Non-Hermitian Systems Overview

• Non-Hermitian systems are physical and mathematical frameworks where operators are not equal to their Hermitian conjugates, enabling complex eigenvalues and non-unitary evolution.
• They exhibit unique spectral phenomena such as exceptional points and the non-Hermitian skin effect, fundamentally altering bulk-boundary correspondence.
• Advanced methods like biorthogonal formalism and pseudo-Hermiticity underpin their analysis, with applications in photonics, quantum circuits, and cold atom simulations.

Non-Hermitian systems are physical or mathematical systems whose governing operators—typically Hamiltonians—are not equal to their Hermitian conjugate ($H \neq H^\dagger$). While Hermiticity enforces real eigenvalues and unitary evolution in closed quantum systems, non-Hermitian systems admit complex eigenvalues and non-unitary evolution, and arise naturally when modeling open, dissipative, or interacting systems with gain/loss, non-reciprocal dynamics, or effective descriptions via postselection or engineered environments. The breakdown of Hermiticity leads to an array of unprecedented phenomena including exceptional points, the non-Hermitian skin effect, new types of topological invariants, and generalized symmetry classifications. This article details the mathematical structures, symmetry classes, transport and topological properties, many-body implications, and physical realization pathways for non-Hermitian systems.

1. Mathematical Structure, Symmetries, and Pseudo-Hermiticity

Non-Hermitian Hamiltonians can possess rich algebraic structures. Unlike Hermitian matrices, their spectra can be entirely real, complex, or feature real–complex transitions. The biorthogonal basis formalism is central: given a diagonalizable non-Hermitian $H$, one defines right eigenvectors $H|\psi_n\rangle = E_n|\psi_n\rangle$ and left eigenvectors $H^\dagger|\phi^n\rangle = E_n^*|\phi^n\rangle$, with $\langle\phi^m|\psi_n\rangle = \delta_{mn}$. This underpinning enables the construction of invariant inner products and correct physical observables.

A critical subclass is pseudo-Hermitian Hamiltonians, satisfying $H^\dagger = \eta H \eta^{-1}$ for some invertible Hermitian metric operator $\eta$. This property guarantees spectra that are either real or organized in complex conjugate pairs. Pseudo-Hermiticity generalizes 𝒫𝒯-symmetry (parity-time symmetry) and connects with the existence of a positive-definite metric, which (when constructed, for instance, via Krein space theory) restores a form of unitarity and supplies consistent probability interpretations for observables (Ramirez et al., 2018).

Symmetry classifications in non-Hermitian systems can deviate dramatically from the familiar Altland-Zirnbauer (AZ) scheme. For example, time-reversal symmetry operators $\Theta$ with $\Theta^2 = +1$ do not induce Kramers degeneracy in Hermitian systems, but in non-Hermitian systems with pseudo-Hermiticity, a split-quaternion algebraic structure underpins a generalized Kramers degeneracy—even when $\Theta^2 = +1$—giving rise to new degeneracy types absent from Hermitian settings (Sato et al., 2011).

2. Spectral Properties, Exceptional Points, and Skin Effects

Non-Hermiticity enables generic spectral phenomena forbidden in Hermitian systems. Two central effects are:

• Exceptional Points (EPs): Isolated degeneracies in parameter space where not only eigenvalues but also eigenvectors coalesce, causing the Hamiltonian to become defective and non-diagonalizable. Encircling EPs in parameter space exchanges eigenstates (“eigenvalue braiding”) and accumulates nontrivial complex Berry phase; the spectrum resolves into multi-sheeted Riemann surfaces (Zhang et al., 2019).
• Non-Hermitian Skin Effect (NHSE): Under open boundary conditions, an extensive number of eigenstates (“skin modes”) localize at one or both boundaries, in contrast with the extended bulk states in Hermitian systems. This is analytically traced to the determinant of the transfer matrix ($\det T\neq 1$) or the modification of the generalized Brillouin zone (complex $k$) in band theory (Kunst et al., 2018, Yokomizo et al., 2019).

A representative skin effect can be seen in the Hatano-Nelson model: asymmetric hopping produces a spectrum loop in the complex plane encircling a basepoint, leading to a nontrivial winding number classifying the NHSE. Moreover, dynamic skin effects—such as acceleration and amplification of wavepackets—emerge from an interplay between skin localization and Hermitian wavepacket spreading, producing inelastic boundary scattering (Li et al., 2022).

3. Topological Phases and Bulk–Boundary Correspondence

Non-Hermitian systems realize topological phases that can have no Hermitian analogues, with new invariants possible even for single-band 1D models. Central principles include:

• Complex Gap Concept: Instead of real energy gaps, topological classification depends on the existence of a “point gap” (the spectrum avoids a reference point in the complex energy plane) or “line gap” (the real or imaginary energy axis is gapped). For a one-dimensional Hamiltonian $H(k)$, topology is encoded in the spectral winding:

$$w = \int_{-\pi}^\pi \frac{dk}{2\pi i} \partial_k \ln \det H(k)$$

(Gong et al., 2018, Yokomizo et al., 2019). This invariant is robust under continuous deformations that keep the point gap open.

• Bulk–Boundary Correspondence Modifications: Unlike Hermitian topological insulators, non-Hermitian systems with NHSE defy the conventional bulk–boundary correspondence: the number and localization of edge modes depend critically on boundary conditions; skin modes invalidate the link between bulk Bloch invariants and edge observables unless calculations are performed using generalized (non-Bloch) Brillouin zones or transfer matrix approaches (Kunst et al., 2018, Yokomizo et al., 2019).
• Symmetry Indicators and Doubling: By mapping a non-Hermitian Hamiltonian to a “doubled” chiral symmetric Hermitian Hamiltonian,

$$\tilde{H}(k)=\begin{pmatrix} 0 & H(k) \ H^\dagger(k) & 0 \end{pmatrix}$$

one imports Hermitian topology—and its powerful symmetry indicator machinery—into the non-Hermitian context. This mapping enables efficient diagnosis of topological phases subject to crystalline symmetries and identifies indicator groups for non-Hermitian systems as identical to those in doubled Hermitian cases (Shiozaki et al., 2021).

• Permutation Group and Multiband Topology: In higher-dimensional and multiband non-Hermitian systems, the state exchange (braiding) induced by adiabatic transport along non-contractible loops is classified by permutation group cycles rather than familiar Chern numbers or $\mathbb{Z}$ windings; pseudo-Hermitian lines (PHLs) can induce nontrivial permutation cycles in the absence of EPs (Ryu et al., 28 May 2024).

4. Non-Hermitian Many-Body Physics

Many-body extensions of non-Hermitian models raise unique questions:

• Generalized Aufbau Principle: The many-body energy spectrum can be built by filling single-particle levels ordered by the real parts of their complex energies, in both fermionic and bosonic models (Sun et al., 2023). In the Hatano-Nelson model, NHSE persists in all many-body eigenstates under open boundary conditions, inducing “anomalous Bose-Einstein condensation” where the ground state is simultaneously sharply localized in position and momentum, challenging standard interpretations of the uncertainty principle.
• ETH Generalization: The eigenstate thermalization hypothesis (ETH), foundational to quantum statistical mechanics, can be extended to non-Hermitian systems by careful treatment of biorthogonal eigenstates, yielding scaling and statistical predictions for local observables in random-matrix and interacting models such as the disordered Hatano-Nelson chain. New terms arising from nonorthogonality must be explicitly subtracted to recover ETH-like Gaussian statistics, and deviations are found in the localized regime (Roy et al., 2023).
• Non-Hermitian Green's Functions and Physical Constraints: For open or interacting many-body systems, formulating effective non-Hermitian Hamiltonians in terms of Matsubara Green’s functions mandates careful analytic structure: the anti-Hermitian part of the self-energy must be combined with a signum function of Matsubara frequency to preserve correct analytic and causal properties (i.e., $i\,\text{sgn}(\omega_n)\,\Gamma$), as opposed to direct insertion of $i\Gamma$ (Kleger et al., 28 Aug 2025). Pseudo-Hermitian quantum mechanics, wherein $H^\dagger = \eta H \eta^{-1}$ for a positive-definite $\eta$, provides one consistent framework for unitary time evolution and alters the structure of distribution function, observables, and response functions.

5. Geometric Phases, Monopoles, and Dynamical Invariants

• Complex Berry Phases and Hall Admittance: In non-Hermitian systems, the Berry connection and curvature become complex-valued quantities, impacting observable transport signatures. The imaginary part of the geometric phase leads to a quantum Hall susceptance, generalizing the quantum Hall conductance to a complex-valued “admittance”:

$$Y = \sigma_H + iB = \frac{e^2}{h}(\gamma^R + i\gamma^I)$$

where $B$ is the quantum Hall susceptance, signifying capacitance or inductance depending on its sign (Fan et al., 2021).

• Monopoles, Branch Cuts, and Gauge Structure: Monopoles associated with geometric curvature in non-Hermitian systems are not strictly tied to band degeneracies (EPs) but also crucially depend on branch cut choices in the Riemann surface of eigenvalues, leading to Möbius-type switching of eigenstates. Despite arbitrariness in branch cuts, topological invariants (e.g., Chern numbers) retain $GL(l,\mathbb{C})$-gauge invariance (Zhang et al., 2019).
• Canonical and Hamiltonian Formulations: Non-Hermitian linear systems admit Hamiltonian and Lagrangian formulations using biorthogonal canonical conjugate pairs. Conserved quantities and adiabatic invariants can be rigorously constructed (e.g., generalized Noether charges, occupation numbers), and all such results reduce to their Hermitian counterparts when $H$ is Hermitian (Zhang, 2023).

6. Physical Realizations and Experimental Signatures

Physical realization of non-Hermitian systems encompasses a broad set of platforms:

• Optics and Photonics: Arrays with balanced gain/loss, engineered non-reciprocal coupling, or complex refraction indices realize non-Hermitian tight-binding Hamiltonians, exceptional points, and skin effects. Photonic crystals with lossy constituents provide multiband architectures supporting PHL-induced topological effects and state braiding without the need for fine-tuned EPs (Ryu et al., 28 May 2024).
• Quantum Circuits and Meta-materials: Non-reciprocal elements (isolators or diodes) in electrical circuits, synthetic lattices, or acoustic settings allow for direct implementation and measurement of spectral and transport features unique to non-Hermitian settings, such as the direct observation of skin modes or topological transitions induced by tunable non-Hermitian parameters (Long et al., 2021).
• Cold Atoms and Quantum Simulation: Disordered and interacting variants of the Hatano-Nelson model are within experimental reach via quantum-gas microscopes, enabling the observation of non-Hermitian ETH, many-body NHSE, and the breakdown of Hermitian–boson–fermion correspondence (Roy et al., 2023, Sun et al., 2023).

7. Outlook and Open Directions

Research in non-Hermitian systems is advancing rapidly, propelled by both fundamental and application-oriented motivations:

• Classification of topological phases extends AZ and crystalline schemes using new invariants, permutation group theory, and generalized symmetry indicators (Gong et al., 2018, Shiozaki et al., 2021, Ryu et al., 28 May 2024).
• The interplay of pseudo-Hermiticity, physical implementability (e.g., preservation of causality), and experimental control over non-Hermitian parameters is revealing new types of phases and transitions (Kleger et al., 28 Aug 2025).
• Many-body and thermodynamic phenomena (NHSE in large systems, anomalous condensation, non-Hermitian ETH) suggest potential for new states of matter and optimized quantum metrology (Sun et al., 2023).
• Time-dependent (parametric or driven) non-Hermitian systems and their invariants, as well as possible connections to generalized quantum thermodynamics, are active research directions (Frith, 2020).

This field is characterized by an ongoing unification of disparate mathematical methods—Lie algebras, transfer matrices, group theory, K-theory, and biorthogonal calculus—under a single conceptual umbrella, with broad experimental applicability and foundational implications for the structure of quantum theory in open and engineered settings.