Supersymmetric SYK model and random matrix theory

TL;DR

The paper analyzes how N=1 supersymmetry affects random-matrix classifications of the SYK model. It shows that while the original SYK follows GOE/GUE/GSE according to N mod 8, the SUSY extension shifts to extended AZ classes determined by the supercharge and Witten index, with H following Wishart-like spectra due to H=Q^2. Numerically, exact diagonalization confirms the predicted DoS shapes, eightfold level statistics, and dip-ramp-plateau structures in spectral form factors, with degeneracy-dependent plateau heights. The results reveal a structured interplay between supersymmetry and random-matrix universality and point to fruitful directions including higher SUSY, holography, and condensed-matter symmetry classifications.

Abstract

In this paper, we investigate the effect of supersymmetry on the symmetry classification of random matrix theory ensembles. We mainly consider the random matrix behaviors in the $\mathcal{N}=1$ supersymmetric generalization of the Sachdev-Ye-Kitaev (SYK) model, a toy model for the two-dimensional quantum black hole with supersymmetric constraint. Some analytical arguments and numerical results are given to show that the statistics of the supersymmetric SYK model could be interpreted as random matrix theory ensembles, with a different eight-fold classification from the original SYK model and some new features. The time-dependent evolution of the spectral form factor is also investigated, where predictions from random matrix theory are governing the late time behavior of the chaotic Hamiltonian with supersymmetry.

Figures (9)

• Figure 1: The density of states for the original SYK model Hamiltonian (left), the supersymmetric SYK Hamiltonian (middle), and the supersymmetric SYK supercharge operators treated as Hamiltonian (right) by exact diagonalization. The densities of states from $N=10$ to $N=28$ are plotted in colors from light blue to dark blue. The eigenvalues have been rescaled by $E(Q)/NJ$, while the density of states has also been rescaled to match the normalization that the integration should be 1.
• Figure 2: The theoretical Wigner surmises for three different standard ensembles. The lower (blue), middle (red) and higher (green) curves are corresponding to GOE, GUE and GSE universal class respectively.
• Figure 3: The nearest-neighbor level spacing distribution for the Hamiltonian of the $\mathcal{N}=1$ supersymmetric SYK model for different $N$. The lower (blue), middle (red), and higher (green) curves are theoretical predictions of Wigner surmises from GOE, GUE, and GSE, respectively. The black dashed curves are distributions for all $r$s from a large number of samples.
• Figure 4: The nearest-neighbor level spacing distribution for the supercharge matrix $Q$ of $\mathcal{N}=1$ supersymmetric SYK model for different $N$. The lower (blue), middle (red), and higher (green) curves are the theoretical prediction of Wigner surmises from GOE, GUE, and GSE, respectively. The black dashed curves are distributions for all $r$s from a large number of samples.
• Figure 5: The spectral form factors $g(t)$, $g_c(t)$ and $g_d(t)$ in the supersymmetric SYK model with $J_{\mathcal{N}=1}=1$, $\beta=0,5,10$ respectively.