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

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.

42 pages, 12 figures. v2: refs added

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Cited by in corpus (6)

• Dispersive SYK model: band structure and quantum chaos
• SYK Models and SYK-like Tensor Models with Global Symmetry
• Correlators in the $\mathcal{N}=2$ Supersymmetric SYK Model
• Supersymmetric SYK Model: Bi-local Collective Superfield/Supermatrix Formulation
• Towards a Finite-$N$ Hologram
• Note on global symmetry and SYK model