Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs

Abstract

In this paper we explain the relation between the free energy of the SYK model for N Majorana fermions with a random q-body interaction and the moments of its spectral density. The high temperature expansion of the free energy gives the cumulants of the spectral density. Using that the cumulants are extensive we find the p dependence of the 1/N² correction of the 2p-th moments obtained in [1]. Conversely, the 1/N² corrections to the moments give the correction (even q) to the β⁶ coefficient of the high temperature expansion of the free energy for arbitrary q. Our result agrees with the 1/q³ correction obtained by Tarnopolsky using a mean field expansion. These considerations also lead to a more powerful method for solving the moment problem and intersection-graph enumeration problems. We take advantage of this and push the moment calculation to 1/N³ order and find surprisingly simple enumeration identities for intersection graphs. The 1/N³ corrections to the moments, give corrections to the β⁸ coefficient (for even q) of the high temperature expansion of the free energy which have not been calculated before. Results for odd q, where the SYK “Hamiltonian” is the supercharge of a supersymmetric theory are discussed as well.