Approximation of the Sachev-Ye-Kitaev Model by Free Fermion Variational States

The Sachdev-Ye-Kitaev model is a zero dimensional model of fermions subject to fully random two fermion interactions. This model has been of considerable interest in both the condensed matter and high energy communities, primarily due to its emergent conformal symmetry in the thermodynamic limit. Strangely, in this limit, the fermion correlators of this model also satisfiy Wick's Theorem, suggesting that there is something "free" about this highly interacting model. Motivated by this observation, we use the Gibbs-Delbruck variational principle to numerically compute the closest free approximation to a ground state or thermal density matrix of a sample of the Sachdev-Ye-Kitaev ensemble. Despite a rich structure to these free fermion approximations, we find that they are exceedingly poor approximations on average in the thermodynamic limit. Some discussion will also be given on how to reconcile this fact with the previously mentioned properties of the free fermion correlators.