Hamiltonian Exceptional Point

• Hamiltonian exceptional points are non-Hermitian degeneracies where eigenvalues and eigenvectors merge into Jordan blocks, marking phase transitions and non-analytic spectral behavior.
• They are diagnosed using Puiseux expansions and discriminant analysis, offering insights into eigenvalue branch switching and enhanced sensitivity near the singularity.
• Experimental realizations in photonics and cavity systems underscore their role in organizing topological features and governing non-adiabatic dynamics.

A Hamiltonian exceptional point is a non-Hermitian degeneracy at which eigenvalues and their eigenvectors coalesce, the Hamiltonian becomes defective, and the appropriate local normal form is Jordan rather than diagonal. In current usage, the term covers finite-dimensional non-Hermitian matrices, effective Hamiltonians extracted from driven or dissipative dynamics, and operator-theoretic settings with continuous spectra. Across these contexts, Hamiltonian exceptional points organize phase transitions, branch-point topology, generalized completeness, and non-adiabatic transport, while remaining conceptually distinct from Liouvillian degeneracies of full master equations (Wiersig, 2020, Andrianov et al., 2011, Ghosh et al., 25 Feb 2026).

1. Formal definition and local spectral structure

For a parameter-dependent non-Hermitian Hamiltonian $H(\lambda)$, an exceptional point of order $m$ is a parameter value $\lambda_{\mathrm{EP}}$ at which $m$ eigenvalues and their eigenvectors coalesce, so that $H(\lambda_{\mathrm{EP}})$ is non-diagonalizable and contains an $m\times m$ Jordan block. In the standard Jordan-chain language, one has an eigenvector $|u_0\rangle$ and associated vectors $|u_j\rangle$ satisfying

$$(H_{\mathrm{EP}}-E_{\mathrm{EP}})|u_0\rangle=0,\qquad (H_{\mathrm{EP}}-E_{\mathrm{EP}})|u_j\rangle=|u_{j-1}\rangle .$$

The algebraic multiplicity then exceeds the geometric multiplicity, which is the defining hallmark of defectiveness (Wiersig, 2020, Yuce, 2019).

Near an EP, the eigenvalue branches are generically non-analytic. For an EP2, the local behavior is of square-root type, $$E_\pm(\epsilon)\approx E_{\mathrm{EP}}+c\,\epsilon^{1/2}$$; more generally, an EP of order $m$0 gives $m$1. This Puiseux structure is the basis of the familiar sheet-exchange behavior under parameter encircling, and it sharply distinguishes exceptional points from Hermitian diabolic points, where eigenvectors remain linearly independent and the crossing is conical rather than defective (Wiersig, 2020, Znojil, 19 Feb 2026).

In quasi-Hermitian formulations, the same phenomenon can be stated in metric language. A Hamiltonian $m$2 is physically admissible only when there exists a positive-definite metric $m$3 such that $m$4. The EP is then the boundary of the domain where observability holds: as $m$5, the metric ceases to remain positive-definite or becomes singular, while the Hamiltonian approaches a Jordan form. In this sense, an EP is not merely a spectral degeneracy but also a boundary of the physical Hilbert-space structure (Znojil, 19 Feb 2026).

2. Exceptional points for Hamiltonian operators with continuous spectrum

In operator theory, exceptional points need not be confined to discrete spectra. For one-dimensional non-Hermitian Schrödinger operators on $m$6, an EP can lie at the boundary of the continuous spectrum or be embedded inside it. In this setting, the operator-theoretic definition uses a normalizable eigenfunction $m$7 together with a Jordan chain of associated functions $m$8,

$m$9

and completeness is recovered only after constructing biorthogonal resolutions of identity with distributional correction terms and, in some cases, non-normalizable associated functions (Andrianov et al., 2011).

A notable feature of continuous-spectrum EPs is that multiplicity bifurcates into inequivalent indices. The analysis of the model family $\lambda_{\mathrm{EP}}$0 distinguishes $\lambda_{\mathrm{EP}}$1, the maximal number of linearly independent normalizable eigenfunctions and associated functions; $\lambda_{\mathrm{EP}}$2, the maximal number of eigenfunctions and formal associated functions entering the biorthogonal resolution of identity; and $\lambda_{\mathrm{EP}}$3, the order of the Green-function pole. These indices agree for EPs outside the continuum but not for EPs on or inside the continuous spectrum. For the boundary EP at $\lambda_{\mathrm{EP}}$4 in $\lambda_{\mathrm{EP}}$5, one has $\lambda_{\mathrm{EP}}$6; for the embedded EP at $\lambda_{\mathrm{EP}}$7, one has $\lambda_{\mathrm{EP}}$8 (Andrianov et al., 2011).

The resolvent reflects this distinction directly. At a boundary EP of multiplicity $\lambda_{\mathrm{EP}}$9, the Green function has a branching singularity with a pole of order $m$0 in $m$1; for an embedded EP, the Green function has an ordinary pole whose order can exceed the Jordan-chain length entering the resolution of identity. This is one reason that Hamiltonian EPs in the continuum cannot be reduced to the finite-dimensional notion of algebraic multiplicity alone (Andrianov et al., 2011).

A complementary example is provided by the real, pseudo-Hermitian von Neumann–Wigner-type Hamiltonian $m$2, where EPs occur at real energies inside the continuous spectrum. There the two unnormalized Jost solutions coalesce at $m$3, the Hamiltonian acquires Jordan cycles of generalized bound-state eigenfunctions embedded in the continuum, the regular scattering eigenfunction vanishes at the EP, and the irregular scattering eigenfunction develops a double pole. Despite this, the scattering matrix remains a regular analytic function of $m$4, while the regular sector evolves unitarily and the generalized sector evolves pseudounitarily (Hernández et al., 2015).

3. Symmetry constraints and topological organization

Pseudo-Hermiticity imposes sharp selection rules on Hamiltonian EP formation. If a Hamiltonian admits a global pseudo-metric $m$5 satisfying $m$6, then every real-energy level carries a $m$7 index

$m$8

This index is conserved while the eigenvalue remains real, and only levels with opposite indices can form second-order EPs. In the corresponding two-level reduction, same-index crossings are locally Hermitian and do not produce EP2s, whereas opposite-index crossings admit the required non-Hermitian off-diagonal structure. The same framework also yields a criterion for EP3 formation: the middle level must carry the index opposite to its two neighbors, producing the “same–opposite–same” pattern (Starkov et al., 2023).

Sublattice symmetry produces a different constraint. Because the spectrum is forced to appear in $m$9 pairs, one might expect only even-order zero-energy EPs. That expectation is incorrect: odd-order zero-energy EPs can occur as mixed objects. In the $H(\lambda_{\mathrm{EP}})$0 four-band case, an EP3 with Jordan form $H(\lambda_{\mathrm{EP}})$1 arises as a limit of EP4 and double-EP2 families, and its eigenvector-coalescence pattern depends on the direction of approach in parameter space. The associated “quantum distance” to the target eigenvector therefore shows enhanced, direction-dependent sensitivity (Yang et al., 2022).

For isolated EPs in two-dimensional band structures, homotopy theory promotes this local algebra into a global topological classification. An isolated EP of order $H(\lambda_{\mathrm{EP}})$2 is characterized by a braid in the braid group $H(\lambda_{\mathrm{EP}})$3, obtained by following the eigenenergies along a loop in the Brillouin zone. The discriminant winding equals the writhe of the corresponding closed braid, and the crossing number gives the number of bulk Fermi arcs emanating from the EP. A non-Hermitian no-go theorem then constrains all EP configurations on a two-dimensional lattice: the ordered product of local braids must lie in the commutator subgroup, which in particular forces the total discriminant invariant over the Brillouin zone to vanish (Hu et al., 2021).

A different topological viewpoint appears at higher-order EPNs. For the unidirectional tight-binding chains $H(\lambda_{\mathrm{EP}})$4 and $H(\lambda_{\mathrm{EP}})$5, both of which are similar to a size-$H(\lambda_{\mathrm{EP}})$6 Jordan block, the relevant invariant is not a spectral gap but adiabatic equivalence of Jordan classes. $H(\lambda_{\mathrm{EP}})$7 and $H(\lambda_{\mathrm{EP}})$8 are not adiabatically equivalent, and an interface between them supports degenerate zero-energy exceptional states localized at the junction. Because the underlying non-Hermiticity is generated by asymmetric tunneling rather than gain/loss, these exceptional states can also exist in closed systems, including rings (Yuce, 2019).

4. Dynamics, encircling, and the distinction between Hamiltonian and Liouvillian EPs

Hamiltonian exceptional points govern the spectrum of an effective Hamiltonian, but open-system dynamics is controlled by a Liouvillian superoperator. These are not interchangeable objects. In the Lindblad setting with stochastic Hamiltonian perturbations, resolvability of spectral splittings is determined by the effective Hamiltonian $H(\lambda_{\mathrm{EP}})$9, whereas dynamical stability is determined by the Liouvillian $m\times m$0. A central relation is that a Hamiltonian EP of order $m\times m$1 generically places the coherent Liouvillian at an EP of order $m\times m$2. For an HEP2, small noise then produces cubic-root splittings of Liouvillian eigenvalues, so the same configuration that enhances spectral sensitivity can also move the Liouvillian spectral abscissa into the unstable half-plane (Wiersig, 2020).

This distinction becomes explicit in cavity optomechanics. In the unconditional Lindblad dynamics, the Liouvillian EP occurs at

$m\times m$3

and is independent of the thermal phonon occupancy $m\times m$4. In contrast, the conditional no-jump evolution is governed by a non-Hermitian Hamiltonian with enhanced mechanical damping $m\times m$5, leading to a Hamiltonian EP at

$m\times m$6

The thermofield construction then yields a hybrid family of exceptional points interpolating continuously between the two limits, and in the weak-quantum-jump regime the Hamiltonian EP is shifted only at second order in the interpolation parameter (Ghosh et al., 25 Feb 2026).

The dynamical implications of encircling a Hamiltonian EP can also be formulated directly in complex time. For a driven two-level problem, analytic continuation of the adiabatic perturbation integral reduces the dynamics to singularities in the complex-time plane, called transition points, where the adiabatic energies become degenerate. In a time-symmetric encircling protocol, the relevant transition-point layout changes from an off-axis complex-conjugate pair to a pair on the imaginary axis, and the observable dynamics switches accordingly between Rabi oscillations and rapid adiabatic passage. The EP thus controls not only static sheet permutation but also which steepest-descent contributions survive in the physical transition amplitude (Kapralova-Zdanska, 2021).

5. Analytical and numerical diagnostics

The algebraic definition of a Hamiltonian EP can be implemented through the secular determinant. If $m\times m$7, then a repeated root satisfies $m\times m$8 and $m\times m$9. A practical simplification is to compute the discriminant of the secular polynomial with respect to $|u_0\rangle$0; its zeros are precisely the parameter values where eigenvalues coalesce. For truncated Hamiltonian operators, this reduces EP finding to the roots of a polynomial in the control parameter alone. This strategy was applied to the particle in a box with polynomial potential, the periodic Mathieu equation, the Stark effect in a polar rigid rotor, and the Stark effect in a polar symmetric top (Amore et al., 2019).

In bosonic quadratic systems, the appropriate object is often the Hopfield–Bogoliubov matrix rather than the original Hermitian quadratic Hamiltonian. For a single bosonic mode, the appearance of an EP in the Hopfield–Bogoliubov matrix is equivalent to the vanishing of its determinant and to a gap closing. In multimode systems this equivalence fails in general: a zero determinant may signal a degenerate point without eigenvector coalescence, and an EP need not coincide with that degenerate point. In the asymmetric two-mode problem analyzed in this framework, increasing the two-photon driving produces the re-entrant sequence normal phase $|u_0\rangle$1 superradiant phase $|u_0\rangle$2 normal phase $|u_0\rangle$3 superradiant phase, precisely because the EP and the degenerate point do not always coincide (Xie et al., 2021).

For larger non-Hermitian lattices, direct Jordan analysis is often supplemented by conditioning diagnostics. In the generalized non-Hermitian Rice–Mele model, EPs are identified numerically by the condition number of the eigenvector matrix, which diverges as the smallest singular value tends to zero; the resulting loci are then confirmed by explicit Jordan decomposition. This strategy is particularly useful under open boundary conditions, where the quantization problem is transcendental and analytic EP equations are unavailable in general (Martinez-Strasser et al., 23 Jun 2026).

6. Experimental and model realizations

A direct experimental realization of a Hamiltonian EP was reported in cavity magnon–polaritons. A yttrium–iron–garnet sphere coupled to a two-port microwave cavity is described, under coherent perfect absorption, by an effective non-Hermitian Hamiltonian whose discriminant vanishes at the EP. In the PT-balanced, on-resonance case, the eigenfrequencies are

$|u_0\rangle$4

so the Hamiltonian EP occurs at $|u_0\rangle$5. Experimentally, coherent-perfect-absorption dips merge at this point, and the setup provides a direct spectroscopic distinction between a Hamiltonian EP and a scattering singularity (Zhang et al., 2017).

Temporal non-Hermitian photonics extends the same concept into Floquet space. In a photonic time crystal, the reduced two-sideband Hamiltonian

$|u_0\rangle$6

has eigenvalues $|u_0\rangle$7, so the EP condition is $|u_0\rangle$8, giving $|u_0\rangle$9 and $|u_j\rangle$0 for real parameters. In that setting the EP is accompanied by mode exchange, a biorthogonal Berry phase of magnitude $|u_j\rangle$1, and a square-root perturbation response that survives comparison with a linewidth-matched Hermitian reference in a Cramér–Rao-bound analysis (Tripathi et al., 2 Dec 2025).

The generalized non-Hermitian Rice–Mele chain with balanced gain/loss and next-nearest-neighbor hopping provides a complementary lattice realization. Under periodic boundary conditions, the EP condition depends only on the nontrivial Pauli-vector part of the Bloch Hamiltonian, so the next-nearest-neighbor hopping $|u_j\rangle$2 does not move the bulk EP loci; they appear as lines and ellipses in parameter space. Under open boundary conditions, this independence is broken: $|u_j\rangle$3 shifts the energies of existing EPs, creates new EPs, and introduces an open-boundary-only topological gap closing at $|u_j\rangle$4. The same model remains reciprocal, and the non-Hermitian skin effect is absent (Martinez-Strasser et al., 23 Jun 2026).

Exceptional points also structure nonlinear and many-body regimes. In the PT-symmetric pseudo-Hermitian Ising chain with staggered imaginary longitudinal field, analytically computable parity indices determine which many-body levels can meet at EP2s and how EP3s arise from coalescing EP2 pairs (Starkov et al., 2023). In a driven-dissipative nonlinear photonic dimer, the EP of the effective two-resonator Hamiltonian separates split-resonance and split-dissipation regimes and organizes dissipative Kerr soliton generation on both sides of the EP (Komagata et al., 2021).

7. Conceptual clarifications and open problems

Several recurrent misconceptions are explicitly contradicted by the literature. A Hamiltonian exceptional point does not require an open gain–loss platform: unidirectional hopping alone can generate EPNs and exceptional interface states in closed systems, even on rings (Yuce, 2019). Nor is algebraic multiplicity by itself a sufficient descriptor once continuous spectra are present: the operator-theoretic indices $|u_j\rangle$5, $|u_j\rangle$6, and $|u_j\rangle$7 need not coincide, and completeness may depend on distributional terms supported at infinity (Andrianov et al., 2011).

Zero-energy EPs also require care. Under sublattice symmetry, odd-order zero-energy EPs can occur as mixed limits of higher- and lower-order even EP families, rather than as regular isolated odd-order degeneracies (Yang et al., 2022). In topological superconductors without chiral symmetry, EPs of the off-diagonal block no longer encode Majorana wavefunctions directly, although complex-momentum solutions of $|u_j\rangle$8 can still be used in the counting formula for zero modes (Mandal, 2015).

Several structural questions remain open in the papers surveyed here. One is the general relation among the multiplicity indices $|u_j\rangle$9, $$(H_{\mathrm{EP}}-E_{\mathrm{EP}})|u_0\rangle=0,\qquad (H_{\mathrm{EP}}-E_{\mathrm{EP}})|u_j\rangle=|u_{j-1}\rangle .$$0, and $$(H_{\mathrm{EP}}-E_{\mathrm{EP}})|u_0\rangle=0,\qquad (H_{\mathrm{EP}}-E_{\mathrm{EP}})|u_j\rangle=|u_{j-1}\rangle .$$1 for non-Hermitian operators with continuous spectra beyond the specific models constructed so far (Andrianov et al., 2011). Another is the extension of braid and knot classifications from isolated two-dimensional EPs to higher-dimensional singular sets and the associated “singular knots” (Hu et al., 2021). A further practical problem is optimization: enhanced Hamiltonian sensitivity near an EP competes with Liouvillian instability, linewidth broadening, and noise, so any useful device must be designed with the Hamiltonian and Liouvillian spectra treated separately rather than conflated (Wiersig, 2020).