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.

💡 Research Summary

The manuscript studies the spectral form factor (SFF) of a quadratic Sachdev‑Ye‑Kitaev model built from R‑para‑particles (R‑PSYK₂) when the random couplings are drawn from various random‑matrix ensembles. The authors extend earlier work that considered only the Gaussian Unitary Ensemble (GUE) to all three Gaussian ensembles—GUE, Gaussian Orthogonal Ensemble (GOE), Gaussian Symplectic Ensemble (GSE)—and to the three circular ensembles—Circular Unitary Ensemble (CUE), Circular Orthogonal Ensemble (COE), Circular Symplectic Ensemble (CSE).

The R‑para‑particles are defined by a four‑index R‑matrix that satisfies the constant Yang‑Baxter equation and a unitarity condition guaranteeing ψ⁺ = (ψ⁻)†. Special choices of the R‑matrix reproduce ordinary bosons (R = +δ) and fermions (R = −δ). The Hamiltonian is
 H = m ∑{i,j,a}(h{ij} − μ δ_{ij}) ψ⁺{i,a} ψ⁻{j,a},
where h_{ij} is a random Hermitian (or unitary) matrix drawn from the chosen ensemble. By diagonalising h_{ij} the model becomes a free‑particle system with single‑particle energies ε_i, and the many‑body spectrum is determined solely by the degeneracies d_n of the n‑particle level, which are encoded in a single‑mode partition function z_R(x)=∑_n d_n x^n. Two concrete examples are examined: z_R(x)=1+mx (Example A) and z_R(x)=1+mx+x² (Example B).

The SFF is defined as
 𝒦(t)=⟨|Z(β+it)|²⟩/⟨Z(β)⟩²,
with Z(β)=∑_j∑_n d_n e^{−β n(ε_j−μ)}. After averaging over the random matrix distribution, the problem reduces to evaluating integrals of the form ⟨∏_j G(ε_j)⟩ where G depends on the chosen observable (partition function or SFF). The joint probability density for the eigenvalues of the Gaussian ensembles is
 P_b(ε₁,…,ε_N)∝exp