Quadratic Quantum Chaos in Two-Body Systems

• Quadratic quantum chaos is defined by the emergence of chaotic behavior from quadratic two-body Hamiltonians that exhibit operator spreading and spectral universality.
• Spectral diagnostics such as level spacing ratios and spectral form factors reveal universal random matrix behavior and level repulsion in these systems.
• Operator growth measures like Krylov complexity and out-of-time-ordered correlators illustrate information scrambling, bridging integrable and maximally chaotic regimes.

Quadratic quantum chaos refers to the emergence and characterization of quantum chaotic phenomena in systems governed by quadratic (two-body) Hamiltonians, particularly those whose single- or few-body interactions and operator structure give rise to spectral and dynamical signatures typically associated with quantum chaos. Unlike classical chaos, which arises from nonlinear equations of motion, quadratic quantum chaos emerges from the interplay of quadratic operator structure, randomness, and information processing in many-body quantum systems. It is a concept that bridges purely integrable quadratic models and the strongly interacting, maximally chaotic models, highlighting regimes where even minimal Hamiltonian complexity gives rise to rich signatures of chaos and information scrambling.

1. Archetypes of Quadratic Quantum Chaos: Hamiltonians and Model Construction

Quadratic quantum chaos is most meaningfully explored in models where the Hamiltonian consists of two-body interactions with random coefficients, such as SYK-like Hamiltonians in spin or bosonic systems. The Spin-SYK$_2$ Hamiltonian serves as a canonical archetype:

$$H_{\text{Spin}\,\mathrm{SYK}_2} = \frac{1}{\sqrt{2N_{\text{spin}}}} \sum_{1 \leq i_1 < i_2 \leq 2N_{\text{spin}}} i^{\eta_{i_1i_2}} J_{i_1i_2} O_{i_1} O_{i_2},$$

where $O_j$ are local Pauli operators (alternating $\sigma^x$ and $\sigma^y$), $J_{i_1i_2}$ are independent Gaussian random couplings, and $\eta_{i_1i_2}$ ensures Hermiticity. Quadratic quantum chaos sharply distinguishes itself from the integrable quadratic fermionic Hamiltonians (e.g., free Fermi gases, quadratic Majorana models) in both eigenvalue statistics and operator spreading, despite the absence of high-order interactions.

The minimal (genuine) two-body spin models, where all self-site terms are removed, display a remarkably rich structure: in contrast to quadratic fermionic systems (which show integrability at $q=2$), the corresponding bosonic models exhibit genuine quantum chaos (Basu et al., 4 Sep 2025).

2. Spectral Diagnostics: Random Matrix Theory (RMT) and Level Statistics

A principal diagnostic of quantum chaos is the comparison of spectral statistics to RMT predictions. Level spacing ratios for consecutive eigenvalues,

$$r_n^{(k)} = \min\left\{ \frac{s_{n+k}^{(k)}}{s_n^{(k)}}, \frac{s_n^{(k)}}{s_{n+k}^{(k)}} \right\}, \quad s_n^{(k)} = \lambda_{n+k} - \lambda_n,$$

are compared against universal distributions. For GOE, GUE, and GSE, the analytic form [see Eq. (4) in (Basu et al., 4 Sep 2025)] is

$$P_\beta^{(1)}(r) = Z_\beta \frac{(r+r^2)^\beta}{(1 + r + r^2)^{1+3\beta/2}},$$

with Dyson index $\beta=1,2,4$ corresponding to GOE, GUE, and GSE, and $Z_\beta$ a normalization. The Spin-SYK$_2$ ensemble exhibits level repulsion and RMT scaling, showing that even quadratic interactions generate spectra consistent with universal chaotic ensembles.

Long-range spectral correlations are probed by the spectral form factor

$$\mathrm{SFF}(t) = \left| \frac{\operatorname{Tr}\left( e^{-(\beta + it)H} \right)}{\operatorname{Tr}(e^{-\beta H})} \right|^2,$$

which displays a characteristic dip, linear ramp, and saturation (plateau) at late times in chaotic systems, confirming the presence of long-range eigenvalue correlations.

3. Operator Growth and Information Scrambling

Quadratic quantum chaos manifests in operator dynamics, where time evolution of initially local operators becomes highly complex. Krylov complexity $K(t)$ quantifies the spread of an operator $O(t)$ in the Krylov basis: $K(t) = \sum_n n |\phi_n(t)|^2,$ where the operator is expanded as

$O(t) = \sum_n i^n \phi_n(t) \mathcal{O}_n,$

with $\mathcal{O}_n$ generated recursively via the Lanczos algorithm. In chaotic quadratic models, Krylov complexity initially grows exponentially, then linearly, and finally saturates, with the peak height encoding the underlying RMT symmetry class.

Higher-order out-of-time-ordered correlators (OTOCs) are also computed: $$F_p^{(i,j)}(t) = \sum_{n=1}^{p} \frac{1}{n} \langle (O_i(0) O_j(t))^n \rangle^2,$$ where $F_p$ decays at late times as operators "become free" in the sense of free probability, a signature of maximal scrambling in quantum chaos (Basu et al., 4 Sep 2025).

4. Eigenstate Structure: Fractal Dimensions and Stabilizer Rényi Entropy

Quantum chaos is reflected in eigenstate ergodicity properties. The fractal dimension $D_\alpha$ for a mid-spectrum eigenstate $|\Psi\rangle$ measures its spread in a basis $\{|\psi_i\rangle\}$: $$I_\alpha = \sum_i |c_i|^{2\alpha}, \qquad D_\alpha = \frac{1}{1-\alpha} \frac{1}{\log_2 d} \log_2 I_\alpha,$$ where $|\Psi\rangle = \sum_i c_i |\psi_i\rangle$ and $d = 2^N$. Haar-random (fully ergodic) states have $D_\alpha \to 1$; lower values indicate multifractality or eigenstate localization. The models studied exhibit weakly ergodic mid-spectrum eigenstates whose fractal dimensions converge to Haar values only in the infinite-size limit.

Stabilizer Rényi Entropy (SRE) measures the nonstabilizerness ("magic") of a pure state $|\psi\rangle$,

$$\mathcal{M}_\alpha(|\psi\rangle) = \frac{1}{1-\alpha} \log_2 \left( \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} |\langle \psi | P | \psi \rangle|^{2\alpha} \right),$$

with $\mathcal{P}_N$ the $N$-qubit Pauli group. For Haar-random states, $$M_2^{\text{Haar}} = -\log_2\left( \frac{4}{3 + 2^N} \right) \to 1$$ per qubit for $N\to\infty$. Quadratic quantum chaotic models show SRE approaching this Haar limit at large system sizes, though with significant finite-size corrections (Basu et al., 4 Sep 2025).

5. Comparison with Integrable and Maximally Chaotic Systems

Quadratic fermionic models (e.g., integrable SYK$_2$) and free-fermion chains are integrable, exhibiting Poisson spectral statistics and negligible operator spreading—no signatures of chaos in level statistics, Krylov complexity, or eigenstate structure (multifractal dimensions and SRE remain well below Haar values).

In contrast, maximally chaotic models (e.g., SYK$_4$) achieve full RMT statistics, ballistic operator spreading, and maximally high SRE at modest system sizes. The quadratic bosonic models presented provide a "minimal" route to quantum chaos, with finite-size deviations from Haar limits that shrink slowly with increasing $N$. This "weakly chaotic" regime is characterized by level repulsion, universal spectral form factors, operator entanglement growth, and eigenstates that eventually approach ergodicity in the thermodynamic limit.

6. Implications, Applications, and Outlook

Minimal quadratic models with random two-body interactions demonstrate that quantum chaos does not require high-order interactions or the complexity of SYK$_q$ for large $q$. The bosonic Spin-SYK$_2$ models are both theoretically tractable and experimentally accessible (e.g., in cold atoms, trapped ions, superconducting circuits), making them promising candidates for probing signatures of chaos, information scrambling, and ergodicity in near-term quantum devices.

The emergence of chaotic spectral and dynamical behavior from minimal Hamiltonian ingredients places quadratic quantum chaos at a unique crossroads: it deepens understanding of the connection between operator structure, physical symmetries, and information-theoretic complexity, and it provides a resource-efficient avenue for exploring the transition from integrability to chaos and for benchmarking quantum devices on analogs of RMT universality (Basu et al., 4 Sep 2025).

Table: Key Spectral and Dynamical Diagnostics in Quadratic Quantum Chaos

Diagnostic	Quadratic Bosonic Model	Integrable Quadratic Fermion	Maximally Chaotic (SYK$_4$)
Level Spacing	GUE/GOE	Poisson	GUE/GOE
Krylov Complexity	Linear growth, peak	Saturates early	Linear growth, Haar peak
Cumulative OTOC	Late-time decay	Plateaus	Maximum decay
Fractal Dimension	$\to 1$ as $N \to \infty$	$\ll 1$	$\approx 1$
SRE (mid-spectrum)	Approaches Haar	Low	Haar

This synthesis demonstrates that even the simplest two-body quadratic bosonic Hamiltonians can generate rich and universal manifestations of quantum chaos, with important consequences for quantum information dynamics, operator spreading, and the emergence of ergodic behavior in many-body quantum systems.