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/N 2 correction of the 2p-th moments obtained in [1]. Conversely, the 1/N 2 corrections to the moments give the correction (even q) to the β 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 β 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.

Towards a full solution of the large N double-scaled SYK model

“…One of the observations in [56] is that, in the normalization that sets trH 2 = 1, the 2k-th cumulant κ 2k of the SYK partition function is suppressed by N 1−k at finite p, which become higher powers of λ in our conventions. This implies that at finite p, infinitesimal λ, the Momenta Two-stage limit Double-scaling limit M 0 Table 1: Low order comparison between the iterated limit results of [56] and the predictions of chord expansions in the double-scaling limit. All expressions are compatible with the existence of a unified double expansion in p −1 and λ.…”

“…• A two-stage limit in which first N → ∞, p fixed, and then p → ∞. This is the standard discussion in SYK [5,56,57].…”

“…From the explicit computation in equations (3.12) and (3.14) in [56], which are a 1/N expansion at finite p, we extract, in our notation, To see where 1/p corrections come from, we can proceed as follows [25].…”

Wilson loops in terms of color invariants

“…Therefore, the procedure we propose is the following: first, group all connected chord diagrams, according to their intersection graphs. For each intersection graph, evaluate the modified color factor by computing it for any of the 4 Intersection graphs of chord diagrams have appeared recently in discussions of the SYK model [53,54]. Figure 5: The first column displays all the non-isomorphic intersection graphs, up to four loops (graphs with four dots).…”

Spectral fluctuations in the Sachdev-Ye-Kitaev model

“…The choice of letters V and E comes from "vertices" and "edges" of the corresponding intersection graphs, see[62] and[66]. In those references the letter G was primarily used to denote intersection graphs, but it should not cause confusion in the present context.…”