Non-Hermitian Hamiltonians in Quantum Systems

• Non-Hermitian Hamiltonians are quantum operators defined by H ≠ H† that extend standard quantum mechanics through pseudo-Hermiticity, enabling real or complex spectra.
• They exhibit biorthogonal eigenbases and exceptional points, providing new insights into topological transitions and non-unitary dynamics in open systems.
• Applications range from quantum optics to condensed matter, using techniques like Dyson maps to simulate gain/loss mechanisms and model dissipative processes.

A non-Hermitian Hamiltonian, defined by the non-self-adjointness condition $H \neq H^\dagger$, generalizes the conventional notion of quantum Hamiltonians and has emerged as a central operator in open quantum systems, quantum optics, quantum fluid dynamics, and quantum field theory. Non-Hermitian Hamiltonians can exhibit entirely real spectra, possess biorthogonal eigenbases, display new symmetry and degeneracy structures, and encode non-unitary dynamics arising from gain, loss, or interaction with environments. Their study provides sophisticated frameworks for generalizing quantum theory, uncovering novel topological phenomena, and modeling dissipative dynamical processes.

1. Pseudo-Hermiticity and Real Spectra

A central distinction of non-Hermitian Hamiltonians in physics is the possibility of possessing a real spectrum via pseudo-Hermiticity. An operator %%%%1%%%% is pseudo-Hermitian if there exists a bounded, invertible, Hermitian metric $\eta$ such that $H^\dagger = \eta H \eta^{-1}$ (Ramirez et al., 2018, Li et al., 2011, Fernández, 2015, Kecita et al., 2 Oct 2025, Miao et al., 2012). In this case, $H$ and $H^\dagger$ are isospectral, and the generalized inner product $$\langle\psi | \phi\rangle_\eta = \langle\psi|\eta|\phi\rangle$$ ensures positivity and unitarity within the $\eta$-Hilbert space. These operators admit biorthonormal right and left eigenbases, and their eigenvalues are either real or occur in complex-conjugate pairs.

Explicit constructions, such as similarity transformations $H = U h U^{-1}$ for invertible $U$ and Hermitian $h$, guarantee the reality of the spectrum and establish the connection with pseudo-Hermiticity via $\eta = (U^{-1})^\dagger U^{-1}$ (Fernández, 2015, Miao et al., 2012).

2. Biorthogonal Bases and Krein Spaces

For diagonalizable non-Hermitian Hamiltonians, the right and left eigenvectors form biorthonormal sets, $\{|\phi_j\rangle\}$ (right) and $\{|\psi_j\rangle\}$ (left), satisfying $H|\phi_j\rangle = E_j|\phi_j\rangle$, $H^\dagger|\psi_j\rangle = E_j^*|\psi_j\rangle$, with $\langle\psi_i | \phi_j\rangle = \delta_{ij}$ (Ramirez et al., 2018, Bebiano et al., 2020). When the spectrum is entirely real, the metric operator can be constructed by summing the projectors onto left eigenstates. When complex-conjugate pairs occur, Krein space theory yields indefinite metrics, which can then be refined to positive-definite operators by separating positive and negative eigenspaces, yielding the Krein metric $S_K = S_+ - S_-$.

The Krein space approach enables consistent definitions of norms, observables, and expectations, underpinning pseudo-unitarity of time evolution (i.e., $U^\dagger_\eta U = I$ with $\eta$-adjoint $U^\dagger_\eta$), and guarantees conservation of generalized probability (Ramirez et al., 2018).

3. Symmetries, Degeneracies, and Split-Quaternion Structures

Non-Hermitian Hamiltonians admit an extended symmetry landscape compared to the Hermitian case. There are two central symmetry mechanisms:

• Commutation: $[H, A] = 0$ (unitary or antiunitary $A$) signifies a symmetry, often yielding conservation laws for bilinear observables (Martínez et al., 2018, Simón et al., 2018).
• Pseudo-Hermiticity intertwining: $A H = H^\dagger A$, which creates spectral pairing between $H$ and $H^\dagger$, often leading to complex-conjugate pairs and generalized conservation laws (Martínez et al., 2018).

Importantly, for non-Hermitian systems, time-reversal symmetry ($\Theta$) and particle-hole symmetry ($C$) can generate new types of degeneracies. In cases with $\Theta^2=+1$ and pseudo-Hermiticity $\{ \eta, \Theta \} = 0$, split-quaternion algebra emerges, resulting in generalized Kramers degeneracy: every eigenvalue is at least two-fold degenerate, even for integer-spin systems (where Hermitian Hamiltonians would not produce such degeneracy) (Sato et al., 2011). Particle-hole symmetry induces doublets with eigenvalues $E$ and $-E$, again deriving from split-quaternion structure rather than ordinary quaternion algebra.

4. Spectral Structure: Real, Complex, and Exceptional Points

The spectrum of a non-Hermitian Hamiltonian is classified by the presence of real eigenvalues, complex conjugate pairs, and exceptional points (EPs). EPs are non-Hermitian degeneracies where both eigenvalues and eigenvectors coalesce; this is in contrast to the Hermitian case where only eigenvalues coincide. EPs are associated with non-diagonalizable Jordan block structure and lead to non-exponential temporal evolution, e.g., linear-in-$t$ modulations (Ramirez et al., 2018, Shen et al., 2017).

At transitions between parameter regimes (for instance, in PT-symmetric systems), the spectrum can pass from entirely real (unbroken PT symmetry) to complex-conjugate pairs (broken PT). At critical values, EPs mediate topological transitions and give rise to intermediate phases with bands that cannot be separated in the complex-energy plane (Shen et al., 2017).

5. Time-Dependent Non-Hermitian Hamiltonians and Gauge Structures

For time-dependent non-Hermitian Hamiltonians $H(t)$, the formalism generalizes via time-dependent Dyson maps $\eta(t)$ and similarity transformations. An infinite chain of gauge-linked, non-observable non-Hermitian Hamiltonians can be constructed, all physically equivalent under global gauge connections (when the chain of Hermitian partners differs only by time-dependent c-numbers). Imposing certain observability criteria (by enforcing Schrödinger-type evolution for the Dyson map) collapses the chain to a single observable non-Hermitian Hamiltonian and removes gauge freedom (Luiz et al., 2017, Kecita et al., 2 Oct 2025).

The transformation strategy $$H(t) \rightarrow F(t) \rightarrow \mathcal{H}_0^{PH} \rightarrow h$$ (with $F(t)$ a suitable time-dependent unitary and $h$ a Hermitian partner) concretely relates non-Hermitian evolution to a standard Hermitian problem, enabling real spectra, exact solution construction, and physically consistent uncertainty relations under the dynamically evolving metric $\tilde{\eta}(t)$ (Kecita et al., 2 Oct 2025).

6. Dynamics, Stability, and Quantum Statistical Properties

The dynamics of open quantum systems modeled by non-Hermitian Hamiltonians feature non-unitary evolution due to the anti-Hermitian part, which physically encodes environmental coupling, gain, or loss. The master equation for the normalized density matrix $\rho(t)$ is nonlinear and may trigger instabilities wherein pure states evolve into mixed states driven by quantum fluctuations and the anti-Hermitian term $\Gamma$ (Zloshchastiev, 2015). Purity, measured by $P(t)=\textrm{Tr}(\rho^2)$, evolves according to the characteristic exponent $\Lambda$; the sign and magnitude of $\Gamma$ determines whether pure states are Lyapunov-stable or undergo singular decoherence (Zloshchastiev, 2015, Sergi et al., 2016).

Entropy functionals, such as the non-Hermitian linear entropy $$S_L^{NH}(t) = 1 - \textrm{Tr}(\sigma^2(t))/\textrm{Tr}\sigma(t)$$ (with $\sigma$ the non-normalized density), better capture the loss or gain of quantum information under sinks or sources than the standard $S_L = 1-\textrm{Tr}(\rho^2)$ (Sergi et al., 2016).

7. Topological Band Theory and Boundary Phenomena

Non-Hermitian band theory generalizes the concept of energy bands to the complex plane, supporting separable and isolated bands, novel vorticity and Chern invariants, and topological edge phenomena even in dissipative systems (Shen et al., 2017). In 1D, half-integer winding (vorticity) invariants classify the effects of energy band dispersion, yielding new topological phases not present in Hermitian systems. In 2D, the non-Hermitian Chern number is built from left and right eigenbases and traces Berry curvature via biorthogonal integrals.

Non-Hermitian systems are acutely sensitive to boundary conditions, leading to phenomena such as the non-Hermitian skin effect, where a macroscopic number of states accumulates at the boundaries (Kawabata et al., 2020, Shen et al., 2017). In symplectic classes, Kramers degeneracy due to reciprocity requires modification of the usual Brillouin zone and generates a $\mathbb{Z}_2$ skin effect protected by symmetry (Kawabata et al., 2020). Topological transitions occur via exceptional points, hybrid points, and exceptional rings, governed by winding and Pfaffian-based invariants.

The extended classification for frequency-dependent non-Hermitian Hamiltonians yields 54 symmetry classes, distinguishing winding in momentum, frequency, and combined variables through refined K-theory constructions (Kotz et al., 2023).

8. Physical Interpretation, Modeling, and Applications

Non-Hermitian Hamiltonians arise fundamentally as effective descriptions for open quantum systems: the "scattering center" of a Hermitian system embedded in infinite leads acquires an equivalent non-Hermitian Hamiltonian, with the eigenstate structure matching the physical center's amplitudes (Jin et al., 2011). This mapping holds even for broken PT or general non-PT cases and explains the appearance of real eigenvalues via engineered gain and loss or via coupling strengths, bypassing the need for abstract metric operators.

Applications span quantum optics (where non-Hermitian effective Hamiltonians model lossy or amplifying cavities and circuits), non-Hermitian band theory for topological insulators, quantum control via generalized time-evolution schemes (Magnus expansions), and simulation protocols in photonics employing symplectic Bogoliubov transformations (Wakefield et al., 2023, Mulian, 14 May 2025).

Table: Key Non-Hermitian Hamiltonian Properties and Formalisms

Concept	Main Feature	Supporting arXiv Paper Ids
Pseudo-Hermiticity	Real spectrum via existence of metric $\eta$	(Ramirez et al., 2018, Fernández, 2015, Li et al., 2011)
Biorthonormality	Distinct left/right eigenbases, generalized inner product	(Bebiano et al., 2020, Ramirez et al., 2018)
Split-quaternion symmetry	Generalized Kramers degeneracy for $\Theta^2=+1$	(Sato et al., 2011)
Krein space theory	Indefinite metric; construction of positive metric	(Ramirez et al., 2018)
Exceptional points (EPs)	Coalescence of eigenvalues/eigenvectors, non-exponential dynamics	(Ramirez et al., 2018, Shen et al., 2017)
Non-Hermitian band topology	Complex energy bands, vorticity/Chern invariants, $\mathbb{Z}_2$ skin effect	(Shen et al., 2017, Kawabata et al., 2020, Kotz et al., 2023)
Time-dependent Dyson maps	Infinite gauge chain, observability criteria	(Luiz et al., 2017, Kecita et al., 2 Oct 2025)
Quantum-statistical entropy	Dynamics of purity, non-Hermitian entropy functionals	(Zloshchastiev, 2015, Sergi et al., 2016)
Symplectic transformation	Simulation via Bogoliubov circuits, decomposition	(Wakefield et al., 2023)

Non-Hermitian Hamiltonians, through pseudo-Hermitian formalism, generalized symmetry structures, Krein space methods, and topological invariant extensions, have significantly expanded the operator-theoretic and application landscape of quantum theory in open, non-equilibrium, and topologically nontrivial systems.