Frame potential of Brownian SYK model of Majorana and Dirac fermions

We consider the Brownian SYK, i.e. a system of $N$ Majorana (Dirac) fermions with a white-noise $q$-body interaction term. We focus on the dynamics of the Frame potentials, a measure of the scrambling and chaos, given by the moments of the overlap between two independent realisations of the model. By means of a Keldysh path-integral formalism, we compute its early and late-time value. We show that, for $q>2$, the late time path integral saddle point correctly reproduces the saturation to the value of the Haar frame potential. On the contrary, for $q=2$, the model is quadratic and consistently we observe saturation to the Haar value in the restricted space of Gaussian states (Gaussian Haar). The latter is characterised by larger system size corrections that we correctly capture by counting the Goldstone modes of the Keldysh saddle point. Finally, in the case of Dirac fermions, we highlight and resolve the role of the global $U(1)$ symmetry.