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.
Highlights
: 1. We extend the symmetry classification of SYK models with N = 0 and 1 SUSY that were focused on the 4-body interaction Hamiltonian [66,67,68] to generic q-body interactions
This means that, while they show the standard Gaussian Unitary Ensemble (GUE)/Gaussian Orthogonal Ensemble (GOE)/Gaussian Symplectic Ensemble (GSE) level correlations in the bulk of the spectrum, their level density exhibits a universal shape near the origin, different for each symmetry class. (Such a property is absent in the Wigner-Dyson classes since the spectrum is translationally invariant after unfolding and there is no special point in the spectrum.) The physical significance of such near-zero eigenvalues depends on the specific context in which random matrix theory (RMT) is used
Level correlations in the bulk Here we report on the first numerical analysis of the bulk statistics of energy levels for the N = 0 SYK model with q = 6 via exact diagonalization to test table 3
Summary
To set the stage for our later discussion focused on the supersymmetric SYK model, we begin with a pedagogical summary of the symmetry classification scheme for a generic Hamiltonian, known as the Altland-Zirnbauer theory [75,76,77]. (Some authors count them as 12 by distinguishing subclasses more carefully, as we will describe later.) The salient feature pertinent to those post-Dyson classes is a spectral mirror symmetry: the energy levels are symmetric about the origin ( called “hard edge”). This means that, while they show the standard GUE/GOE/GSE level correlations in the bulk of the spectrum (i.e., sufficiently far away from the edges of the energy band), their level density exhibits a universal shape near the origin, different for each symmetry class.
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