Exact moments of the Sachdev-Ye-Kitaev model up to order $1/N^2$

TL;DR

This work analytically computes all moments $M_{2p}$ of the SYK model's spectral density up to $O(1/N^2)$ and demonstrates that the $O(1/N^2)$ corrections are governed by the total number of triangular loops in the intersection graphs of Wick contractions. By proving that $O(1/N)$ corrections vanish and reducing $1/N^2$ corrections to a triangle-counting problem, the authors derive explicit moment corrections for both even and odd $q$ and connect them to exact results at $q=1$ and $q=2$. They also provide exact contraction-diagram formulas and use these to obtain corrected spectral densities, including detailed even/odd $q$ expressions and edge behavior, showing that the Q-Hermite approximation remains surprisingly accurate at finite $N$. The results illuminate why the Q-Hermite density closely matches SYK spectra even for modest $N$ and suggest that higher-order corrections are tied to simple graph-theoretic structures, potentially enabling all-orders insights into the SYK moment problem and its holographic connections.

Abstract

We analytically evaluate the moments of the spectral density of the $q$-body Sachdev-Ye-Kitaev (SYK) model, and obtain order $1/N^2$ corrections for all moments, where $N$ is the total number of Majorana fermions. To order $1/N$, moments are given by those of the weight function of the Q-Hermite polynomials. Representing Wick contractions by rooted chord diagrams, we show that the $1/N^2$ correction for each chord diagram is proportional to the number of triangular loops of the corresponding intersection graph, with an extra grading factor when $q$ is odd. Therefore the problem of finding $1/N^2$ corrections is mapped to a triangle counting problem. Since the total number of triangles is a purely graph-theoretic property, we can compute them for the $q=1$ and $q=2$ SYK models, where the exact moments can be obtained analytically using other methods, and therefore we have solved the moment problem for any $q$ to $1/N^2$ accuracy. The moments are then used to obtain the spectral density of the SYK model to order $1/N^2$. We also obtain an exact analytical result for all contraction diagrams contributing to the moments, which can be evaluated up to eighth order. This shows that the Q-Hermite approximation is accurate even for small values of $N$.

Figures (16)

• Figure 1: Three contraction diagrams contributing to the sixth moment.
• Figure 2: Intersection graphs corresponding to the contraction diagrams of figure \ref{['fig1']} in the same order from the left to the right. (a) has $V=3$ and $E=1$, (b) has $V=3$ and $E=2$, (c) has $V=3$ and $E=3$.
• Figure 3: Venn diagrams with three index sets. Each index set is represented by a circle, containing $q$ elements. The box is the set of all possible values an index can take, which has cardinality $N$. The box is partitioned into eight regions.
• Figure 4: Venn diagrams with three index sets. There are eight regions, each labeled by its own cardinality.
• Figure 5: The $N$-dependence of the sixth moment of the eigenvalue density of the SYK model for $q =1,\; 3,\; 5,\; 7$ (left) and $q =2,\; 4,\; 5,\; 8$ (right). We compare the exact result (solid curve) to the Q-Hermite result (dashed).

Theorems & Definitions (1)

• Theorem 1