Spectral form factor of quadratic $R$-para-particle SYK model with Random Matrix Coupling

TL;DR

This work analyzes the spectral form factor (SFF) of the quadratic $R$-para-particle SYK$_2$ model across Gaussian and circular random-matrix ensembles. It develops and combines two analytic approaches—the coherent-state method and the cluster-function framework—for Gaussian ensembles and leverages exact results in circular ensembles, especially for CUE, to reveal a robust GUE--CUE correspondence in the ramp regime via $\mathcal{K}_{\text{GUE}}(2t) \approx \mathcal{K}_{\text{CUE}}(t)$ for $1 \ll t \ll N$, with analogous behavior in symplectic ensembles under time rescaling. The paper provides detailed expressions for ramp coefficients, plateau times, and transfer-matrix structures, and outlines methods and partial results for GOE/COE, highlighting the utility of circular ensembles as benchmarks for chaos in quadratic models. Overall, the results establish a tractable, ensemble-wide framework for understanding chaotic spectral statistics in minimal SYK-like systems and offer guidance for exploring more complex interacting variants. The findings have potential implications for diagnosing chaos in engineered quantum systems and for benchmarking analytical techniques in random-matrix theory applied to many-body quantum chaos.

Abstract

This paper investigates the spectral form factor (SFF) of the quadratic $R$-para-particle Sachdev-Ye-Kitaev ($R$-PSYK$_2$) model with various random matrix ensemble couplings. We generalize previous work on Gaussian Unitary Ensemble (GUE) couplings to all three Gaussian ensembles (GUE, GOE, GSE) and three circular ensembles (CUE, COE, CSE). Through analytical and numerical methods, we establish precise correspondences between GUE and CUE results, demonstrating their SFFs satisfy $\mathcal{K}_{\text{GUE}}(2t) \approx \mathcal{K}_{\text{CUE}}(t)$ in the time regime $1 \ll t \ll N$. For the symplectic ensembles, we observe similar behavior with appropriate time rescaling, while we only provide the calculation method for the orthogonal ensembles.

Figures (9)

• Figure 1: Numerical results for the SFF of two GUE instances are presented, where we set the scaling time $t_S = 2N$. In the large $N$ limit, the theoretic prediction implies that $\frac{1}{N} \log \mathcal{K}$ depends linearly on $t$ within the time regime $t_p \ll t \ll N$. For finite $N$, however, oscillations obscure the numerical results. One may average over multiple $N$ values to reveal a distinct linear growth regime. In this work, we instead leverage the similarity between GUE and CUE to perform an exact analytical verification.
• Figure 2: Plots of $\rho_{\pm}$ and the growth rates for Example A.
• Figure 3: Comparing the SFF of CUE and GUE with $z_R(x) = 1 + mx$. The GUE data is generated by Eq. \ref{['eq:GUE-SFF-num']}, and the CUE data is obtained from Eq. \ref{['eq:CUE-SFF-recur']} with $N = 500$. Scaling time $t_S=2 N$ for GUE and $t_S=N$ for CUE.
• Figure 4: Plot of the SFF for the CUE PSYK model with $z_R(x) = 1 + mx$. The scatter points for the CUE case were generated using the exact formulas in Eq. \ref{['eq:CUE-SFF-recur']} and Eq. \ref{['eq:CUE-SFF-t1']}. The straight lines are plots of Eq. \ref{['eq:CUE-lines']}.
• Figure 5: Comparing SFFs of CUE and GUE with $z_R(x)=1+mx+x^2$. Here data of GUE is generated by Eq. (\ref{['eq:GUE-SFF-num']}) and the data of CUE is calculated numerically via Eq. (\ref{['eq:CUE-rn']}). Scaling time $t_S=2 N$ for GUE and $t_S=N$ for CUE.