Non-Hermitian Quantum Systems

• Non-Hermitian quantum systems are defined by Hamiltonians that capture open system dynamics with loss, gain, and complex energy spectra.
• They exhibit unique phenomena such as exceptional points, biorthogonal eigenstates, and altered thermodynamic and topological properties.
• Methodological advances like semiclassical quantization and non-Hermitian quantum geometry drive cutting-edge research in these systems.

Non-Hermitian quantum systems are quantum mechanical systems whose effective Hamiltonians are not Hermitian, $H \neq H^\dagger$, reflecting the presence of openness, loss, or gain, and often encode probabilistic or "leaky" evolution characteristic of interactions with environmental reservoirs. Such systems exhibit phenomena — including complex eigenvalue spectra, biorthogonal quantum state structures, exceptional points, non-trivial quantum geometry, and modified thermodynamics — that fundamentally lie outside the scope of conventional Hermitian quantum physics.

1. Theoretical Foundations: Open Systems and Non-Hermitian Hamiltonians

A realistic quantum system $S$ is generically coupled to an environment or "continuum" of decay/scattering channels $C$, and its reduced dynamics are necessarily non-unitary. The effective description of $S$ is formalized by eliminating the environmental degrees of freedom, resulting in an energy-dependent, often explicitly non-Hermitian, system Hamiltonian $H_{\text{eff}}$: $$H_{\text{eff}}(E) = H_{PP} + H_{PQ}(E-H_{QQ})^{-1}H_{QP}$$ Here, $P$ and $Q$ are orthogonal projectors into system and environment, respectively; $H_{PP}$ is Hermitian, while the second term introduces complex contributions encoding both Lamb-type energy shifts and finite-state lifetimes (decay widths) (Rotter, 2017, Hatano et al., 15 Feb 2026).

The spectrum of $H_{\text{eff}}$ is generically complex:

$S$0

where $S$1 is the resonance position, and $S$2 is the total decay width. The eigenstates associated with $S$3 are resonant (decaying), anti-resonant (growing), or bound states as categorized by their complex eigenvalues and boundary conditions, with outgoing-wave (Siegert) conditions yielding poles outside $S$4, i.e., outside the usual Hilbert space representation (Hatano et al., 15 Feb 2026).

2. Biorthogonal Structure and Non-Hermitian Quantum Geometry

Because non-Hermitian Hamiltonians $S$5 satisfy $S$6, right and left eigenstates are distinct and form a biorthogonal system: $S$7 Biorthogonal geometry underpins all quantum measurement and evolution protocols in this context. Observable expectation values are defined biorthogonally, e.g., for zero-temperature ground state: $S$8 (Guo et al., 2023, Bebiano et al., 2020). Biorthogonality enables a consistent extension of the Fubini–Study metric, Berry connection, and quantum geometric tensor (QGT) to non-Hermitian settings (Behrends et al., 17 Mar 2025, Das et al., 14 Jun 2026, Montag et al., 8 Dec 2025, Pal, 24 Jul 2025).

Quantum geometry in non-Hermitian systems is inherently richer. The QGT becomes a complex tensor decomposing into quantum metric (real symmetric) and non-Hermitian Berry curvature (imaginary antisymmetric) parts. The generalized (mixed) Berry connections and gauge-invariant quantities such as $S$9 encode the geometric corrections arising solely from the non-orthogonality of the eigenstates (Behrends et al., 17 Mar 2025, Montag et al., 8 Dec 2025). This framework unifies geometric responses, susceptibility, wave packet dynamics, and measurable localization.

3. Exceptional Points and Topological Phenomena

A hallmark of non-Hermitian quantum systems is the presence of exceptional points (EPs)—parameters where multiple eigenvalues and eigenvectors coalesce, rendering $C$0 non-diagonalizable. EPs are branch points of the spectrum and produce nontrivial topology: encircling an EP, eigenstates exchange identity and may accumulate geometric (topological) phases (Rotter, 2017, Zhang et al., 2018, Das et al., 14 Jun 2026). Near an EP, eigenvalue trajectories exhibit square-root singularities, and the quantum geometric tensor and Berry curvature show universal divergence scalings ($C$1 for the metric, $C$2 for the curvature) (Das et al., 14 Jun 2026).

Non-Hermitian analogues of Chern numbers and winding numbers classify topological phases in both single-particle and many-body systems, with non-Hermitian bands supporting quantized invariants constructed from biorthogonal connections and curvatures (Matraszek et al., 29 Dec 2025, Fan et al., 2021). Phenomena such as the non-Hermitian skin effect, worldline winding in quantum Monte Carlo, and the breakdown of bulk-boundary correspondence manifest as direct physical consequences of such topological structures (Hu et al., 2023). The theory naturally generalizes quantum Hall conductance to a full quantum Hall admittance with intrinsic capacitance or inductance, depending on system parameters (Fan et al., 2021).

4. Quantum Phases, Statics, and Statistical Mechanics

In the non-Hermitian regime, the classification of many-body quantum phases diverges from the Hermitian paradigm. The ground-state structure is inherently composite, associated with a pair of left and right ground states, and composite quantum phases (CSPT) arise when these belong to different Hermitian-equivalence classes. The presence of line gaps (gaps avoiding lines in the complex plane) is essential for phase robustness (Guo et al., 2023).

Statistical mechanics of non-Hermitian systems requires a positive-definite path-independent conserved quantity $C$3 in addition to $C$4 to define a physical equilibrium state. The equilibrium density operator must generalize to

$C$5

with the partition function and response inheriting non-Boltzmann features dependent on $C$6. Finite relaxation times require that the "abnormal index" $C$7 not exceed the non-Hermitian strength $C$8 (Cao et al., 2023). Steady-state solutions and thermal behaviors may differ critically from Hermitian cases, especially when the spectrum is partially or fully complex.

Information thermodynamics in non-Hermitian quantum systems reveals that the entropy production rate $C$9 can be negative, enabling the construction of single-bath information engines, violating the conventional (Kelvin–Planck) bound. Key informational state variables, in particular the non-Hermitian information content $S$0, serve as new order parameters for phase transitions inaccessible via conventional partition functions (Cao et al., 2024).

5. Quantum Dynamics, Decoherence, and Time Dependence

Non-Hermitian systems display distinct dynamical properties due to non-unitarity. The time-dependent Schrödinger equation requires generalization — the energy observable is not given by $S$1 but by a properly defined $S$2 with an instantaneous metric $S$3 (Frith, 2020). In this framework, the time-evolution operator is pseudo-unitary with respect to the generalized metric, and the broken $S$4-symmetric regime becomes physically regularized under time dependence.

Decoherence and entanglement generation in finite-dimensional and many-body non-Hermitian systems are characterized by mixedness and entanglement timescales, which are explicitly determined by the anti-Hermitian part of $S$5 and probe-state expectation values. The linear entropy growth time, $S$6, and the entanglement timescale, $S$7, provide efficient diagnostic tools for quantifying decoherence and transition to mixed (entropy-generating) states in open or effective non-Hermitian settings (Pires et al., 2022).

6. Quantum Geometric Bounds, Response, and Physical Constraints

The quantum geometric tensor in non-Hermitian systems admits universal operator inequalities and metric bounds. For a biorthogonal band, the quantum metric is positive semidefinite, while the full QGT (including mixed LR/RL sectors) is bounded by the product of norms of response operators and the minimal complex spectral gap: $S$8 Here, $S$9 is the real gap and $H_{\text{eff}}$0 the complex gap (decay width) (Matraszek et al., 29 Dec 2025). These bounds directly constrain optical weights, conductivities, and more generally the linear response in driven and open systems, including those described by Lindblad master equations.

Universal constraints propagate to topological invariants: non-Hermitian Chern numbers and optical weights are always strictly bounded from above and below, such that topological edge features remain observable even for complex spectra.

7. Methodological Advances and Future Directions

Complex semiclassical theory extends Bohr–Sommerfeld quantization, Hamilton–Jacobi equations, and closed-orbit theory into the complex domain. Phase-space variables, orbits, and energy become complexified, with quantization conditions imposed by closed loops in complexified phase space. This approach enables direct computation of complex spectra, wavefunctions, and phase transitions in larger and higher-dimensional non-Hermitian systems, as well as semiclassical predictions of topological phase boundaries and skin effects (Yang et al., 2023).

Mapping between Hermitian and non-Hermitian systems at quantum criticality reveals that although some spectral and entanglement features can be preserved under similarity or dilation transformations, generically non-Hermitian systems realize fundamentally distinct conformal field theories, including non-unitary ones with negative central charge (Hsieh et al., 2022).

Major open research areas include robust numerical diagonalization of large non-Hermitian operators, the classification of exceptional points in many-body and multi-channel contexts, the thermodynamics of steady-state preparation, and the interplay of non-Hermitian geometry with experimental measurement protocols.

---

Key references: (Rotter, 2017, Hatano et al., 15 Feb 2026, Behrends et al., 17 Mar 2025, Guo et al., 2023, Das et al., 14 Jun 2026, Montag et al., 8 Dec 2025, Matraszek et al., 29 Dec 2025, Pires et al., 2022, Frith, 2020, Cao et al., 2024, Hsieh et al., 2022, Yang et al., 2023, Fan et al., 2021, Hu et al., 2023, Kleger et al., 28 Aug 2025, Zhang et al., 2018, Pal, 24 Jul 2025, Cao et al., 2023, Moulopoulos, 2018, Bebiano et al., 2020).