Matter Coupling of Dirac Matter in the Context of the SYK Model: Non-Gaussian Random Couplings and Bulk Mass Deformations

TL;DR

This work extends the SYK model by coupling Dirac matter to JT gravity, incorporating non-Gaussian disorder and bulk fermion mass. By deriving the boundary description and analyzing chaos indicators (adjacent gap ratio and spectral form factor) alongside entanglement entropy, it shows that the random-matrix symmetry class remains unchanged, while non-Gaussianity and mass shift dynamical timescales such as the SFF ramp and entanglement saturation. The results reveal a robust large-$N$ holographic structure (Type II$_1$ algebra at saturation) that can be perturbed toward Type II$_\infty$ with richer bulk couplings, emphasizing how matter content shapes information dynamics without altering late-time universality. These insights inform holographic duals of matter–gravity systems and provide a concrete, computable arena for exploring how non-Gaussian disorder and bulk masses influence quantum chaos and spacetime emergence.

Abstract

We elaborate further on the matter coupling of Dirac matter in the SYK framework, incorporating non-Gaussian coupling distributions and bulk fermion mass effects. Our study analyzes quartic matter couplings generated by a non-Gaussian distribution as an illustrative example. The introduction of bulk-fermion mass alters the boundary coupling between the Dirac and Majorana fermions. The averaged adjacent gap ratio is sensitive to the distribution of random couplings, which remains independent of the Hamiltonian's symmetry. The generalization of the SYK model to non-Gaussian distributions and the inclusion of bulk fermion mass remain qualitatively similar to the Gaussian and massless cases. Key deviations are observed only in the time scales for the linear ramp in the spectral form factor and the saturation of entanglement entropy.

Figures (6)

• Figure 1: Probability distribution for the coupling $g_{i_1i_2,j}$\ref{['eqn:quarticDistribution2']} for various $N$ and $C$. The distribution does not depend on the choice of $M$. $10^5$ samples are collected with $10^4$ Monte Carlo steps between each sample. The horizontal axis is the real part or imaginary part of the coupling, which shows the same distribution.
• Figure 2: Averaged adjacent gap ratio for the entire energy spectrum after removing degeneracy. $2^{24}$ eigenvalues are used. Here and thereafter, $10^{4}$ MC steps are taken between each sample for $C>0$.
• Figure 3: Spectral form factor $g(t)$ for SYK$2\chi$\ref{['HSYK']} and SYK$3\chi$\ref{['eqn:SYK3chi']}. Only the contribution of the even-parity sector has been computed for simplicity. For $M=2$, results for $N=16,20,24$ are plotted. For $M=4$, results for $N=12,16,20$ are plotted. The late-time values decrease as $N$ is increased. $2^{25-N/2-M}$ samples are used for SYK$2\chi$ while 128 samples are used for SYK$3\chi$.
• Figure 4: Normalized density of states ($\int \rho(E)dE=1$) for SYK2$\chi$\ref{['HSYK']} (solid line) and SYK$3\chi$\ref{['eqn:SYK3chi']} (dashed line). $2^{25-N/2-M}$ ($2^{20-N/2-M}$) samples are used for SYK$2\chi$ (SYK$3\chi$).
• Figure 5: Dynamical behavior of the averaged entanglement entropy for SYK$2\chi$ and SYK$3\chi$. $2^{20-N/2-M}$ samples are used.