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

## TL;DR

This paper classifies the symmetry properties of various SYK models with different supersymmetries using random matrix theory, revealing new classes, confirming quantum chaos, and introducing a novel SUSY-like model with hybrid features.

## Contribution

It provides a complete RMT-based symmetry classification of SYK models with $
=0,1,2$ supersymmetry, including new classes and a novel SUSY-like model with hybrid properties.

## Key findings

- RMT classes for $
=0,1,2$ SYK models are fully classified.
- Numerical evidence confirms quantum chaotic behavior in $
=1$ and $
=2$ SYK models.
- A new SUSY-like model without SUSY exhibits hybrid properties of known SYK 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.