Exceptional Spectral Phases in Non-Hermitian Systems

• Exceptional spectral phases are regimes in non-Hermitian systems where eigenvalues and eigenvectors coalesce, leading to robust topology-driven singularities.
• They use Puiseux-law expansions and winding invariants to produce ultrasensitive spectral responses and anomalous dynamical behaviors in various physical platforms.
• The topological classification of these phases enables enhanced sensing, non-exponential relaxation, and novel phase transitions in open quantum and classical systems.

An exceptional spectral phase is a parameter regime in which the spectrum of a non-Hermitian (or non-diagonalizable) operator displays robust, topology-driven singularities—exceptional points (EPs) of arbitrary order—characterized by the coalescence of both eigenvalues and eigenvectors. This regime is marked by non-analytic parameter dependence (Puiseux law), ultrasensitive spectral response, anomalous dynamical behavior, and, typically, nontrivial topological invariants (vorticity, Berry phase, resultant winding). Exceptional spectral phases arise in a broad array of physical settings—ranging from energy spectra of open quantum systems and Liouvillian superoperators, to generalized susceptibilities of Hermitian many-body systems, and to classical or quantum wave systems subject to dissipation or instability. The associated topological and dynamical phenomena have concrete consequences for phase transitions, sensing, non-exponential relaxation, and Fermi surface topology.

1. Foundational Concepts: Exceptional Points and Exceptional Spectral Phases

Exceptional points (EPs) are parameter values of a non-Hermitian operator $H(\lambda)$ at which two or more eigenvalues and their corresponding eigenvectors coalesce, leading to a non-diagonalizable Jordan block structure. An exceptional spectral phase is an extended region in parameter space containing EPs of given order (typically $n \geq 2$) and governed by the unique non-Hermitian topology and algebraic properties of these degeneracies.

• Mathematical Structure: At an EP$n$ (order $n$ exceptional point), the characteristic polynomial and its first $n-1$ derivatives share a common root, and the Hamiltonian is similar to a single $n \times n$ Jordan block. In the parameter region surrounding the EP, the eigenvalue splitting is governed by a Puiseux expansion:
  $$
  \lambda_h(\varkappa) = \lambda^{(0)} + \sum{m \geq 1} \lambda^{(m)} \left(e^{2\pi i (h-1)/n} \varkappa^{1/n}\right)^m, \quad h=1,\dots,n,
  $$
  resulting in anomalously sensitive dependence to perturbations [2511.17067].

• Topological Invariants: The winding of eigenvalues/eigenvectors or of resultant vectors around EPs encodes integer or $\mathbb{Z}_2$ invariants (vorticity, Berry phase, resultant winding number), which characterize the global phase structure and stability of the exceptional spectral phase [2412.15323, 2511.17067].

• Physical Realizations: Exceptional spectral phases appear in black hole quasinormal mode spectra [2511.17067], Liouvillian spectra of dissipative quantum spin systems [2202.09337, 2602.01375], susceptibility matrices in correlated fermion systems [2307.00849], non-Hermitian band structures [2412.15323], and open photonic or quantum optical systems [2510.20571, 2009.00930].

2. Topology and Invariants in Exceptional Spectral Phases

The presence and nature of an exceptional spectral phase is determined by robust topological invariants inherited from the structure of EPs:

• Vorticity and Berry Phase: For a second-order EP, a closed loop in parameter space encircling the EP accumulates a half-integer vorticity ($\nu = \pm 1/2$) and a non-Hermitian Berry phase ($\gamma = \pi$); loops not enclosing the EP yield $\nu=0$, $\gamma=0$ [2511.17067].

• Resultant Vector and Winding: In non-Hermitian band theory, the resultant vector $\mathbf{R}(\mathbf{k})$ is constructed from the resultants of the characteristic polynomial and its derivatives, with its winding around EP$_n$ capturing the topological charge. The associated "resultant winding number" $W$ is an integer topological invariant, linked via the “doubling theorem” to global constraints in the Brillouin zone [2412.15323].

• Abelian and $\mathbb{Z}_2$ invariants: For systems with local and non-local symmetries (pseudo-Hermiticity, PT, etc.), the exceptional spectral phase can be classified by either Abelian (AIII, Chern number) or $\mathbb{Z}_2$ invariants. These invariants protect Fermi-arc connectivity and bulk singularities [2412.15323].

3. Dynamical and Spectroscopic Manifestations

Exceptional spectral phases have direct dynamical and spectroscopic consequences:

• Ultrasensitive Response: Near an EP of order $q$, the spectral response and pseudospectrum contour size scale as $\epsilon^{1/q}$, in contrast to linear scaling for ordinary eigenvalues; this produces enhanced susceptibility to small perturbations [2511.17067, 2304.00764].

• Non-exponential Relaxation: In Liouvillian spectra with EPs, time evolution manifests non-exponential decay—e.g., linear-in-time corrections ($t e^{-\gamma t}$) or higher-order polynomial prefactors—directly tied to the rank of the Jordan block [2202.09337, 2602.01375, 2511.21825].

• Spectral Anomalies: Spectroscopic observables, such as emission spectra in driven-dissipative spin models, display super-Lorentzian line shapes at EPs, visible for generic initial conditions but suppressed in steady-state fluorescence due to state-selection rules [2602.01375].

• Mode Mixing and Template Design: In gravitational-wave black hole spectroscopy, adiabatic parameter cycles encircling an EL (exceptional line) result in topological exchange of quasinormal modes and accumulation of non-trivial Berry phase; phenomenological templates must accommodate Puiseux-type nonanalyticity [2511.17067].

Context	Observable effect	Quantitative scaling / signature
Black hole QNM	Mode mixing, $\delta\omega \sim \epsilon^{1/2}$	Half-integer vorticity, Berry phase $\pi$
Liouvillian dynamics	Linear-in-time corrections	$t^\alpha e^{-\gamma t}$, $\alpha > 0$ near EP
Density wave ARPES	t$^2$ algebraic decay, Fermi arcs	Cusp singularities and threading
Susceptibility in Fermi-Hubbard	Topologically protected instability region	Real-$\lambda$ corridor bounded by EPs

4. Physical Realizations and Models

Exceptional spectral phases are realized in a variety of physical platforms:

• Open Quantum Systems and Liouvillian Spectra: Dissipative collective spin models exhibit a division between a "normal" phase and an exceptional Liouvillian spectral phase comprised entirely of second-order EPs in the thermodynamic limit [2202.09337]. Signatures include anomalous non-exponential relaxation and divergent eigenvalue density at the phase boundary. Spectroscopic detection is enhanced in non-steady-state preparations [2602.01375].

• Non-Hermitian Band Theory: Multifold EPs in non-Hermitian Hamiltonians produce topologically protected Fermi arcs and monodromy in both energy and eigenvector structure. The Abelian resultant winding number classifies the exceptional spectral phase, with concrete examples in 2D and 3D band structures [2412.15323].

• Quantum Optics and Dissipative Baths: Emitter relaxation in quantum electrodynamics platforms coupled to lossy photonic lattices can undergo transitions between oscillatory and overdamped decay modes, governed by "virtual" EPs (on the second Riemann sheet) and associated exceptional spectral phase boundaries [2510.20571].

• Correlated Fermion Systems: In Fermi-Hubbard models, non-Hermitian symmetries of the susceptibility matrix lead to EPs at which eigenvalues become real, thus stabilizing phase-separation instabilities over finite chemical potential intervals associated with the exceptional spectral phase [2307.00849].

• Driven-Dissipative Cavity Systems: Optical parametric oscillators display spectral phase transitions with square-root scaling and divergent susceptibility at criticality, mirroring the response near a second-order EP, and supporting both second-order and first-order scenarios via coupling [2009.00930].

5. Mathematical Frameworks for Topological Classification

The mathematical underpinnings of exceptional spectral phases build on and generalize topological band theory:

• Resultant Construction: The resultant vector $R_j(k) = \mathrm{Res}(\chi(\lambda; k), \partial_\lambda^j \chi(\lambda; k))$ for $j=1, \dots, n-1$ provides a map from parameter or momentum space to $\mathbb{C}^{n-1}$, whose vanishing locates EP$_n$ [2412.15323].

• Resultant Hamiltonian and Tenfold Way: By packing $R_j(k)$ into a "resultant Hamiltonian," one connects the topological classification of EPs to the tenfold way (Class AIII in general, Class A under certain symmetries), allowing the use of winding numbers, Chern numbers, and $\mathbb{Z}_2$ invariants.

• Doubling Theorem and Mayer–Vietoris: The sum of winding numbers of all EPs in the full torus of momentum (or parameter) space must vanish, reflecting global spectral constraints akin to Poincaré–Hopf and the Mayer–Vietoris sequence in cohomology [2412.15323].

6. Experimental and Practical Manifestations

The recognition and exploitation of exceptional spectral phases underpins several proposed and realized experimental phenomena:

• Dynamical Phase Transitions: Distinct spectral behavior (non-exponential relaxation, symmetry-breaking, collapse/revival, critical slowing down) can be accessed and controlled through parameter tuning across the exceptional phase boundaries [2207.01862, 2212.12403, 2510.20571].

• Spectroscopy and Sensing: Super-Lorentzian features in emission spectra, enhanced Petermann factors, and critical sensitivity to perturbations afford new modalities for quantum sensing and metrology, particularly when operated near higher-order EPs [2304.00764, 2602.01375].

• Criticality Beyond Hermitian Paradigms: The critical lines delimiting exceptional spectral phases can support quantum phase transitions without Hermitian analogs—e.g., dissipative quantum phase transitions in spin systems, or boundary time crystals emerging from the collapse of Liouvillian eigenmodes [2202.09337].

• Template and Signal Design: In fields such as gravitational-wave data analysis, the nonanalyticity and ultrasensitivity enforced by the exceptional spectral phase demand new template expansion schemes incorporating Puiseux laws and topological constraints [2511.17067].

7. Outlook and Significance

Exceptional spectral phases represent a unifying concept linking non-Hermitian topology, dynamical criticality, and parameter-sensitive spectral restructuring. Their mathematical structure generalizes concepts from Hermitian topological band theory, with branching, monodromy, and robust invariants dictating the qualitative and quantitative features observed in experiment and theory. The emergence of exceptional spectral phases across black hole spectroscopy, open quantum systems, condensed matter, and photonic platforms highlights their fundamental and versatile role in quantum and classical physics [2412.15323, 2511.17067, 2307.00849, 2602.01375].

Their study continues to inform both the search for enhanced sensing strategies and the deeper classification of non-equilibrium and non-Hermitian phases, bridging abstract algebraic topology, semiclassical dynamics, and experimentally accessible observables.