Revealing nonperturbative effects in the SYK model

We study the large $N$ saddle points of two SYK chains coupled by an interaction that is nonlocal in Euclidean time. We start from analytic treatment of the free case with $q=2$ and perform the numerical study of the interacting case $q=4$. We show that in both cases there is a nontrivial phase structure with infinite number of phases. Every phase correspond to a saddle point in the non-interacting two-replica SYK. The nontrivial saddle points have non-zero value of the replica-nondiagonal correlator in the sense of quasi-averaging, when the coupling between replicas is turned off. Thus, the nonlocal interaction between replicas provides a protocol for turning the nonperturbatively subleading effects in SYK into non-equilibrium configurations which dominate at large $N$. For comparison we also study two SYK chains with local interaction for $q=2$ and $q=4$. We show that the $q=2$ model also has a similar phase structure, whereas in the $q=4$ model, dual to the traversable wormhole, the phase structure is different.