We investigate the nonperturbative effects in the Sadchev–Ye–Kitaev (SYK) model at large $$N$$ . The replica-nondiagonal solutions are considered in the SYK model with an integer number of replicas $$M > 1$$ . An important property of such solutions is the fact that they realize spontaneous breaking of the time translation group on $$M$$ replicas of the thermal circle. It is demonstrated in this study that this spontaneous symmetry breaking can be described in terms of the Bogolyubov quasi-averages, if we consider the system of nonlocally interacting replicas of the SYK model. The inclusion of the interaction of the replicas in this system transforms the subleading replica-nondiagonal saddle points from a pure SYK to thermodynamic phases, thereby generating a nontrivial phase structure. We investigate this phase structure exactly analytically in the SYK2 model and numerically in the SYK4 model.