Combinatorial aspects of the Sachdev-Ye-Kitaev model

TL;DR

This paper reviews the combinatorial structure of the SYK model, extends it to flavored and colored variants using tensor techniques, and analyzes the impact of non-Gaussian couplings on its large N behavior.

Contribution

It provides a combinatorial proof of melonic dominance in the SYK model, generalizes the model with flavors and colors, and studies the effects of non-Gaussian couplings using tensor methods.

Findings

Diagrammatic proof of melonic dominance

Identification of leading and next-to-leading order diagrams in colored SYK

Non-Gaussian couplings modify the variance of the coupling distribution

Abstract

The Sachdev-Ye-Kitaev (SYK) model is a model of $q$ interacting fermions whose large N limit is dominated by melonic graphs. In this review we first present a diagrammatic proof of that result by direct, combinatorial analysis of its Feynman graphs. Gross and Rosenhaus have then proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, these flavors can be seen as reminiscent of the colors used in random tensor theory. Applying modern tools from random tensors to such a colored SYK model, all leading and next-to-leading orders diagrams of the 2-point and 4-point functions in the large $N$ expansion can be identified. We then study the effect of non-Gaussian average over the random couplings in a complex, colored version of the SYK model. Using a Polchinski-like equation and random tensor Gaussian universality, we show that the effect of this non-Gaussian averaging leads to a modification of the variance of the Gaussian distribution of couplings at leading order in $N$. We then derive the form of the effective action to all orders.