates under open boundary conditions. We analyze the band degeneracy and defectiveness of two typical 1D non-Hermitian models. We obtain the unusual presence and absence of the exceptional points in the generalized non-Hermitian Su-Schrieffer-Heeger model under open boundary conditions. Beyond the general theory, we discover that infernal points exist in 1D non-Hermitian systems, where the energy spectra under open boundary conditions converge on some discrete energy values. We analyze two relevant 1D non-Hermitian models with the existence of infernal points. Moreover, we generalize the infernal points to the infernal knots in four-dimensional systems. The general theory and the infernal points of non-Hermitian systems developed in this paper are also valid in Hermitian systems.
Abstract Exceptional points are interesting physical phenomena in non-Hermitian physics at which the eigenvalues are degenerate and the eigenvectors coalesce. In this paper, we find that in projected non-Hermitian two-level systems (sub-systems under projecting partial Hilbert space) the singularities of exceptional points (EPs) is due to basis defectiveness rather than energy degeneracy or state coalescence. This leads to the discovery of extended exceptional points (EEPs). For EEPs, more subtle structures (e.g. the so-called Bloch peach), additional classification, and ‘hidden’ quantum phase transitions are explored. By using the topologically protected sub-space from two edge states in the non-Hermitian Su–Schrieffer–Heeger model as an example, we illustrate the physical properties of different types of EEPs.
The effect of repeated measurements on unitary evolution of a quantum system at regular intervals has been studied picking a specific example of a quantum, particle moving on a lattice of three sites. Dividing the system into two subsystems, we obtain the survival probability of the particle (i.e. particle not being detected) by time evolution followed by projection onto one subsystem. Using this evolution operator we obtain the expression for effective non-Hermitian Hamiltonian whose dynamics could be mapped to a general firs-order perturbation approach. We obtained a good argument between three approaches followed for calculating the survival probability, namely, the first order perturbative approach, the exact non-hermitian evolution and repeated the non-hermitian measurements interrupting the unitary evolution. Further we obtained the non-Hermitian degeneracies called Exceptional Points using the effective Hamiltonian. Finally, the presence of exceptional point is verified by decomposing the initial state into eigenstates of effective Hamiltonian and observing the behavior of the coefficients of decomposition while traversing through each parameter on which the exceptional point depends.
Up to these days, the popular PT-symmetric imaginary cubic oscillator did not find any consistent probabilistic quantum-mechanical interpretation because its Hamiltonian has been shown, by mathematicians, intrinsic-exceptional-point (IEP) singular. In the paper we explain why there is even no reasonable small-perturbation-based regularization of the similar unacceptable (i.e., IEP-singular) quantum models. The explanation is based on a partial formal analogy of the IEP singularity with the conventional exceptional point (EP). What is important is that we are able to construct a simplified NNN by NNN-matrix (and exactly solvable) toy-model Hamiltonian admitting the asymptotic (i.e., high-excitation) EP-related wave-function degeneracy which, in some sense (i.e., in the limit of large NNN) mimics several aspects of its IEP analogue. In this comparison, the difference is that the regularization of the EP singularities is possible (using an ad hoc perturbation of size O(1/N){\cal O}(1/N)O(1/N)) while an analogous regularization of the IEP singularity is not (we have to consider N→∞N \to \inftyN→∞).