Complexity of Quadratic Quantum Chaos (2509.04075v1)

Abstract: We investigate minimal two-body Hamiltonians with random interactions that generate spectra resembling those of Gaussian random matrices, a phenomenon we term quadratic quantum chaos. Unlike integrable two-body fermionic systems, the corresponding hard-core boson models exhibit genuinely chaotic dynamics, closely paralleling the Sachdev-Ye-Kitaev (SYK) model in its spin representation. This chaotic behavior is diagnosed through spectral statistics and measures of operator growth, including Krylov complexity and the late-time decay of higher-order out-of-time-ordered correlators (OTOCs); the latter reveals the emergence of freeness in the sense of free probability. Moreover, the fractal dimension and Stabilizer Renyi entropy of a representative mid-spectrum eigenstate show finite-size deviations yet converge toward Haar-randomness as the system size increases. This convergence, constrained by local interactions, highlights the "weakly chaotic" character of these eigenstates. Owing to its simplicity and bosonic nature, these minimal models may constitute promising and resource-efficient candidates for probing quantum chaos and information scrambling on near-term quantum devices.

• The paper demonstrates that minimal quadratic spin SYK models exhibit robust quantum chaos through RMT statistics and a dip–ramp–plateau spectral form factor.
• It employs detailed spectral analysis and operator growth diagnostics, such as Lanczos coefficients and Krylov complexity, to delineate chaos emergence.
• Eigenstate studies indicate weak ergodicity with finite-size deviations from Haar randomness, converging in the thermodynamic limit.

Complexity and Quantum Chaos in Quadratic Spin SYK Models

Introduction and Motivation

The paper of quantum chaos in many-body systems has traditionally focused on models with highly nonlocal interactions, such as the Sachdev-Ye-Kitaev (SYK) model, which features all-to-all random $q$-body couplings among Majorana fermions. While the SYK model is maximally chaotic and exhibits universal random matrix theory (RMT) statistics, its dense interaction structure and fermionic nonlocality present significant challenges for both classical simulation and quantum hardware implementation. This work addresses the question of whether minimal, fermion-free, and resource-efficient models can exhibit similar haLLMarks of quantum chaos, focusing on quadratic (two-body) spin SYK models constructed from local Pauli operators.

The central result is that even these minimal two-body spin models, with random couplings and hard-core bosonic statistics, display robust signatures of quantum chaos. This is in stark contrast to the integrable nature of the quadratic fermionic SYK$_2$ model. The analysis leverages spectral diagnostics, operator growth measures (Krylov complexity, OTOCs), and eigenstate properties (fractal dimension, stabilizer Rényi entropy) to establish the chaotic character and ergodicity properties of these models.

Model Construction and Symmetries

The quadratic spin SYK model is defined on $N_{\mathrm{Spin}}$ sites, with local operators

$$O_{2a-1} = \mathbb{I}^{\otimes (a-1)} \otimes \sigma^x_a \otimes \mathbb{I}^{\otimes (N_{\mathrm{Spin}}-a)}, \quad O_{2a} = \mathbb{I}^{\otimes (a-1)} \otimes \sigma^y_a \otimes \mathbb{I}^{\otimes (N_{\mathrm{Spin}}-a)}$$

for $a = 1, \ldots, N_{\mathrm{Spin}}$. The Hamiltonian is

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

where $J_{i_1 i_2}$ are i.i.d. Gaussian random variables and $\eta_{i_1 i_2}$ ensures Hermiticity. The model admits a $\mathbb{Z}_2$ symmetry generated by $$\Gamma^3 = \bigotimes_{i=1}^{N_{\mathrm{Spin}}} \sigma^z_i$$, which partitions the Hilbert space into even and odd parity sectors.

A further refinement, the "genuine" (g)Spin-SYK model, removes self-site (on-site) terms, retaining only true two-body interactions. Both variants are analyzed, with the gSpin-SYK model providing a cleaner probe of two-body chaos.

Spectral Diagnostics: RMT Statistics and Spectral Form Factor

The density of states (DOS) for both Spin-SYK$_2$ and gSpin-SYK$_2$ models is Gaussian, consistent with RMT predictions.

Figure 1: The Density of States (DOS) of the (g)Spin SYK models for various $N_{\mathrm{Spin}}$.

Short-range spectral statistics, such as the distribution of nearest-neighbor and next-to-nearest level spacing ratios, are in excellent agreement with the universal RMT ensembles (GUE or GOE, depending on $N_{\mathrm{Spin}}$ parity and model variant). The mean spacing ratios and the full distributions match the analytic forms for the corresponding Dyson index.

Figure 2: The nearest ($k = 1$) and next-to-nearest ($k = 2$) level spacing ratio distributions for the Spin SYK$_2$ and (g)Spin SYK$_2$ models, showing universality class dependence on $N_{\mathrm{Spin}}$.

Long-range spectral rigidity is probed via the spectral form factor (SFF), which exhibits the characteristic dip–ramp–plateau structure of chaotic systems. The presence of a linear ramp, followed by a plateau at late times, is a robust indicator of long-range eigenvalue correlations and quantum chaos.

Figure 3: Time evolution of the Spectral Form Factor (SFF) for the (g)Spin SYK$_2$ models, showing a clear ramp indicative of chaos.

Operator Growth: Krylov Complexity and OTOCs

Operator growth is analyzed using the Krylov basis generated by repeated commutators with the Hamiltonian. The Lanczos coefficients $b_n$ exhibit an initial linear growth regime, followed by saturation due to finite-size effects. This linear growth is a haLLMark of chaotic dynamics and is absent in integrable systems.

Figure 4: The behavior of Lanczos coefficients $b_n$ for Spin SYK$_2$ (solid) and (g)Spin SYK$_2$ (dashed) models, showing linear growth and finite-size saturation.

The associated Krylov complexity displays early-time exponential growth, a pronounced peak, and eventual saturation. The peak height and saturation value are sensitive to the universality class and system size, but the qualitative structure is robust across model variants.

Figure 5: Time evolution of the Krylov spread complexity for the (g)Spin SYK$_2$ models, with a pronounced peak signaling chaos.

The cumulative out-of-time-ordered correlator (OTOC) decays to zero at late times, consistent with the emergence of operator "freeness" in the sense of free probability theory. This indicates that, under chaotic evolution, local operators become statistically independent of their initial configurations.

Figure 6: Cumulative OTOC for Spin SYK$_2$ and gSpin SYK$_2$ with a local operator, decaying to zero at late times.

Eigenstate Properties: Fractal Dimension and Stabilizer Rényi Entropy

The ergodicity of mid-spectrum eigenstates is quantified via the fractal dimension $D_\alpha$ and the stabilizer Rényi entropy (SRE) $\mathcal{M}_\alpha$. For Haar-random states, $D_\alpha = 1$ and SRE saturates the analytic Haar value. In the Spin SYK$_2$ model, both quantities exhibit systematic finite-size deviations from the Haar benchmark, indicating "weak ergodicity." As $N_{\mathrm{Spin}}$ increases, the deviations decrease, but full ergodicity is only achieved in the thermodynamic limit.

Figure 7: The fractal dimension $D_{\alpha}$ for a single mid-spectrum eigenstate of the Spin SYK$_2$ model as a function of $1/N_{\mathrm{Spin}}$, compared to Haar-random predictions.

Figure 8: The Stabilizer Rényi entropy $\mathcal{M}_{\alpha}$, rescaled by the number of spins, for a single mid-spectrum eigenstate of the Spin SYK$_2$ model as a function of $1/N_{\mathrm{Spin}}$, compared to Haar-random values.

This weakly ergodic behavior is a direct consequence of the local structure of the model and distinguishes it from both integrable systems (which remain far from Haar-randomness) and fully chaotic models with all-to-all interactions.

Minimal Ingredients for Chaos and Generalizations

The appendices demonstrate that cross-interactions between $\sigma_x$ and $\sigma_y$ are essential for chaos; models with only $\sigma_x$–$\sigma_x$ or $\sigma_y$–$\sigma_y$ interactions are integrable (Poisson statistics), while inclusion of both types yields Wigner-Dyson statistics. The analysis is extended to $q=3,4$ models, where similar chaotic features are observed, with quantitative differences in the growth rates and saturation values of Lanczos coefficients and Krylov complexity.

Implications and Future Directions

The identification of quadratic, bosonic, and resource-efficient models exhibiting robust quantum chaos has significant implications for both theory and experiment. These models are amenable to classical simulation for moderate sizes and are well-suited for implementation on near-term quantum devices, where hardware constraints favor local, two-body interactions and bosonic statistics.

Theoretically, the results clarify the minimal ingredients required for quantum chaos and provide a platform for studying the transition from integrability to chaos in many-body systems. The observed weak ergodicity of eigenstates at finite size suggests a nuanced landscape between integrable and fully chaotic phases, with implications for thermalization and information scrambling.

Future work should address the extension to models with all three Pauli matrices, higher-dimensional qudits, and non-Hermitian or open-system generalizations. The scaling of the critical sparsity threshold for chaos, the behavior of ground states versus mid-spectrum states, and the resource-theoretic properties (e.g., magic) of eigenstates in these models are open questions. The connection to holographic duality and the possibility of realizing maximally chaotic behavior at finite temperature in quadratic models also merit further investigation.

Conclusion

Quadratic spin SYK models with random two-body interactions provide a minimal and accessible setting for quantum chaos, exhibiting RMT spectral statistics, operator growth consistent with the operator-growth hypothesis, and emergent operator freeness. While mid-spectrum eigenstates are only weakly ergodic at finite size, they approach Haar-randomness in the thermodynamic limit. These models offer a promising route for probing quantum chaos and information scrambling in both theoretical and experimental contexts, with broad implications for quantum simulation, complexity theory, and the paper of many-body quantum dynamics.