Eigenstate Thermalisation in the conformal Sachdev-Ye-Kitaev model: an analytic approach

TL;DR

The paper addresses whether closed quantum systems thermalise within the conformal sector of the SYK model by analytic means. ItDevelops a bilocal collective-field framework to compute heavy-light three-point functions and, via the six-point fermion correlator, extracts OPE coefficients that obey ETH-like structure in the regime $m,n\gg k$, with a diagonal function $f_k(\bar E)$ scaling as $\propto(2\bar E)^{2k}$ and off-diagonal terms suppressed as $e^{-\sigma\ln 2}$. In the large-$q$ limit, planar diagrams dominate and a Generalised Free Theory captures the same ETH behaviour, suggesting a broader mechanism for thermalisation beyond interacting chaos. The results provide analytic evidence for ETH in the conformal SYK sector and raise questions about the bulk interpretation in AdS$_2$ gravity and the role of Schwarzian modes, motivating further study of heavy operators and the full dynamical theory including Schwarzian contributions. Overall, the work advances understanding of microscopic thermalisation in holographic quantum systems and highlights the special role of conformal dynamics in SYK-like models.

Abstract

The Sachdev-Ye-Kitaev (SYK) model provides an uncommon example of a chaotic theory that can be analysed analytically. In the deep infrared limit, the original model has an emergent conformal (reparametrisation) symmetry that is broken both spontaneously and explicitly. The explicit breaking of this symmetry comes about due to pseudo-Nambu-Goldstone modes that are not exact zero-modes of the model. In this paper, we study a version of the model which preserves the reparametrisation symmetry at all length scales. We study the heavy-light correlation functions of the operators in the conformal spectrum of the theory. The three point functions of such operators allow us to demonstrate that matrix elements of primaries ${\cal O}_n$ of the CFT$_1$ take the form postulated by the Eigenstate Thermalisation Hypothesis. We also discuss the implications of these results for the states in AdS$_2$ gravity dual.

Figures (6)

• Figure 1: Schwinger-Dyson equation for the exact propagator ${\cal G}(\tau_1,\tau_2;\tau_3,\tau_4)$. For the theory formulated in terms of Majorana fermions this is a four-point function.
• Figure 2: Schwinger-Dyson equation determining the three-point vertex ${\cal V}_{(3)}$. We note that there are two kinds of contributions on the right-hand side, planar ones and non-planar ones. For the theory formulated in terms of Majorana fermions this is a six-point function.
• Figure 3: The operator-state map in 1D CFT proceeds via the map $S^1 \rightarrow S^0\times \mathbb{R}$ introduced in Eq. \ref{['eq.S1toS0crossRMap']}. The semi circle in red is mapped to one copy of $\mathbb{R}$ shown in red, situated at $\sigma=0$, while the blue semi-circle is mapped to the blue copy of $\mathbb{R}$, situated at $\sigma = -\pi$. The Hamiltonian $\frac{\partial}{\partial\tau}$ is mapped to $\cos\theta \frac{\partial}{\partial\theta}$ and thus generates radial evolution as shown around the south pole. Modulo the subtlety of mapping to the disconnected $S^0\times \mathbb{R}$ this is the analog of radial quantisation in higher dimensional CFT.
• Figure 4: Left: Diagonal values of the OPE coefficients $\left|c^{(1)}_{mkn}\right|$ become smaller as we take the limit $m,n\gg k$. Right: Diagonal values of the OPE coefficients $c^{(2)}_{mkn}$ become larger as we take the limit $m,n\gg k$, thereby demonstrating the dominance of contact diagrams over the planar diagrams in this limit. Note that the y-axis of the right figure is plotted on a logarithmic scale.
• Figure 5: Left: The ratio ${\left|c^{(1)}_{m1n}\right|}/{c^{(2)}_{m1n}}$ is largely suppressed even for $k=1$ as $m$ or $n$ are increased. Right: Increasing $k$ accentuates the supression even further (here $k=3$).