Exceptional Points in Momentum Space

• Exceptional points in momentum space are spectral singularities in non-Hermitian band structures where eigenvalues and eigenvectors coalesce, leading to unique topological phenomena.
• They manifest in experimental platforms like photonic lattices and Rashba semiconductors, giving rise to observable effects such as Fermi arcs, chiral edge modes, and enhanced sensing.
• Recent advances have refined the classification, detection, and manipulation of EPs through analytic continuation and topological invariants, driving innovations in quantum and classical systems.

Exceptional points (EPs) in momentum space are singularities of non-Hermitian band structures where two or more bands simultaneously coalesce in both eigenvalues and eigenvectors. These spectral degeneracies, which are strictly forbidden in Hermitian systems except in the presence of extra symmetries, have far-reaching physical consequences in topological phases, open quantum system dynamics, and quantum and classical wave propagation. In systems where the effective Hamiltonians parametrically depend on momentum (or "quasi-momentum" in periodic contexts), the location, type, and topology of EPs encode a wealth of information about the system’s spectral, dynamical, and topological properties. Recent work has greatly expanded the theoretical classification of EPs in momentum space, has clarified criteria for their robust appearance and manipulation, and has connected their existence to experimentally accessible phenomena such as Fermi arcs, chiral edge states, enhanced sensing, and novel topological transitions.

1. Definition, Basic Properties, and Mathematical Structure

In a non-Hermitian Hamiltonian $H(\mathbf{k})$ depending on momentum $\mathbf{k}$, an exceptional point (EP) is a point in complexified momentum space where two or more eigenvalues $E_j(\mathbf{k})$ coincide and the Hamiltonian becomes defective, i.e., there are fewer linearly independent eigenvectors than eigenvalues. For a two-band model, the EP is characterized by the simultaneous solution of

$$\det[H(\mathbf{k})-\lambda I]=0, \qquad \frac{\partial}{\partial \lambda} \det[H(\mathbf{k})-\lambda I]=0,$$

so that both the eigenvalue and its derivative vanish, indicating non-diagonalizability. For $n$-band models, an EP$_n$ (order $n$ EP) occurs when $H$ is similar to an $n\times n$ Jordan block: all $n$ eigenvalues coalesce, and there is only a single eigenvector.

In momentum-space band structures, EPs generically manifest as branch points of the complex energy surfaces $E_j(\mathbf{k})$, with $E_j(\mathbf{k})$ and $E_{j'}(\mathbf{k})$ exchanging branches when $\mathbf{k}$ encircles the EP. For two-band systems, this results in a square-root energy topology; for higher-order EPs, fractional-root ($\sim (k-k_{\text{EP}})^{1/n}$) dispersion emerges near the defect.

The presence of EPs can lead to dramatic physical effects, including the breakdown of adiabaticity, nonorthogonality-induced Petermann factor divergences, sensitivity enhancement, and topologically protected defect modes.

2. Classification: Generic and Symmetry-Protected EPs, Dimensionality, and Topological Invariants

EPs are classified according to the number of coalescing eigenvalues (order $n$), the presence (or lack) of protecting symmetries, and the codimension (number of independent real tuning parameters).

• Generic EP$_n$s: In a generic $n$-band non-Hermitian system, an EP$_n$ requires tuning $2n-2$ parameters. In momentum space, this means generically EP$_n$s are zero-dimensional (points) in $2n-2$ dimensions. For example, a generic EP2 (second order, typical in two-band models) is a point in the two-dimensional complex momentum (or real $k_x$, $k_y$, and other parameters, e.g., non-Hermiticity strength).
• Symmetry-Protected EP$_n$s: The presence of symmetries such as PT, chiral, pseudo-Hermiticity, or CP symmetries can reduce the codimension. For PT symmetry, the characteristic polynomial can be made real, lowering the codimension to $n-1$: a PT-symmetric EP3 appears robustly in two (not four) parameters, e.g., in the two-dimensional Brillouin zone. Generalized chiral symmetry in non-Hermitian Lieb lattices produces flat bands and modifies the scaling and topology of degeneracies (Mandal et al., 2021, Yoshida et al., 13 Sep 2024).

A systematic topological classification is provided by resultant vectors constructed from the characteristic polynomial $P(\lambda,\mathbf{k})$: the vanishing of resultants $r_j(\mathbf{k})$ ($j=1,...,n-1$) defines the locus of EP$_n$s. Mapping a normalized resultant vector on $S^{2n-3}$ (or $S^{n-2}$ for symmetry-protected cases) enables the definition of winding numbers as topological invariants $W_{2n-3}$, which protect EP$_n$s against generic perturbations, with "doubling theorems" constraining the sum of winding numbers in a periodic Brillouin zone to vanish (Yoshida et al., 13 Sep 2024).

3. Physical Realizations: Model Systems and Experimental Signatures

A broad range of systems, including photonic lattices, microcavity lasers, topolectrical circuits, spin-orbit-coupled semiconductors, and exciton-polariton condensates, realize EPs in momentum space:

• Chiral Majorana Systems: In chiral topological superconductors, the analytic continuation of momentum yields EPs corresponding via $ik_\perp\to -z$ to the decay exponents of Majorana bound states. The imaginary part of the EP determines the localization length, and sign changes at topological transitions signal the appearance or disappearance of zero modes. A formula based on counting sign changes of $\mathrm{Im}(k_{EP})$ quantitatively matches winding number invariants (Mandal, 2015).
• Spin-Orbit Semiconductors: In non-Hermitian Rashba models, coupling to ferromagnets produces self-energies leading to EP rings in momentum space; the EPs serve as boundaries for bulk Fermi arc structures and manifest as large enhancements in the spectral function. Coalescence of spin projections at the EP provides an observable NH fingerprint (Cayao, 2023).
• Photonic and Exciton-Polariton Systems: EPs are engineered via gain-loss profiles, unidirectional coupling, or periodic driving. Their signatures include the direct transition between spectral singlets and triplets (Downing et al., 23 Apr 2025), the splitting of Dirac points into paired EPs accompanied by Fermi arcs in synthetic OAM or real-space momenta (Yang et al., 2022, Nguyen et al., 2023), and the emergence of chiral vortex states at EPs (Gao et al., 2017).
• Long-Range-Coupled Lattices and Braiding: Modified SSH-type lattices with $n$th-order non-Hermitian couplings exhibit "energy knotting" and nodal link structures, classifiable via braid indices. Topological phase transitions between different braiding phases are mediated by single (Type-1) or multiple (Type-2) EPs, with direct impact on the non-Hermitian skin effect and localization properties (Rafi-Ul-Islam et al., 5 Jul 2024).

4. Symmetry, Topological Defects, and Higher-Order EPs

Physical symmetries exert profound influence on the topology and abundance of EPs:

• PT Symmetry: PT-symmetric models enable robust, dimension-reduced EPs and can stabilize higher-order EPs (EP3, EP4) at lower codimension than generic cases. For example, in a three-band system with PT symmetry, a 3EP requires only two parameters, and the local spectrum near the EP exhibits scaling $E\sim k^{1/3}$, with characteristic Fermi arcs and surface terminations of lower-order EP lines (Mandal et al., 2021).
• Chiral/Pseudo-Hermitian Symmetry: In systems with generalized chiral symmetry, triple degeneracies can split into EP3s with $E\sim k^{1/2}$ (rather than cube-root behavior), robust flat bands, and "arc degeneracy" structures (Mandal et al., 2021).
• Non-Orientable Momentum Spaces: If the momentum space is endowed with non-orientable topology (e.g., a Klein Bottle Brillouin Zone), the adiabatic permutation of eigenstates around EPs leads to orientation-dependent braid representations and Berry phase accumulation becomes locally but not globally well-defined. This directly modifies the topological classification of EP defects (Ryu et al., 16 Apr 2025).

5. Physical Consequences and Observable Phenomena

EPs in momentum space induce a broad spectrum of measurable effects and dynamical signatures:

• Fermi Arcs and Enhanced Spectral Weight: EPs serve as endpoints for lines or arcs of coalescing eigenvalues (Fermi arcs) in the complex band structure. These regions manifest as enhanced response or density of states in ARPES or optical spectroscopy (Cayao, 2023, Yang et al., 2022).
• Topological Phase Transitions: The change in the sign of $\mathrm{Im}(k_{EP})$ upon tuning a Hamiltonian parameter is directly associated with topologically protected phase transitions, e.g., the transition from trivial to nontrivial phases in topological superconductors (Mandal, 2015).
• Non-Hermitian Skin Effect and Braiding: The braiding topology of eigenvalues, characterized by knotting/linking in $(\mathrm{Re} E, \mathrm{Im} E, k)$ space, ignites robust skin effect phenomena: eigenstates become exponentially localized at system edges, with the localization pattern (unipolar or bipolar) directly linked to the braiding index of the EP phase (Rafi-Ul-Islam et al., 5 Jul 2024).
• Chiral and Asymmetric Scattering: Scattering near an EP may display perfect transmission and asymmetric reflection depending on the incident direction, a property seen in order-1 Floquet $\pi$ EPs in periodically driven lattices (Zhu, 14 Jul 2024).
• Response Enhancement: Sensitivity to small perturbations (quantified by the "spectral response strength" or Petermann factor) can dramatically diverge as an EP of order $n$ is approached, especially as the system is tuned towards higher-order EPs (Wiersig, 2023).
• Experimental Control via Nonlinearities and Synthetic Dimensions: EP locations can be tuned with saturable nonlinearities (Gu et al., 24 Feb 2024), the introduction of unidirectional coupling (Hayenga et al., 2019), or by navigating "hybrid dimensions" (e.g., spatial momentum plus gain/loss strength), producing novel Dirac EPs (Rivero et al., 2022).

6. Methodologies for Identification, Classification, and Experimental Detection

• Analytic Continuation: EPs are found by analytically continuing real momentum variables ($k$) into the complex plane and solving for points where the determinant of a relevant block (e.g., $\det \mathcal{A}(k)=0$ in chiral Hamiltonians) vanishes and the Hamiltonian is non-diagonalizable (Mandal, 2015).
• Resultant Polynomials and Topological Mapping: The use of resultant maps to spheres of appropriate dimension provides both a detection criterion and a means to compute topological invariants ($W_{2n-3}$, $W_{n-2}$) protecting EPs (Yoshida et al., 13 Sep 2024).
• Direct Imaging: Techniques such as cross-polarization resonant scattering mapping in photonic crystals allow direct momentum-resolved detection of EPs and their associated arcs in a single shot, providing robust experimental validation (Nguyen et al., 2023).
• Exceptional Perturbation Theory: Perturbative methods, projecting onto degenerate subspaces and exploiting biorthogonal eigenvectors, give analytic predictions for the emergence, propagation ("fans"), and mutual annihilation (coalescence into higher-order EPs) in interacting quantum systems (Schäfer et al., 2022).
• Floquet and Hybrid Parameter Approaches: Periodic driving and synthetic dimensions allow for dynamic control and realization of Floquet EPs and EPs in spaces combining real and synthetic variables (Longhi, 2017, Rivero et al., 2022).

7. Outlook and Open Directions

Exceptional points in momentum space represent a foundational feature of non-Hermitian band theory, offering both topological rigidity and unusual physical effects absent from Hermitian physics. The continued development of their classification via resultant winding numbers and symmetry-based codimension arguments (Yoshida et al., 13 Sep 2024), their realization in synthetic and non-orientable spaces (Ryu et al., 16 Apr 2025), and their exploitation in highly sensitive devices, topological lasers, and quantum simulators, push the frontier of non-Hermitian quantum materials and photonics.

Outstanding challenges include a full classification of multifold EPs in interacting many-body systems; the interplay of EPs with disorder, nonlinearities, and strong correlations; and the controlled manipulation of high-order and symmetry-protected EPs for practical device engineering. The fundamental work on the stability, topology, and physical consequences of EPs continues to drive advances in both theory and experimental implementations across quantum and classical platforms.