Chaos exponents of SYK traversable wormholes

In this paper we study the chaos exponent, the exponential growth rate of the out-of-time-ordered four point functions, in a two coupled SYK models which exhibits a first order phase transition between the high temperature black hole phase and the low temperature gapped phase interpreted as a traversable wormhole. We see that as the temperature decreases the chaos exponent exhibits a discontinuous fall-off from the value of order the universal bound $2\pi/\beta$ at the critical temperature of the phase transition, which is consistent with the expected relation between black holes and strong chaos. Interestingly, the chaos exponent is small but non-zero even in the wormhole phase. This is surprising but consistent with the observation on the decay rate of the two point function [arXiv:2003.03916], and we found the chaos exponent and the decay rate indeed obey the same temperature dependence in this regime. We also studied the chaos exponent of a closely related model with single SYK term, and found that the chaos exponent of this model is always greater than that of the two coupled model in the entire parameter space.