Comments on the Sachdev-Ye-Kitaev model

TL;DR

The paper analyzes the Sachdev‑Ye‑Kitaev model at large N, deriving the Schwinger‑Dyson equations for the two‑point function and exposing an emergent reparametrization symmetry that is spontaneously and explicitly broken in the infrared. It computes the four‑point function through ladder diagrams, diagonalizes the conformal kernel via SL(2,R) Casimir eigenfunctions, and isolates the dominant enhanced contribution from the h=2 reparameterization sector, which saturates the chaos bound in the chaos limit. A Schwarzian effective action is derived for the low‑energy reparameterizations, linking the specific heat and finite‑temperature free energy to the density of states and to the chaos dynamics. The authors also discuss a bulk interpretation in near‑AdS2/NCFT1 language, suggesting a tower of bulk states (low‑tension strings) and exploring implications for holography and scrambling near extremal black holes. Overall, the work provides a tractable, large‑N platform connecting quantum chaos, conformal symmetry breaking, and holographic ideas with potential bulk realizations involving dilaton gravity and stringy excitations.

Abstract

We study a quantum mechanical model proposed by Sachdev, Ye and Kitaev. The model consists of $N$ Majorana fermions with random interactions of a few fermions at a time. It it tractable in the large $N$ limit, where the classical variable is a bilocal fermion bilinear. The model becomes strongly interacting at low energies where it develops an emergent conformal symmetry. We study two and four point functions of the fundamental fermions. This provides the spectrum of physical excitations for the bilocal field. The emergent conformal symmetry is a reparametrization symmetry, which is spontaneously broken to $SL(2,R)$, leading to zero modes. These zero modes are lifted by a small residual explicit breaking, which produces an enhanced contribution to the four point function. This contribution displays a maximal Lyapunov exponent in the chaos region (out of time ordered correlator). We expect these features to be universal properties of large $N$ quantum mechanics systems with emergent reparametrization symmetry. This article is largely based on talks given by Kitaev \cite{KitaevTalks}, which motivated us to work out the details of the ideas described there.

Figures (16)

• Figure 1: Diagrams representing corrections to the two point function, for the $q=4$ case. The free two point function is given by the straight line. The first correction involves also an average over disorder, which is represented by a dashed line. We have also indicated a couple more diagrams that also contribute at leading order in $N$.
• Figure 2: Equations that define the summation of the leading large $N$ contributions, for the $q=4$ case. The solid circle represents the one particle irreducible contributions. The dotted circle represents the full two point function. This is a graphical representation of the equations in (\ref{['fulltwo']}).
• Figure 3: Diagrams representing the $1/N$ term in the index-averaged four point function, for the $q = 4$ case. One should also include the diagrams with $(\tau_3\leftrightarrow \tau_4)$ and a relative minus sign. The propagators here are the dressed two point functions discussed above.
• Figure 4: The $(n{+}1)$-rung ladder ${\mathcal{F}} _{n+1}$ can be generated from the $n$-rung ladder by "multiplication" with the kernel $K$, shown in blue. We call the vertical propagators a "rung" and the horizontal ones a "rail".
• Figure 5: The symmetry of the $\chi>1$ correlator under $\chi\rightarrow \frac{\chi}{\chi-1}$ is manifest as $\theta\rightarrow -\theta$ after mapping to the circle.