Stabilizer entropy and entanglement complexity in the Sachdev-Ye-Kitaev model

The Sachdev-Ye-Kitaev (SYK) model is of paramount importance for the understanding of both strange metals and a microscopic theory of two-dimensional gravity. We study the interplay between stabilizer R\'enyi entropy and entanglement entropy in both the ground state and highly excited states of the $\mathrm{SYK}\ensuremath{-}4+\mathrm{SYK}\ensuremath{-}2$ model, interpolating the highly chaotic four-body interactions model with the integrable two-body interactions one. The interplay between these quantities is also assessed through universal statistics of the entanglement spectrum and its antiflatness. We find that SYK-4 is indeed characterized by a complex pattern of both entanglement and nonstabilizer resources, while SYK-2 is nonuniversal and not complex. We discuss the fragility and robustness of these features depending on the interpolation parameter.