Complete random matrix classification of SYK models with $\mathcal{N}=0$, $1$ and $2$ supersymmetry

TL;DR

<3-5 sentence high-level summary> This work provides a complete symmetry classification of SYK models with ${\cal N}=0$, ${\cal N}=1$, and ${\cal N}=2$ SUSY within the Altland-Zirnbauer random-matrix framework, extending previous analyses to generic $q$-body interactions and validating results with extensive numerics. It demonstrates distinct RMT classes for each SUSY level, establishes hard-edge universality for SUSY cases, and unveils a comprehensive $Q\overline{Q}$/$\overline{Q}Q$-based classification for the ${\cal N}=2$ model across fermion-number sectors, including analytic zero-mode counts. A bridging ${\cal N}=0$-to-${\cal N}=2$ toy model is introduced to illuminate spectral structure, while a complete spectral-density analysis and bulk-versus-edge statistics reinforce the presence of quantum chaotic dynamics. The findings sharpen the understanding of spectral fluctuations in SUSY-SYK systems and have potential implications for holography and low-energy quantum gravity models.

Abstract

We present a complete symmetry classification of the Sachdev-Ye-Kitaev (SYK) model with $\mathcal{N}=0$, $1$ and $2$ supersymmetry (SUSY) on the basis of the Altland-Zirnbauer scheme in random matrix theory (RMT). For $\mathcal{N}=0$ and $1$ we consider generic $q$-body interactions in the Hamiltonian and find RMT classes that were not present in earlier classifications of the same model with $q=4$. We numerically establish quantitative agreement between the distributions of the smallest energy levels in the $\mathcal{N}=1$ SYK model and RMT. Furthermore, we delineate the distinctive structure of the $\mathcal{N}=2$ SYK model and provide its complete symmetry classification based on RMT for all eigenspaces of the fermion number operator. We corroborate our classification by detailed numerical comparisons with RMT and thus establish the presence of quantum chaotic dynamics in the $\mathcal{N}=2$ SYK model. We also introduce a new SYK-like model without SUSY that exhibits hybrid properties of the $\mathcal{N}=1$ and $\mathcal{N}=2$ SYK models and uncover its rich structure both analytically and numerically.

Figures (12)

• Figure 1: Statistical distribution of the ratio $r$ of two consecutive level spacings for the ${\cal N}=0$ SYK model with $q=6$. The number of realizations used for averaging was $10^3$ for $N_m=16$, $10^2$ for $N_m=18$ and $20$, and 10 for $N_m=22$. The blue lines are surmises for the RMT classes in table \ref{['tb:rmtSYK01']}.
• Figure 2: Distributions of the eigenvalues of $H$ with smallest absolute values in the ${\cal N}=0$ SYK model with $q=6$ and $J=1$, compared with the predictions (solid lines) of the RMT classes in table \ref{['tb:rmtSYK01']}. The number of independent random samples is $10^4$ for each plot. The small deviations from RMT for the third nonzero eigenvalue are interpreted to be effects of finite $N_m$.
• Figure 3: Distribution of the ratio $r$ of two consecutive level spacings in the ${\cal N}=1$ SYK model with ${\hat{q}}=5$. The number of realizations used for averaging was $10^3$ for $N_m=16$, $100$ for $N_m=18$ and $22$, and $200$ for $N_m=20$. The blue lines are surmises for the RMT classes in table \ref{['tb:RMTQ00']}.
• Figure 4: Distributions of the smallest 3 eigenvalues of $Q$ in \ref{['eq:QdefN1']} in the ${\cal N}=1$ SYK model with ${\hat{q}}=3$ and $J=1$, compared with the predictions (solid lines) of the RMT classes in table \ref{['tb:RMTQ01']}. The number of independent random samples is $10^4$ for each plot. As in figure \ref{['fg:PminN0']}, the small deviations from RMT for $\lambda_3$ are interpreted to be effects of finite $N_m$.
• Figure 5: Same as figure \ref{['fg:PminN1q3']} but for ${\hat{q}}=5$ and compared with the RMT predictions in table \ref{['tb:RMTQ00']}.