Thouless time for mass-deformed SYK

TL;DR

This work investigates the onset of random-matrix-like dynamics in the mass-deformed SYK model by comparing the connected unfolded SFF and the Gaussian-filtered SFF as chaos probes. It finds that the chaotic/integrable transition is not uniform across the spectrum: the tail of the spectrum (near-ground states) shows a transition at κ ≈ 1—in qualitative agreement with OTOC-based chaos—whereas the bulk requires κ ≈ 15 for the transition, aligning with traditional RMT diagnostics. The study shows that the two probes are complementary, with the connected unfolded SFF sensitive to early-time chaos and the Gaussian-filtered SFF capturing late-time, bulk spectral properties. These results motivate using the connected unfolded SFF as a possible link between OTOC chaos and BGS/RMT chaos, while highlighting unfolding subtleties and the necessity of large-N analysis for robust conclusions.

Abstract

We study the onset of RMT dynamics in the mass-deformed SYK model (i.e. an SYK model deformed by a quadratic random interaction) in terms of the strength of the quadratic deformation. We use as chaos probes both the connected unfolded Spectral Form Factor (SFF) as well as the Gaussian-filtered SFF, which has been recently introduced in the literature. We show that they detect the chaotic/integrable transition of the mass-deformed SYK model at different values of the mass deformation: the Gaussian-filtered SFF sees the transition for large values of the mass deformation; the connected unfolded SFF sees the transition at small values. The latter is in qualitative agreement with the transition as seen by the OTOCs. We argue that the chaotic/integrable deformation affect the energy levels inhomogeneously: for small values of the mass deformation only the low-lying states are modified while for large values of the mass deformation also the states in the bulk of the spectrum move to the integrable behavior.

Figures (22)

• Figure 1: Left: the full, not-unfolded, SFF for the case of $N=30$ and for increasing values of $\kappa$. Right: the full, unfolded, SFF for the case of $N=30$ and for the same values of $\kappa$. In both cases, the average is performed over $M = 100$ ensembles.
• Figure 2: The connected unfolded SFF for $N=28, \, 30$ and $\beta=0, \, 5\times 10^{-5}, \, 10^{-4}, \, 10^{-3}$. The average is performed over $M=603$ realizations for $N=28$ and $M=312$ realizations for $N=30$ respectively.
• Figure 3: The connected unfolded SFF for $N=32$ and $\beta=0, \, 5\times 10^{-5}, \, 10^{-4}, \, 10^{-3}$. The average is performed over $M=150$ realizations.
• Figure 4: The connected unfolded SFF for large $\kappa$. The average is performed over $M=201$ realizations for $N=28$ and $M=101$ realizations for $N=30$.
• Figure 5: The level spacing distribution, averaged over $M=312$ realizations for the $N = 30$ Hamiltonian with $\kappa=0, \, 0.5, \, 5$. The distributions are computed on subsectors of the spectra including $L$ levels starting from the ground state.