Exceptional Deficiency in Non-Hermitian Systems

• Exceptional Deficiency (ED) is a non-Hermitian singularity where two large N-dimensional eigenspaces fully merge, resulting in extensive Jordan block coalescence.
• ED exhibits unique spectral engineering, anomalous non-Hermitian skin effects, and hybrid propagation-localization dynamics, as demonstrated in coupled lattice models and active mechanical systems.
• Its rigorous algebraic and spectral formulation, along with off-ED sensitivity enhancement, opens new avenues for broadband sensing and robust mode control in complex systems.

Exceptional deficiency (ED) is a non-Hermitian singularity in which two eigenspaces of identical but arbitrarily large dimension completely coalesce and their spectral continua coincide. Introduced as a generalization of the exceptional point (EP), ED replaces isolated level coalescence by high-dimensional coalescence: in a $2N\times 2N$ system, the Hamiltonian loses exactly $N$ dimensions of its would-be Hilbert space and its Jordan normal form consists of $N$ Jordan blocks of size $2$ (Li et al., 16 Apr 2025). Subsequent work connected ED to anomalous manifestations of the non-Hermitian skin effect (NHSE), synergistic localization-propagation dynamics, and a sensing regime in which strict operation at ED preserves exponential sensitivity but amplifies noise with a fractional-order law, motivating controlled operation slightly away from the singular point (Li et al., 3 Jun 2026).

1. Conceptual placement within non-Hermitian singularities

An EP is a non-Hermitian singularity at which a small number of eigenvalues and their eigenvectors coalesce, rendering the Hamiltonian defective by $O(1)$ dimension and reducible to a single Jordan block of size $2$ or higher. ED is presented as its direct generalization to two entire eigenspaces of arbitrarily large dimension. At ED, not just two levels but two $N$-dimensional subspaces align and their spectra coincide over a continuum of energies (Li et al., 16 Apr 2025).

This development sits alongside a broader reorientation in non-Hermitian theory toward defectiveness rather than degeneracy alone. In the two-level setting, Wang et al. argued that the universal feature of exceptionality is basis defectiveness rather than energy degeneracy or state coalescence, and introduced general exceptional points (GEPs) classified as M-GEPs, B-GEPs, and H-GEPs according to which non-unitary similarity becomes singular (Wang et al., 2021). ED differs in scale and structure: it is not a two-level singularity with a defective matrix basis, but a many-state coalescence in which entire eigenspaces coincide (Li et al., 16 Apr 2025).

A plausible implication is that ED extends the organizing role of defectiveness from isolated Jordan singularities to continuum-scale spectral structures. In that sense, ED is not merely a higher-order EP, but a defect-induced regime in which spectral coincidence, localization, and dynamics become collective rather than level-specific.

2. Algebraic and spectral formulation

The canonical algebraic setting for ED is the block-triangular Hamiltonian

$$H \;=\; \begin{pmatrix} h_1 & K\ 0 & h_2 \end{pmatrix}, \qquad h_{1,2}\in\mathbb{C}^{N\times N},\;\;K\in\mathbb{C}^{N\times N}.$$

Its characteristic polynomial factorizes as

$$\det\bigl(H - E\,I_{2N}\bigr) \;=\;\det\bigl(h_1 - E\,I_N\bigr)\;\det\bigl(h_2 - E\,I_N\bigr).$$

ED occurs if and only if

$\sigma(h_1)\;=\;\sigma(h_2),$

so that for every eigenvalue $N$0 in the common spectrum both blocks are singular (Li et al., 16 Apr 2025).

Under this condition, all of the right-eigenvectors of $N$1 lie in an $N$2-dimensional invariant subspace, with no nontrivial eigenvectors supported in the second block. The total defect of $N$3 is therefore $N$4, and the Jordan decomposition takes the form

$N$5

with

$N$6

Each $N$7 coalesces one one-dimensional eigenvector of $N$8 and one of $N$9, so the loss of geometric multiplicity is extensive rather than isolated (Li et al., 16 Apr 2025).

This algebraic criterion is unusually transparent. The off-diagonal block $N$0 does not alter the factorized spectrum, but once spectral equality is imposed it forces complete eigenspace coalescence. The singularity is therefore controlled jointly by one-way coupling and exact spectral alignment.

3. Lattice realizations and spectral engineering

A concrete realization of ED uses two coupled Su–Schrieffer–Heeger-type chains, labeled $N$1 and $N$2, with one chain Hermitian and the other non-Hermitian. In momentum space, the Bloch Hamiltonians are

$N$3

where

$N$4

and

$N$5

These parameters encode intra-cell hoppings, detuning, and the non-Hermitian asymmetry $N$6 (Li et al., 16 Apr 2025).

Under periodic boundary conditions (PBC), one finds

$N$7

independent of $N$8. Under open boundary conditions (OBC), each chain separately has real spectrum. By tuning

$N$9

one enforces

$2$0

so the entire continua of OBC bands coincide and the block-triangular structure forces ED (Li et al., 16 Apr 2025).

This spectral engineering is central to the physical meaning of ED. The relevant coincidence is not a discrete degeneracy at selected momenta, but equality of full OBC spectra. The singularity therefore survives as a continuum-scale condition in finite lattices.

4. NHSE, anomalous bulk-edge behavior, and ED dynamics

In non-Hermitian lattices, the NHSE is governed by nonzero winding of the PBC spectrum around a reference energy, and the conventional bulk-edge correspondence states that nontrivial spectral winding implies extensive skin modes under OBC. ED produces a marked departure from that pattern. Systems I and II in the one-way-coupled SSH setting have the same PBC spectrum and the same OBC spectrum, but completely opposite OBC localization (Li et al., 16 Apr 2025).

At ED, non-Bloch band theory gives the factorized characteristic equation

$2$1

Imposing a common OBC energy $2$2 yields

$2$3

System I is dominated by $2$4, so all modes are extended, whereas system II is dominated by $2$5, so all modes are skin modes. ED therefore transcends the usual winding-skin relation (Li et al., 16 Apr 2025).

ED also reshapes real-time evolution. For OBC Hamiltonian $2$6,

$2$7

and

$2$8

In system I, an initial excitation on the non-Hermitian chain $2$9 produces transient skin growth together with eventual propagation through the nondefective Hermitian subspace; this was termed Skin-Effect-Amplified Propagation (SEAP). In system II, an excitation on chain $O(1)$0 partly converts into skin modes on chain $O(1)$1, amplifies upon reflection, and eventually clings to an edge with stronger localization than a pure NHSE would produce; this was termed Propagation-Enhanced Skin Effect (PESE) (Li et al., 16 Apr 2025).

These dynamical signatures show that ED is not only a spectral singularity. It is also a mechanism for hybridizing propagation and localization through defective-eigenspace structure.

5. Experimental realization in active mechanical lattices

ED has been experimentally observed using active mechanical lattices built from eight-cell double-chain arrays of torsional oscillators. Each site is a servo-controlled rotor with natural frequency approximately $O(1)$2 Hz. Reciprocal spring couplings implement the Hermitian SSH chain, while nonreciprocal one-way couplings are realized by real-time feedback driving of selected motors (Li et al., 16 Apr 2025).

In steady-state measurements, a single site is driven harmonically at frequency $O(1)$3, and the integrated amplitude

$O(1)$4

is recorded. The experiments distinguish extended localization from skin localization in agreement with the theoretical predictions. Transient wavepacket measurements also reproduce the SEAP and PESE signatures despite realistic dissipation and parameter detuning (Li et al., 16 Apr 2025).

The mechanical implementation is significant because it demonstrates that ED is not confined to idealized matrix models. The phenomena associated with continuum-scale eigenspace coalescence, including anomalous NHSE behavior and mixed propagation-localization dynamics, persist in a platform with finite size, losses, and actuation constraints.

6. Off-ED sensing, disorder tolerance, and noise regularization

A later development recast ED as a sensing resource in a double-chain Hatano–Nelson model with unidirectional interlayer coupling. In the real-space basis $O(1)$5, the unperturbed ED Hamiltonian is

$O(1)$6

with a weak reverse coupling introduced through

$O(1)$7

The chain blocks satisfy

$O(1)$8

$O(1)$9

with $2$0 and $2$1 for perfect OBC spectral alignment. The ED limit is $2$2, whereas the Off-ED regime corresponds to small but finite $2$3 (Li et al., 3 Jun 2026).

Under OBCs, the system at ED has a macroscopically degenerate complex spectrum and a pronounced NHSE. The sensitivity to a target perturbation in the reverse channel remains exponential in system size,

$2$4

or equivalently

$2$5

The abstract reports that this exponential sensitivity is robust across a six-order-of-magnitude detuning range. In the Off-ED regime,

$2$6

the exponential scaling is preserved, albeit with a slightly smaller prefactor than at strict ED (Li et al., 3 Jun 2026).

The central sensing issue is the sensitivity-noise trade-off. Noise is introduced as

$2$7

and the signal-to-noise ratio is defined from the signal-induced and noise-induced spectral splittings. At strict ED, the $2$8 Jordan blocks produce fractional-order noise amplification,

$2$9

The SNR then saturates at an $N$0-independent suboptimal plateau once the perturbation leaves the tiny linear regime. Off ED, a finite $N$1 lifts each Jordan block into two simple eigenvalues separated by $N$2, yielding

$N$3

This restores a linear scaling law and produces an SNR enhancement of several orders of magnitude while preserving the exponential sensitivity scaling (Li et al., 3 Jun 2026).

Robustness was analyzed through diagonal disorder,

$N$4

with $N$5 a Bernoulli$N$6 mask and $N$7 uniform. Eigenspace alignment was quantified by the cosine similarity

$N$8

At strict ED, $N$9 exactly, independent of large disorder mask $$H \;=\; \begin{pmatrix} h_1 & K\ 0 & h_2 \end{pmatrix}, \qquad h_{1,2}\in\mathbb{C}^{N\times N},\;\;K\in\mathbb{C}^{N\times N}.$$0 up to a critical $$H \;=\; \begin{pmatrix} h_1 & K\ 0 & h_2 \end{pmatrix}, \qquad h_{1,2}\in\mathbb{C}^{N\times N},\;\;K\in\mathbb{C}^{N\times N}.$$1; numerically, up to disorder strength $$H \;=\; \begin{pmatrix} h_1 & K\ 0 & h_2 \end{pmatrix}, \qquad h_{1,2}\in\mathbb{C}^{N\times N},\;\;K\in\mathbb{C}^{N\times N}.$$2, even at $$H \;=\; \begin{pmatrix} h_1 & K\ 0 & h_2 \end{pmatrix}, \qquad h_{1,2}\in\mathbb{C}^{N\times N},\;\;K\in\mathbb{C}^{N\times N}.$$3, the ensemble-averaged $$H \;=\; \begin{pmatrix} h_1 & K\ 0 & h_2 \end{pmatrix}, \qquad h_{1,2}\in\mathbb{C}^{N\times N},\;\;K\in\mathbb{C}^{N\times N}.$$4 with small variance. Beyond $$H \;=\; \begin{pmatrix} h_1 & K\ 0 & h_2 \end{pmatrix}, \qquad h_{1,2}\in\mathbb{C}^{N\times N},\;\;K\in\mathbb{C}^{N\times N}.$$5, eigenspace coalescence breaks down. In the Off-ED regime, $$H \;=\; \begin{pmatrix} h_1 & K\ 0 & h_2 \end{pmatrix}, \qquad h_{1,2}\in\mathbb{C}^{N\times N},\;\;K\in\mathbb{C}^{N\times N}.$$6 drops sharply as disorder grows, which was described as algebraic fragility of perfect coalescence. The corresponding restructuring threshold is

$$H \;=\; \begin{pmatrix} h_1 & K\ 0 & h_2 \end{pmatrix}, \qquad h_{1,2}\in\mathbb{C}^{N\times N},\;\;K\in\mathbb{C}^{N\times N}.$$7

beyond which the NHSE is quenched and exponential sensitivity is lost (Li et al., 3 Jun 2026).

The broader significance of ED was framed in two directions. First, because ED involves continuous spectral coalescence, it offers broadband analogues of EP-enhanced sensing, mode control, and lasing. Second, future work was identified in exploring ED in more general coupling geometries, classifying higher-dimensional topological invariants, and realizing ED in photonics, acoustics, electronics, and other platforms (Li et al., 16 Apr 2025).