Analytic approach to Eigenstate thermalization (ETH) in the SYK model and the Schwarzian theory

The SYK model provides an uncommon example of a theory where Eigenstate Thermalization Hypothesis (ETH) can be verified in analytically. In this talk I will discuss this model in the deep infrared limit where the theory has an emergent conformal (reparametrization) symmetry that is broken both spontaneously and explicitly. To study the validity of ETH, we compute the heavy- light correlation functions of operators in the conformal spectrum of the theory. We compute these correlation functions with and without the contribution of the low energy (Schwarzian) modes, which are known to be the origin of the chaotic behaviour in this theory. In considering the contributions of the Schwarzian modes we find a weaker form of ETH: while the heavy operator insertions increase the effective temperature perceived by the light insertions, this effective temperature is proportional to the background temperature and goes to zero with the background temperature. In the case where Schwarzian modes aren’t considered, we find ETH in limit in which the weight of the heavy operators approach infinity. I will also discuss implications of these results for the states in AdS2 gravity dual.