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

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.