Periodic Table of the Ordinary and Supersymmetric Sachdev-Ye-Kitaev Models

We develop a unified minimal scheme to classify quantum chaos in the Sachdev-Ye-Kitaev (SYK) and supersymmetric (SUSY) SYK models and also work out the structure of the energy levels in one periodic table. The SYK with even q-body or SUSY SYK with odd q-body interaction, with N even or odd number of sites, are put on an equal footing in the minimal Hilbert space; N (mod 8), q (mod 4) double Bott periodicity, and a reflection condition are identified. Exact diagonalizations (EDs) are performed to study both the bulk energy level statistics and hard-edge behaviors. Excellent agreements between the ED results and the symmetry classifications are demonstrated. Our compact and systematic methods can be transformed to map out more complicated periodic tables of SYK models with more degrees of freedom, tensor models, or symmetry protected topological phases.