Exceptional Point with Non-Zero Residue

• Exceptional point with non-zero residue refers to a non-Hermitian degeneracy where eigenvalues coalesce and the leading singular coefficient remains nonzero.
• This residue governs Puiseux-law resonance splitting, quantifies spectral response strength, and influences observable scattering corrections in experimental setups.
• Residue-based perturbation theory employs generalized Wigner–Smith operators to bridge mathematical defectiveness and physical response in open systems.

An exceptional point with non-zero residue is a non-Hermitian spectral degeneracy in which the defective singularity is accompanied by a nonvanishing coefficient of the leading singular term in a Green’s function, scattering-derived quantity, or response function. In the recent literature, that coefficient is not merely a bookkeeping device: it controls Puiseux-law resonance splitting, quantifies spectral response strength, and can appear directly as an experimentally measurable correction to standard dispersion relations (Wang et al., 7 Aug 2025, Wiersig, 2023, Liu, 30 Apr 2026). At the same time, the literature also contains explicit zero-residue constructions, so “exceptional point with non-zero residue” denotes a specific analytic realization of EP physics rather than a tautological synonym for defectiveness (Andrianov et al., 2011).

1. Defining the singular object

At an exceptional point (EP), a non-Hermitian operator becomes defective: the algebraic multiplicity exceeds the geometric multiplicity, and eigenvalues and eigenvectors coalesce. In perturbative and resolvent-based treatments, the relevant singular object is not a simple Hermitian projector but a higher-order pole or branch-point contribution. For an order-$n$ EP, Jan Wiersig formulates the Green’s function near the EP as

$G(E) \sim \frac{W}{(E - E^{(0)})^n},$

with $W$ a rank-1 operator, and defines the spectral response strength as

$$\xi = \|W\| = \sqrt{\operatorname{Tr}[W^\dagger W]}.$$

In this language, a non-zero residue means that the coefficient $W$ of the leading divergent term does not vanish, so the EP produces enhanced sensitivity to generic perturbations (Wiersig, 2023).

A closely related scattering formulation appears in generalized Wigner–Smith theory. There, the residue is extracted not from $G(E)$ directly but from a function of the generalized Wigner–Smith operator evaluated at the degenerate frequency. The key claim is that this residue determines the amplitude and orientation of the EP splitting in the complex-frequency plane (Wang et al., 7 Aug 2025).

A third realization arises in linear response. In a PT-symmetric open dimer, when a reflection-coefficient pole enters the upper half-plane across the EP, the residue of that pole fixes the magnitude and Lorentzian profile of the Kramers–Kronig residual. Kejun Liu’s analysis makes the residue itself an observable of the causality-breaking transition (Liu, 30 Apr 2026).

Framework	Singular quantity	Role of the non-zero residue
Generalized Wigner–Smith theory	Residue of $[i\,\mathrm{tr}\,\mathbf{Q}_{\alpha}]^{1/N}$ at $\omega_p$	Sets size and orientation of Puiseux splitting
Green’s-function theory	Coefficient $W$ of $(E-E^{(0)})^{-n}$	Determines spectral response strength $G(E) \sim \frac{W}{(E - E^{(0)})^n},$0
Open-system reflection response	Residue $G(E) \sim \frac{W}{(E - E^{(0)})^n},$1 of an upper-half-plane pole	Fixes Lorentzian Kramers–Kronig residual

This comparison indicates that “residue” is framework-specific, but across these formulations it serves the same structural role: it is the leading coefficient that converts defectiveness into a quantitatively predictive response law.

2. Residue-based perturbation theory from generalized Wigner–Smith operators

The most explicit modern formulation is the residue-based perturbation theory developed for diabolic points (DPs) and exceptional points (EPs) using generalized Wigner–Smith (GWS) operators (Wang et al., 7 Aug 2025). The GWS operator is defined by

$G(E) \sim \frac{W}{(E - E^{(0)})^n},$2

where $G(E) \sim \frac{W}{(E - E^{(0)})^n},$3 is the scattering matrix and $G(E) \sim \frac{W}{(E - E^{(0)})^n},$4 is an external parameter.

At an EP, small generic perturbations lead to the Puiseux expansion

$G(E) \sim \frac{W}{(E - E^{(0)})^n},$5

with $G(E) \sim \frac{W}{(E - E^{(0)})^n},$6 the degenerate frequency, $G(E) \sim \frac{W}{(E - E^{(0)})^n},$7 the Jordan-block size, and $G(E) \sim \frac{W}{(E - E^{(0)})^n},$8. The central result is that the coefficient $G(E) \sim \frac{W}{(E - E^{(0)})^n},$9 is expressed through a residue evaluated at the EP: $W$0 The paper states that at an EP the trace of the GWS operator has a pole of order $W$1 at the degenerate frequency $W$2, and that the residue is non-zero at an EP. That non-zero residue is what enables the characteristic non-analytic response. It provides both the size and the orientation of the splitting in the complex plane, so it is the coefficient that turns the abstract Jordan structure into a measurable scattering prediction (Wang et al., 7 Aug 2025).

The significance of the formulation lies in its data economy. The theory is presented as accurately predicting degenerate resonance splitting using only scattering data, and the paper reports validation by analytic Hamiltonian models and numerical electromagnetic simulations, with excellent agreement across a range of cases. This suggests a route from EP identification to precision tuning and inverse design without reconstructing the full internal Hamiltonian (Wang et al., 7 Aug 2025).

3. Why non-zero residue distinguishes EPs from ordinary degeneracies

The contrast with a diabolic point is fundamental. At a DP, the system is diagonalizable, the algebraic multiplicity equals the geometric multiplicity, and eigenfrequencies split linearly and independently under perturbation. Each mode has its own ordinary Taylor expansion. The GWS-based residue does not encode a collective fractional-power law there; one must treat each mode separately and analyze the residues individually (Wang et al., 7 Aug 2025).

At an EP, by contrast, the non-zero residue is tied to a higher-order pole and a collective Puiseux response. The degeneracy does not resolve into independent analytic branches. Instead, the degenerate resonance splits into $W$3 branches with scaling $W$4. The residue’s phase sets the angular disposition of those branches, and its magnitude sets their scale (Wang et al., 7 Aug 2025).

A common misconception is that any spectral degeneracy with large sensitivity is “exceptional” in the same sense. The residue-based comparison refutes that simplification. DPs can be sensitive, but their residues only capture ordinary analytic shifts. EPs are distinguished by a non-analytic branch structure, and the non-zero residue is precisely the coefficient governing that non-analyticity (Wang et al., 7 Aug 2025).

Another misconception is that an EP with diverging response strength must exhibit diverging eigenvalues. Wiersig’s analysis shows the opposite can occur: the spectral response strength may diverge as one approaches a higher-order EP along an exceptional surface, while the energy eigenvalues themselves remain finite because of subtle cancellations (Wiersig, 2023).

4. Spectral response strength, higher-order EPs, and Petermann-factor divergence

Wiersig extends EP response theory to the general case in which the Hilbert-space dimension exceeds the order of the EP (Wiersig, 2023). For a perturbation $W$5, the eigenvalue splitting near an order-$W$6 EP obeys

$W$7

where $W$8 is the spectral response strength. The coefficient $W$9 entering the Green’s-function pole is obtained by residue calculus: $$\xi = \|W\| = \sqrt{\operatorname{Tr}[W^\dagger W]}.$$0 The non-zero residue criterion is explicit in this framework: if $$\xi = \|W\| = \sqrt{\operatorname{Tr}[W^\dagger W]}.$$1, then $$\xi = \|W\| = \sqrt{\operatorname{Tr}[W^\dagger W]}.$$2, and the EP exhibits enhanced sensitivity to perturbations; if $$\xi = \|W\| = \sqrt{\operatorname{Tr}[W^\dagger W]}.$$3 vanishes, then $$\xi = \|W\| = \sqrt{\operatorname{Tr}[W^\dagger W]}.$$4, which the paper characterizes as a non-generic scenario (Wiersig, 2023).

The same work connects this residue-controlled response to the Petermann factor. Near the EP,

$$\xi = \|W\| = \sqrt{\operatorname{Tr}[W^\dagger W]}.$$5

so the divergence of Petermann factors near EPs is linked to the same analytic coefficient that governs the spectral response strength (Wiersig, 2023). This places non-zero residue at the center of a broader non-Hermitian sensitivity theory: it controls perturbative splitting, bounds response amplitude, and explains the divergence of eigenvector nonorthogonality measures.

The paper also presents a numerical contour-integration scheme for extracting $$\xi = \|W\| = \sqrt{\operatorname{Tr}[W^\dagger W]}.$$6, and reports errors below $$\xi = \|W\| = \sqrt{\operatorname{Tr}[W^\dagger W]}.$$7. That result is technically significant because it shows that residue calculus is not confined to low-dimensional toy Hamiltonians; it can be implemented robustly in larger systems and on exceptional surfaces approaching higher-order EPs (Wiersig, 2023).

5. Observable consequences in scattering, causality, and radiative coupling

In open systems, a non-zero residue can be read out directly from linear response. In the PT-symmetric open dimer studied by Kejun Liu, a single pole of the reflection coefficient migrates into the upper half-plane as the gain-loss parameter crosses the exceptional point. At the same transition, the Blaschke winding number jumps from $$\xi = \|W\| = \sqrt{\operatorname{Tr}[W^\dagger W]}.$$8 to $$\xi = \|W\| = \sqrt{\operatorname{Tr}[W^\dagger W]}.$$9, and the standard Kramers–Kronig reconstruction acquires a Lorentzian residual fixed by the pole residue (Liu, 30 Apr 2026). The corrected dispersion relation is

$W$0

so the residue $W$1 appears as an additive, measurable term rather than as an abstract spectral coefficient. The same work reports the scaling

$W$2

for the $W$3-norm of the Kramers–Kronig violation in the single-port geometry, with the violation strongest at threshold and weaker deeper in the broken phase (Liu, 30 Apr 2026).

A different open-system route to EPs with non-zero residue is provided by retardation-induced radiative coupling. In an optical dimer of spheres, exceptional points arise not only in PT-symmetric gain-loss configurations but also in symmetric configurations of identical spheres, because the coupling is frequency-dependent and complex. The paper states that in both PT-symmetric and symmetric cases the residue at the exceptional point can be nonzero, and attributes this to the radiative, open-system origin of the non-Hermiticity and the nonlinear frequency dependence of the coupling (Dmitriev et al., 2023). This broadens the notion of “EP with non-zero residue” beyond balanced gain and loss: open radiative channels alone can generate the requisite analytic structure.

These examples show that non-zero residue is not merely a spectral diagnostic. It can govern directly observable lineshapes, residual dispersion, and gain-driven or radiation-driven breakdowns of standard response relations.

6. Mathematical subtleties, completeness, and scope

The mathematical status of the residue at an EP is subtler than the simple-pole residue of a Hermitian isolated eigenvalue. A review of non-Hermitian topology emphasizes that the biorthogonal projector

$W$4

becomes ill-defined at an EP because the denominator vanishes as the left and right eigenvectors become self-orthogonal; correspondingly, the Green’s function acquires a square-root or higher-root singularity rather than an ordinary simple pole (Ding et al., 2022). This indicates that “non-zero residue at an EP” should be interpreted through the appropriate higher-order or branch-point formalism, not by naive transplantation of Hermitian spectral theory.

Continuous-spectrum models make the distinction explicit. In Sokolov’s construction of resolutions of identity for non-Hermitian Hamiltonians with exceptional points in the continuous spectrum, there are cases in which the residue at the exceptional point vanishes, making contour deformation around the point immaterial. In more general settings, however, non-zero residues generate explicit correction terms associated with the Jordan block and its chain of associated functions, some normalizable and some not (Andrianov et al., 2011). The presence or absence of a non-zero residue therefore affects completeness relations, contour prescriptions, and the role of associated functions in the rigged Hilbert space.

Not all EP analyses compute residues explicitly. In the generalized non-Hermitian Rice–Mele model with balanced gain and loss and next-nearest-neighbor hopping, exceptional points are identified numerically via the condition number of the eigenvector matrix and confirmed by Jordan decomposition, while the paper does not provide an explicit calculation of the Green’s-function or spectral-projector residue at the EP (Martinez-Strasser et al., 23 Jun 2026). This clarifies the scope of the concept: non-zero residue is a powerful analytic classifier and response coefficient, but EP detection can also proceed through Jordan structure, condition numbers, winding numbers, or dynamical probes.

Taken together, these works establish a precise meaning for an exceptional point with non-zero residue. It is an EP for which the leading singular coefficient in the relevant analytic object does not vanish, and that coefficient governs perturbative splitting, response amplification, or observable residual structure. The recent shift from abstract defectiveness to residue-based quantification has made EP sensitivity calculable from scattering data, robustly computable by contour integration, and directly testable in open-system experiments (Wang et al., 7 Aug 2025, Wiersig, 2023, Liu, 30 Apr 2026).