An introduction to the SYK model

TL;DR

The notes present the SYK model as a solvable, strongly coupled large $N$ system with melonic diagram dominance, enabling explicit computation of correlation functions and exposing a nearly conformal infrared structure. The framework extends to tensor models, yielding a bilocal action for $G$ and $\Sigma$ that sums melon diagrams at leading order and supports systematic $1/N$ corrections. In the infrared, a soft reparametrization mode leads to the Schwarzian action and a connection to $AdS_2$/dilaton gravity, while the spectrum of bilinears $O_h$ organizes higher-point functions through conformal blocks. Correlation functions are structured by conformal blocks with a density $\rho(h)$ and kernel eigenvalues $k(h)$, enabling a consistent decomposition into single- and double-trace exchanges and extending to higher-point functions. The applications to AdS/CFT and strange metals illustrate SYK’s potential to illuminate holography and non-Fermi liquid transport, while highlighting the challenges of formulating a complete bulk dual and the limitations of the large $N$ all-to-all framework.

Abstract

These notes are a short introduction to the Sachdev-Ye-Kitaev model. We discuss: SYK and tensor models as a new class of large N quantum field theories, the near-conformal invariance in the infrared, the computation of correlation functions, generalizations of SYK, and applications to AdS/CFT and strange metals.

Figures (6)

• Figure 1: The self-energy for the $O(N)$ vector model, in terms of the propagator. Iterating gives a sum of bubble diagrams.
• Figure 2: The self-energy for SYK and tensor models, in terms of the propagator. Iterating gives a sum of melon diagrams.
• Figure 3: (a) The four-point function is a sum of ladder diagrams. (b) The kernel that adds rungs to the ladder. Each line denotes the full propagator, so it is actually dressed by melons.
• Figure 4: (a) The fermion six-point function. (b) A contribution to the eight-point function.
• Figure 5: A comparison of SYK to matrix models and vector models. One comment is that for $\mathcal{N}=4$ the anomalous dimensions are large only for the non-supersymmetry protected operators (the half-BPS operators are dual to Kaluza-Klein modes on the $S^5$; such operators are ignored in the table). Another, relating to the gravity description, is that it is only at large 't Hooft coupling, when the stringy modes become very massive, that one can say that the bulk is Einstein gravity. This is to be contrasted with the bulk dual of SYK, where there is no limit in which the tower of bulk fields decouple; their mass is of order-one. Finally, in Vasiliev theory, the spin two field (graviton) is related by symmetry to the other higher spin fields.