Andreev Exceptional Points in Superconducting Systems

• Andreev exceptional points are non-Hermitian degeneracies where both eigenvalues and eigenvectors coalesce in superconducting hybrid systems.
• They arise from dissipative couplings in open superconducting devices, leading to complex energy spectra and measurable resonance width bifurcation.
• These points delineate transitions between real-gap and imaginary-gap phases and are linked to Majorana physics and non-Hermitian topological behavior.

Andreev exceptional points are non-Hermitian spectral degeneracies in the complex spectrum of Andreev bound states or Andreev quasi-bound states of open superconducting hybrid systems. In these systems, coupling to dissipative leads or normal reservoirs renders the effective Bogoliubov–de Gennes or Green-function description non-Hermitian, so that Andreev levels acquire complex energies and can undergo coalescence of both eigenvalues and eigenvectors. Recent work identifies such exceptional points in multiterminal superconductor-normal junctions, in trivial normal–superconductor junctions hosting Majorana dark states, in non-Hermitian $p$-wave Josephson junctions, and in Josephson junctions formed by minimal Kitaev chains, with the superconducting phase difference, dissipation strength, magnetic-field orientation, and non-Hermiticity distribution acting as primary control parameters (Solow et al., 1 Apr 2025, San-Jose et al., 2014, Li et al., 30 Jul 2025, Cayao et al., 22 Jun 2026).

1. Concept and defining structure

The defining feature of an exceptional point is that not only the eigenvalues but also the eigenvectors of a non-Hermitian Hamiltonian coalesce, so the propagator or effective Hamiltonian becomes non-diagonalizable. In the Andreev context, this occurs in the spectrum of subgap states generated by superconducting coherence, typically in open junction geometries where dissipation enters as an imaginary self-energy or an imaginary potential. A standard parametrization of open-system poles is

$\epsilon_p = E_p - i\Gamma_p,$

where $E_p$ is the energy and $\Gamma_p$ is the decay rate into the reservoir (San-Jose et al., 2014).

This structure is distinct from ordinary Hermitian level crossings. In a Hermitian junction, Andreev levels may cross or anticross on the real axis. In an open superconducting device, the same levels generally move in the complex plane, and an exceptional point marks the transition from a regime dominated by energetic splitting to one dominated by dissipative splitting. In the multiterminal superconducting-normal setting, this distinction is made explicit by the statement that robust exceptional points separate phases with a real gap from phases with an imaginary gap, the latter corresponding to a difference in lifetimes or broadenings rather than a purely energetic splitting (Solow et al., 1 Apr 2025).

A closely related formulation appears in open normal–superconductor junctions, where two quasibound Andreev levels can bifurcate into two quasibound Majorana zero modes after crossing an exceptional point. In that setting, charge-conjugation symmetry enforces the pole pairing $\epsilon_p \leftrightarrow -\epsilon_p^*$, and purely imaginary zero-energy poles acquire a special status in the open-system spectrum (San-Jose et al., 2014). This suggests that Andreev exceptional points are best understood not as isolated algebraic curiosities, but as bifurcation points in the non-Hermitian evolution of superconducting resonances.

2. Effective non-Hermitian descriptions in superconducting junctions

A minimal realization is the single spinful level coupled to two superconducting leads and a spin-polarized normal lead, with a noncollinear magnetic field and a weakly coupled normal probe. In the superconducting infinite-gap limit, the effective Hamiltonian is

$$H_\mathrm{eff} = \begin{pmatrix} \varepsilon_\uparrow & B_x & \gamma\cos\frac{\phi}{2} & 0 \ B_x & \varepsilon_\downarrow & 0 & \gamma\cos\frac{\phi}{2} \ \gamma\cos\frac{\phi}{2} & 0 & -\varepsilon_\downarrow^* & B_x \ 0 & \gamma\cos\frac{\phi}{2} & B_x & -\varepsilon_\uparrow^* \end{pmatrix},$$

with

$$\varepsilon_{\uparrow/\downarrow} = \varepsilon \pm B\cos\theta + i\Gamma_{\uparrow/\downarrow}, \qquad B_x = B\sin\theta,$$

so dissipation via the normal lead enters as imaginary parts of the energies (Solow et al., 1 Apr 2025).

In one-dimensional $p$-wave non-Hermitian Josephson junctions, the non-Hermitian term is a local imaginary potential,

$U(x) = -iV\delta(x),$

which represents dissipative coupling to an environment through continuous loss of quasiparticles. The resulting Andreev spectrum follows from a secular equation in which the dissipation strength enters through $Z \equiv \tfrac{mV}{\hbar^2 k_F}$, and the explicit spectrum contains square-root branch points characteristic of exceptional points (Li et al., 30 Jul 2025).

In Josephson junctions formed by minimal Kitaev chains, non-Hermiticity is generated by coupling quantum dots in the chains to normal metallic reservoirs. In the wide-band limit, the effective retarded self-energy is frequency-independent,

$\epsilon_p = E_p - i\Gamma_p,$0

and the energies are determined by

$\epsilon_p = E_p - i\Gamma_p,$1

For the $\epsilon_p = E_p - i\Gamma_p,$2-site problem considered there, this becomes an $\epsilon_p = E_p - i\Gamma_p,$3th-order polynomial in $\epsilon_p = E_p - i\Gamma_p,$4 (Cayao et al., 22 Jun 2026).

These formulations share a common architecture: the Andreev sector is no longer described by a closed Hermitian bound-state problem, but by a non-Hermitian effective theory in which coupling to leads, reservoirs, or dissipative barriers converts sharp subgap levels into finite-lifetime resonances. A plausible implication is that “Andreev exceptional point” denotes a structural property of open superconducting spectra rather than a platform-specific mechanism.

3. Protection mechanisms and symmetry classes

The most developed classification in the Andreev setting distinguishes topologically-protected, or robust, exceptional points from symmetry-protected, or fragile, exceptional points. In the multiterminal single-level model, robust exceptional points arise generically for a range of parameters and are enforced by non-Hermitian particle-hole symmetry, denoted PHS$\epsilon_p = E_p - i\Gamma_p,$5, with the system placed in non-Hermitian symmetry class D$\epsilon_p = E_p - i\Gamma_p,$6. In a rotated basis aligned with the magnetic field, the relevant reduced Hamiltonian is

$\epsilon_p = E_p - i\Gamma_p,$7

with topological invariant

$\epsilon_p = E_p - i\Gamma_p,$8

The corresponding exceptional points mark boundaries between distinct non-Hermitian topological phases characterized by real-gap or imaginary-gap spectra (Solow et al., 1 Apr 2025).

Fragile exceptional points in the same model appear only for fine-tuned parameters, for example strictly $\epsilon_p = E_p - i\Gamma_p,$9, where an additional symmetry is present. For $E_p$0, a representative block is

$E_p$1

and when $E_p$2 the Hamiltonian obeys class BDI symmetry with

$E_p$3

If the symmetry is broken, for instance by detuning $E_p$4, the exceptional point disappears (Solow et al., 1 Apr 2025).

A related classification appears in the minimal Kitaev-chain Josephson junction. There, zero-energy exceptional points are protected by non-Hermitian particle-hole symmetry in class D$E_p$5, characterized by a $E_p$6 point-gap invariant, whereas finite-real-energy exceptional points are protected by accidental non-Hermitian parity-time symmetry, also supporting a $E_p$7 invariant for one-dimensional phase space (Cayao et al., 22 Jun 2026). In the $E_p$8-wave non-Hermitian Josephson junction, the two exceptional points carry opposite topological charges and cannot be gapped unless they meet at $E_p$9 and annihilate due to the particle-hole symmetry $\Gamma_p$0 (Li et al., 30 Jul 2025).

These results establish that Andreev exceptional points are not uniformly “protected” in a single sense. Their stability may derive from non-Hermitian particle-hole symmetry, from fine-tuned BDI structure, or from accidental parity-time symmetry, depending on the platform and spectral location.

4. Spectral manifestations and experimental signatures

The clearest signatures arise in spectroscopy of Andreev states. In the multiterminal superconducting-normal system, the spectral function

$\Gamma_p$1

reveals Andreev resonances as peaks, and its poles correspond to the eigenvalues of the effective Hamiltonian. Exceptional points appear as mergers or splittings of resonance lines in the spectral function as a function of control parameters such as the phase difference $\Gamma_p$2, and these features can be tracked in differential conductance maps versus voltage or phase. In the imaginary-gap phase reached after crossing an exceptional point, one resonance becomes much broader than the other, so width bifurcation is directly visible in the spectra (Solow et al., 1 Apr 2025).

In minimal Kitaev-chain Josephson junctions, local and nonlocal conductance are proposed as detection channels. Zero-real-energy exceptional points tend to appear as robust, often very broad, dispersions or conductance peaks pinned at zero bias for certain phase ranges, whereas finite-energy exceptional points manifest as splittings or circular structures in conductance maps associated with higher-energy Andreev bound states. The sensitivity depends on where conductance is measured: near the sweet spot, local conductance on outer quantum dots best reveals zero-energy exceptional points, while outside that regime nonlocal conductance becomes particularly informative (Cayao et al., 22 Jun 2026).

The following platform summary organizes the experimentally emphasized signatures.

Platform	Non-Hermitian source	Primary signature
Multiterminal SNS with spinful level	Spin-dependent normal lead	Merger/splitting and width bifurcation of Andreev resonances
Trivial NS junction with helical normal region	Reservoir coupling in open NS junction	Bifurcation of quasibound Andreev levels into zero-energy Majorana modes
$\Gamma_p$3-wave non-Hermitian Josephson junction	Local imaginary barrier $\Gamma_p$4	Phase-tunable exceptional points symmetric about $\Gamma_p$5
Minimal Kitaev-chain Josephson junction	Normal-reservoir self-energies	Local and nonlocal conductance signatures of zero- and finite-energy EPs

A recurrent conclusion is that spectroscopy resolves exceptional points more directly than integrated transport observables. This is stated explicitly for multiterminal junctions, where spectroscopy provides a much clearer probe than the Josephson current (Solow et al., 1 Apr 2025).

5. Supercurrent, Josephson response, and common misconceptions

A common expectation is that exceptional points should produce sharp anomalies in the Josephson response. The recent Andreev literature does not support that expectation as a universal rule. Exact transport calculations for the multiterminal superconducting-normal system show that systems with and without exceptional points may have similar or even identical Josephson currents once dissipation is treated correctly, and any enhancement is attributed to dissipation rather than to the exceptional point itself (Solow et al., 1 Apr 2025).

The one-dimensional $\Gamma_p$6-wave non-Hermitian Josephson junction reaches the same conclusion by a different route. There the supercurrent is obtained directly from inelastic Andreev reflection amplitudes, and the supercurrent varies continuously as a function of $\Gamma_p$7 across the exceptional points. No enhancement of critical current is observed. As dissipation increases, the critical current decreases almost linearly, and the current-phase relation becomes more sinusoidal than in the Hermitian limit (Li et al., 30 Jul 2025).

These results matter because early intuition from naive non-Hermitian Hamiltonian treatments could suggest singular current enhancement at an exceptional point. The more careful conclusion is narrower: Andreev exceptional points are spectral singularities in the complex subgap structure, but they need not induce singularities in dc supercurrent. A plausible implication is that experimental identification should prioritize phase-resolved spectroscopy, local and nonlocal conductance, or direct mapping of resonance widths over Josephson critical-current anomalies.

6. Relation to Majorana physics and non-Hermitian topology

The relation between Andreev exceptional points and Majorana physics is central but not uniform across models. In trivial normal–superconductor junctions strongly coupled to a helical normal region, exceptional points provide the route by which two quasibound Andreev levels bifurcate into two quasibound Majorana zero modes. Beyond the exceptional point, one Majorana escapes into the reservoir while the other becomes a nondecaying Majorana dark state localized at the NS junction. In the Andreev limit, this state exhibits zero energy, self-conjugation, a $\Gamma_p$8-Josephson effect, and non-Abelian braiding statistics, even though the superconducting bulk is topologically trivial (San-Jose et al., 2014).

This open-system mechanism differs from the bulk-defect correspondence familiar from chiral topological superconductors. In systems with chiral symmetry, exceptional points in complex momentum space can be associated with chiral Majorana zero modes localized at defects or boundaries, and the sign changes of the imaginary parts of the relevant complex momenta track topological phase transitions and localization-delocalization of the zero modes (Mandal, 2015). The paper also emphasizes that this strict exceptional-point correspondence does not persist in systems without chiral symmetry, although complex solutions of $\Gamma_p$9 can still be used for mode counting (Mandal, 2015).

Andreev exceptional points therefore sit at an interface between superconducting spectroscopy, open-system topology, and Majorana phenomenology. In multiterminal superconducting-normal systems they mark boundaries between non-Hermitian topological phases of Andreev resonances (Solow et al., 1 Apr 2025). In minimal Kitaev-chain Josephson junctions they can generate exceptional lines that enclose protected two-dimensional zero-real-energy areas, producing stable topological states that do not exist in the Hermitian field (Cayao et al., 22 Jun 2026). This suggests that the Andreev sector offers a concrete mesoscopic platform in which non-Hermitian topology is not merely appended to superconductivity, but realized directly through dissipative superconducting quasibound states.