Hermitian Quadratic Bosonic Model

• The Hermitian Quadratic Bosonic Model is a framework describing bosonic systems with quadratic Hamiltonians, emphasizing stability, dynamic evolution, and spectral structure.
• It utilizes pseudo-Hermiticity and Bogoliubov diagonalization to analyze critical behavior, edge phenomena, and phase transitions under both closed and open system conditions.
• The model bridges quantum optics, condensed matter, and field theory by addressing non-Hermitian effective dynamics and mapping topological invariants in bosonic systems.

A Hermitian Quadratic Bosonic Model (QBH) is a mathematical and physical framework for describing bosonic systems whose Hamiltonians are quadratic in bosonic creation and annihilation operators. These models are pivotal in quantum optics, condensed matter, and quantum field theory, as they capture phenomena ranging from stability and dynamical instabilities to topological phase transitions and edge phenomena. Although the underlying Hamiltonian is Hermitian, the dynamical evolution is governed by an effective non-Hermitian operator, leading to rich spectral structures and critical behavior.

1. Structure of Quadratic Bosonic Hamiltonians

A general quadratic bosonic Hamiltonian is expressed as

$$H = \sum_{i,j} A_{ij} (b_i^\dag b_j + b_j b_i^\dag) + \sum_{i,j} (B_{ij} b_i b_j + B^*_{ij} b_i^\dag b_j^\dag)$$

where $b_i, b_i^\dag$ are bosonic annihilation and creation operators. The matrix $A$ is Hermitian, and $B$ is symmetric. Mapping this into the Nambu formalism,

$$Z^T = (b_1, b_2, \ldots, b_1^\dag, b_2^\dag, \ldots)$$

the Hamiltonian can be rewritten as

$H = \frac{1}{2} Z^\dag H Z$

The dynamical matrix driving the Heisenberg evolution is

$$\mathcal{H} = M H = \begin{pmatrix} A & -B^* \ B & -A^* \end{pmatrix}$$

where $M$ encodes the bosonic commutation structure (Rossignoli et al., 2014).

2. Pseudo-Hermiticity and Stability

While $H$ itself is Hermitian, the evolution operator $\mathcal{H}$ is generally pseudo-Hermitian: $\mathcal{H}^\dagger = M \mathcal{H} M^{-1}$ Pseudo-Hermiticity underpins dynamical stability analysis. All eigenvalues real $\Rightarrow$ bounded, quasiperiodic dynamics; complex eigenvalues $\Rightarrow$ exponentially growing/decaying modes (Fernández, 2015, Flynn et al., 2020). If the quadratic form is positive definite, canonical Bogoliubov diagonalization yields proper bosonic modes. For indefinite forms, "non-standard" boson-like operators—defined via non-unitary transformations—are essential for diagonalization

$Z = W Z',\quad W M W = M$

Such operators may not be mutual adjoints, and correspond to non-Hermitian coordinates and momenta (Rossignoli et al., 2014).

Dynamic stability is categorized:

• Real, positive eigenvalues: stable, periodic evolution
• Real, negative eigenvalues: Hamiltonian unbounded below but evolution still bounded
• Complex eigenvalues: dynamical instability (exponential growth/decay)

In non-diagonalizable cases (Jordan block structure), time evolution can be polynomially in time multiplied by exponentials, leading to unconventional instabilities.

3. Spectral Analysis and Phase Transitions

Eigenmodes and spectrum are determined by solving the generalized eigenvalue problem

$\mathcal{H} \, W = M W \lambda$

The spectrum typically falls into distinct regimes (Garcia et al., 2017):

• Harmonic: discrete eigenvalues, biorthogonal eigenstates, Gaussian vacua
• Coherent-like: continuous, two-fold degenerate complex spectrum; normalizability lost for one branch
• Non-diagonalizable/critical: mixing between creation and annihilation leads to modes analogous to coordinate/momentum operators (continuous spectrum).

Exceptional points (EPs), defined as parameter values where eigenvectors coalesce and $\mathcal{H}$ is non-diagonalizable, separate these regimes. EPs mediate quantum phase transitions, dividing parameter space into regions characterized by harmonic oscillator (HO) or inverted harmonic oscillator (IHO) effective descriptions (He et al., 26 Feb 2025). Second-order intensity correlations $g^{(2)}$, evaluated along quench dynamics, serve as sharp witnesses of such transitions.

4. Topological and Symmetry Classification

Topological phases in quadratic bosonic systems arise via the structure of the dynamical matrix, which—while Hermitian in closed systems—can be written as

$H_{dyn}(k) = \tau_3 H(k)$

where $\tau_3 = \sigma_3 \otimes I$. The symmetry properties (time-reversal, particle-hole, chiral) of $H_{dyn}$ or, equivalently, the effective Hamiltonian in the Lindblad formalism for open systems, yield a periodic (tenfold) table classification fully analogous to the Altland–Zirnbauer (AZ) scheme for fermions (Lieu, 2018, Zhou et al., 2019, He et al., 2021).

Bernard-LeClair (BL) symmetry classes generalize Hermitian classification: quadratic bosonic systems respect pseudo-Hermiticity (Q symmetry) and a particle–hole-like involution (K symmetry). In practice:

• BL symmetry protects edge instabilities (e.g., imaginary edge modes in bosonic SSH chains)
• Topological invariants (Chern number, $\mathbb{Z}_2$ index) are inherited from effective single-particle blocks, sometimes "doubled" or uniquely constrained by bosonic commutation (Zhou et al., 2019)

Coupling to a reservoir via non-Hermitian Lindblad operators may break protecting symmetries, changing classification and associated invariants. The interplay of topology and dynamics manifests in chiral amplification, Möbius phase winding, and bulk-boundary correspondence (Estake et al., 20 Aug 2025).

5. Renormalization and Self-Adjointness

In infinite-dimensional settings, rigorous definition of quadratic bosonic Hamiltonians requires careful handling of operator domains and renormalization (Dereziński, 2016). Self-adjointness necessitates the positivity of the "free" operator and a norm bound on the pairing term

$\| h^{-1/2} g h^{-1/2} \| < 1$

In local quantum field theory (QFT), infinite renormalizations are performed by subtracting divergent loop energies, formally yielding well-defined ground state energies ("vacuum energy" or Casimir energy)

$$E_{ren} = \inf \left( H_W - E_0 - E_1 - E_2 \right)$$

Explicit formulas for the infimum are available in terms of traces over one-particle operators and symplectic transformations.

The Shale condition (Hilbert–Schmidt norm of off-diagonal block) guarantees that Bogoliubov transformations are unitarily implementable.

6. Number Conservation Duality and Computational Stability

Pairing terms (number-non-conserving quadratic interactions) lead to parametric instability. When the system is dynamically stable (real spectrum), it is always possible to construct a dual, unitarily equivalent number-conserving quadratic bosonic Hamiltonian via a canonical transformation constructed from the positive-definite pseudo-Hermitian metric $S$ (Flynn et al., 2020). This mapping ensures that the physical behavior (including topological invariants, as computed via generalized Berry curvature using the Krein metric) is preserved in the dual.

Stability under perturbations is quantitatively assessed via relative perturbation theory for the generalized quadratic eigenvalue problem

$\left( \lambda^2 M + \lambda C + K \right)x = 0$

with relative perturbation bounds depending explicitly on condition numbers, separation metrics, and normalized matrix variations

$$\left|\frac{\widetilde{\lambda} - \lambda}{\lambda}\right| \leq \frac{\kappa_2(X)\kappa_2(\widetilde{X})(\|A^{\dagger}\delta A\|_2 + \|B^{-1}\delta B\|_2)}{|x^H B \widetilde{x}|}$$

Such bounds are sharp even in challenging, nearly defective regimes encountered in indefinite damping bosonic models (Benner et al., 2016).

7. Open Systems and Dissipative Dynamics

In Markovian open quantum systems, Lindblad master equations describe evolution. For bosonic models, with quadratic Hamiltonians and linear/quadratic Hermitian Lindblad dissipators, one can solve for the full many-body spectrum via third quantization, transforming the Liouvillian to Jordan (or block-triangular) canonical form (Barthel et al., 2021). The covariance matrix evolution is closed for Gaussian states, and the existence of steady states is linked to the spectral properties of the dynamical generator:

• All symplectic eigenvalues within permissible range $\Rightarrow$ steady state exists
• Criticality/dissipative phase transitions signaled by closing of Liouvillian gap or non-diagonalizability

Quadratic dissipators drive the system away from Gaussianity but the correlation hierarchy remains closed. These tools underpin rigorous analyses of stability, decoherence, and criticality in quadratic Hermitian bosonic models.

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The Hermitian Quadratic Bosonic Model thus encompasses a universal framework for analyzing the spectrum, stability, nonlinear dynamics, and topological properties of bosonic systems—both closed and open, Hermitian or effectively non-Hermitian. This central architecture facilitates the paper of quantum phase transitions, edge phenomena, amplification effects, particle-hole duality, and their engineering in photonic, cold-atom, superconducting, and field-theoretical platforms.