Exploring the Spectral Edge in SYK Models

TL;DR

This work shows that edge-universal spectral statistics, described by Airy-RMT behavior, control the low-temperature quenched entropy in the structured SYK model and in $\ N=2$ SUSY wormholes. Through exact diagonalization and analysis of level-spacing near the spectral edge, the authors demonstrate that the two lowest-energy gaps produce a quenched entropy with a power-law decay $S_Q(\beta) \sim \beta^{-(1+\upbeta)}$, with the Dyson index $\upbeta$ determined by GOE, GUE, or GSE symmetry classes, and that this universality persists despite the models' non-random structure. In the SUSY case, the LMRS (BPS-projected) operator spectrum near the edge similarly dictates $S_Q(k)$ as a function of operator insertions $k$, with scaling exponents set by edge statistics and symmetry; various operator projections yield GOE, GUE, and GSE-like behaviors, including regimes with $S_Q \sim k^{-1}$, $k^{-3}$, or $k^{-5}$. Collectively, the results indicate that random-matrix edge universality robustly informs quenched entropies in chaotic many-body systems and offers a bridge to gravitational interpretations, while also highlighting symmetry-dependent deviations from pure RMT in structured models.

Abstract

Previous work on Jackiw-Teitelboim (JT) gravity has shown that, at low temperatures, the annealed entropy becomes negative and departs from the quenched entropy. From the perspective of the random-matrix theory (RMT) dual of JT gravity, this effect is encoded in the continuous spectrum at the spectral edge that is universally described by the Airy model. At low temperature, the quenched entropy exhibits a power law dependence determined by the symmetry class of the RMT ensemble. Here we study the same question in the Sachdev-Ye-Kitaev (SYK) model which possesses much more structure than RMT. Through numerical simulations, we find that the level spacing statistics of the SYK model match the relevant RMT ensembles even near the spectral edge, thus leading to an agreement with the RMT prediction for the power-law behaviour of the quenched entropy at low temperatures. We also show similar effects in supersymmetric wormholes filled with matter, which is modeled by the $\mathcal N = 2$ supersymmetric SYK model. Numerically extracting the spectral edge properties of the BPS operators allows us to compute the quenched entanglement entropy of the wormhole in the large particle number limit.

Figures (20)

• Figure 1: The scaled annealed entropy $S_A(\beta) / N$ for $N = 8, 10, 12, 14, 16, 18$ and $20$ averaged over 3000 instances. Here we consider the full Hamiltonian without any subdivision.
• Figure 2: The distribution of the spacing between the two lowest eigenvalues of the Full $H_{SYK}$ at $N = 16$ obtained by sampling over $5 \times 10^5$ instances. The gap is rescaled by its average.
• Figure 3: The distribution of the spacing between the two lowest eigenvalues of $H_{SYK}$ in the even (left) and odd (right) fermion parity sectors at $N = 16$ obtained by sampling over $10^6$ instances. The gap in each case is rescaled by the mean value.
• Figure 4: (Left): Comparison of $S_Q(\beta)$ averaged over $10^6$ Hamiltonians to the entropies of 400 random instances (red), showing the dominance of unlikely cases with small gaps between the two lowest eigenvalues. (Right): The relationship between $\log S_Q(\beta)$ and $\log \beta$. The slope of the line is approximately $- 2$, which demonstrates the $\beta$ dependence of $S_Q(\beta)$ in Eq. \ref{['GOE Sq']}.
• Figure 5: $\rho (x_2 - x_1)$ rescaled by its average for $H_{SYK}$ in the even fermion parity sector at $N = 10$ (left) and 14 (right) obtained by sampling over $2 \times 10^{6}$ instances respectively. $\rho (x_2 - x_1)$ is identical in the odd fermion parity sector due to the particle-hole symmetry.