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

TL;DR

This paper explores the large N expansion of the SYK model's free energy and moments, linking spectral density cumulants to intersection graph enumeration, and extends calculations to higher orders revealing new identities.

Contribution

It introduces a novel method connecting spectral moments with intersection graph enumeration and computes higher-order corrections in the SYK model.

Findings

Derived the $p$ dependence of $1/N^2$ corrections to moments.

Established a new enumeration identity for intersection graphs.

Calculated $1/N^3$ corrections to moments and free energy coefficients.

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 $1/N^2$ correction of the $2p$-th moments obtained in 1801.02696. Conversely, the $1/N^2$ corrections to the moments give the correction (even $q$) to the $\beta^6$ coefficient of the high temperature expansion of the free energy for arbitrary $q$. Our result agrees with the $1/q^3$ 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 $1/N^3$ order and find surprisingly simple enumeration identities for intersection graphs. The $1/N^3$ corrections to the moments, give corrections to the $\beta^8$ 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.